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Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Algebraic structures
Group-like
• Group
• Semigroup / Monoid
• Rack and quandle
• Quasigroup and loop
• Abelian group
• Magma
• Lie group
Group theory
Ring-like
• Ring
• Rng
• Semiring
• Near-ring
• Commutative ring
• Domain
• Integral domain
• Field
• Division ring
• Lie ring
Ring theory
Lattice-like
• Lattice
• Semilattice
• Complemented lattice
• Total order
• Heyting algebra
• Boolean algebra
• Map of lattices
• Lattice theory
Module-like
• Module
• Group with operators
• Vector space
• Linear algebra
Algebra-like
• Algebra
• Associative
• Non-associative
• Composition algebra
• Lie algebra
• Graded
• Bialgebra
• Hopf algebra
Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
Introduction and definition
Motivation
In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as Lp spaces.)
Formal definition
Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M such that for all r, s in R and x, y in M, we have
1. $r\cdot (x+y)=r\cdot x+r\cdot y$
2. $(r+s)\cdot x=r\cdot x+s\cdot x$
3. $(rs)\cdot x=r\cdot (s\cdot x)$
4. $1\cdot x=x.$
The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation · : M × R → M.
Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.[1]
An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S.
If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.
Examples
• If K is a field, then K-vector spaces (vector spaces over K) and K-modules are identical.
• If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms.
• The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
• The decimal fractions (including negative ones) form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
• If R is any ring and n a natural number, then the cartesian product Rn is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
• If Mn(R) is the ring of n × n matrices over a ring R, M is an Mn(R)-module, and ei is the n × n matrix with 1 in the (i, i)-entry (and zeros elsewhere), then eiM is an R-module, since reim = eirm ∈ eiM. So M breaks up as the direct sum of R-modules, M = e1M ⊕ ... ⊕ enM. Conversely, given an R-module M0, then M0⊕n is an Mn(R)-module. In fact, the category of R-modules and the category of Mn(R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rn is an Mn(R)-module.
• If S is a nonempty set, M is a left R-module, and MS is the collection of all functions f : S → M, then with addition and scalar multiplication in MS defined pointwise by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), MS is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of NM).
• If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C∞(X). The set of all smooth vector fields defined on X form a module over C∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C∞(X)-modules and the category of vector bundles over X are equivalent.
• If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules.
• If R is a ring, we can define the opposite ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.
• Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra.
• If R and S are rings with a ring homomorphism φ : R → S, then every S-module M is an R-module by defining rm = φ(r)m. In particular, S itself is such an R-module.
Submodules and homomorphisms
Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or more explicitly an R-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right R-module) is in N.
If X is any subset of an R-module M, then the submodule spanned by X is defined to be $ \langle X\rangle =\,\bigcap _{N\supseteq X}N$ where N runs over the submodules of M which contain X, or explicitly $ \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}$, which is important in the definition of tensor products.[2]
The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N1, N2 of M such that N1 ⊂ N2, then the following two submodules are equal: (N1 + U) ∩ N2 = N1 + (U ∩ N2).
If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if for any m, n in M and r, s in R,
$f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)$.
This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map.
A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.
The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.[3] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.
Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod (see category of modules).
Types of modules
Finitely generated
An R-module M is finitely generated if there exist finitely many elements x1, ..., xn in M such that every element of M is a linear combination of those elements with coefficients from the ring R.
Cyclic
A module is called a cyclic module if it is generated by one element.
Free
A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.
Projective
Projective modules are direct summands of free modules and share many of their desirable properties.
Injective
Injective modules are defined dually to projective modules.
Flat
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness.
Torsionless
A module is called torsionless if it embeds into its algebraic dual.
Simple
A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.[4]
Semisimple
A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.
Indecomposable
An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules).
Faithful
A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.
Torsion-free
A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0.
Noetherian
A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
Artinian
An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
Graded
A graded module is a module with a decomposition as a direct sum M = ⨁x Mx over a graded ring R = ⨁x Rx such that RxMy ⊂ Mx+y for all x and y.
Uniform
A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.
Further notions
Relation to representation theory
A representation of a group G over a field k is a module over the group ring k[G].
If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted EndZ(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to EndZ(M).
Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. Such a representation R → EndZ(M) may also be called a ring action of R on M.
A representation is called faithful if and only if the map R → EndZ(M) is injective. In terms of modules, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Every abelian group is a faithful module over the integers or over some ring of integers modulo n, Z/nZ.
Generalizations
A ring R corresponds to a preadditive category R with a single object. With this understanding, a left R-module is just a covariant additive functor from R to the category Ab of abelian groups, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.
Modules over commutative rings can be generalized in a different direction: take a ringed space (X, OX) and consider the sheaves of OX-modules (see sheaf of modules). These form a category OX-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X).
One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.
Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.
See also
• Group ring
• Algebra (ring theory)
• Module (model theory)
• Module spectrum
• Annihilator
Notes
1. Dummit, David S. & Foote, Richard M. (2004). Abstract Algebra. Hoboken, NJ: John Wiley & Sons, Inc. ISBN 978-0-471-43334-7.
2. Mcgerty, Kevin (2016). "ALGEBRA II: RINGS AND MODULES" (PDF).
3. Ash, Robert. "Module Fundamentals" (PDF). Abstract Algebra: The Basic Graduate Year.
4. Jacobson (1964), p. 4, Def. 1
References
• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3
• Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8
External links
• "Module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• module at the nLab
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| Wikipedia |
Maplet
A maplet or maplet arrow (symbol: ↦, commonly pronounced "maps to") is a symbol consisting of a vertical line with a rightward-facing arrow. It is used in mathematics and in computer science to denote functions (the expression x ↦ y is also called a maplet). One example of use of the maplet is in Z notation, a formal specification language used in software development.[1]
↦
Maplet
In the Unicode character set, the maplet is at the point U+21A6.[2]
See also
• Arrow notation – e.g., $x\mapsto x+1$, also known as map
References
1. Mikušiak, Luboš; Miroslav Adámy; Thomas Seidmann (1997). "Publishing formal specifications in Z notation on world wide web". TAPSOFT '97: Theory and Practice of Software Development. Lecture Notes in Computer Science. Vol. 1214. pp. 871–874. doi:10.1007/BFb0030650. ISBN 978-3-540-62781-4.
2. Unicode Character 'RIGHTWARDS ARROW FROM BAR' (U+21A6)
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If and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence),[1] and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false.
↔⇔≡⟺
Logical symbols representing iff
In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q.[2] Some authors regard "iff" as unsuitable in formal writing;[3] others consider it a "borderline case" and tolerate its use.[4] In logical formulae, logical symbols, such as $\leftrightarrow $ and $\Leftrightarrow $,[5] are used instead of these phrases; see § Notation below.
Definition
The truth table of P $\Leftrightarrow $ Q is as follows:[6][7]
Truth table
P Q P $\Rightarrow $ Q P $\Leftarrow $ Q P $\Leftrightarrow $ Q
TTTTT
TFFTF
FTTFF
FFTTT
It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.[8]
Usage
Notation
The corresponding logical symbols are "$\leftrightarrow $", "$\Leftrightarrow $",[5] and $\equiv $,[9] and sometimes "iff". These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas (e.g., in metalogic). In Łukasiewicz's Polish notation, it is the prefix symbol $E$.[10]
Another term for the logical connective, i.e., the symbol in logic formulas, is exclusive nor.
In TeX, "if and only if" is shown as a long double arrow: $\iff $ via command \iff or \Longleftrightarrow.[11]
Proofs
In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction "(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts—that is, because "iff" is truth-functional, "P iff Q" follows if P and Q have been shown to be both true, or both false.
Origin of iff and pronunciation
Usage of the abbreviation "iff" first appeared in print in John L. Kelley's 1955 book General Topology.[12] Its invention is often credited to Paul Halmos, who wrote "I invented 'iff,' for 'if and only if'—but I could never believe I was really its first inventor."[13]
It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". However, in the preface of General Topology, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and euphony demands something less I use Halmos' 'iff'". The authors of one discrete mathematics textbook suggest:[14] "Should you need to pronounce iff, really hang on to the 'ff' so that people hear the difference from 'if'", implying that "iff" could be pronounced as [ɪfː].
Usage in definitions
Technically, definitions are "if and only if" statements; some texts — such as Kelley's General Topology — follow the strict demands of logic, and use "if and only if" or iff in definitions of new terms.[15] However, this logically correct usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including English Wikipedia articles) follow the linguistic convention to interpret "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover").[16]
Distinction from "if" and "only if"
• "Madison will eat the fruit if it is an apple." (equivalent to "Only if Madison will eat the fruit, can it be an apple" or "Madison will eat the fruit ← the fruit is an apple")
This states that Madison will eat fruits that are apples. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. All that is known for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficient condition for Madison to eat the fruit.
• "Madison will eat the fruit only if it is an apple." (equivalent to "If Madison will eat the fruit, then it is an apple" or "Madison will eat the fruit → the fruit is an apple")
This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat all the apples she is given.
• "Madison will eat the fruit if and only if it is an apple." (equivalent to "Madison will eat the fruit ↔ the fruit is an apple")
This statement makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit.
Sufficiency is the converse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. "not"). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways:
P is sufficient for Q
Q is necessary for P
¬Q is sufficient for ¬P
¬P is necessary for ¬Q
As an example, take the first example above, which states P→Q, where P is "the fruit in question is an apple" and Q is "Madison will eat the fruit in question". The following are four equivalent ways of expressing this very relationship:
If the fruit in question is an apple, then Madison will eat it.
Only if Madison will eat the fruit in question, is it an apple.
If Madison will not eat the fruit in question, then it is not an apple.
Only if the fruit in question is not an apple, will Madison not eat it.
Here, the second example can be restated in the form of if...then as "If Madison will eat the fruit in question, then it is an apple"; taking this in conjunction with the first example, we find that the third example can be stated as "If the fruit in question is an apple, then Madison will eat it; and if Madison will eat the fruit, then it is an apple".
In terms of Euler diagrams
• A is a proper subset of B. A number is in A only if it is in B; a number is in B if it is in A.
• C is a subset but not a proper subset of B. A number is in B if and only if it is in C, and a number is in C if and only if it is in B.
Euler diagrams show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "P→Q" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.
More general usage
Iff is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
The elements of X are all and only the elements of Y means: "For any z in the domain of discourse, z is in X if and only if z is in Y."
See also
• Equivalence relation
• Logical biconditional
• Logical equality
• Logical equivalence
• Polysyllogism
References
1. Copi, I. M.; Cohen, C.; Flage, D. E. (2006). Essentials of Logic (Second ed.). Upper Saddle River, NJ: Pearson Education. p. 197. ISBN 978-0-13-238034-8.
2. Weisstein, Eric W. "Iff." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Iff.html Archived 13 November 2018 at the Wayback Machine
3. E.g. Daepp, Ulrich; Gorkin, Pamela (2011), Reading, Writing, and Proving: A Closer Look at Mathematics, Undergraduate Texts in Mathematics, Springer, p. 52, ISBN 9781441994790, While it can be a real time-saver, we don't recommend it in formal writing.
4. Rothwell, Edward J.; Cloud, Michael J. (2014), Engineering Writing by Design: Creating Formal Documents of Lasting Value, CRC Press, p. 98, ISBN 9781482234312, It is common in mathematical writing
5. Peil, Timothy. "Conditionals and Biconditionals". web.mnstate.edu. Archived from the original on 24 October 2020. Retrieved 4 September 2020.
6. p <=> q Archived 18 October 2016 at the Wayback Machine. Wolfram|Alpha
7. If and only if, UHM Department of Mathematics, archived from the original on 5 May 2000, retrieved 16 October 2016, Theorems which have the form "P if and only Q" are much prized in mathematics. They give what are called "necessary and sufficient" conditions, and give completely equivalent and hopefully interesting new ways to say exactly the same thing.
8. "XOR/XNOR/Odd Parity/Even Parity Gate". www.cburch.com. Archived from the original on 7 April 2022. Retrieved 22 October 2019.
9. Weisstein, Eric W. "Equivalent". mathworld.wolfram.com. Archived from the original on 3 October 2020. Retrieved 4 September 2020.
10. "Jan Łukasiewicz > Łukasiewicz's Parenthesis-Free or Polish Notation (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Archived from the original on 9 August 2019. Retrieved 22 October 2019.
11. "LaTeX:Symbol". Art of Problem Solving. Archived from the original on 22 October 2019. Retrieved 22 October 2019.
12. General Topology, reissue ISBN 978-0-387-90125-1
13. Nicholas J. Higham (1998). Handbook of writing for the mathematical sciences (2nd ed.). SIAM. p. 24. ISBN 978-0-89871-420-3.
14. Maurer, Stephen B.; Ralston, Anthony (2005). Discrete Algorithmic Mathematics (3rd ed.). Boca Raton, Fla.: CRC Press. p. 60. ISBN 1568811667.
15. For instance, from General Topology, p. 25: "A set is countable iff it is finite or countably infinite." [boldface in original]
16. Krantz, Steven G. (1996), A Primer of Mathematical Writing, American Mathematical Society, p. 71, ISBN 978-0-8218-0635-7
External links
Wikimedia Commons has media related to If and only if.
• "Tables of truth for if and only if". Archived from the original on 5 May 2000.
• Language Log: "Just in Case"
• Southern California Philosophy for philosophy graduate students: "Just in Case"
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| Wikipedia |
Partial function
In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function.
More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set.
A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total.
When arrow notation is used for functions, a partial function $f$ from $X$ to $Y$ is sometimes written as $f:X\rightharpoonup Y,$ $f:X\nrightarrow Y,$ or $f:X\hookrightarrow Y.$ However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.
Specifically, for a partial function $f:X\rightharpoonup Y,$ and any $x\in X,$ one has either:
• $f(x)=y\in Y$ (it is a single element in Y), or
• $f(x)$ is undefined.
For example, if $f$ is the square root function restricted to the integers
$f:\mathbb {Z} \to \mathbb {N} ,$ defined by:
$f(n)=m$ if, and only if, $m^{2}=n,$ $m\in \mathbb {N} ,n\in \mathbb {Z} ,$
then $f(n)$ is only defined if $n$ is a perfect square (that is, $0,1,4,9,16,\ldots $). So $f(25)=5$ but $f(26)$ is undefined.
Basic concepts
A partial function arises from the consideration of maps between two sets X and Y that may not be defined on the entire set X. A common example is the square root operation on the real numbers $\mathbb {R} $: because negative real numbers do not have real square roots, the operation can be viewed as a partial function from $\mathbb {R} $ to $\mathbb {R} .$ The domain of definition of a partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y. In the example of the square root operation, the set S consists of the nonnegative real numbers $[0,+\infty ).$
The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see Halting problem.
In case the domain of definition S is equal to the whole set X, the partial function is said to be total. Thus, total partial functions from X to Y coincide with functions from X to Y.
Many properties of functions can be extended in an appropriate sense of partial functions. A partial function is said to be injective, surjective, or bijective when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively.
Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective.[1]
An injective partial function may be inverted to an injective partial function, and a partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to a bijective partial function.
The notion of transformation can be generalized to partial functions as well. A partial transformation is a function $f:A\rightharpoonup B,$ where both $A$ and $B$ are subsets of some set $X.$[1]
Function spaces
For convenience, denote the set of all partial functions $f:X\rightharpoonup Y$ from a set $X$ to a set $Y$ by $[X\rightharpoonup Y].$ This set is the union of the sets of functions defined on subsets of $X$ with same codomain $Y$:
$[X\rightharpoonup Y]=\bigcup _{D\subseteq X}[D\to Y],$
the latter also written as $ \bigcup _{D\subseteq {X}}Y^{D}.$ In finite case, its cardinality is
$|[X\rightharpoonup Y]|=(|Y|+1)^{|X|},$
because any partial function can be extended to a function by any fixed value $c$ not contained in $Y,$ so that the codomain is $Y\cup \{c\},$ an operation which is injective (unique and invertible by restriction).
Discussion and examples
The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents a function since every element on the left-hand set is associated with exactly one element in the right hand set.
Natural logarithm
Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function.
Subtraction of natural numbers
Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:
$f:\mathbb {N} \times \mathbb {N} \rightharpoonup \mathbb {N} $
$f(x,y)=x-y.$
It is defined only when $x\geq y.$
Bottom element
In denotational semantics a partial function is considered as returning the bottom element when it is undefined.
In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested.
In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function.
In category theory
In category theory, when considering the operation of morphism composition in concrete categories, the composition operation $\circ \;:\;\hom(C)\times \hom(C)\to \hom(C)$ is a function if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \operatorname {ob} (C) has one element. The reason for this is that two morphisms $f:X\to Y$ and $g:U\to V$ can only be composed as $g\circ f$ if $Y=U,$ that is, the codomain of $f$ must equal the domain of $g.$
The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps.[2] One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science."[3]
The category of sets and partial bijections is equivalent to its dual.[4] It is the prototypical inverse category.[5]
In abstract algebra
Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).[6]
The set of all partial functions (partial transformations) on a given base set, $X,$ forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on $X$), typically denoted by ${\mathcal {PT}}_{X}.$[7][8][9] The set of all partial bijections on $X$ forms the symmetric inverse semigroup.[7][8]
Charts and atlases for manifolds and fiber bundles
Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps.
The reason for the use of partial functions instead of functions is to permit general global topologies to be represented by stitching together local patches to describe the global structure. The "patches" are the domains where the charts are defined.
See also
Function
x ↦ f (x)
Examples of domains and codomains
• $X$ → $\mathbb {B} $, $\mathbb {B} $ → $X$, $\mathbb {B} ^{n}$ → $X$
• $X$ → $\mathbb {Z} $, $\mathbb {Z} $ → $X$
• $X$ → $\mathbb {R} $, $\mathbb {R} $ → $X$, $\mathbb {R} ^{n}$ → $X$
• $X$ → $\mathbb {C} $, $\mathbb {C} $ → $X$, $\mathbb {C} ^{n}$ → $X$
Classes/properties
• Constant
• Identity
• Linear
• Polynomial
• Rational
• Algebraic
• Analytic
• Smooth
• Continuous
• Measurable
• Injective
• Surjective
• Bijective
Constructions
• Restriction
• Composition
• λ
• Inverse
Generalizations
• Partial
• Multivalued
• Implicit
• space
• Analytic continuation – Extension of the domain of an analytic function (mathematics)
• Multivalued function – Generalized mathematical function
• Densely defined operator – Function that is defined almost everywhere (mathematics)
References
1. Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
2. Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.
3. Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science & Business Media. p. 290. ISBN 978-1-4419-0615-1.
4. Francis Borceux (1994). Handbook of Categorical Algebra: Volume 2, Categories and Structures. Cambridge University Press. p. 289. ISBN 978-0-521-44179-7.
5. Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups. World Scientific. p. 55. ISBN 978-981-4407-06-9.
6. Peter Burmeister (1993). "Partial algebras – an introductory survey". In Ivo G. Rosenberg; Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9.
7. Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American Mathematical Soc. p. xii. ISBN 978-0-8218-0272-4.
8. Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5.
9. Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4.
• Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9.
• Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.
• Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Company, New York.
| Wikipedia |
Uniqueness quantification
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition.[1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!"[2] or "∃=1". For example, the formal statement
$\exists !n\in \mathbb {N} \,(n-2=4)$
may be read as "there is exactly one natural number $n$ such that $n-2=4$".
Proving uniqueness
The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, $a$ and $b$) must be equal to each other (i.e. $a=b$).
For example, to show that the equation $x+2=5$ has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds:
$3+2=5.$
To establish the uniqueness of the solution, one would then proceed by assuming that there are two solutions, namely $a$ and $b$, satisfying $x+2=5$. That is,
$a+2=5{\text{ and }}b+2=5.$
By transitivity of equality,
$a+2=b+2.$
Subtracting 2 from both sides then yields
$a=b.$
which completes the proof that 3 is the unique solution of $x+2=5$.
In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.
An alternative way to prove uniqueness is to prove that there exists an object $a$ satisfying the condition, and then to prove that every object satisfying the condition must be equal to $a$.
Reduction to ordinary existential and universal quantification
Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula $\exists !xP(x)$ to mean
$\exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)),$
which is logically equivalent to
$\exists x\,(P(x)\wedge \forall y\,(P(y)\to y=x)).$
An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is
$\exists x\,P(x)\wedge \forall y\,\forall z\,[(P(y)\wedge P(z))\to y=z].$
Another equivalent definition, which has the advantage of brevity, is
$\exists x\,\forall y\,(P(y)\leftrightarrow y=x).$
Generalizations
The uniqueness quantification can be generalized into counting quantification (or numerical quantification[3]). This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.[4]
Uniqueness depends on a notion of equality. Loosening this to some coarser equivalence relation yields quantification of uniqueness up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in category theory are defined to be unique up to isomorphism.
The exclamation mark $!$ !} can be also used as a separate quantification symbol, so $(\exists !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))$, where $(!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)$. E.g. it can be safely used in the replacement axiom, instead of $\exists !$ !} .
See also
• Essentially unique
• One-hot
• Singleton (mathematics)
• Uniqueness theorem
References
1. Weisstein, Eric W. "Uniqueness Theorem". mathworld.wolfram.com. Retrieved 2019-12-15.
2. "2.5 Uniqueness Arguments". www.whitman.edu. Retrieved 2019-12-15.
3. Helman, Glen (August 1, 2013). "Numerical quantification" (PDF). persweb.wabash.edu. Retrieved 2019-12-14.
4. This is a consequence of the compactness theorem.
Bibliography
• Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press International. p. 199.
• Andrews, Peter B. (2002). An introduction to mathematical logic and type theory to truth through proof (2. ed.). Dordrecht: Kluwer Acad. Publ. p. 233. ISBN 1-4020-0763-9.
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| Wikipedia |
Function composition
In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[nb 1]
"Ring operator" redirects here. Not to be confused with operator ring or operator assistance.
Function
x ↦ f (x)
Examples of domains and codomains
• $X$ → $\mathbb {B} $, $\mathbb {B} $ → $X$, $\mathbb {B} ^{n}$ → $X$
• $X$ → $\mathbb {Z} $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \mathbb {Z} → $X$
• $X$ → $\mathbb {R} $, $\mathbb {R} $ → $X$, $\mathbb {R} ^{n}$ → $X$
• $X$ → $\mathbb {C} $, $\mathbb {C} $ → $X$, $\mathbb {C} ^{n}$ → $X$
Classes/properties
• Constant
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• Rational
• Algebraic
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• Continuous
• Measurable
• Injective
• Surjective
• Bijective
Constructions
• Restriction
• Composition
• λ
• Inverse
Generalizations
• Partial
• Multivalued
• Implicit
• space
The notation g ∘ f is read as "g of f ", "g after f ", "g circle f ", "g round f ", "g about f ", "g composed with f ", "g following f ", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions is a chaining process in which the output of function f feeds the input of function g.
The composition of functions is a special case of the composition of relations, sometimes also denoted by $\circ $. As a result, all properties of composition of relations are true of composition of functions,[1] such as the property of associativity.
Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.[2]
Examples
• Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}, and g = {(1, 2), (2, 3), (3, 1), (4, 2)}, then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, as shown in the figure.
• Composition of functions on an infinite set: If f: R → R (where R is the set of all real numbers) is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then:
(f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and
(g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)3.
• If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t.
Properties
The composition of functions is always associative—a property inherited from the composition of relations.[1] That is, if f, g, and h are composable, then f ∘ (g ∘ h) = (f ∘ g) ∘ h.[3] Since the parentheses do not change the result, they are generally omitted.
In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter.[nb 2] Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g. For example, the composition g ∘ f of the functions f : R → (−∞,+9] defined by f(x) = 9 − x2 and g : [0,+∞) → R defined by $g(x)={\sqrt {x}}$ can be defined on the interval [−3,+3].
The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, |x| + 3 = |x + 3| only when x ≥ 0. The picture shows another example.
The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that (f ∘ g)−1 = g−1∘ f−1.[4]
Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula.[3]
Composition monoids
Main article: Transformation monoid
Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) a composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup[5] or symmetric semigroup[6] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.[7])
If the transformations are bijective (and thus invertible), then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley's theorem, essentially says that any group is in fact just a subgroup of a permutation group (up to isomorphism).[8]
The set of all bijective functions f: X → X (called permutations) forms a group with respect to function composition. This is the symmetric group, also sometimes called the composition group.
In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup.[9]
Functional powers
Main article: Iterated function
If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2. That is:
(f ∘ f)(x) = f(f(x)) = f 2(x)
(f ∘ f ∘ f)(x) = f(f(f(x))) = f 3(x)
(f ∘ f ∘ f ∘ f)(x) = f(f(f(f(x)))) = f 4(x)
More generally, for any natural number n ≥ 2, the nth functional power can be defined inductively by f n = f ∘ f n−1 = f n−1 ∘ f, a notation introduced by Hans Heinrich Bürmann[10][11] and John Frederick William Herschel.[12][10][13][11] Repeated composition of such a function with itself is called iterated function.
• By convention, f 0 is defined as the identity map on f 's domain, idX.
• If even Y = X and f: X → X admits an inverse function f −1, negative functional powers f −n are defined for n > 0 as the negated power of the inverse function: f −n = (f −1)n.[12][10][11]
Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x).[11] For trigonometric functions, usually the latter is meant, at least for positive exponents.[11] For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan ≠ 1/tan.
In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2.
More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n can be defined as gm.
Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter; in this case, such a system is called a flow, specified through solutions of Schröder's equation. Iterated functions and flows occur naturally in the study of fractals and dynamical systems.
To avoid ambiguity, some mathematicians choose to use ∘ to denote the compositional meaning, writing f∘n(x) for the n-th iterate of the function f(x), as in, for example, f∘3(x) meaning f(f(f(x))). For the same purpose, f[n](x) was used by Benjamin Peirce[14][11] whereas Alfred Pringsheim and Jules Molk suggested nf(x) instead.[15][11][nb 3]
Alternative notations
Many mathematicians, particularly in group theory, omit the composition symbol, writing gf for g ∘ f.[16]
In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf " for "f(x)" and "(xf)g" for "g(f(x))".[17] This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because function composition is not necessarily commutative (e.g. matrix multiplication). Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.
Mathematicians who use postfix notation may write "fg", meaning first apply f and then apply g, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f ; g" for this,[18] thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the Z notation the ⨾ character is used for left relation composition.[19] Since all functions are binary relations, it is correct to use the [fat] semicolon for function composition as well (see the article on composition of relations for further details on this notation).
Composition operator
Main article: Composition operator
Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as
$C_{g}f=f\circ g.$
Composition operators are studied in the field of operator theory.
In programming languages
Function composition appears in one form or another in numerous programming languages.
Multivariate functions
Partial composition is possible for multivariate functions. The function resulting when some argument xi of the function f is replaced by the function g is called a composition of f and g in some computer engineering contexts, and is denoted f |xi = g
$f|_{x_{i}=g}=f(x_{1},\ldots ,x_{i-1},g(x_{1},x_{2},\ldots ,x_{n}),x_{i+1},\ldots ,x_{n}).$
When g is a simple constant b, composition degenerates into a (partial) valuation, whose result is also known as restriction or co-factor.[20]
$f|_{x_{i}=b}=f(x_{1},\ldots ,x_{i-1},b,x_{i+1},\ldots ,x_{n}).$
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, ..., gn, the composition of f with g1, ..., gn, is the m-ary function
$h(x_{1},\ldots ,x_{m})=f(g_{1}(x_{1},\ldots ,x_{m}),\ldots ,g_{n}(x_{1},\ldots ,x_{m})).$
This is sometimes called the generalized composite or superposition of f with g1, ..., gn.[21] The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen projection functions. Here g1, ..., gn can be seen as a single vector/tuple-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.[22]
A set of finitary operations on some base set X is called a clone if it contains all projections and is closed under generalized composition. Note that a clone generally contains operations of various arities.[21] The notion of commutation also finds an interesting generalization in the multivariate case; a function f of arity n is said to commute with a function g of arity m if f is a homomorphism preserving g, and vice versa i.e.:[21]
$f(g(a_{11},\ldots ,a_{1m}),\ldots ,g(a_{n1},\ldots ,a_{nm}))=g(f(a_{11},\ldots ,a_{n1}),\ldots ,f(a_{1m},\ldots ,a_{nm})).$
A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.[21]
Generalizations
Composition can be generalized to arbitrary binary relations. If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition R∘S is the relation defined as {(x, z) ∈ X × Z : ∃y ∈ Y. (x, y) ∈ R ∧ (y, z) ∈ S}. Considering a function as a special case of a binary relation (namely functional relations), function composition satisfies the definition for relation composition. A small circle R∘S has been used for the infix notation of composition of relations, as well as functions. When used to represent composition of functions $(g\circ f)(x)\ =\ g(f(x))$ however, the text sequence is reversed to illustrate the different operation sequences accordingly.
The composition is defined in the same way for partial functions and Cayley's theorem has its analogue called the Wagner–Preston theorem.[23]
The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[24] The structures given by composition are axiomatized and generalized in category theory with the concept of morphism as the category-theoretical replacement of functions. The reversed order of composition in the formula (f ∘ g)−1 = (g−1 ∘ f −1) applies for composition of relations using converse relations, and thus in group theory. These structures form dagger categories.
Typography
The composition symbol ∘ is encoded as U+2218 ∘ RING OPERATOR (∘, ∘); see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written \circ.
See also
• Cobweb plot – a graphical technique for functional composition
• Combinatory logic
• Composition ring, a formal axiomatization of the composition operation
• Flow (mathematics)
• Function composition (computer science)
• Function of random variable, distribution of a function of a random variable
• Functional decomposition
• Functional square root
• Higher-order function
• Infinite compositions of analytic functions
• Iterated function
• Lambda calculus
Notes
1. Some authors use f ∘ g : X → Z, defined by (f ∘ g )(x) = g(f(x)) instead. This is common when a postfix notation is used, especially if functions are represented by exponents, as, for instance, in the study of group actions. See Dixon, John D.; Mortimer, Brian (1996). Permutation groups. Springer. p. 5. ISBN 0-387-94599-7.
2. The strict sense is used, e.g., in category theory, where a subset relation is modelled explicitly by an inclusion function.
3. Alfred Pringsheim's and Jules Molk's (1907) notation nf(x) to denote function compositions must not be confused with Rudolf von Bitter Rucker's (1982) notation nx, introduced by Hans Maurer (1901) and Reuben Louis Goodstein (1947) for tetration, or with David Patterson Ellerman's (1995) nx pre-superscript notation for roots.
References
1. Velleman, Daniel J. (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1-139-45097-3.
2. "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28.
3. Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.
4. Rodgers, Nancy (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp. 359–362. ISBN 978-0-471-37122-9.
5. Hollings, Christopher (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 334. ISBN 978-1-4704-1493-1.
6. Grillet, Pierre A. (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
7. Dömösi, Pál; Nehaniv, Chrystopher L. (2005). Algebraic Theory of Automata Networks: An introduction. SIAM. p. 8. ISBN 978-0-89871-569-9.
8. Carter, Nathan (2009-04-09). Visual Group Theory. MAA. p. 95. ISBN 978-0-88385-757-1.
9. Ganyushkin, Olexandr; Mazorchuk, Volodymyr (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 24. ISBN 978-1-84800-281-4.
10. Herschel, John Frederick William (1820). "Part III. Section I. Examples of the Direct Method of Differences". A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. Archived from the original on 2020-08-04. Retrieved 2020-08-04. (NB. Inhere, Herschel refers to his 1813 work and mentions Hans Heinrich Bürmann's older work.)
11. Cajori, Florian (1952) [March 1929]. "§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions". A History of Mathematical Notations. Vol. 2 (3rd corrected printing of 1929 issue, 2nd ed.). Chicago, USA: Open court publishing company. pp. 108, 176–179, 336, 346. ISBN 978-1-60206-714-1. Retrieved 2016-01-18. […] §473. Iterated logarithms […] We note here the symbolism used by Pringsheim and Molk in their joint Encyclopédie article: "2logb a = logb (logb a), …, k+1logb a = logb (klogb a)."[a] […] §533. John Herschel's notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in the Philosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that since d2 x, Δ3 x, Σ2 x mean dd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we write d−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he used fn(x), f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst, Burmann, has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan−1, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."[b] […] §535. Persistence of rival notations for inverse function.— […] The use of Herschel's notation underwent a slight change in Benjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."[c] […] §537. Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x · sin x; second,[d] sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x · log x and log (log x) are of frequent occurrence in analysis. […] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. […] (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
12. Herschel, John Frederick William (1813) [1812-11-12]. "On a Remarkable Application of Cotes's Theorem". Philosophical Transactions of the Royal Society of London. London: Royal Society of London, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall. 103 (Part 1): 8–26 [10]. doi:10.1098/rstl.1813.0005. JSTOR 107384. S2CID 118124706.
13. Peano, Giuseppe (1903). Formulaire mathématique (in French). Vol. IV. p. 229.
14. Peirce, Benjamin (1852). Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.{{cite book}}: CS1 maint: location missing publisher (link)
15. Pringsheim, Alfred; Molk, Jules (1907). Encyclopédie des sciences mathématiques pures et appliquées (in French). Vol. I. p. 195. Part I.
16. Ivanov, Oleg A. (2009-01-01). Making Mathematics Come to Life: A Guide for Teachers and Students. American Mathematical Society. pp. 217–. ISBN 978-0-8218-4808-1.
17. Gallier, Jean (2011). Discrete Mathematics. Springer. p. 118. ISBN 978-1-4419-8047-2.
18. Barr, Michael; Wells, Charles (1998). Category Theory for Computing Science (PDF). p. 6. Archived from the original (PDF) on 2016-03-04. Retrieved 2014-08-23. (NB. This is the updated and free version of book originally published by Prentice Hall in 1990 as ISBN 978-0-13-120486-7.)
19. ISO/IEC 13568:2002(E), p. 23
20. Bryant, R. E. (August 1986). "Logic Minimization Algorithms for VLSI Synthesis" (PDF). IEEE Transactions on Computers. C-35 (8): 677–691. doi:10.1109/tc.1986.1676819. S2CID 10385726.
21. Bergman, Clifford (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 79–80, 90–91. ISBN 978-1-4398-5129-6.
22. Tourlakis, George (2012). Theory of Computation. John Wiley & Sons. p. 100. ISBN 978-1-118-31533-0.
23. Lipscomb, S. (1997). Symmetric Inverse Semigroups. AMS Mathematical Surveys and Monographs. p. xv. ISBN 0-8218-0627-0.
24. Hilton, Peter; Wu, Yel-Chiang (1989). A Course in Modern Algebra. John Wiley & Sons. p. 65. ISBN 978-0-471-50405-4.
External links
• "Composite function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.
| Wikipedia |
Radical symbol
In mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as
${\sqrt {11}},$
while the nth root of x is written as
${\sqrt[{n}]{x}}.$
It is also used for other meanings in more advanced mathematics, such as the radical of an ideal.
In linguistics, the symbol is used to denote a root word.
Principal square root
Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root#Principal square root of a complex number.
Origin
The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "ج" (ǧīm), which is the first letter in the Arabic word "جذر" (jadhir, meaning "root").[1] However, Leonhard Euler[2] believed it originated from the letter "r", the first letter of the Latin word "radix" (meaning "root"), referring to the same mathematical operation.
The symbol was first seen in print without the vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician. In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.[3]
Encoding
The Unicode and HTML character codes for the radical symbols are:
ReadCharacterUnicode[4]XMLURLHTML[5]
Square root√U+221A√ or √%E2%88%9A√ or √
Cube root∛U+221B∛ or ∛%E2%88%9B
Fourth root∜U+221C∜ or ∜%E2%88%9C
However, these characters differ in appearance from most mathematical typesetting by omitting the overline connected to the radical symbol, which surrounds the argument of the square root function. The OpenType math table allows adding this overline following the radical symbol.
Legacy encodings of the square root character U+221A include:
• 0xC3 in Mac OS Roman and Mac OS Cyrillic
• 0xFB (Alt+251) in Code page 437 and Code page 866 (but not Code page 850) on DOS and the Windows console
• 0xD6 in the Symbol font encoding[6]
• 02-69 (7-bit 0x2265, SJIS 0x81E3, EUC 0xA2E5) in Japanese JIS X 0208[7]
• 01-78 (EUC/UHC 0xA1EE) in Korean Wansung code[8]
• 01-44 (EUC 0xA1CC) in Mainland Chinese GB 2312 or GBK[9]
• Traditional Chinese: 0xA1D4 in Big5[10][11] or 1-2235 (kuten 01-02-21, EUC 0xA2B5 or 0x8EA1A2B5) in CNS 11643[11][12]
The Symbol font displays the character without any vinculum whatsoever; the overline may be a separate character at 0x60.[13] The JIS,[14] Wansung[15] and CNS 11643[11][16] code charts include a short overline attached to the radical symbol, whereas the GB 2312[17] and GB 18030 charts do not.[18]
Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in the Miscellaneous Technical block.[19] This was used in contexts where box-drawing characters are used, such as in the technical character set of DEC terminals, to join up with box drawing characters on the line above to create the vinculum.[20]
In LaTeX the square root symbol may be generated by the \sqrt macro,[21] and the square root symbol without the overline may be generated by the \surd macro.
References
1. "Language Log: Ab surd". Retrieved 22 June 2012.
2. Leonhard Euler (1755). Institutiones calculi differentialis (in Latin).
3. Cajori, Florian (2012) [1928], A History of Mathematical Notations, vol. I, Dover, p. 208, ISBN 978-0-486-67766-8
4. Unicode Consortium (2022-09-16). "Mathematical Operators" (PDF). The Unicode Standard (15.0 ed.). Retrieved 2023-07-16.
5. Web Hypertext Application Technology Working Group (2023-07-14). "Named Character References". HTML Living Standard. Retrieved 2023-07-16.
6. Apple Computer (2005-04-05) [1995-04-15]. Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later. Unicode Consortium. SYMBOL.TXT.
7. Unicode Consortium (2015-12-02) [1994-03-08]. JIS X 0208 (1990) to Unicode. JIS0208.TXT.
8. Unicode Consortium (2011-10-14) [1995-07-24]. Unified Hangeul(KSC5601-1992) to Unicode table. KSC5601.TXT.
9. IBM (2002). "windows-936-2000". International Components for Unicode.
10. Unicode Consortium (2015-12-02) [1994-02-11]. BIG5 to Unicode table (complete). BIG5.TXT.
11. "[√] 1-2235". Word Information. National Development Council.
12. IBM (2014). "euc-tw-2014". International Components for Unicode.
13. IBM. Code Page 01038 (PDF). Archived from the original (PDF) on 2015-07-08.
14. ISO/IEC JTC 1/SC 2 (1992-07-13). Japanese Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-168.
15. Korea Bureau of Standards (1988-10-01). Korean Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-149.
16. ECMA (1994). Chinese Standard Interchange Code (CSIC) - Set 1 (PDF). ITSCJ/IPSJ. ISO-IR-171.
17. China Association for Standardization (1980). Coded Chinese Graphic Character Set for Information Interchange (PDF). ITSCJ/IPSJ. ISO-IR-58.
18. Standardization Administration of China (2005). Information Technology—Chinese coded character set. p. 8. GB 18030-2005.
19. Unicode Consortium (2022-09-16). "Miscellaneous Technical" (PDF). The Unicode Standard (15.0 ed.). Retrieved 2023-07-16.
20. Williams, Paul Flo (2002). "DEC Technical Character Set (TCS)". VT100.net. Retrieved 2023-07-16.
21. Braams, Johannes; et al. (2023-06-01). "The LATEX 2ε Sources" (PDF) (2023-06-01 Patch Level 1 ed.). § ltmath.dtx: Math Environments. Retrieved 2023-07-16.
| Wikipedia |
Right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or $\pi $/2 radians[1] corresponding to a quarter turn.[2] If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles.[3] The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.
Types of angles
2D angles
Right
Interior
Exterior
2D angle pairs
Adjacent
Vertical
Complementary
Supplementary
Transversal
3D angles
Dihedral
Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles,[4] making the right angle basic to trigonometry.
Etymology
The meaning of right in right angle possibly refers to the Latin adjective rectus 'erect, straight, upright, perpendicular'. A Greek equivalent is orthos 'straight; perpendicular' (see orthogonality).
In elementary geometry
A rectangle is a quadrilateral with four right angles. A square has four right angles, in addition to equal-length sides.
The Pythagorean theorem states how to determine when a triangle is a right triangle.
Symbols
In Unicode, the symbol for a right angle is U+221F ∟ RIGHT ANGLE (∟). It should not be confused with the similarly shaped symbol U+231E ⌞ BOTTOM LEFT CORNER (⌞, ⌞). Related symbols are U+22BE ⊾ RIGHT ANGLE WITH ARC (⊾), U+299C ⦜ RIGHT ANGLE VARIANT WITH SQUARE (⦜), and U+299D ⦝ MEASURED RIGHT ANGLE WITH DOT (⦝).[5]
In diagrams, the fact that an angle is a right angle is usually expressed by adding a small right angle that forms a square with the angle in the diagram, as seen in the diagram of a right triangle (in British English, a right-angled triangle) to the right. The symbol for a measured angle, an arc, with a dot, is used in some European countries, including German-speaking countries and Poland, as an alternative symbol for a right angle.[6]
Euclid
Right angles are fundamental in Euclid's Elements. They are defined in Book 1, definition 10, which also defines perpendicular lines. Definition 10 does not use numerical degree measurements but rather touches at the very heart of what a right angle is, namely two straight lines intersecting to form two equal and adjacent angles.[7] The straight lines which form right angles are called perpendicular.[8] Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle).[9] Two angles are called complementary if their sum is a right angle.[10]
Book 1 Postulate 4 states that all right angles are equal, which allows Euclid to use a right angle as a unit to measure other angles with. Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions. Saccheri gave a proof as well but using a more explicit assumption. In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork. One may argue that, even if postulate 4 can be proven from the preceding ones, in the order that Euclid presents his material it is necessary to include it since without it postulate 5, which uses the right angle as a unit of measure, makes no sense.[11]
Conversion to other units
A right angle may be expressed in different units:
• 1/4 turn
• 90° (degrees)
• π/2 radians
• 100 grad (also called grade, gradian, or gon)
• 8 points (of a 32-point compass rose)
• 6 hours (astronomical hour angle)
Rule of 3-4-5
Throughout history, carpenters and masons have known a quick way to confirm if an angle is a true "right angle". It is based on the most widely known Pythagorean triple (3, 4, 5) and so called the "rule of 3-4-5". From the angle in question, running a straight line along one side exactly 3 units in length, and along the second side exactly 4 units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly 5 units in length. This measurement can be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem ("The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides").
Thales' theorem
Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
Alternative construction if P outside of the half-line h and the distance A to P' is small (B is freely selectable),
animation at the end with pause 10 s
Main article: Thales' theorem
Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle.
Two application examples in which the right angle and the Thales' theorem are included (see animations).
See also
Wikimedia Commons has media related to Right angles.
• Cartesian coordinate system
• Types of angles
References
1. "Right Angle". Math Open Reference. Retrieved 26 April 2017.
2. Wentworth p. 11
3. Wentworth p. 8
4. Wentworth p. 40
5. Unicode 5.2 Character Code Charts Mathematical Operators, Miscellaneous Mathematical Symbols-B
6. Müller-Philipp, Susanne; Gorski, Hans-Joachim (2011). Leitfaden Geometrie [Handbook Geometry] (in German). Springer. ISBN 9783834886163.
7. Heath p. 181
8. Heath p. 181
9. Heath p. 181
10. Wentworth p. 9
11. Heath pp. 200–201 for the paragraph
• Wentworth, G.A. (1895). A Text-Book of Geometry. Ginn & Co.
• Euclid, commentary and trans. by T. L. Heath Elements Vol. 1 (1908 Cambridge) Google Books
Authority control: National
• Germany
| Wikipedia |
Internal and external angles
In geometry, an angle of a polygon is formed by two adjacent sides. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.
"Interior angle" redirects here. For interior angles on the same side of the transversal, see Transversal line.
Types of angles
2D angles
Right
Interior
Exterior
2D angle pairs
Adjacent
Vertical
Complementary
Supplementary
Transversal
3D angles
Dihedral
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex.
In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1]: pp. 261-264
Properties
• The sum of the internal angle and the external angle on the same vertex is π radians (180°).
• The sum of all the internal angles of a simple polygon is π(n−2) radians or 180(n–2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on.
• The sum of the external angles of any simple convex or non-convex polygon, if only one of the two external angles is assumed at each vertex, is 2π radians (360°).
• The measure of the exterior angle at a vertex is unaffected by which side is extended: the two exterior angles that can be formed at a vertex by extending alternately one side or the other are vertical angles and thus are equal.
Extension to crossed polygons
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360°) revolutions one undergoes by walking around the perimeter of the polygon. In other words, the sum of all the exterior angles is 2πk radians or 360k degrees. Example: for ordinary convex polygons and concave polygons, k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.
References
1. Posamentier, Alfred S., and Lehmann, Ingmar. The Secrets of Triangles, Prometheus Books, 2012.
External links
• Internal angles of a triangle
• Interior angle sum of polygons: a general formula - Provides an interactive Java activity that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons.
| Wikipedia |
Monus
In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the $\mathop {\dot {-}} $ symbol to distinguish it from the standard subtraction operator.
Notation
glyph Unicode name Unicode code point[1] HTML character entity reference HTML/XML numeric character references TeX
∸ DOT MINUS U+2238 ∸ \dot -
− MINUS SIGN U+2212 − − -
Definition
Let $(M,+,0)$ be a commutative monoid. Define a binary relation $\leq $ on this monoid as follows: for any two elements $a$ and $b$, define $a\leq b$ if there exists an element $c$ such that $a+c=b$. It is easy to check that $\leq $ is reflexive[2] and that it is transitive.[3] $M$ is called naturally ordered if the $\leq $ relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements $a$ and $b$, a unique smallest element $c$ exists such that $a\leq b+c$, then M is called a commutative monoid with monus[4]: 129 and the monus $a\mathop {\dot {-}} b$ of any two elements $a$ and $b$ can be defined as this unique smallest element $c$ such that $a\leq b+c$.
An example of a commutative monoid that is not naturally ordered is $(\mathbb {Z} ,+,0)$, the commutative monoid of the integers with usual addition, as for any $a,b\in \mathbb {Z} $ there exists $c$ such that $a+c=b$, so $a\leq b$ holds for any $a,b\in \mathbb {Z} $, so $\leq $ is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.[5]
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[6]) is a semiring where the commutative monoid induced by the addition operator is naturally ordered. When this monoid is a commutative monoid with monus, the semiring is called a semiring with monus, or m-semiring.
Examples
If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under $a+b=a\vee b$ and $a\mathop {\dot {-}} b=a\wedge \neg b$.[4]: 129
Natural numbers
The natural numbers including 0 form a commutative monoid with monus, with their ordering being the usual order of natural numbers and the monus operator being a saturating variant of standard subtraction, variously referred to as truncated subtraction,[7] limited subtraction, proper subtraction, doz (difference or zero),[8] and monus.[9] Truncated subtraction is usually defined as[7]
$a\mathop {\dot {-}} b={\begin{cases}0&{\mbox{if }}a<b\\a-b&{\mbox{if }}a\geq b,\end{cases}}$
where − denotes standard subtraction. For example, 5 − 3 = 2 and 3 − 5 = −2 in regular subtraction, whereas in truncated subtraction 3 ∸ 5 = 0. Truncated subtraction may also be defined as[9]
$a\mathop {\dot {-}} b=\max(a-b,0).$
In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[7]
${\begin{aligned}P(0)&=0\\P(S(a))&=a\\a\mathop {\dot {-}} 0&=a\\a\mathop {\dot {-}} S(b)&=P(a\mathop {\dot {-}} b).\end{aligned}}$
A definition that does not need the predecessor function is:
${\begin{aligned}a\mathop {\dot {-}} 0&=a\\0\mathop {\dot {-}} b&=0\\S(a)\mathop {\dot {-}} S(b)&=a\mathop {\dot {-}} b.\end{aligned}}$
Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[7] Truncated subtraction is also used in the definition of the multiset difference operator.
Properties
The class of all commutative monoids with monus form a variety.[4]: 129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:
${\begin{aligned}a+(b\mathop {\dot {-}} a)&=b+(a\mathop {\dot {-}} b),\\(a\mathop {\dot {-}} b)\mathop {\dot {-}} c&=a\mathop {\dot {-}} (b+c),\\(a\mathop {\dot {-}} a)&=0,\\(0\mathop {\dot {-}} a)&=0.\\\end{aligned}}$
Notes
1. Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
2. taking $c$ to be the neutral element of the monoid
3. if $a\leq b$ with witness $d$ and $b\leq c$ with witness $d'$ then $d+d'$ witnesses that $a\leq c$
4. Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
5. M.Monet (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". Mathematics Stack Exchange. Retrieved 2016-10-14.
6. Semirings for breakfast, slide 17
7. Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
8. Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8.
9. Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice (eds.). Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. Vol. 1101. Springer. p. 522. ISBN 3-540-61463-X.
| Wikipedia |
Wreath product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
Given two groups $A$ and $H$ (sometimes known as the bottom and top[1]), there exist two variations of the wreath product: the unrestricted wreath product $A{\text{ Wr }}H$ and the restricted wreath product $A{\text{ wr }}H$. The general form, denoted by $A{\text{ Wr}}_{\Omega }H$ or $A{\text{ wr}}_{\Omega }H$ respectively, requires that $H$ acts on some set $\Omega $; when unspecified, usually $\Omega =H$ (a regular wreath product), though a different $\Omega $ is sometimes implied. The two variations coincide when $A$, $H$, and $\Omega $ are all finite. Either variation is also denoted as $A\wr H$ (with \wr for the LaTeX symbol) or A ≀ H (Unicode U+2240).
The notion generalizes to semigroups and is a central construction in the Krohn–Rhodes structure theory of finite semigroups.
Definition
Let $A$ be a group and let $H$ be a group acting on a set $\Omega $ (on the left). The direct product $A^{\Omega }$ of $A$ with itself indexed by $\Omega $ is the set of sequences ${\overline {a}}=(a_{\omega })_{\omega \in \Omega }$ in $A$ indexed by $\Omega $, with a group operation given by pointwise multiplication. The action of $H$ on $\Omega $ can be extended to an action on $A^{\Omega }$ by reindexing, namely by defining
$h\cdot (a_{\omega })_{\omega \in \Omega }:=(a_{h^{-1}\cdot \omega })_{\omega \in \Omega }$
for all $h\in H$ and all $(a_{\omega })_{\omega \in \Omega }\in A^{\Omega }$.
Then the unrestricted wreath product $A{\text{ Wr}}_{\Omega }H$ of $A$ by $H$ is the semidirect product $A^{\Omega }\rtimes H$ with the action of $H$ on $A^{\Omega }$ given above. The subgroup $A^{\Omega }$ of $A^{\Omega }\rtimes H$ is called the base of the wreath product.
The restricted wreath product $A{\text{ wr}}_{\Omega }H$ is constructed in the same way as the unrestricted wreath product except that one uses the direct sum as the base of the wreath product. In this case, the base consists of all sequences in $A$ with finitely-many non-identity entries.
In the most common case, $\Omega =H$, and $H$ acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by $A{\text{ Wr }}H$ and $A{\text{ wr }}H$ respectively. This is called the regular wreath product.
Notation and conventions
The structure of the wreath product of A by H depends on the H-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention to the circumstances.
• In literature A≀ΩH may stand for the unrestricted wreath product A WrΩ H or the restricted wreath product A wrΩ H.
• Similarly, A≀H may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H.
• In literature the H-set Ω may be omitted from the notation even if Ω ≠ H.
• In the special case that H = Sn is the symmetric group of degree n it is common in the literature to assume that Ω = {1,...,n} (with the natural action of Sn) and then omit Ω from the notation. That is, A≀Sn commonly denotes A≀{1,...,n}Sn instead of the regular wreath product A≀SnSn. In the first case the base group is the product of n copies of A, in the latter it is the product of n! copies of A.
Properties
Agreement of unrestricted and restricted wreath product on finite Ω
Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted A WrΩ H and the restricted wreath product A wrΩ H agree if the H-set Ω is finite. In particular this is true when Ω = H is finite.
Subgroup
A wrΩ H is always a subgroup of A WrΩ H.
Cardinality
If A, H and Ω are finite, then
|A≀ΩH| = |A||Ω||H|.[2]
Universal embedding theorem
Main article: Universal embedding theorem
Universal embedding theorem: If G is an extension of A by H, then there exists a subgroup of the unrestricted wreath product A≀H which is isomorphic to G.[3] This is also known as the Krasner–Kaloujnine embedding theorem. The Krohn–Rhodes theorem involves what is basically the semigroup equivalent of this.[4]
Canonical actions of wreath products
If the group A acts on a set Λ then there are two canonical ways to construct sets from Ω and Λ on which A WrΩ H (and therefore also A wrΩ H) can act.
• The imprimitive wreath product action on Λ × Ω.
If ((aω),h) ∈ A WrΩ H and (λ,ω′) ∈ Λ × Ω, then
$((a_{\omega }),h)\cdot (\lambda ,\omega '):=(a_{h(\omega ')}\lambda ,h\omega ').$
• The primitive wreath product action on ΛΩ.
An element in ΛΩ is a sequence (λω) indexed by the H-set Ω. Given an element ((aω), h) ∈ A WrΩ H its operation on (λω) ∈ ΛΩ is given by
$((a_{\omega }),h)\cdot (\lambda _{\omega }):=(a_{h^{-1}\omega }\lambda _{h^{-1}\omega }).$
Examples
• The Lamplighter group is the restricted wreath product ℤ2≀ℤ.
• ℤm≀Sn (Generalized symmetric group).
The base of this wreath product is the n-fold direct product
ℤmn = ℤm × ... × ℤm
of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric group Sn of degree n is given by
φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).[5]
• S2≀Sn (Hyperoctahedral group).
The action of Sn on {1,...,n} is as above. Since the symmetric group S2 of degree 2 is isomorphic to ℤ2 the hyperoctahedral group is a special case of a generalized symmetric group.[6]
• The smallest non-trivial wreath product is ℤ2≀ℤ2, which is the two-dimensional case of the above hyperoctahedral group. It is the symmetry group of the square, also called Dih4, the dihedral group of order 8.
• Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group Spn. Then P is isomorphic to the iterated regular wreath product Wn = ℤp ≀ ℤp≀...≀ℤp of n copies of ℤp. Here W1 := ℤp and Wk := Wk−1≀ℤp for all k ≥ 2.[7][8] For instance, the Sylow 2-subgroup of S4 is the above ℤ2≀ℤ2 group.
• The Rubik's Cube group is a normal subgroup of index 12 in the product of wreath products, (ℤ3≀S8) × (ℤ2≀S12), the factors corresponding to the symmetries of the 8 corners and 12 edges.
• The Sudoku validity preserving transformations (VPT) group contains the double wreath product (S3 ≀ S3) ≀ S2, where the factors are the permutation of rows/columns within a 3-row or 3-column band or stack (S3), the permutation of the bands/stacks themselves (S3) and the transposition, which interchanges the bands and stacks (S2). Here, the index sets Ω are the set of bands (resp. stacks) (|Ω| = 3) and the set {bands, stacks} (|Ω| = 2). Accordingly, |S3 ≀ S3| = |S3|3|S3| = (3!)4 and |(S3 ≀ S3) ≀ S2| = |S3 ≀ S3|2|S2| = (3!)8 × 2.
• Wreath products arise naturally in the symmetry group of complete rooted trees and their graphs. For example, the repeated (iterated) wreath product S2 ≀ S2 ≀ ... ≀ S2 is the automorphism group of a complete binary tree.
References
1. Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998), "Wreath products", Notes on Infinite Permutation Groups, Lecture Notes in Mathematics, Berlin, Heidelberg: Springer, pp. 67–76, doi:10.1007/bfb0092558, ISBN 978-3-540-49813-1, retrieved 2021-05-12
2. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)
3. M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", Acta Sci. Math. 14, pp. 69–82 (1951)
4. J D P Meldrum (1995). Wreath Products of Groups and Semigroups. Longman [UK] / Wiley [US]. p. ix. ISBN 978-0-582-02693-3.
5. J. W. Davies and A. O. Morris, "The Schur Multiplier of the Generalized Symmetric Group", J. London Math. Soc. (2), 8, (1974), pp. 615–620
6. P. Graczyk, G. Letac and H. Massam, "The Hyperoctahedral Group, Symmetric Group Representations and the Moments of the Real Wishart Distribution", J. Theoret. Probab. 18 (2005), no. 1, 1–42.
7. Joseph J. Rotman, An Introduction to the Theory of Groups, p. 176 (1995)
8. L. Kaloujnine, "La structure des p-groupes de Sylow des groupes symétriques finis", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 65, pp. 239–276 (1948)
External links
• Wreath product in Encyclopedia of Mathematics.
• Some Applications of the Wreath Product Construction. Archived 21 February 2014 at the Wayback Machine
| Wikipedia |
≬
Wikipedia does not currently have an article on "≬", but its sister project Wiktionary does:
Read the Wiktionary entry on "≬"
You can also:
• Search for ≬ in Wikipedia to check for alternative titles or spellings.
• Start the ≬ article, using the Article Wizard if you wish, or add a request for it; but please remember that Wikipedia is not a dictionary.
wiktionary:≬
| Wikipedia |
Double turnstile
In logic, the symbol ⊨, ⊧ or $\models $ is called the double turnstile. It is often read as "entails", "models", "is a semantic consequence of" or "is stronger than".[1] It is closely related to the turnstile symbol $\vdash $, which has a single bar across the middle, and which denotes syntactic consequence (in contrast to semantic).
Meaning
The double turnstile is a binary relation. It has several different meanings in different contexts:
• To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denote that if every sentence on the left is true, the sentence on the right must be true, e.g. $\Gamma \vDash \varphi $. This usage is closely related to the single-barred turnstile symbol which denotes syntactic consequence.
• To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denote that the structure is a model for (or satisfies) the set of sentences, e.g. ${\mathcal {A}}\models \Gamma $. This is typically done inductively along with restricting the range of a variable assignment, a function mapping each variable symbol to a value in ${\mathcal {A}}$ it might hold. [2]
• In this context, the semantic consequence in the previous list can be stated as "For a given model ${\mathcal {A}}$, if ${\mathcal {A}}\models \Gamma $ then ${\mathcal {A}}\vDash \varphi $".
• To denote a tautology, $\vDash \varphi $. which is to say that the expression $\varphi $ is a semantic consequence of the empty set.
• You can also use this symbol as follows: ⊭ to denote the statement 'does not entail'.
Typography
In TeX, the turnstile symbols $\vDash $ and $\models $ are obtained from the commands \vDash and \models respectively.
In Unicode it is encoded at U+22A8 ⊨ TRUE (⊨, ⊨) , and the opposite of it is U+22AD ⊭ NOT TRUE (⊭) .
In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and is capable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial on using this package.
See also
• List of logic symbols
• List of mathematical symbols
• Turnstile ⊢
References
1. Nederpelt, Rob (2004). "Chapter 7: Strengthening and weakening". Logical Reasoning: A First Course (3rd revised ed.). King's College Publications. p. 62. ISBN 0-9543006-7-X.
2. Open Logic Project, First-order logic (p.7). Accessed 4 January 2022.
Common logical symbols
∧ or &
and
∨
or
¬ or ~
not
→
implies
⊃
implies,
superset
↔ or ≡
iff
|
nand
∀
universal
quantification
∃
existential
quantification
⊤
true,
tautology
⊥
false,
contradiction
⊢
entails,
proves
⊨
entails,
therefore
∴
therefore
∵
because
Philosophy portal
Mathematics portal
| Wikipedia |
Ordered set operators
In mathematical notation, ordered set operators indicate whether an object precedes or succeeds another. These relationship operators are denoted by the unicode symbols U+227A-F, along with symbols located unicode blocks U+228x through U+22Ex.
Mathematical Operators[1]
Official Unicode Consortium code chart (PDF)
0123456789ABCDEF
U+227x ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ≺ ≻ ≼ ≽ ≾ ≿
U+228x ⊀ ⊁ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉ ⊊ ⊋ ⊌ ⊍ ⊎ ⊏
U+22Bx ⊰ ⊱ ⊲ ⊳ ⊴ ⊵ ⊶ ⊷ ⊸ ⊹ ⊺ ⊻ ⊼ ⊽ ⊾ ⊿
U+22Dx ⋐ ⋑ ⋒ ⋓ ⋔ ⋕ ⋖ ⋗ ⋘ ⋙ ⋚ ⋛ ⋜ ⋝ ⋞ ⋟
U+22Ex ⋠ ⋡ ⋢ ⋣ ⋤ ⋥ ⋦ ⋧ ⋨ ⋩ ⋪ ⋫ ⋬ ⋭ ⋮ ⋯
Notes
1.^ As of Unicode version 7.0
Examples
• The relationship x precedes y is written x ≺ y. The relation x precedes or is equal to y is written x ≼ y.
• The relationship x succeeds (or follows) y is written x ≻ y. The relation x succeeds or is equal to y is written x ≽ y.
Use in political science
Political scientists use order relations typically in the context of an agent's choice, for example the preferences of a voter over several political candidates.
• x ≺ y means that the voter prefers candidate y over candidate x.
• x ∼ y means the voter is indifferent between candidates x and y.
• x ≲ y means the voter is indifferent or prefers candidate y.[1]
References
1. Cooley, Brandon. "Ordered Sets" (PDF) (Lecture note for: Introduction to Mathematics for Political Science (2019) at Princeton University). pp. 2–3. Retrieved 2021-05-11.{{cite web}}: CS1 maint: url-status (link)
See also
• Order theory
• Partially ordered set
• Directional symbols
• Polynomial-time reduction
• Wolfram Mathworld: precedes and succeeds
| Wikipedia |
Normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup)[1] is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup $N$ of the group $G$ is normal in $G$ if and only if $gng^{-1}\in N$ for all $g\in G$ and $n\in N.$ The usual notation for this relation is $N\triangleleft G.$
"Invariant subgroup" redirects here. Not to be confused with Fully invariant subgroup.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of $G$ are precisely the kernels of group homomorphisms with domain $G,$ which means that they can be used to internally classify those homomorphisms.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.[2]
Definitions
A subgroup $N$ of a group $G$ is called a normal subgroup of $G$ if it is invariant under conjugation; that is, the conjugation of an element of $N$ by an element of $G$ is always in $N.$[3] The usual notation for this relation is $N\triangleleft G.$
Equivalent conditions
For any subgroup $N$ of $G,$ the following conditions are equivalent to $N$ being a normal subgroup of $G.$ Therefore, any one of them may be taken as the definition.
• The image of conjugation of $N$ by any element of $G$ is a subset of $N,$[4] i.e., $gNg^{-1}\subseteq N$ for all $g\in G$.
• The image of conjugation of $N$ by any element of $G$ is equal to $N,$[4] i.e., $gNg^{-1}=N$ for all $g\in G$.
• For all $g\in G,$ the left and right cosets $gN$ and $Ng$ are equal.[4]
• The sets of left and right cosets of $N$ in $G$ coincide.[4]
• Multiplication in $G$ preserves the equivalence relation "is in the same left coset as". That is, for every $g,g',h,h'\in G$ satisfying $gN=g'N$ and $hN=h'N$, we have $(gh)N=(g'h')N.$
• There exists a group on the set of left cosets of $N$ where multiplication of any two left cosets $gN$ and $hN$ yields the left coset $(gh)N$. (This group is called the quotient group of $G$ modulo $N$, denoted $G/N$.)
• $N$ is a union of conjugacy classes of $G.$[2]
• $N$ is preserved by the inner automorphisms of $G.$[5]
• There is some group homomorphism $G\to H$ whose kernel is $N.$[2]
• There exists a group homomorphism $\phi :G\to H$ whose fibers form a group where the identity element is $N$ and multiplication of any two fibers $\phi ^{-1}(h_{1})$ and $\phi ^{-1}(h_{2})$ yields the fiber $\phi ^{-1}(h_{1}h_{2})$. (This group is the same group $G/N$ mentioned above.)
• There is some congruence relation on $G$ for which the equivalence class of the identity element is $N$.
• For all $n\in N$ and $g\in G,$ the commutator $[n,g]=n^{-1}g^{-1}ng$ is in $N.$
• Any two elements commute modulo the normal subgroup membership relation. That is, for all $g,h\in G,$ $gh\in N$ if and only if $hg\in N.$
Examples
For any group $G,$ the trivial subgroup $\{e\}$ consisting of just the identity element of $G$ is always a normal subgroup of $G.$ Likewise, $G$ itself is always a normal subgroup of $G.$ (If these are the only normal subgroups, then $G$ is said to be simple.)[6] Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup $[G,G].$[7][8] More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup.[9]
If $G$ is an abelian group then every subgroup $N$ of $G$ is normal, because $gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng.$ More generally, for any group $G$, every subgroup of the center $Z(G)$ of $G$ is normal in $G$. (In the special case that $G$ is abelian, the center is all of $G$, hence the fact that all subgroups of an abelian group are normal.) A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.[10]
A concrete example of a normal subgroup is the subgroup $N=\{(1),(123),(132)\}$ of the symmetric group $S_{3},$ consisting of the identity and both three-cycles. In particular, one can check that every coset of $N$ is either equal to $N$ itself or is equal to $(12)N=\{(12),(23),(13)\}.$ On the other hand, the subgroup $H=\{(1),(12)\}$ is not normal in $S_{3}$ since $(123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123).$[11] This illustrates the general fact that any subgroup $H\leq G$ of index two is normal.
As an example of a normal subgroup within a matrix group, consider the general linear group $\mathrm {GL} _{n}(\mathbf {R} )$ of all invertible $n\times n$ matrices with real entries under the operation of matrix multiplication and its subgroup $\mathrm {SL} _{n}(\mathbf {R} )$ of all $n\times n$ matrices of determinant 1 (the special linear group). To see why the subgroup $\mathrm {SL} _{n}(\mathbf {R} )$ is normal in $\mathrm {GL} _{n}(\mathbf {R} )$, consider any matrix $X$ in $\mathrm {SL} _{n}(\mathbf {R} )$ and any invertible matrix $A$. Then using the two important identities $\det(AB)=\det(A)\det(B)$ and $\det(A^{-1})=\det(A)^{-1}$, one has that $\det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1$, and so $AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )$ as well. This means $\mathrm {SL} _{n}(\mathbf {R} )$ is closed under conjugation in $\mathrm {GL} _{n}(\mathbf {R} )$, so it is a normal subgroup.[lower-alpha 1]
In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.[12]
The translation group is a normal subgroup of the Euclidean group in any dimension.[13] This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin.
Properties
• If $H$ is a normal subgroup of $G,$ and $K$ is a subgroup of $G$ containing $H,$ then $H$ is a normal subgroup of $K.$[14]
• A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8.[15] However, a characteristic subgroup of a normal subgroup is normal.[16] A group in which normality is transitive is called a T-group.[17]
• The two groups $G$ and $H$ are normal subgroups of their direct product $G\times H.$
• If the group $G$ is a semidirect product $G=N\rtimes H,$ then $N$ is normal in $G,$ though $H$ need not be normal in $G.$
• If $M$ and $N$ are normal subgroups of an additive group $G$ such that $G=M+N$ and $M\cap N=\{0\}$, then $G=M\oplus N.$[18]
• Normality is preserved under surjective homomorphisms;[19] that is, if $G\to H$ is a surjective group homomorphism and $N$ is normal in $G,$ then the image $f(N)$ is normal in $H.$
• Normality is preserved by taking inverse images;[19] that is, if $G\to H$ is a group homomorphism and $N$ is normal in $H,$ then the inverse image $f^{-1}(N)$ is normal in $G.$
• Normality is preserved on taking direct products;[20] that is, if $N_{1}\triangleleft G_{1}$ and $N_{2}\triangleleft G_{2},$ then $N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}.$
• Every subgroup of index 2 is normal. More generally, a subgroup, $H,$ of finite index, $n,$ in $G$ contains a subgroup, $K,$ normal in $G$ and of index dividing $n!$ called the normal core. In particular, if $p$ is the smallest prime dividing the order of $G,$ then every subgroup of index $p$ is normal.[21]
• The fact that normal subgroups of $G$ are precisely the kernels of group homomorphisms defined on $G$ accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,[22] a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.
Lattice of normal subgroups
Given two normal subgroups, $N$ and $M,$ of $G,$ their intersection $N\cap M$and their product $NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}$ are also normal subgroups of $G.$
The normal subgroups of $G$ form a lattice under subset inclusion with least element, $\{e\},$ and greatest element, $G.$ The meet of two normal subgroups, $N$ and $M,$ in this lattice is their intersection and the join is their product.
The lattice is complete and modular.[20]
Normal subgroups, quotient groups and homomorphisms
If $N$ is a normal subgroup, we can define a multiplication on cosets as follows:
$\left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N.$
This relation defines a mapping $G/N\times G/N\to G/N.$ To show that this mapping is well-defined, one needs to prove that the choice of representative elements $a_{1},a_{2}$ does not affect the result. To this end, consider some other representative elements $a_{1}'\in a_{1}N,a_{2}'\in a_{2}N.$ Then there are $n_{1},n_{2}\in N$ such that $a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}.$ It follows that
$a_{1}'a_{2}'N=a_{1}n_{1}a_{2}n_{2}N=a_{1}a_{2}n_{1}'n_{2}N=a_{1}a_{2}N,$
where we also used the fact that $N$ is a normal subgroup, and therefore there is $n_{1}'\in N$ such that $n_{1}a_{2}=a_{2}n_{1}'.$ This proves that this product is a well-defined mapping between cosets.
With this operation, the set of cosets is itself a group, called the quotient group and denoted with $G/N.$ There is a natural homomorphism, $f:G\to G/N,$ given by $f(a)=aN.$ This homomorphism maps $N$ into the identity element of $G/N,$ which is the coset $eN=N,$[23] that is, $\ker(f)=N.$
In general, a group homomorphism, $f:G\to H$ sends subgroups of $G$ to subgroups of $H.$ Also, the preimage of any subgroup of $H$ is a subgroup of $G.$ We call the preimage of the trivial group $\{e\}$ in $H$ the kernel of the homomorphism and denote it by $\ker f.$ As it turns out, the kernel is always normal and the image of $G,f(G),$ is always isomorphic to $G/\ker f$ (the first isomorphism theorem).[24] In fact, this correspondence is a bijection between the set of all quotient groups of $G,G/N,$ and the set of all homomorphic images of $G$ (up to isomorphism).[25] It is also easy to see that the kernel of the quotient map, $f:G\to G/N,$ is $N$ itself, so the normal subgroups are precisely the kernels of homomorphisms with domain $G.$[26]
See also
Operations taking subgroups to subgroups
• Normalizer
• Conjugate closure
• Normal core
Subgroup properties complementary (or opposite) to normality
• Malnormal subgroup
• Contranormal subgroup
• Abnormal subgroup
• Self-normalizing subgroup
Subgroup properties stronger than normality
• Characteristic subgroup
• Fully characteristic subgroup
Subgroup properties weaker than normality
• Subnormal subgroup
• Ascendant subgroup
• Descendant subgroup
• Quasinormal subgroup
• Seminormal subgroup
• Conjugate permutable subgroup
• Modular subgroup
• Pronormal subgroup
• Paranormal subgroup
• Polynormal subgroup
• C-normal subgroup
Related notions in algebra
• Ideal (ring theory)
Notes
1. In other language: $\det $ is a homomorphism from $\mathrm {GL} _{n}(\mathbf {R} )$ to the multiplicative subgroup $\mathbf {R} ^{\times }$, and $\mathrm {SL} _{n}(\mathbf {R} )$ is the kernel. Both arguments also work over the complex numbers, or indeed over an arbitrary field.
References
1. Bradley 2010, p. 12.
2. Cantrell 2000, p. 160.
3. Dummit & Foote 2004.
4. Hungerford 2003, p. 41.
5. Fraleigh 2003, p. 141.
6. Robinson 1996, p. 16.
7. Hungerford 2003, p. 45.
8. Hall 1999, p. 138.
9. Hall 1999, p. 32.
10. Hall 1999, p. 190.
11. Judson 2020, Section 10.1.
12. Bergvall et al. 2010, p. 96.
13. Thurston 1997, p. 218.
14. Hungerford 2003, p. 42.
15. Robinson 1996, p. 17.
16. Robinson 1996, p. 28.
17. Robinson 1996, p. 402.
18. Hungerford 2013, p. 290.
19. Hall 1999, p. 29.
20. Hungerford 2003, p. 46.
21. Robinson 1996, p. 36.
22. Dõmõsi & Nehaniv 2004, p. 7.
23. Hungerford 2003, pp. 42–43.
24. Hungerford 2003, p. 44.
25. Robinson 1996, p. 20.
26. Hall 1999, p. 27.
Bibliography
• Bergvall, Olof; Hynning, Elin; Hedberg, Mikael; Mickelin, Joel; Masawe, Patrick (16 May 2010). "On Rubik's Cube" (PDF). KTH. {{cite journal}}: Cite journal requires |journal= (help)
• Cantrell, C.D. (2000). Modern Mathematical Methods for Physicists and Engineers. Cambridge University Press. ISBN 978-0-521-59180-5.
• Dõmõsi, Pál; Nehaniv, Chrystopher L. (2004). Algebraic Theory of Automata Networks. SIAM Monographs on Discrete Mathematics and Applications. SIAM.
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
• Fraleigh, John B. (2003). A First Course in Abstract Algebra (7th ed.). Addison-Wesley. ISBN 978-0-321-15608-2.
• Hall, Marshall (1999). The Theory of Groups. Providence: Chelsea Publishing. ISBN 978-0-8218-1967-8.
• Hungerford, Thomas (2003). Algebra. Graduate Texts in Mathematics. Springer.
• Hungerford, Thomas (2013). Abstract Algebra: An Introduction. Brooks/Cole Cengage Learning.
• Judson, Thomas W. (2020). Abstract Algebra: Theory and Applications.
• Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. ISBN 978-1-4612-6443-9. Zbl 0836.20001.
• Thurston, William (1997). Levy, Silvio (ed.). Three-dimensional geometry and topology, Vol. 1. Princeton Mathematical Series. Princeton University Press. ISBN 978-0-691-08304-9.
• Bradley, C. J. (2010). The mathematical theory of symmetry in solids : representation theory for point groups and space groups. Oxford New York: Clarendon Press. ISBN 978-0-19-958258-7. OCLC 859155300.
Further reading
• I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.
External links
• Weisstein, Eric W. "normal subgroup". MathWorld.
• Normal subgroup in Springer's Encyclopedia of Mathematics
• Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year
• Timothy Gowers, Normal subgroups and quotient groups
• John Baez, What's a Normal Subgroup?
| Wikipedia |
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:
$A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {{a}_{ji}}}$
For matrices with symmetry over the real number field, see Symmetric matrix.
or in matrix form:
$A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.$
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix $A$ is denoted by $A^{\mathsf {H}},$ then the Hermitian property can be written concisely as
$A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}$
Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are $A^{\mathsf {H}}=A^{\dagger }=A^{\ast },$ although in quantum mechanics, $A^{\ast }$ typically means the complex conjugate only, and not the conjugate transpose.
Alternative characterizations
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
Equality with the adjoint
A square matrix $A$ is Hermitian if and only if it is equal to its adjoint, that is, it satisfies
$\langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,$
for any pair of vectors $\mathbf {v} ,\mathbf {w} ,$ where $\langle \cdot ,\cdot \rangle $ denotes the inner product operation.
This is also the way that the more general concept of self-adjoint operator is defined.
Reality of quadratic forms
An $n\times {}n$ matrix $A$ is Hermitian if and only if
$\langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad \mathbf {v} \in \mathbb {C} ^{n}.$
Spectral properties
A square matrix $A$ is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.
Applications
Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a$ of an operator ${\hat {A}}$ on some quantum state $|\psi \rangle $ is one of the possible measurement outcomes of the operator, which necessitates the need for operators with real eigenvalues.
Examples and solutions
In this section, the conjugate transpose of matrix $A$ is denoted as $A^{\mathsf {H}},$ the transpose of matrix $A$ is denoted as $A^{\mathsf {T}}$ and conjugate of matrix $A$ is denoted as ${\overline {A}}.$
See the following example:
${\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}$
The diagonal elements must be real, as they must be their own complex conjugate.
Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices and their generalizations. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,[1][2] which results in skew-Hermitian matrices.
Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix $A$ equals the product of a matrix with its conjugate transpose, that is, $A=BB^{\mathsf {H}},$ then $A$ is a Hermitian positive semi-definite matrix. Furthermore, if $B$ is row full-rank, then $A$ is positive definite.
Properties
Main diagonal values are real
The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.
Proof
By definition of the Hermitian matrix
$H_{ij}={\overline {H}}_{ji}$
so for i = j the above follows.
Only the main diagonal entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their off-diagonal elements, as long as diagonally-opposite entries are complex conjugates.
Symmetric
A matrix that has only real entries is symmetric if and only if it is Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.
Proof
$H_{ij}={\overline {H}}_{ji}$ by definition. Thus $H_{ij}=H_{ji}$ (matrix symmetry) if and only if $H_{ij}={\overline {H}}_{ij}$ ($H_{ij}$ is real).
So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit $i,$ then it becomes Hermitian.
Normal
Every Hermitian matrix is a normal matrix. That is to say, $AA^{\mathsf {H}}=A^{\mathsf {H}}A.$
Proof
$A=A^{\mathsf {H}},$ so $AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.$
Diagonalizable
The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A with dimension n are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an orthogonal basis of Cn consisting of n eigenvectors of A.
Sum of Hermitian matrices
The sum of any two Hermitian matrices is Hermitian.
Proof
$(A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},$
as claimed.
Inverse is Hermitian
The inverse of an invertible Hermitian matrix is Hermitian as well.
Proof
If $A^{-1}A=I,$ then $I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}},$ so $A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}$ as claimed.
Associative product of Hermitian matrices
The product of two Hermitian matrices A and B is Hermitian if and only if AB = BA.
Proof
$(AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}\ {\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.$
Thus $(AB)^{\mathsf {H}}=AB$ if and only if $AB=BA.$
Thus An is Hermitian if A is Hermitian and n is an integer.
ABA Hermitian
If A and B are Hermitian, then ABA is also Hermitian.
Proof
$(ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA$
vHAv is real for complex v
For an arbitrary complex valued vector v the product $\mathbf {v} ^{\mathsf {H}}A\mathbf {v} $ is real because of $\mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.$ This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system, e.g. total spin, which have to be real.
Complex Hermitian forms vector space over R
The Hermitian complex n-by-n matrices do not form a vector space over the complex numbers, C, since the identity matrix In is Hermitian, but i In is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n2-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:
$E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})$
together with the set of matrices of the form
${\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$
and the matrices
${\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$
where $i$ denotes the imaginary unit, $i={\sqrt {-1}}~.$
An example is that the four Pauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over R.
Eigendecomposition
If n orthonormal eigenvectors $\mathbf {u} _{1},\dots ,\mathbf {u} _{n}$ of a Hermitian matrix are chosen and written as the columns of the matrix U, then one eigendecomposition of A is $A=U\Lambda U^{\mathsf {H}}$ where $UU^{\mathsf {H}}=I=U^{\mathsf {H}}U$ and therefore
$A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},$
where $\lambda _{j}$ are the eigenvalues on the diagonal of the diagonal matrix $\Lambda .$
Singular values[3]
The singular values of $A$ are the absolute values of its eigenvalues:
Since $A$ has an eigendecomposition $A=U\Lambda U^{H}$, where $U$ is a unitary matrix (its columns are orthonormal vectors; see above), a singular value decomposition of $A$ is $A=U|\Lambda |{\text{sgn}}(\Lambda )U^{H}$, where $|\Lambda |$ and ${\text{sgn}}(\Lambda )$ are diagonal matrices containing the absolute values $|\lambda |$ and signs ${\text{sgn}}(\lambda )$ of $A$'s eigenvalues, respectively. $\operatorname {sgn}(\Lambda )U^{H}$ is unitary, since the columns of $U^{H}$ are only getting multiplied by $\pm 1$. $|\Lambda |$ contains the singular values of $A$, namely, the absolute values of its eigenvalues.
Real determinant
The determinant of a Hermitian matrix is real:
Proof
$\det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}$ Therefore if $A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.$
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
Decomposition into Hermitian and skew-Hermitian matrices
Additional facts related to Hermitian matrices include:
• The sum of a square matrix and its conjugate transpose $\left(A+A^{\mathsf {H}}\right)$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $\left(A-A^{\mathsf {H}}\right)$ is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B. This is known as the Toeplitz decomposition of C.[4]: 227
$C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)$
Rayleigh quotient
Main article: Rayleigh quotient
In mathematics, for a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient[5] $R(M,\mathbf {x} ),$ is defined as:[4]: p. 234 [6]
$R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.$
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose $\mathbf {x} ^{\mathsf {H}}$ to the usual transpose $\mathbf {x} ^{\mathsf {T}}.$ $R(M,c\mathbf {x} )=R(M,\mathbf {x} )$ for any non-zero real scalar $c.$ Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
It can be shown[4] that, for a given matrix, the Rayleigh quotient reaches its minimum value $\lambda _{\min }$ (the smallest eigenvalue of M) when $\mathbf {x} $ is $\mathbf {v} _{\min }$ (the corresponding eigenvector). Similarly, $R(M,\mathbf {x} )\leq \lambda _{\max }$ and $R(M,\mathbf {v} _{\max })=\lambda _{\max }.$
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, $\lambda _{\max }$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to M associates the Rayleigh quotient R(M, x) for a fixed x and M varying through the algebra would be referred to as "vector state" of the algebra.
See also
• Complex symmetric matrix – Matrix equal to its transposePages displaying short descriptions of redirect targets
• Haynsworth inertia additivity formula – Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix
• Hermitian form – Generalization of a bilinear formPages displaying short descriptions of redirect targets
• Normal matrix – Matrix that commutes with its conjugate transpose
• Schur–Horn theorem – Characterizes the diagonal of a Hermitian matrix with given eigenvalues
• Self-adjoint operator – Linear operator equal to its own adjoint
• Skew-Hermitian matrix – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix)
• Unitary matrix – Complex matrix whose conjugate transpose equals its inverse
• Vector space – Algebraic structure in linear algebra
References
1. Frankel, Theodore (2004). The Geometry of Physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7.
2. Physics 125 Course Notes at California Institute of Technology
3. Trefethan, Lloyd N.; Bau, III, David (1997). Numerical linear algebra. Philadelphia, PA, USA: SIAM. p. 34. ISBN 0-89871-361-7.
4. Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402.
5. Also known as the Rayleigh–Ritz ratio; named after Walther Ritz and Lord Rayleigh.
6. Parlet B. N. The symmetric eigenvalue problem, SIAM, Classics in Applied Mathematics,1998
External links
• "Hermitian matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
• "Hermitian Matrices". MathPages.com.
Matrix classes
Explicitly constrained entries
• Alternant
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| Wikipedia |
Exclusive or
Exclusive or or exclusive disjunction or exclusive alternation, also known as non-equivalence which is the negation of equivalence, is a logical operation that is true if and only if its arguments differ (one is true, the other is false).[1]
Exclusive or
XOR
Truth table$(0110)$
Logic gate
Normal forms
Disjunctive${\overline {x}}\cdot y+x\cdot {\overline {y}}$
Conjunctive$({\overline {x}}+{\overline {y}})\cdot (x+y)$
Zhegalkin polynomial$x\oplus y$
Post's lattices
0-preservingyes
1-preservingno
Monotoneno
Affineyes
It is symbolized by the prefix operator $J$[2]: 16 and by the infix operators XOR (/ˌɛks ˈɔːr/, /ˌɛks ˈɔː/, /ˈksɔːr/ or /ˈksɔː/), EOR, EXOR, ${\dot {\vee }}$, ${\overline {\vee }}$, ${\underline {\vee }}$, ⩛, $\oplus $, $\nleftrightarrow $ and $\not \equiv $.
It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both" or "either one or the other". This could be written as "A or B, but not, A and B".
XOR is equivalent to logical inequality (NEQ) since it is true only when the inputs are different (one is true, and one is false). The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same, which is equivalent to logical equality (EQ).
Since it is associative, it may be considered to be an n-ary operator which is true if and only if an odd number of arguments are true. That is, a XOR b XOR ... may be treated as XOR(a,b,...).
Definition
The truth table of $A\oplus B$ shows that it outputs true whenever the inputs differ:
$A$ $B$ $A\oplus B$
FalseFalseFalse
FalseTrueTrue
TrueFalseTrue
TrueTrueFalse
Equivalences, elimination, and introduction
Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction $p\nleftrightarrow q$, also denoted by $p\operatorname {?} q$ or $Jpq$, can be expressed in terms of the logical conjunction ("logical and", $\wedge $), the disjunction ("logical or", $\lor $), and the negation ($\lnot $) as follows:
${\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}$
The exclusive disjunction $p\nleftrightarrow q$ can also be expressed in the following way:
${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}$
This representation of XOR may be found useful when constructing a circuit or network, because it has only one $\lnot $ operation and small number of $\wedge $ and $\lor $ operations. A proof of this identity is given below:
${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}$
It is sometimes useful to write $p\nleftrightarrow q$ in the following way:
${\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}$
or:
${\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}$
This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof.
The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence.
In summary, we have, in mathematical and in engineering notation:
${\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}$
Negation of the operator
The spirit of De Morgan's laws can be applied, we have:
$\lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.$
Relation to modern algebra
Although the operators $\wedge $ (conjunction) and $\lor $ (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:
The systems $(\{T,F\},\wedge )$ and $(\{T,F\},\lor )$ are monoids, but neither is a group. This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring.
However, the system using exclusive or $(\{T,F\},\oplus )$ is an abelian group. The combination of operators $\wedge $ and $\oplus $ over elements $\{T,F\}$ produce the well-known two-element field $\mathbb {F} _{2}$. This field can represent any logic obtainable with the system $(\land ,\lor )$ and has the added benefit of the arsenal of algebraic analysis tools for fields.
More specifically, if one associates $F$ with 0 and $T$ with 1, one can interpret the logical "AND" operation as multiplication on $\mathbb {F} _{2}$ and the "XOR" operation as addition on $\mathbb {F} _{2}$:
${\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}$
The description of a Boolean function as a polynomial in $\mathbb {F} _{2}$, using this basis, is called the function's algebraic normal form.[3]
Exclusive or in natural language
Disjunction is often understood exclusively in natural languages. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.[4][5]
1. Mary is a singer or a poet.
However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.[4]
2. Mary is either a singer or a poet or both.
3. Nobody ate either rice or beans.
Examples such as the above have motivated analyses of the exclusivity inference as pragmatic conversational implicatures calculated on the basis of an inclusive semantics. Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity. However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it.[4]
This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French soit... soit.[4]
Alternative symbols
The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen:
• $+$ was used by George Boole in 1847.[6] Although Boole used $+$ mainly on classes, he also considered the case that $x,y$ are propositions in $x+y$, and at the time $+$ is a connective. Furthermore, Boole used it exclusively. Although such use doesn't show the relationship between inclusive disjunction (for which $\vee $ is almost fixedly used nowadays) and exclusive disjunction, and may also bright about confusions with its other uses, some classical and modern textbooks still keep such use.[7][8]
• ${\overline {\vee }}$ was used by Christine Ladd-Franklin in 1883.[9] Strictly speaking, Ladd used $A\operatorname {\overline {\vee }} B$ to express "$A$ is-not $B$" or "No $A$ is $B$", i.e., used ${\overline {\vee }}$ as exclusions, while implicitly ${\overline {\vee }}$ has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".
• $\not =$, denoting the negation of equivalence, was used by Ernst Schröder in 1890,[10]: 307 Although the usage of $=$ as equivalence could be dated back to George Boole in 1847,[6] during the 40 years after Boole, his followers, such as Charles Sanders Peirce, Hugh MacColl, Giuseppe Peano and so on, didn't use $\not =$ as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
• $\circ $ was used by Giuseppe Peano in 1894: "$a\circ b=a-b\,\cup \,b-a$. The sign $\circ $ corresponds to Latin aut; the sign $\cup $ to vel."[11]: 10 Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use $\cup $ as inclusive disjunction.
• $\vee \vee $ was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.[12]: 76
• $\oplus $ was used by Claude Shannon in 1938.[13] Shannon borrowed the symbol as exclusive disjunction from Edward Vermilye Huntington in 1904.[14] Huntington borrowed the symbol from Gottfried Wilhelm Leibniz in 1890 (the original date is not definitely known, but almost certainly it's written after 1685; and 1890 is the publishing time).[15] While both Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction (logical sum) too, and in 1933 used $+$ as inclusive disjunction.[16]
• $\not \equiv $, also denoting the negation of equivalence, was used by Alonzo Church in 1944.[17]
• $J$ (as a prefix operator, $J\phi \psi $) was used by Józef Maria Bocheński in 1949.[2]: 16 Somebody[18] may mistake that it's Jan Łukasiewicz who is the first to use $J$ for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929[19] nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which $J$ for exclusive disjunction appeared at the first time. Bocheński's usage of $J$ as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.
• ^, the caret, has been used in several programming languages to denote the bitwise exclusive or operator, beginning with C[20] and also including C++, C#, D, Java, Perl, Ruby, PHP and Python.
• The symmetric difference of two sets $S$ and $T$, which may be interpreted as their elementwise exclusive or, has variously been denoted as $S\ominus T$, $S\mathop {\triangledown } T$, or $S\mathop {\vartriangle } T$.[21]
Properties
Commutativity: yes
$A\oplus B$ $\Leftrightarrow $ $B\oplus A$
$\Leftrightarrow $
Associativity: yes
$~A$ $~~~\oplus ~~~$ $(B\oplus C)$ $\Leftrightarrow $ $(A\oplus B)$ $~~~\oplus ~~~$ $~C$
$~~~\oplus ~~~$ $\Leftrightarrow $ $\Leftrightarrow $ $~~~\oplus ~~~$
Distributivity:
The exclusive or doesn't distribute over any binary function (not even itself), but logical conjunction distributes over exclusive or. $C\land (A\oplus B)=(C\land A)\oplus (C\land B)$ (Conjunction and exclusive or form the multiplication and addition operations of a field GF(2), and as in any field they obey the distributive law.)
Idempotency: no
$~A~$ $~\oplus ~$ $~A~$ $\Leftrightarrow $ $~0~$ $\nLeftrightarrow $ $~A~$
$~\oplus ~$ $\Leftrightarrow $ $\nLeftrightarrow $
Monotonicity: no
$A\rightarrow B$ $\nRightarrow $ $(A\oplus C)$ $\rightarrow $ $(B\oplus C)$
$\nRightarrow $ $\Leftrightarrow $ $\rightarrow $
Truth-preserving: no
When all inputs are true, the output is not true.
$A\land B$ $\nRightarrow $ $A\oplus B$
$\nRightarrow $
Falsehood-preserving: yes
When all inputs are false, the output is false.
$A\oplus B$ $\Rightarrow $ $A\lor B$
$\Rightarrow $
Walsh spectrum: (2,0,0,−2)
Non-linearity: 0
The function is linear.
If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2.
Computer science
Bitwise operation
Main article: Bitwise operation
Exclusive disjunction is often used for bitwise operations. Examples:
• 1 XOR 1 = 0
• 1 XOR 0 = 1
• 0 XOR 1 = 1
• 0 XOR 0 = 0
• 11102 XOR 10012 = 01112 (this is equivalent to addition without carry)
As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two n-bit strings is identical to the standard vector of addition in the vector space $(\mathbb {Z} /2\mathbb {Z} )^{n}$.
In computer science, exclusive disjunction has several uses:
• It tells whether two bits are unequal.
• It is an optional bit-flipper (the deciding input chooses whether to invert the data input).
• It tells whether there is an odd number of 1 bits ($A\oplus B\oplus C\oplus D\oplus E$ is true if and only if an odd number of the variables are true), which is equal to the parity bit returned by a parity function.
In logical circuits, a simple adder can be made with an XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output.
On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) instead of loading and storing the value zero.
In simple threshold-activated neural networks, modeling the XOR function requires a second layer because XOR is not a linearly separable function.
Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems.
Exclusive-or is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR).
Similarly, XOR can be used in generating entropy pools for hardware random number generators. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.[22]
XOR is used in RAID 3–6 for creating parity information. For example, RAID can "back up" bytes 100111002 and 011011002 from two (or more) hard drives by XORing the just mentioned bytes, resulting in (111100002) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing 011011002 is lost, 100111002 and 111100002 can be XORed to recover the lost byte.[23]
XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow.
XOR can be used to swap two numeric variables in computers, using the XOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice.
XOR linked lists leverage XOR properties in order to save space to represent doubly linked list data structures.
In computer graphics, XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes.
Encodings
It is also called "not left-right arrow" (\nleftrightarrow) in LaTeX-based markdown ($\nleftrightarrow $). Apart from the ASCII codes, the operator is encoded at U+22BB ⊻ XOR (⊻) and U+2295 ⊕ CIRCLED PLUS (⊕, ⊕), both in block mathematical operators.
See also
• Material conditional • (Paradox)
• Affirming a disjunct
• Ampheck
• Controlled NOT gate
• Disjunctive syllogism
• Inclusive or
• Involution
• List of Boolean algebra topics
• Logical graph
• Logical value
• Propositional calculus
• Rule 90
• XOR cipher
• XOR gate
• XOR linked list
Notes
1. Germundsson, Roger; Weisstein, Eric. "XOR". MathWorld. Wolfram Research. Retrieved 17 June 2015.
2. Bocheński, J. M. (1949). Précis de logique mathématique (PDF) (in French). The Netherlands: F. G. Kroonder, Bussum, Pays-Bas. Translated as Bocheński, J. M. (1959). A Precis of Mathematical Logic. Translated by Bird, O. Dordrecht, Holland: D. Reidel Publishing Company. doi:10.1007/978-94-017-0592-9.
3. Joux, Antoine (2009). "9.2: Algebraic normal forms of Boolean functions". Algorithmic Cryptanalysis. CRC Press. pp. 285–286. ISBN 9781420070033.
4. Aloni, Maria (2016). "Disjunction". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2016 ed.). Metaphysics Research Lab, Stanford University. Retrieved 2020-09-03.
5. Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See Chapter 3, "The First Myth of 'Or'":
Jennings, R. E. (1994). The Genealogy of Disjunction. New York: Oxford University Press.
6. Boole, G. (1847). The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17.
7. Enderton, H. (2001) [1972]. A Mathematical Introduction to Logic (2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51.
8. Rautenberg, W. (2010) [2006]. A Concise Introduction to Mathematical Logic (3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3.
9. Ladd, Christine (1883). "On the Algebra of Logic". In Peirce, C. S. (ed.). Studies in Logic by Members of the Johns Hopkins University. Boston: Little, Brown & Company. pp. 17–71.
10. Schröder, E. (1890). Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band (in German). Leipzig: Druck und Verlag B. G. Teubner. Reprinted by Thoemmes Press in 2000.
11. Peano, G. (1894). Notations de logique mathématique. Introduction au formulaire de mathématique. Turin: Fratelli Bocca. Reprinted in Peano, G. (1958). Opere Scelte, Volume II. Roma: Edizioni Cremonese. pp. 123–176.
12. ГРАДШТЕЙН, И. С. (1959) [1936]. ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ (in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ. Translated as Gradshtein, I. S. (1963). Direct and Converse Theorems: The Elements of Symbolic Logic. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press.
13. Shannon, C. E. (1938). "A Symbolic Analysis of Relay and Switching Circuits" (PDF). Transactions of the American Institute of Electrical Engineers. 57 (12): 713–723. doi:10.1109/T-AIEE.1938.5057767. hdl:1721.1/11173. S2CID 51638483.
14. Huntington, E. V. (1904). "Sets of Independent Postulates for the Algebra of Logic". Transactions of the American Mathematical Society. 5 (3): 288–309.
15. Leibniz, G. W. (1890) [16??/17??]. Gerhardt, C. I. (ed.). Die philosophischen Schriften, Siebter Band (in German). Berlin: Weidmann. p. 237. Retrieved 7 July 2023.
16. Huntington, E. V. (1933). "New Sets of Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica". Transactions of the American Mathematical Society. 35 (1): 274–304.
17. Church, A. (1996) [1944]. Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37.
18. Craig, Edward (1998). Routledge Encyclopedia of Philosophy, Volume 8. Taylor & Francis. p. 496. ISBN 978-0-41507310-3.
19. Łukasiewicz, Jan (1929). Elementy logiki matematycznej [Elements of Mathematical Logic] (in Polish) (1 ed.). Warsaw, Poland: Państwowe Wydawnictwo Naukowe.
20. Kernighan, Brian W.; Ritchie, Dennis M. (1978). "2.9: Bitwise logical operators". The C Programming Language. Prentice-Hall. pp. 44–46.
21. Weisstein, Eric W. "Symmetric Difference". MathWorld.
22. Davies, Robert B (28 February 2002). "Exclusive OR (XOR) and hardware random number generators" (PDF). Retrieved 28 August 2013.
23. Nobel, Rickard (26 July 2011). "How RAID 5 actually works". Retrieved 23 March 2017.
External links
Wikimedia Commons has media related to Exclusive disjunction.
Look up exclusive or or XOR in Wiktionary, the free dictionary.
• All About XOR
• Proofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University
Common logical connectives
• Tautology/True $\top $
• Alternative denial (NAND gate) $\uparrow $
• Converse implication $\leftarrow $
• Implication (IMPLY gate) $\rightarrow $
• Disjunction (OR gate) $\lor $
• Negation (NOT gate) $\neg $
• Exclusive or (XOR gate) $\not \leftrightarrow $
• Biconditional (XNOR gate) $\leftrightarrow $
• Statement (Digital buffer)
• Joint denial (NOR gate) $\downarrow $
• Nonimplication (NIMPLY gate) $\nrightarrow $
• Converse nonimplication $\nleftarrow $
• Conjunction (AND gate) $\land $
• Contradiction/False $\bot $
Philosophy portal
| Wikipedia |
Logical NOR
In Boolean logic, logical NOR or non-disjunction or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both of p and q are false. It is logically equivalent to $\neg (p\lor q)$ and $\neg p\land \neg q$, where the symbol $\neg $ signifies logical negation, $\lor $ signifies OR, and $\land $ signifies AND.
Logical NOR
NOR
Definition${\overline {x+y}}$
Truth table$(0001)$
Logic gate
Normal forms
Disjunctive${\overline {x}}\cdot {\overline {y}}$
Conjunctive${\overline {x}}\cdot {\overline {y}}$
Zhegalkin polynomial$1\oplus x\oplus y\oplus xy$
Post's lattices
0-preservingno
1-preservingno
Monotoneno
Affineno
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Non-disjunction is usually denoted as $\downarrow $ or ${\overline {\vee }}$ or $X$ (prefix) or $\operatorname {NOR} $.
As with its dual, the NAND operator (also known as the Sheffer stroke—symbolized as either $\uparrow $, $\mid $ or $/$), NOR can be used by itself, without any other logical operator, to constitute a logical formal system (making NOR functionally complete).
The computer used in the spacecraft that first carried humans to the moon, the Apollo Guidance Computer, was constructed entirely using NOR gates with three inputs.[1]
Definition
The NOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false. In other words, it produces a value of false if and only if at least one operand is true.
Truth table
The truth table of $P\downarrow Q$ is as follows:
$P$ $Q$ $P\downarrow Q$
TrueTrueFalse
TrueFalseFalse
FalseTrueFalse
FalseFalseTrue
Logical equivalences
The logical NOR $\downarrow $ is the negation of the disjunction:
$P\downarrow Q$ $\Leftrightarrow $ $\neg (P\lor Q)$
$\Leftrightarrow $ $\neg $
Alternative notations and names
Peirce is the first to show the functional completeness of non-disjunction while he doesn't publish his result.[2][3] Peirce used ${\overline {\curlywedge }}$ for non-conjunction and $\curlywedge $ for non-disjunction (in fact, what Peirce himself used is $\curlywedge $ and he didn't introduce ${\overline {\curlywedge }}$ while Peirce's editors made such disambiguated use).[3] Peirce called $\curlywedge $ as ampheck (from Ancient Greek ἀμφήκης, amphēkēs, "cutting both ways").[3]
In 1911, Stamm was the first to publish a description of both non-conjunction (using $\sim $, the Stamm hook), and non-disjunction (using $*$, the Stamm star), and showed their functional completeness.[4]Zach, R. (2023-02-18). "Sheffer stroke before Sheffer: Edward Stamm". Retrieved 2023-07-02.</ref> Note that most uses in logical notation of $\sim $ use this for negation.
In 1913, Sheffer described non-disjunction and showed its functional completeness. Sheffer used $\mid $ for non-conjunction, and $\wedge $ for non-disjunction.
In 1935, Webb described non-disjunction for $n$-valued logic, and use $\mid $ for the operator. So some people call it Webb operator,[5] Webb operation[6] or Webb function.[7]
In 1940, Quine also described non-disjunction and use $\downarrow $ for the operator.[8] So some people call the operator Peirce arrow or Quine dagger.
In 1944, Church also described non-disjunction and use ${\overline {\vee }}$ for the operator.[9]
In 1954, Bocheński used $X$ in $Xpq$ for non-disjunction in Polish notation.[10]
Properties
Logical NOR does not possess any of the five qualities (truth-preserving, false-preserving, linear, monotonic, self-dual) required to be absent from at least one member of a set of functionally complete operators. Thus, the set containing only NOR suffices as a complete set.
Other Boolean operations in terms of the logical NOR
NOR has the interesting feature that all other logical operators can be expressed by interlaced NOR operations. The logical NAND operator also has this ability.
Expressed in terms of NOR $\downarrow $, the usual operators of propositional logic are:
$\neg P$ $\Leftrightarrow $ $P\downarrow P$
$\neg $ $\Leftrightarrow $
$P\rightarrow Q$ $\Leftrightarrow $ ${\Big (}(P\downarrow P)\downarrow Q{\Big )}$ $\downarrow $ ${\Big (}(P\downarrow P)\downarrow Q{\Big )}$
$\Leftrightarrow $ $\downarrow $
$P\land Q$ $\Leftrightarrow $ $(P\downarrow P)$ $\downarrow $ $(Q\downarrow Q)$
$\Leftrightarrow $ $\downarrow $
$P\lor Q$ $\Leftrightarrow $ $(P\downarrow Q)$ $\downarrow $ $(P\downarrow Q)$
$\Leftrightarrow $ $\downarrow $
See also
• Bitwise NOR
• Boolean algebra
• Boolean domain
• Boolean function
• Functional completeness
• NOR gate
• Propositional logic
• Sole sufficient operator
• Sheffer stroke as symbol for the logical NAND
References
1. Hall, Eldon C. (1996). Journey to the Moon: The History of the Apollo Guidance Computer. Reston, Virginia, USA: American Institute of Aeronautics and Astronautics. p. 196. ISBN 1-56347-185-X.
2. Peirce, C. S. (1933) [1880]. "A Boolian Algebra with One Constant". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 13–18.
3. Peirce, C. S. (1933) [1902]. "The Simplest Mathematics". In Hartshorne, C.; Weiss, P. (eds.). Collected Papers of Charles Sanders Peirce, Volume IV The Simplest Mathematics. Massachusetts: Harvard University Press. pp. 189–262.
4. Stamm, Edward Bronisław [in Polish] (1911). "Beitrag zur Algebra der Logik". Monatshefte für Mathematik und Physik (in German). 22 (1): 137–149. doi:10.1007/BF01742795. S2CID 119816758.
5. Webb, Donald Loomis (May 1935). "Generation of any n-valued logic by one binary operation". Proceedings of the National Academy of Sciences. USA: National Academy of Sciences.
6. Vasyukevich, Vadim O. (2011). "1.10 Venjunctive Properties (Basic Formulae)". Written at Riga, Latvia. Asynchronous Operators of Sequential Logic: Venjunction & Sequention — Digital Circuits Analysis and Design. Lecture Notes in Electrical Engineering (LNEE). Vol. 101 (1st ed.). Berlin / Heidelberg, Germany: Springer-Verlag. p. 20. doi:10.1007/978-3-642-21611-4. ISBN 978-3-642-21610-7. ISSN 1876-1100. LCCN 2011929655. p. 20: Historical background […] Logical operator NOR named Peirce arrow and also known as Webb-operation. (xiii+1+123+7 pages) (NB. The back cover of this book erroneously states volume 4, whereas it actually is volume 101.)
7. Freimann, Michael; Renfro, Dave L.; Webb, Norman (2018-05-24) [2017-02-10]. "Who is Donald L. Webb?". History of Science and Mathematics. Stack Exchange. Archived from the original on 2023-05-18. Retrieved 2023-05-18.
8. Quine, W. V (1981) [1940]. Mathematical Logic (Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45.
9. Church, A. (1996) [1944]. Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37.
10. Bocheński, J. M. (1954). Précis de logique mathématique (in French). Netherlands: F. G. Kroonder, Bussum, Pays-Bas. p. 11.
External links
• Media related to Logical NOR at Wikimedia Commons
Common logical connectives
• Tautology/True $\top $
• Alternative denial (NAND gate) $\uparrow $
• Converse implication $\leftarrow $
• Implication (IMPLY gate) $\rightarrow $
• Disjunction (OR gate) $\lor $
• Negation (NOT gate) $\neg $
• Exclusive or (XOR gate) $\not \leftrightarrow $
• Biconditional (XNOR gate) $\leftrightarrow $
• Statement (Digital buffer)
• Joint denial (NOR gate) $\downarrow $
• Nonimplication (NIMPLY gate) $\nrightarrow $
• Converse nonimplication $\nleftarrow $
• Conjunction (AND gate) $\land $
• Contradiction/False $\bot $
Philosophy portal
| Wikipedia |
Diamond operator
In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix (a b
c δ
)
in Γ0(N) where δ ≈ d mod N. The diamond operators form an abelian group and commute with the Hecke operators.
Unicode
In Unicode, the diamond operator is represented by the character U+22C4 ⋄ DIAMOND OPERATOR.[1]
Notes
1. "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-22.
References
• Diamond, Fred; Shurman, Jerry (2005), A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Berlin, New York: Springer-Verlag, ISBN 978-0-387-23229-4, MR 2112196
| Wikipedia |
Floor and ceiling functions
In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).[1]
Floor and ceiling functions
Floor function
Ceiling function
For example (floor), ⌊2.4⌋ = 2, ⌊−2.4⌋ = −3, and for ceiling; ⌈2.4⌉ = 3, and ⌈−2.4⌉ = −2.
Historically, the floor of x has been–and still is–called the integral part or integer part of x, often denoted [x] (as well as a variety of other notations).[2] However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers.
For n an integer, ⌊n⌋ = ⌈n⌉ = [n] = n.
Examples
x Floor ⌊x⌋ Ceiling ⌈x⌉ Fractional part {x}
2 2 2 0
2.4 2 3 0.4
2.9 2 3 0.9
−2.7 −3 −2 0.3
−2 −2 −2 0
Notation
The integral part or integer part of a number (partie entière in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula.
Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).[3] This remained the standard[4] in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations ⌊x⌋ and ⌈x⌉.[5][6] (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.
In some sources, boldface or double brackets ⟦x⟧ are used for floor, and reversed brackets ⟧x⟦ or ]x[ for ceiling.[7][8]
The fractional part is the sawtooth function, denoted by {x} for real x and defined by the formula
{x} = x − ⌊x⌋[9]
For all x,
0 ≤ {x} < 1.
These characters are provided in Unicode:
• U+2308 ⌈ LEFT CEILING (⌈, ⌈)
• U+2309 ⌉ RIGHT CEILING (⌉, ⌉)
• U+230A ⌊ LEFT FLOOR (⌊, ⌊)
• U+230B ⌋ RIGHT FLOOR (⌋, ⌋)
In the LaTeX typesetting system, these symbols can be specified with the \lceil, \rceil, \lfloor, and \rfloor commands in math mode, and extended in size using \left\lceil, \right\rceil, \left\lfloor, and \right\rfloor as needed.
Some authors define [x] as the round-toward-zero function, so [2.4] = 2 and [−2.4] = −2, and call it the "integer part". This is truncation to zero significant digits.
Definition and properties
Given real numbers x and y, integers m and n and the set of integers $\mathbb {Z} $, floor and ceiling may be defined by the equations
$\lfloor x\rfloor =\max\{m\in \mathbb {Z} \mid m\leq x\},$
$\lceil x\rceil =\min\{n\in \mathbb {Z} \mid n\geq x\}.$
Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
$x-1<m\leq x\leq n<x+1.$
where $\lfloor x\rfloor =m$ and $\lceil x\rceil =n$ may also be taken as the definition of floor and ceiling.
Equivalences
These formulas can be used to simplify expressions involving floors and ceilings.[10]
${\begin{aligned}\lfloor x\rfloor =m&\;\;{\mbox{ if and only if }}&m&\leq x<m+1,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&n-1&<x\leq n,\\\lfloor x\rfloor =m&\;\;{\mbox{ if and only if }}&x-1&<m\leq x,\\\lceil x\rceil =n&\;\;{\mbox{ if and only if }}&x&\leq n<x+1.\end{aligned}}$
In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals.
${\begin{aligned}x<n&\;\;{\mbox{ if and only if }}&\lfloor x\rfloor &<n,\\n<x&\;\;{\mbox{ if and only if }}&n&<\lceil x\rceil ,\\x\leq n&\;\;{\mbox{ if and only if }}&\lceil x\rceil &\leq n,\\n\leq x&\;\;{\mbox{ if and only if }}&n&\leq \lfloor x\rfloor .\end{aligned}}$
These formulas show how adding an integer n to the arguments affects the functions:
${\begin{aligned}\lfloor x+n\rfloor &=\lfloor x\rfloor +n,\\\lceil x+n\rceil &=\lceil x\rceil +n,\\\{x+n\}&=\{x\}.\end{aligned}}$
The above are never true if n is not an integer; however, for every x and y, the following inequalities hold:
${\begin{aligned}\lfloor x\rfloor +\lfloor y\rfloor &\leq \lfloor x+y\rfloor \leq \lfloor x\rfloor +\lfloor y\rfloor +1,\\\lceil x\rceil +\lceil y\rceil -1&\leq \lceil x+y\rceil \leq \lceil x\rceil +\lceil y\rceil .\end{aligned}}$
Monotonicity
Both floor and ceiling functions are the monotonically non-decreasing function:
${\begin{aligned}x_{1}\leq x_{2}&\Rightarrow \lfloor x_{1}\rfloor \leq \lfloor x_{2}\rfloor ,\\x_{1}\leq x_{2}&\Rightarrow \lceil x_{1}\rceil \leq \lceil x_{2}\rceil .\end{aligned}}$
Relations among the functions
It is clear from the definitions that
$\lfloor x\rfloor \leq \lceil x\rceil ,$ with equality if and only if x is an integer, i.e.
$\lceil x\rceil -\lfloor x\rfloor ={\begin{cases}0&{\mbox{ if }}x\in \mathbb {Z} \\1&{\mbox{ if }}x\not \in \mathbb {Z} \end{cases}}$
In fact, for integers n, both floor and ceiling functions are the identity:
$\lfloor n\rfloor =\lceil n\rceil =n.$
Negating the argument switches floor and ceiling and changes the sign:
${\begin{aligned}\lfloor x\rfloor +\lceil -x\rceil &=0\\-\lfloor x\rfloor &=\lceil -x\rceil \\-\lceil x\rceil &=\lfloor -x\rfloor \end{aligned}}$
and:
$\lfloor x\rfloor +\lfloor -x\rfloor ={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\-1&{\text{if }}x\not \in \mathbb {Z} ,\end{cases}}$
$\lceil x\rceil +\lceil -x\rceil ={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\1&{\text{if }}x\not \in \mathbb {Z} .\end{cases}}$
Negating the argument complements the fractional part:
$\{x\}+\{-x\}={\begin{cases}0&{\text{if }}x\in \mathbb {Z} \\1&{\text{if }}x\not \in \mathbb {Z} .\end{cases}}$
The floor, ceiling, and fractional part functions are idempotent:
${\begin{aligned}{\Big \lfloor }\lfloor x\rfloor {\Big \rfloor }&=\lfloor x\rfloor ,\\{\Big \lceil }\lceil x\rceil {\Big \rceil }&=\lceil x\rceil ,\\{\Big \{}\{x\}{\Big \}}&=\{x\}.\end{aligned}}$
The result of nested floor or ceiling functions is the innermost function:
${\begin{aligned}{\Big \lfloor }\lceil x\rceil {\Big \rfloor }&=\lceil x\rceil ,\\{\Big \lceil }\lfloor x\rfloor {\Big \rceil }&=\lfloor x\rfloor \end{aligned}}$
due to the identity property for integers.
Quotients
If m and n are integers and n ≠ 0,
$0\leq \left\{{\frac {m}{n}}\right\}\leq 1-{\frac {1}{|n|}}.$
If n is a positive integer[11]
$\left\lfloor {\frac {x+m}{n}}\right\rfloor =\left\lfloor {\frac {\lfloor x\rfloor +m}{n}}\right\rfloor ,$
$\left\lceil {\frac {x+m}{n}}\right\rceil =\left\lceil {\frac {\lceil x\rceil +m}{n}}\right\rceil .$
If m is positive[12]
$n=\left\lceil {\frac {n}{m}}\right\rceil +\left\lceil {\frac {n-1}{m}}\right\rceil +\dots +\left\lceil {\frac {n-m+1}{m}}\right\rceil ,$
$n=\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {n+1}{m}}\right\rfloor +\dots +\left\lfloor {\frac {n+m-1}{m}}\right\rfloor .$
For m = 2 these imply
$n=\left\lfloor {\frac {n}{2}}\right\rfloor +\left\lceil {\frac {n}{2}}\right\rceil .$
More generally,[13] for positive m (See Hermite's identity)
$\lceil mx\rceil =\left\lceil x\right\rceil +\left\lceil x-{\frac {1}{m}}\right\rceil +\dots +\left\lceil x-{\frac {m-1}{m}}\right\rceil ,$
$\lfloor mx\rfloor =\left\lfloor x\right\rfloor +\left\lfloor x+{\frac {1}{m}}\right\rfloor +\dots +\left\lfloor x+{\frac {m-1}{m}}\right\rfloor .$
The following can be used to convert floors to ceilings and vice versa (m positive)[14]
$\left\lceil {\frac {n}{m}}\right\rceil =\left\lfloor {\frac {n+m-1}{m}}\right\rfloor =\left\lfloor {\frac {n-1}{m}}\right\rfloor +1,$
$\left\lfloor {\frac {n}{m}}\right\rfloor =\left\lceil {\frac {n-m+1}{m}}\right\rceil =\left\lceil {\frac {n+1}{m}}\right\rceil -1,$
For all m and n strictly positive integers:[15]
$\sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {(m-1)(n-1)+\gcd(m,n)-1}{2}},$
which, for positive and coprime m and n, reduces to
$\sum _{k=1}^{n-1}\left\lfloor {\frac {km}{n}}\right\rfloor ={\frac {1}{2}}(m-1)(n-1),$
and similarly for the ceiling and fractional part functions (still for positive and coprime m and n),
$\sum _{k=1}^{n-1}\left\lceil {\frac {km}{n}}\right\rceil ={\frac {1}{2}}(m+1)(n-1),$
$\sum _{k=1}^{n-1}\left\{{\frac {km}{n}}\right\}={\frac {1}{2}}(n-1).$
Since the right-hand side of the general case is symmetrical in m and n, this implies that
$\left\lfloor {\frac {m}{n}}\right\rfloor +\left\lfloor {\frac {2m}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m}{n}}\right\rfloor =\left\lfloor {\frac {n}{m}}\right\rfloor +\left\lfloor {\frac {2n}{m}}\right\rfloor +\dots +\left\lfloor {\frac {(m-1)n}{m}}\right\rfloor .$
More generally, if m and n are positive,
${\begin{aligned}&\left\lfloor {\frac {x}{n}}\right\rfloor +\left\lfloor {\frac {m+x}{n}}\right\rfloor +\left\lfloor {\frac {2m+x}{n}}\right\rfloor +\dots +\left\lfloor {\frac {(n-1)m+x}{n}}\right\rfloor \\=&\left\lfloor {\frac {x}{m}}\right\rfloor +\left\lfloor {\frac {n+x}{m}}\right\rfloor +\left\lfloor {\frac {2n+x}{m}}\right\rfloor +\cdots +\left\lfloor {\frac {(m-1)n+x}{m}}\right\rfloor .\end{aligned}}$
This is sometimes called a reciprocity law.[16]
Nested divisions
For positive integer n, and arbitrary real numbers m,x:[17]
$\left\lfloor {\frac {\lfloor x/m\rfloor }{n}}\right\rfloor =\left\lfloor {\frac {x}{mn}}\right\rfloor $
$\left\lceil {\frac {\lceil x/m\rceil }{n}}\right\rceil =\left\lceil {\frac {x}{mn}}\right\rceil .$
Continuity and series expansions
None of the functions discussed in this article are continuous, but all are piecewise linear: the functions $\lfloor x\rfloor $, $\lceil x\rceil $, and $\{x\}$ have discontinuities at the integers.
$\lfloor x\rfloor $ is upper semi-continuous and $\lceil x\rceil $ and $\{x\}$ are lower semi-continuous.
Since none of the functions discussed in this article are continuous, none of them have a power series expansion. Since floor and ceiling are not periodic, they do not have uniformly convergent Fourier series expansions. The fractional part function has Fourier series expansion[18]
$\{x\}={\frac {1}{2}}-{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$
for x not an integer.
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true value.
Using the formula floor(x) = x − {x} gives
$\lfloor x\rfloor =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}$
for x not an integer.
Applications
Mod operator
For an integer x and a positive integer y, the modulo operation, denoted by x mod y, gives the value of the remainder when x is divided by y. This definition can be extended to real x and y, y ≠ 0, by the formula
$x{\bmod {y}}=x-y\left\lfloor {\frac {x}{y}}\right\rfloor .$
Then it follows from the definition of floor function that this extended operation satisfies many natural properties. Notably, x mod y is always between 0 and y, i.e.,
if y is positive,
$0\leq x{\bmod {y}}<y,$
and if y is negative,
$0\geq x{\bmod {y}}>y.$
Quadratic reciprocity
Gauss's third proof of quadratic reciprocity, as modified by Eisenstein, has two basic steps.[19][20]
Let p and q be distinct positive odd prime numbers, and let
$m={\frac {p-1}{2}},$ $n={\frac {q-1}{2}}.$
First, Gauss's lemma is used to show that the Legendre symbols are given by
$\left({\frac {q}{p}}\right)=(-1)^{\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor }$
and
$\left({\frac {p}{q}}\right)=(-1)^{\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor }.$
The second step is to use a geometric argument to show that
$\left\lfloor {\frac {q}{p}}\right\rfloor +\left\lfloor {\frac {2q}{p}}\right\rfloor +\dots +\left\lfloor {\frac {mq}{p}}\right\rfloor +\left\lfloor {\frac {p}{q}}\right\rfloor +\left\lfloor {\frac {2p}{q}}\right\rfloor +\dots +\left\lfloor {\frac {np}{q}}\right\rfloor =mn.$
Combining these formulas gives quadratic reciprocity in the form
$\left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{mn}=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.$
There are formulas that use floor to express the quadratic character of small numbers mod odd primes p:[21]
$\left({\frac {2}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{4}}\right\rfloor },$
$\left({\frac {3}{p}}\right)=(-1)^{\left\lfloor {\frac {p+1}{6}}\right\rfloor }.$
Rounding
For an arbitrary real number $x$, rounding $x$ to the nearest integer with tie breaking towards positive infinity is given by ${\text{rpi}}(x)=\left\lfloor x+{\tfrac {1}{2}}\right\rfloor =\left\lceil {\tfrac {\lfloor 2x\rfloor }{2}}\right\rceil $; rounding towards negative infinity is given as ${\text{rni}}(x)=\left\lceil x-{\tfrac {1}{2}}\right\rceil =\left\lfloor {\tfrac {\lceil 2x\rceil }{2}}\right\rfloor $.
If tie-breaking is away from 0, then the rounding function is ${\text{ri}}(x)=\operatorname {sgn}(x)\left\lfloor |x|+{\tfrac {1}{2}}\right\rfloor $ (see sign function), and rounding towards even can be expressed with the more cumbersome $\lfloor x\rceil =\left\lfloor x+{\tfrac {1}{2}}\right\rfloor +\left\lceil {\tfrac {2x-1}{4}}\right\rceil -\left\lfloor {\tfrac {2x-1}{4}}\right\rfloor -1$, which is the above expression for rounding towards positive infinity ${\text{rpi}}(x)$ minus an integrality indicator for ${\tfrac {2x-1}{4}}$.
Number of digits
The number of digits in base b of a positive integer k is
$\lfloor \log _{b}{k}\rfloor +1=\lceil \log _{b}{(k+1)}\rceil .$
Number of strings without repeated characters
The number of possible strings of arbitrary length that doesn't use any character twice is given by[22]
$(n)_{0}+\cdots +(n)_{n}=\lfloor en!\rfloor $
where:
• n > 0 is the number of letters in the alphabet (e.g., 26 in English)
• the falling factorial $(n)_{k}=n(n-1)\cdots (n-k+1)$ denotes the number of strings of length k that don't use any character twice.
• n! denotes the factorial of n
• e = 2.718… is Euler's number
For n = 26, this comes out to 1096259850353149530222034277.
Factors of factorials
Let n be a positive integer and p a positive prime number. The exponent of the highest power of p that divides n! is given by a version of Legendre's formula[23]
$\left\lfloor {\frac {n}{p}}\right\rfloor +\left\lfloor {\frac {n}{p^{2}}}\right\rfloor +\left\lfloor {\frac {n}{p^{3}}}\right\rfloor +\dots ={\frac {n-\sum _{k}a_{k}}{p-1}}$
where $ n=\sum _{k}a_{k}p^{k}$ is the way of writing n in base p. This is a finite sum, since the floors are zero when pk > n.
Beatty sequence
The Beatty sequence shows how every positive irrational number gives rise to a partition of the natural numbers into two sequences via the floor function.[24]
Euler's constant (γ)
There are formulas for Euler's constant γ = 0.57721 56649 ... that involve the floor and ceiling, e.g.[25]
$\gamma =\int _{1}^{\infty }\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx,$
$\gamma =\lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}\left(\left\lceil {\frac {n}{k}}\right\rceil -{\frac {n}{k}}\right),$
and
$\gamma =\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\cdots -{\tfrac {1}{15}}\right)+\cdots $
Riemann zeta function (ζ)
The fractional part function also shows up in integral representations of the Riemann zeta function. It is straightforward to prove (using integration by parts)[26] that if $\varphi (x)$ is any function with a continuous derivative in the closed interval [a, b],
$\sum _{a<n\leq b}\varphi (n)=\int _{a}^{b}\varphi (x)\,dx+\int _{a}^{b}\left(\{x\}-{\tfrac {1}{2}}\right)\varphi '(x)\,dx+\left(\{a\}-{\tfrac {1}{2}}\right)\varphi (a)-\left(\{b\}-{\tfrac {1}{2}}\right)\varphi (b).$
Letting $\varphi (n)=n^{-s}$ for real part of s greater than 1 and letting a and b be integers, and letting b approach infinity gives
$\zeta (s)=s\int _{1}^{\infty }{\frac {{\frac {1}{2}}-\{x\}}{x^{s+1}}}\,dx+{\frac {1}{s-1}}+{\frac {1}{2}}.$
This formula is valid for all s with real part greater than −1, (except s = 1, where there is a pole) and combined with the Fourier expansion for {x} can be used to extend the zeta function to the entire complex plane and to prove its functional equation.[27]
For s = σ + it in the critical strip 0 < σ < 1,
$\zeta (s)=s\int _{-\infty }^{\infty }e^{-\sigma \omega }(\lfloor e^{\omega }\rfloor -e^{\omega })e^{-it\omega }\,d\omega .$
In 1947 van der Pol used this representation to construct an analogue computer for finding roots of the zeta function.[28]
Formulas for prime numbers
The floor function appears in several formulas characterizing prime numbers. For example, since $\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor $ is equal to 1 if m divides n, and to 0 otherwise, it follows that a positive integer n is a prime if and only if[29]
$\sum _{m=1}^{\infty }\left(\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor \right)=2.$
One may also give formulas for producing the prime numbers. For example, let pn be the n-th prime, and for any integer r > 1, define the real number α by the sum
$\alpha =\sum _{m=1}^{\infty }p_{m}r^{-m^{2}}.$
Then[30]
$p_{n}=\left\lfloor r^{n^{2}}\alpha \right\rfloor -r^{2n-1}\left\lfloor r^{(n-1)^{2}}\alpha \right\rfloor .$
A similar result is that there is a number θ = 1.3064... (Mills' constant) with the property that
$\left\lfloor \theta ^{3}\right\rfloor ,\left\lfloor \theta ^{9}\right\rfloor ,\left\lfloor \theta ^{27}\right\rfloor ,\dots $
are all prime.[31]
There is also a number ω = 1.9287800... with the property that
$\left\lfloor 2^{\omega }\right\rfloor ,\left\lfloor 2^{2^{\omega }}\right\rfloor ,\left\lfloor 2^{2^{2^{\omega }}}\right\rfloor ,\dots $
are all prime.[31]
Let π(x) be the number of primes less than or equal to x. It is a straightforward deduction from Wilson's theorem that[32]
$\pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {(j-1)!+1}{j}}-\left\lfloor {\frac {(j-1)!}{j}}\right\rfloor \right\rfloor .$
Also, if n ≥ 2,[33]
$\pi (n)=\sum _{j=2}^{n}\left\lfloor {\frac {1}{\sum _{k=2}^{j}\left\lfloor \left\lfloor {\frac {j}{k}}\right\rfloor {\frac {k}{j}}\right\rfloor }}\right\rfloor .$
None of the formulas in this section are of any practical use.[34][35]
Solved problems
Ramanujan submitted these problems to the Journal of the Indian Mathematical Society.[36]
If n is a positive integer, prove that
1. $\left\lfloor {\tfrac {n}{3}}\right\rfloor +\left\lfloor {\tfrac {n+2}{6}}\right\rfloor +\left\lfloor {\tfrac {n+4}{6}}\right\rfloor =\left\lfloor {\tfrac {n}{2}}\right\rfloor +\left\lfloor {\tfrac {n+3}{6}}\right\rfloor ,$
2. $\left\lfloor {\tfrac {1}{2}}+{\sqrt {n+{\tfrac {1}{2}}}}\right\rfloor =\left\lfloor {\tfrac {1}{2}}+{\sqrt {n+{\tfrac {1}{4}}}}\right\rfloor ,$
3. $\left\lfloor {\sqrt {n}}+{\sqrt {n+1}}\right\rfloor =\left\lfloor {\sqrt {4n+2}}\right\rfloor .$
Some generalizations to the above floor function identities have been proven.[37]
Unsolved problem
The study of Waring's problem has led to an unsolved problem:
Are there any positive integers k ≥ 6 such that[38]
$3^{k}-2^{k}\left\lfloor \left({\tfrac {3}{2}}\right)^{k}\right\rfloor >2^{k}-\left\lfloor \left({\tfrac {3}{2}}\right)^{k}\right\rfloor -2$ ?
Mahler[39] has proved there can only be a finite number of such k; none are known.
Computer implementations
In most programming languages, the simplest method to convert a floating point number to an integer does not do floor or ceiling, but truncation. The reason for this is historical, as the first machines used ones' complement and truncation was simpler to implement (floor is simpler in two's complement). FORTRAN was defined to require this behavior and thus almost all processors implement conversion this way. Some consider this to be an unfortunate historical design decision that has led to bugs handling negative offsets and graphics on the negative side of the origin.
A bit-wise right-shift of a signed integer $x$ by $n$ is the same as $\left\lfloor {\frac {x}{2^{n}}}\right\rfloor $. Division by a power of 2 is often written as a right-shift, not for optimization as might be assumed, but because the floor of negative results is required. Assuming such shifts are "premature optimization" and replacing them with division can break software.
Many programming languages (including C, C++,[40][41] C#,[42][43] Java,[44][45] PHP,[46][47] R,[48] and Python[49]) provide standard functions for floor and ceiling, usually called floor and ceil, or less commonly ceiling.[50] The language APL uses ⌊x for floor. The J Programming Language, a follow-on to APL that is designed to use standard keyboard symbols, uses <. for floor and >. for ceiling.[51] ALGOL usesentier for floor.
In Microsoft Excel the floor function is implemented as INT (which rounds down rather than toward zero).[52] The command FLOOR in earlier versions would round toward zero, effectively the opposite of what "int" and "floor" do in other languages. Since 2010 FLOOR has been fixed to round down, with extra arguments that can reproduce previous behavior.[53] The OpenDocument file format, as used by OpenOffice.org, Libreoffice and others, uses the same function names; INT does floor[54] and FLOOR has a third argument that can make it round toward zero.[55]
See also
• Bracket (mathematics)
• Integer-valued function
• Step function
• Modulo operation
Citations
1. Graham, Knuth, & Patashnik, Ch. 3.1
2. 1) Luke Heaton, A Brief History of Mathematical Thought, 2015, ISBN 1472117158 (n.p.)
2) Albert A. Blank et al., Calculus: Differential Calculus, 1968, p. 259
3) John W. Warris, Horst Stocker, Handbook of mathematics and computational science, 1998, ISBN 0387947469, p. 151
3. Lemmermeyer, pp. 10, 23.
4. e.g. Cassels, Hardy & Wright, and Ribenboim use Gauss's notation. Graham, Knuth & Patashnik, and Crandall & Pomerance use Iverson's.
5. Iverson, p. 12.
6. Higham, p. 25.
7. Mathwords: Floor Function.
8. Mathwords: Ceiling Function
9. Graham, Knuth, & Patashnik, p. 70.
10. Graham, Knuth, & Patashink, Ch. 3
11. Graham, Knuth, & Patashnik, p. 73
12. Graham, Knuth, & Patashnik, p. 85
13. Graham, Knuth, & Patashnik, p. 85 and Ex. 3.15
14. Graham, Knuth, & Patashnik, Ex. 3.12
15. Graham, Knuth, & Patashnik, p. 94.
16. Graham, Knuth, & Patashnik, p. 94
17. Graham, Knuth, & Patashnik, p. 71, apply theorem 3.10 with x/m as input and the division by n as function
18. Titchmarsh, p. 15, Eq. 2.1.7
19. Lemmermeyer, § 1.4, Ex. 1.32–1.33
20. Hardy & Wright, §§ 6.11–6.13
21. Lemmermeyer, p. 25
22. OEIS sequence A000522 (Total number of arrangements of a set with n elements: a(n) = Sum_{k=0..n} n!/k!.) (See Formulas.)
23. Hardy & Wright, Th. 416
24. Graham, Knuth, & Patashnik, pp. 77–78
25. These formulas are from the Wikipedia article Euler's constant, which has many more.
26. Titchmarsh, p. 13
27. Titchmarsh, pp.14–15
28. Crandall & Pomerance, p. 391
29. Crandall & Pomerance, Ex. 1.3, p. 46. The infinite upper limit of the sum can be replaced with n. An equivalent condition is n > 1 is prime if and only if $\sum _{m=1}^{\lfloor {\sqrt {n}}\rfloor }\left(\left\lfloor {\frac {n}{m}}\right\rfloor -\left\lfloor {\frac {n-1}{m}}\right\rfloor \right)=1$ .
30. Hardy & Wright, § 22.3
31. Ribenboim, p. 186
32. Ribenboim, p. 181
33. Crandall & Pomerance, Ex. 1.4, p. 46
34. Ribenboim, p. 180 says that "Despite the nil practical value of the formulas ... [they] may have some relevance to logicians who wish to understand clearly how various parts of arithmetic may be deduced from different axiomatzations ... "
35. Hardy & Wright, pp. 344—345 "Any one of these formulas (or any similar one) would attain a different status if the exact value of the number α ... could be expressed independently of the primes. There seems no likelihood of this, but it cannot be ruled out as entirely impossible."
36. Ramanujan, Question 723, Papers p. 332
37. Somu, Sai Teja; Kukla, Andrzej (2022). "On some generalizations to floor function identities of Ramanujan" (PDF). Integers. 22. arXiv:2109.03680.
38. Hardy & Wright, p. 337
39. Mahler, Kurt (1957). "On the fractional parts of the powers of a rational number II". Mathematika. 4 (2): 122–124. doi:10.1112/S0025579300001170.
40. "C++ reference of floor function". Retrieved 5 December 2010.
41. "C++ reference of ceil function". Retrieved 5 December 2010.
42. dotnet-bot. "Math.Floor Method (System)". docs.microsoft.com. Retrieved 28 November 2019.
43. dotnet-bot. "Math.Ceiling Method (System)". docs.microsoft.com. Retrieved 28 November 2019.
44. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018.
45. "Math (Java SE 9 & JDK 9 )". docs.oracle.com. Retrieved 20 November 2018.
46. "PHP manual for ceil function". Retrieved 18 July 2013.
47. "PHP manual for floor function". Retrieved 18 July 2013.
48. "R: Rounding of Numbers".
49. "Python manual for math module". Retrieved 18 July 2013.
50. Sullivan, p. 86.
51. "Vocabulary". J Language. Retrieved 6 September 2011.
52. "INT function". Retrieved 29 October 2021.
53. "FLOOR function". Retrieved 29 October 2021.
54. "Documentation/How Tos/Calc: INT function". Retrieved 29 October 2021.
55. "Documentation/How Tos/Calc: FLOOR function". Retrieved 29 October 2021.
References
• J.W.S. Cassels (1957), An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 45, Cambridge University Press
• Crandall, Richard; Pomerance, Carl (2001), Prime Numbers: A Computational Perspective, New York: Springer, ISBN 0-387-94777-9
• Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren (1994), Concrete Mathematics, Reading Ma.: Addison-Wesley, ISBN 0-201-55802-5
• Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN 978-0-19-853171-5
• Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25
• ISO/IEC. ISO/IEC 9899::1999(E): Programming languages — C (2nd ed), 1999; Section 6.3.1.4, p. 43.
• Iverson, Kenneth E. (1962), A Programming Language, Wiley
• Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, ISBN 3-540-66957-4
• Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN 978-0-8218-2076-6
• Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer, ISBN 0-387-94457-5
• Michael Sullivan. Precalculus, 8th edition, p. 86
• Titchmarsh, Edward Charles; Heath-Brown, David Rodney ("Roger") (1986), The Theory of the Riemann Zeta-function (2nd ed.), Oxford: Oxford U. P., ISBN 0-19-853369-1
External links
Wikimedia Commons has media related to Floor and ceiling functions.
• "Floor function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Štefan Porubský, "Integer rounding functions", Interactive Information Portal for Algorithmic Mathematics, Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic, retrieved 24 October 2008
• Weisstein, Eric W. "Floor Function". MathWorld.
• Weisstein, Eric W. "Ceiling Function". MathWorld.
| Wikipedia |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.
In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry.
Cartesian coordinates
In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.[1] The origin divides each of these axes into two halves, a positive and a negative semiaxis.[2] Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three.[1]
Other coordinate systems
In a polar coordinate system, the origin may also be called the pole. It does not itself have well-defined polar coordinates, because the polar coordinates of a point include the angle made by the positive x-axis and the ray from the origin to the point, and this ray is not well-defined for the origin itself.[3]
In Euclidean geometry, the origin may be chosen freely as any convenient point of reference.[4]
The origin of the complex plane can be referred as the point where real axis and imaginary axis intersect each other. In other words, it is the complex number zero.[5]
See also
• Null vector, an analogous point of a vector space
• Distance from a point to a plane
• Pointed space, a topological space with a distinguished point
• Radial basis function, a function depending only on the distance from the origin
References
1. Madsen, David A. (2001), Engineering Drawing and Design, Delmar drafting series, Thompson Learning, p. 120, ISBN 9780766816343.
2. Pontrjagin, Lev S. (1984), Learning higher mathematics, Springer series in Soviet mathematics, Springer-Verlag, p. 73, ISBN 9783540123514.
3. Tanton, James Stuart (2005), Encyclopedia of Mathematics, Infobase Publishing, ISBN 9780816051243.
4. Lee, John M. (2013), Axiomatic Geometry, Pure and Applied Undergraduate Texts, vol. 21, American Mathematical Society, p. 134, ISBN 9780821884782.
5. Gonzalez, Mario (1991), Classical Complex Analysis, Chapman & Hall Pure and Applied Mathematics, CRC Press, ISBN 9780824784157.
| Wikipedia |
Flatness (mathematics)
In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. (See curvature.)[1]
Flatness in homological algebra and algebraic geometry means, of an object $A$ in an abelian category, that $-\otimes A$ is an exact functor. See flat module or, for more generality, flat morphism.[2]
Character encodings
Character information
Preview⏥
Unicode name FLATNESS
Encodingsdecimalhex
Unicode9189U+23E5
UTF-8226 143 165E2 8F A5
Numeric character reference⏥⏥
See also
• Developable surface
• Flat (mathematics)
References
1. Committee 117, A. C. I. (November 3, 2006). Specifications for Tolerances for Concrete Construction and Materials and Commentary. American Concrete Institute. ISBN 9780870312212 – via Google Books.
2. Ballast, David Kent (March 16, 2007). Handbook of Construction Tolerances. John Wiley & Sons. ISBN 9780471931515 – via Google Books.
| Wikipedia |
Semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, π radians, or a half-turn). It has only one line of symmetry (reflection symmetry).
Semicircle
Areaπr2/2
Perimeter(π+2)r
In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.
By Thales' theorem, any triangle inscribed in a semicircle with a vertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a right angle at the third vertex.
All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.
Uses
A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter).
The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying the Pythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengths a and b.[1]
The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the quadrature of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as a lemma in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.[2]
Equation
The equation of a semicircle with midpoint $(x_{0},y_{0})$ on the diameter between its endpoints and which is entirely concave from below is
$y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}$
If it is entirely concave from above, the equation is
$y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}$
Arbelos
An arbelos is a region in the plane bounded by three semicircles connected at the corners, all on the same side of a straight line (the baseline) that contains their diameters.
See also
• Amphitheater
• Archimedes' twin circles
• Archimedes' quadruplets
• Salinon
• Wigner semicircle distribution
References
1. Euclid's Elements, Book VI, Proposition 13
2. Euclid's Elements, Book VI, Proposition 25
External links
• Weisstein, Eric W. "Semicircle". MathWorld.
| Wikipedia |
Pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around the five points creates a similar symbol referred to as the pentacle,[1] which is used widely by Wiccans and in paganism, or as a sign of life and connections. The word "pentagram" refers only to the five-pointed star, not the surrounding circle of a pentacle.
Pentagrams were used symbolically in ancient Greece and Babylonia. Christians once commonly used the pentagram to represent the five wounds of Jesus.
The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon),[2] from πέντε (pente), "five" + γραμμή (grammē), "line".[3] Pentagram refers to just the star and pentacle refers to the star within the circle specifically although there is some overlap in usage.[4] The word pentalpha is a 17th-century revival of a post-classical Greek name of the shape.[5]
History
Early history
Early pentagrams have been found on Sumerian pottery from Ur circa 3500 BCE, and the five-pointed star was at various times the symbol of Ishtar or Marduk.[6][7]
Pentagram symbols from about 5,000 years ago were found in the Liangzhu culture of China.[9]
The pentagram was known to the ancient Greeks, with a depiction on a vase possibly dating back to the 7th century BCE.[10] Pythagoreanism originated in the 6th century BCE and used the pentagram as a symbol of mutual recognition, of wellbeing, and to recognize good deeds and charity.[11]
From around 300-150 BCE the pentagram stood as the symbol of Jerusalem, marked by the 5 Hebrew letters ירשלם spelling its name.[12]
The word Pentemychos (πεντέμυχος lit. "five corners" or "five recesses")[13] was the title of the cosmogony of Pherecydes of Syros.[14] Here, the "five corners" are where the seeds of Chronos are placed within the Earth in order for the cosmos to appear.[15]
In Neoplatonism, the pentagram was said to have been used as a symbol or sign of recognition by the Pythagoreans, who called the pentagram ὑγιεία hugieia "health"[16]
Middle Ages
The pentagram was used in ancient times as a Christian symbol for the five senses,[17] or of the five wounds of Christ. The pentagram plays an important symbolic role in the 14th-century English poem Sir Gawain and the Green Knight, in which the symbol decorates the shield of the hero, Gawain. The unnamed poet credits the symbol's origin to King Solomon, and explains that each of the five interconnected points represents a virtue tied to a group of five: Gawain is perfect in his five senses and five fingers, faithful to the Five Wounds of Christ, takes courage from the five joys that Mary had of Jesus, and exemplifies the five virtues of knighthood,[18] which are generosity, friendship, chastity, chivalry, and piety.[19]
The North rose of Amiens Cathedral (built in the 13th century) exhibits a pentagram-based motif. Some sources interpret the unusual downward-pointing star as symbolizing the Holy Spirit descending on people.
Renaissance
Heinrich Cornelius Agrippa and others perpetuated the popularity of the pentagram as a magic symbol, attributing the five neoplatonic elements to the five points, in typical Renaissance fashion.
Romanticism
By the mid-19th century, a further distinction had developed amongst occultists regarding the pentagram's orientation. With a single point upwards it depicted spirit presiding over the four elements of matter, and was essentially "good". However, the influential but controversial writer Éliphas Lévi, known for believing that magic was a real science, had called it evil whenever the symbol appeared the other way up.
• "A reversed pentagram, with two points projecting upwards, is a symbol of evil and attracts sinister forces because it overturns the proper order of things and demonstrates the triumph of matter over spirit. It is the goat of lust attacking the heavens with its horns, a sign execrated by initiates."[20]
• "The flaming star, which, when turned upside down, is the heirolgyphic [sic] sign of the goat of black magic, whose head may be drawn in the star, the two horns at the top, the ears to the right and left, the beard at the bottom. It is a sign of antagonism and fatality. It is the goat of lust attacking the heavens with its horns."[21]
• "Let us keep the figure of the Five-pointed Star always upright, with the topmost triangle pointing to heaven, for it is the seat of wisdom, and if the figure is reversed, perversion and evil will be the result."[22]
• Man inscribed in a pentagram, from Heinrich Cornelius Agrippa's De occulta philosophia libri tres. The five signs at the pentagram's vertices are astrological.
• Another pentagram from Agrippa's book. This one has the Pythagorean letters inscribed around the circle.
• The occultist and magician Éliphas Lévi's pentagram, which he considered to be a symbol of the microcosm, or human
Star polygons
• pentagram
• hexagram
• heptagram
• octagram
• enneagram
• decagram
• hendecagram
• dodecagram
The apotropaic (protective) use in German folklore of the pentagram symbol (called Drudenfuss in German) is referred to by Goethe in Faust (1808), where a pentagram prevents Mephistopheles from leaving a room (but did not prevent him from entering by the same way, as the outward pointing corner of the diagram happened to be imperfectly drawn):
Mephistopheles:
I must confess, I'm prevented though
By a little thing that hinders me,
The Druid's-foot on your doorsill–
Faust:
The Pentagram gives you pain?
Then tell me, you Son of Hell,
If that's the case, how did you gain
Entry? Are spirits like you cheated?
Mephistopheles:
Look carefully! It's not completed:
One angle, if you inspect it closely
Has, as you see, been left a little open.[23]
Also protective is the use in Icelandic folklore of a gestured or carved rather than painted pentagram (called smèrhnút in Icelandic), according to 19th century folklorist Jón Árnason:[24]
A butter that comes from the fake vomit is called a fake butter; it looks like something else; but if one makes a sign of a cross over it, or carves a cross on it, or a figure called a buttermilk-knot,* it all explodes into small pieces and becomes like a grain of dross, so that nothing remains of it, except only particles, or it subsides like foam. Therefore it seems more prudent, if a person is offered a horrible butter to eat, or as a fee,[25] to make either mark on it, because a fake butter cannot withstand either a cross mark or a butter-knot.
* The butter-knot is shaped like this:
East Asian symbolism
Wu Xing (Chinese: 五行; pinyin: Wǔ Xíng) are the five phases, or five elements in Taoists Chinese tradition. They are differentiated from the formative ancient Japanese or Greek elements, due to their emphasis on cyclic transformations and change. The five phases are: Fire (火 huǒ), Earth (土 tǔ), Metal (金 jīn), Water (水 shuǐ), and Wood (木 mù). The Wuxing is the fundamental philosophy and doctrine of traditional Chinese Medicine and Acupuncture.[26]
Uses in modern occultism
Based on Renaissance-era occultism, the pentagram found its way into the symbolism of modern occultists. Its major use is a continuation of the ancient Babylonian use of the pentagram as an apotropaic charm to protect against evil forces.[27] Éliphas Lévi claimed that "The Pentagram expresses the mind's domination over the elements and it is by this sign that we bind the demons of the air, the spirits of fire, the spectres of water, and the ghosts of earth."[28] In this spirit, the Hermetic Order of the Golden Dawn developed the use of the pentagram in the lesser banishing ritual of the pentagram, which is still used to this day by those who practice Golden Dawn-type magic.
Aleister Crowley made use of the pentagram in his Thelemic system of magick: an adverse or inverted pentagram represents the descent of spirit into matter, according to the interpretation of Lon Milo DuQuette.[29] Crowley contradicted his old comrades in the Hermetic Order of the Golden Dawn, who, following Levi, considered this orientation of the symbol evil and associated it with the triumph of matter over spirit.
Baháʼí Faith
The five-pointed star is a symbol of the Baháʼí Faith.[30][31] In the Baháʼí Faith, the star is known as the Haykal (Arabic: "temple"), and it was initiated and established by the Báb. The Báb and Bahá'u'lláh wrote various works in the form of a pentagram.[32][33]
The Church of Jesus Christ of Latter-day Saints
The Church of Jesus Christ of Latter-day Saints is theorized to have begun using both upright and inverted five-pointed stars in Temple architecture, dating from the Nauvoo Illinois Temple dedicated on 30 April 1846.[34] Other temples decorated with five-pointed stars in both orientations include the Salt Lake Temple and the Logan Utah Temple. These usages come from the symbolism found in Revelation chapter 12: "And there appeared a great wonder in heaven; a woman clothed with the sun, and the moon under her feet, and upon her head a crown of twelve stars."[35]
Wicca
Typical Neopagan pentagram (circumscribed)
USVA headstone emblem 37
Because of a perceived association with Satanism and occultism, many United States schools in the late 1990s sought to prevent students from displaying the pentagram on clothing or jewelry.[36] In public schools, such actions by administrators were determined in 2000 to be in violation of students' First Amendment right to free exercise of religion.[37]
The encircled pentagram (referred to as a pentacle by the plaintiffs) was added to the list of 38 approved religious symbols to be placed on the tombstones of fallen service members at Arlington National Cemetery on 24 April 2007. The decision was made following ten applications from families of fallen soldiers who practiced Wicca. The government paid the families US$225,000 to settle their pending lawsuits.[38][39]
Satanism
The inverted pentagram is the most notable and widespread symbol of Satanism.
The Sigil of Baphomet, the official insignia of the Church of Satan and LaVeyan Satanism
The inverted pentagram is the symbol used for Satanism, sometimes depicted with the goat's head of Baphomet within it, which originated from the Church of Satan. In some depictions the devil is depicted, like Baphomet, as a goat, therefore the goat and goat's head is a significant symbol throughout Satanism. The inverted pentagram is also used as the logo for The Satanic Temple, which also featured a depiction of Baphomet's head. The Sigil of Baphomet is also adopted by the Joy of Satan Ministries who instead incorporate cuneiform script, attributing it to the earliest use of the pentagram in Sumeria.
Serer religion
The five-pointed star is a symbol of the Serer religion and the Serer people of West Africa. Called Yoonir in their language, it symbolizes the universe in the Serer creation myth, and also represents the star Sirius.[40][41]
Judaism
The pentagram has been used in Judaism since at least 300BCE when it first was used as the stamp of Jerusalem. It is used to represent justice, mercy, and wisdom.
Other modern use
• The pentagram is featured on the national flags of Morocco (adopted 1915) and Ethiopia (adopted 1996 and readopted 2009)
• Morocco's flag
• Ethiopia's flag
• The Order of the Eastern Star, an organization (established 1850) associated with Freemasonry, uses a pentagram as its symbol, with the five isosceles triangles of the points colored blue, yellow, white, green, and red. In most Grand Chapters the pentagram is used pointing down, but in a few, it is pointing up. Grand Chapter officers often have a pentagon inscribed around the star[42](the emblem shown here is from the Prince Hall Association).
• Order of the Eastern Star emblem
• A pentagram is featured on the flag of the Dutch city of Haaksbergen, as well on its coat of arms.
• Flag of Haaksbergen
• A pentagram is featured on the flag of the Japanese city of Nagasaki, as well on its emblem.
• Flag of Nagasaki
Geometry
The pentagram is the simplest regular star polygon. The pentagram contains ten points (the five points of the star, and the five vertices of the inner pentagon) and fifteen line segments. It is represented by the Schläfli symbol {5/2}. Like a regular pentagon, and a regular pentagon with a pentagram constructed inside it, the regular pentagram has as its symmetry group the dihedral group of order 10.
It can be seen as a net of a pentagonal pyramid although with isosceles triangles.
Construction
The pentagram can be constructed by connecting alternate vertices of a pentagon; see details of the construction. It can also be constructed as a stellation of a pentagon, by extending the edges of a pentagon until the lines intersect.
Golden ratio
The golden ratio, φ = (1 + √5) / 2 ≈ 1.618, satisfying
$\varphi =1+2\sin(\pi /10)=1+2\sin 18^{\circ }\,$
$\varphi =1/(2\sin(\pi /10))=1/(2\sin 18^{\circ })\,$
$\varphi =2\cos(\pi /5)=2\cos 36^{\circ }\,$
plays an important role in regular pentagons and pentagrams. Each intersection of edges sections the edges in the golden ratio: the ratio of the length of the edge to the longer segment is φ, as is the length of the longer segment to the shorter. Also, the ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (a side of the pentagon in the pentagram's center) is φ. As the four-color illustration shows:
${\frac {\mathrm {red} }{\mathrm {green} }}={\frac {\mathrm {green} }{\mathrm {blue} }}={\frac {\mathrm {blue} }{\mathrm {magenta} }}=\varphi .$
The pentagram includes ten isosceles triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles are golden triangles. The obtuse isosceles triangle highlighted via the colored lines in the illustration is a golden gnomon.
Trigonometric values
Main article: Exact trigonometric values
${\begin{aligned}\sin {\frac {\pi }{10}}&=\sin 18^{\circ }={\frac {{\sqrt {5}}-1}{4}}={\frac {\varphi -1}{2}}={\frac {1}{2\varphi }}\\[5pt]\cos {\frac {\pi }{10}}&=\cos 18^{\circ }={\frac {\sqrt {2(5+{\sqrt {5}})}}{4}}\\[5pt]\tan {\frac {\pi }{10}}&=\tan 18^{\circ }={\frac {\sqrt {5(5-2{\sqrt {5}})}}{5}}\\[5pt]\cot {\frac {\pi }{10}}&=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\\[5pt]\sin {\frac {\pi }{5}}&=\sin 36^{\circ }={\frac {\sqrt {2(5-{\sqrt {5}})}}{4}}\\[5pt]\cos {\frac {\pi }{5}}&=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}}\\[5pt]\tan {\frac {\pi }{5}}&=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\\[5pt]\cot {\frac {\pi }{5}}&=\cot 36^{\circ }={\frac {\sqrt {5(5+2{\sqrt {5}})}}{5}}\end{aligned}}$
As a result, in an isosceles triangle with one or two angles of 36°, the longer of the two side lengths is φ times that of the shorter of the two, both in the case of the acute as in the case of the obtuse triangle.
Spherical pentagram
Further information: Pentagramma mirificum
A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss.
Three-dimensional figures
Further information: Uniform polyhedron: Icosahedral symmetry
Several polyhedra incorporate pentagrams:
• Pentagrammic prism
• Pentagrammic antiprism
• Pentagrammic crossed-antiprism
• Small stellated dodecahedron
• Great stellated dodecahedron
• Small ditrigonal icosidodecahedron
• Dodecadodecahedron
Higher dimensions
Orthogonal projections of higher dimensional polytopes can also create pentagrammic figures:
4D 5D
The regular 5-cell (4-simplex) has five vertices and 10 edges.
The rectified 5-cell has 10 vertices and 30 edges.
The rectified 5-simplex has 15 vertices, seen in this orthogonal projection as three nested pentagrams.
The birectified 5-simplex has 20 vertices, seen in this orthogonal projection as four overlapping pentagrams.
All ten 4-dimensional Schläfli–Hess 4-polytopes have either pentagrammic faces or vertex figure elements.
Pentagram of Venus
The pentagram of Venus is the apparent path of the planet Venus as observed from Earth. Successive inferior conjunctions of Venus repeat with an orbital resonance of approximately 13:8—that is, Venus orbits the Sun approximately 13 times for every eight orbits of Earth—shifting 144° at each inferior conjunction.[44] The tips of the five loops at the center of the figure have the same geometric relationship to one another as the five vertices, or points, of a pentagram, and each group of five intersections equidistant from the figure's center have the same geometric relationship.
In computer systems
The pentagram has these Unicode code points that enable them to be included in documents:
• U+26E4 ⛤ PENTAGRAM
• U+26E5 ⛥ RIGHT-HANDED INTERLACED PENTAGRAM
• U+26E6 ⛦ LEFT-HANDED INTERLACED PENTAGRAM
• U+26E7 ⛧ INVERTED PENTAGRAM
See also
• Abe no Seimei – Japanese painter
• Christian symbolism – Use of symbols, including archetypes, acts, artwork or events, by Christianity
• Command at Sea insignia
• Enneagram (geometry) – Nine-pointed star polygon
• Five-pointed star – Geometrically a regular concave decagon, is a common ideogram in modern culture
• Heptagram – Star polygon
• Hexagram – Six-pointed star polygon
• Lute of Pythagoras – Self-similar geometric figure
• Medal of Honor – Highest award in the United States Armed Forces
• Pentachoron – the 4-simplex
• Pentagram map – Discrete dynamical system on the moduli space of polygons in the projective plane
• Pentalpha – Puzzle involving stones and a pentagram
• Petersen graph – Cubic graph with 10 vertices and 15 edges
• Ptolemy's theorem – Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle
• Seal of Solomon – Signet ring attributed to the Israelite king Solomon
• Star polygons in art and culture – Polygons as symbolic elements
• Star (heraldry) – In heraldry, any pierced or unpierced star-shaped charge with any number of straight or wavy rays
• Stellated polygons – Extending the elements of a polytope to form a new figure
References
1. Gene Brown (n.d.). "Difference Between Pentagram and Pentacle". Difference Between. Retrieved 29 June 2023.
2. πεντάγραμμον, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; a noun form of adjectival πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines"
3. πέντε, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus; Satan all 3 names mentioned before daylight full γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
4. this usage is borne out by the Oxford English Dictionary, although that work specifies that a circumscription makes the form of a five-pointed star and its etymon post-classical Latin pentaculum [...] A pentagram, esp. one enclosed in a circle; a talisman or magical symbol in the shape of or inscribed with a pentagram. Also, in extended use: any similar magical symbol (freq. applied to a hexagram formed by two intersecting or interlaced equilateral triangles)."
5. πένταλφα, "five Alphas", interpreting the shape as five Α shapes overlapping at 72-degree angles.
6. Budge, Sir E. A. Wallis (1968). Amulets and Talismans. p. 433.
7. Scott, Dustin Jon (2006). "History of the Pentagram". Retrieved 18 May 2021.
8. Allman, G. J., Greek Geometry From Thales to Euclid (1889), p.26.
9. 馬愛平 (23 September 2019). "距今5000年！良渚文物中發現最古老五角星圖案" (in Chinese). China Daily.
10. Coxeter, H.S.M.; Regular Polytopes, 3rd edn, Dover, 1973, p. 114.
11. Ball, W. W. Rouse and Coxeter, H. S. M.; Mathematical Recreations and Essays, 13th Edn., Dover, 1987, p. 176.
12. "Star of David vs. Pentagram: Everything You Need to Know". 17 July 2020.
13. πεντέμυχος, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
14. This is a lost book, but its contents are preserved in Damascius, De principiis, quoted in Kirk and Raven, (1983) [1956], p. 55.
15. "the divine products of Chronos' seed, when disposed in five recesses, were called πεντέμυχος (Pentemychos)" Kirk, Geoffrey Stephen; Raven, John Earle; Schofield, Malcolm (1983) [1957]. The Presocratic Philosophers: A Critical History with a Selection of Texts (2nd, illustrated, revised, reprint ed.). Cambridge University Press. pp. 51–52, 55. ISBN 978-0-521-27455-5. the only other place in Homer where Ortygie [sic] is mentioned is Odyssey V, 123, where Orion, having been carried off by Eos [the dawn], is slain... by Artemis... since solstices would normally be observed at sunrise in summer, and so in the north-east-by-east direction, that is what the phrase might suggest... the dwelling-place of Eos... Aia..
16. Allman, G. J., Greek Geometry From Thales to Euclid, part I (1877), in Hermathena 3.5, pp. 183, 197, citing Iamblichus and the Scholiast on Aristophanes. The pentagram was said to have been so called from Pythagoras himself having written the letters Υ, Γ, Ι, Θ (= /ei/), Α on its vertices.
17. Christian Symbols Ancient and Modern, Child, Heather and Dorothy Colles. New York: Charles Scribner's Sons, 1971, ISBN 0-7135-1960-6.
18. Morgan, Gerald (1979). "The Significance of the Pentangle Symbolism in "Sir Gawain and the Green Knight"". The Modern Language Review. 74 (4): 769–790. doi:10.2307/3728227. JSTOR 3728227.
19. Sir Gawain and the Green Knight, lines 619–665
20. Lévi, Éliphas (1999) [1896 (translated), 1854 (first published)]. Transcendental Magic, its Doctrine and Ritual [Dogme et rituel de la haute magie]. Trans. by A. E. Waite. York Beach: Weiser. OCLC 263626874.
21. Lévi, Éliphas (2002) [1939 (translated), 1859 (first published)]. The Key of the Mysteries [la Clef des grands mystères suivant Hénoch, Abraham, Hermès Trismégiste et Salomon]. Trans. by Aleister Crowley. Boston: Weiser. p. 69. OCLC 49053462.
22. Hartmann, Franz (1895) [1886]. Magic, White and Black (5th ed.). New York: The Path. OCLC 476635673.
23. "Goethe, Johann Wolfgang von (1749–1832) - Faust, Part I: Scenes I to III". www.poetryintranslation.com. Retrieved 25 May 2021.
24. Árnason, Jón (1862). "Töfrabrogð [Magic trick]". Íslenzkar Þjoðsögur og Æfintýri [Icelandic Folktales and Legends] (in Icelandic). Vol. 1. Leipzig: J. C. Hinrich's Bookstore. p. 432. Smèr það, er verður af tilberaspýunni, er kallað tilberasmèr; er það útlits sem annað smèr; en gjöri maður krossmark yfir því, eða risti á það kross, eða mynd þá, er smèrhnútur heitir,* springur það alt í smámola og verður eins og draflakyrníngur, svo ekki sèst eptir af því, nema agnir einar, eða það hjaðnar niður sem froða. Þykir það því varlegra, ef manni er boðið óhrjálegt smèr að borða, eða í gjöld, að gjóra annaðhvort þetta mark á það, því tilberasmèr þolir hvorki krossmark né smjörhnút. / * Smèrhnútur er svo í lögun:
25. In the Middle Ages, butter was used for payment, e.g. rent. See:
• Sexton, Regina (2003). "The Role and Function of Butter in the Diet of the Monk and Penitent in Early Medieval Ireland". In Walker, Harlan (ed.). The Fat of the Land: Proceedings of the Oxford Symposium on Food and Cooking 2002. Bristol: Footwork. pp. 253–269.
26. Chen, Yuan Julian (2014). "Legitimation Discourse and the Theory of the Five Elements in Imperial China". Journal of Song-Yuan Studies. 44 (1): 325–364. doi:10.1353/sys.2014.0000. S2CID 147099574.
27. Schouten, Jan (1968). The Pentagram as Medical Symbol: An Iconological Study. Hes & De Graaf. p. 18. ISBN 978-90-6004-166-6.
28. Waite, Arthur Edward (1886). The Mysteries of Magic: A Digest of the Writings of Eliphas Lévi. London: George Redway. p. 136.
29. DuQuette, Lon Milo (2003). The Magick of Aleister Crowley: A Handbook of the Rituals of Thelema. Weiser Books. pp. 93, 247. ISBN 978-1-57863-299-2.
30. "Bahá'í Reference Library - Directives from the Guardian, Pages 51-52". reference.bahai.org.
31. "The Nine-Pointed Star". bahai-library.com.
32. Moojan Momen (2019). The Star Tablet of the Bab. British Library Blog.
33. Bayat, Mohamad Ghasem (2001). An Introduction to the Súratu'l-Haykal (Discourse of The Temple) in Lights of Irfan, Book 2.
34. See the Nauvoo Temple Archived 17 May 2020 at the Wayback Machine website discussing its architecture, and particularly the page on Nauvoo Temple exterior symbolism Archived 17 May 2020 at the Wayback Machine. Retrieved 16 December 2006.
35. Brown, Matthew B (2002). "Inverted Stars on LDS Temples" (PDF). FAIRLDS.org. Archived from the original (PDF) on 29 February 2008.
36. "Religious Clothing in School", Robinson, B.A., Ontario Consultants on Religious Tolerance, 20 August 1999, updated 29 April 2005. Retrieved 10 February 2006. "ACLU Defends Honor Student Witch Pentacle" (Press release). American Civil Liberties Union of Michigan. 10 February 1999. Archived from the original on 8 November 2003. Retrieved 10 February 2006.{{cite press release}}: CS1 maint: bot: original URL status unknown (link) "Witches and wardrobes: Boy says he was suspended from school for wearing magical symbol" Rouvalis, Cristina; Pittsburgh Post-Gazette, 27 September 2000. Retrieved 10 February 2006.
37. "Federal judge upholds Indiana students' right to wear Wiccan symbols". Associated Press. 1 May 2000. Archived from the original on 30 March 2014. Retrieved 21 September 2007.
38. "Wiccan symbol OK for soldiers' graves". CNN.com. Associated Press. 23 April 2007. Archived from the original on 26 April 2007.
39. "Burial and Memorials: Available Emblems of Belief for Placement on Government Headstones and Markers". United States Department of Veterans Affairs. 3 July 2013. Retrieved 13 January 2014.
40. Gravrand 1990, p. 20.
41. Madiya, Clémentine Faïk-Nzuji (1996). Tracing Memory: A Glossary of Graphic Signs and Symbols in African Art and Culture. Mercury series, no. 71. Hull, Québec: Canadian Museum of Civilization. pp. 27, 155. ISBN 0-660-15965-1.
42. Ritual of the Order of the Eastern Star, 1976
43. Pietrocola, Giorgio (2005). "Tartapelago. Exposure of fractals". Maecla.
44. Baez, John (4 January 2014). "The Pentagram of Venus". Azimuth. Archived from the original on 14 December 2015. Retrieved 7 January 2016.
Bibliography
• Becker, Udo (1994). "Pentagram". The Continuum Encyclopedia of Symbols. Translated by Garmer, Lance W. New York City: Continuum Books. p. 230. ISBN 978-0-8264-0644-6.
• Conway, John Horton; Burgiel, Heidi; Goodman-Strauss, Chaim (April 2008). "Chapter 26, Higher Still: Regular Star-Polytopes". The Symmetries of Things. Wellesley, Massachusetts: A. K. Peters. p. 404. ISBN 978-1-56881-220-5.
• Ferguson, George Wells (1966) [1954]. Signs and Symbols in Christian Art. New York City: Oxford University Press. p. 59. OCLC 65081051.
• Gravrand, Henry (January 1990). La civilisation Sereer, Volume II: Pangool. Nouvelles éditions Africaines du Sénégal (in French). Dakar, Senegal. ISBN 2-7236-1055-1.{{cite book}}: CS1 maint: location missing publisher (link)
• Grünbaum, Branko; Shephard, Geoffrey Colin (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 978-0-7167-1193-3.
• Grünbaum, Branko (1994). "Polyhedra with Hollow Faces". In Bisztriczky, T.; McMullen, P.; Schneider, A.; Weiss, A. Ivić (eds.). Polytopes: Abstract, Convex and Computational. NATO ASI Series C: Mathematical and Physical Sciences. Vol. 440. Dordrecht: Springer Netherlands. pp. 43–70. doi:10.1007/978-94-011-0924-6_3. ISBN 978-94-010-4398-4.
External links
Wikimedia Commons has media related to Pentagrams.
• Weisstein, Eric W. "Pentagram". MathWorld.
• The Pythagorean Pentacle from the Biblioteca Arcana.
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| Wikipedia |
Spherical angle
A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs of great circles on a sphere. It is measured by the angle between the planes containing the arcs (which naturally also contain the centre of the sphere).[1]
Not to be confused with Solid angle.
See also
• Spherical coordinate system
• Spherical trigonometry
• Transcendent angle
References
1. Green, Robin Michael (1985), Spherical Astronomy, Cambridge University Press, p. 3, ISBN 9780521317795.
| Wikipedia |
Tiny and miny
In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equal to {0|{0|-G}} for any game G, whereas miny G (analogously denoted ⧿G) is tiny G's negative, or {{G|0}|0}.
Tiny and miny aren't just abstract mathematical operators on combinatorial games: tiny and miny games do occur "naturally" in such games as toppling dominoes. Specifically, tiny n, where n is a natural number, can be generated by placing two black dominoes outside n + 2 white dominoes.
Tiny games and up have certain curious relational characteristics. Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up. Expansion of ⧾⧾⧾G into its canonical form yields {0|{{0|{{0|{0|-G}}|0}}|0}}. While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑. Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's assertion is also easily verifiable with canonical forms and game trees.
References
• Albert, Michael H.; Nowakowski, Richard J.; Wolfe, David (2007). Lessons in Play: An Introduction to Combinatorial Game Theory. A K Peters, Ltd. ISBN 1-56881-277-9.
• Berlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (2003). Winning Ways for Your Mathematical Plays. A K Peters, Ltd.
| Wikipedia |
Integral symbol
The integral symbol:
∫ (Unicode), $\displaystyle \int $ (LaTeX)
∫
Integral symbol
In UnicodeU+222B ∫ INTEGRAL (∫, ∫)
Graphical variants
$\displaystyle \int $
Different from
Different fromU+017F ſ LONG S
U+0283 ʃ ESH
is used to denote integrals and antiderivatives in mathematics, especially in calculus.
History
Main article: Leibniz's notation
The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings;[1][2] it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686.[3][4] The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.
Typography in Unicode and LaTeX
Fundamental symbol
Main article: Integral calculus
The integral symbol is U+222B ∫ INTEGRAL in Unicode[5] and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (named entity).
The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.
The ∫ symbol is very similar to, but not to be confused with, the letter ʃ ("esh").
Extensions of the symbol
See also: Multiple integral
Related symbols include:[5][6]
Meaning Unicode LaTeX
Double integral ∬ U+222C $\iint $ \iint
Triple integral ∭ U+222D $\iiint $ \iiint
Quadruple integral ⨌ U+2A0C $\iiiint $ \iiiint
Contour integral ∮ U+222E $\oint $ \oint
Clockwise integral ∱ U+2231
Counterclockwise integral ⨑ U+2A11
Clockwise contour integral ∲ U+2232 \varointclockwise
Counterclockwise contour integral ∳ U+2233 \ointctrclockwise
Closed surface integral ∯ U+222F \oiint
Closed volume integral ∰ U+2230 \oiiint
Typography in other languages
In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.[7]
Another difference is in the placement of limits for definite integrals. Generally, in English-language books, limits go to the right of the integral symbol:
$\int _{0}^{5}f(t)\,\mathrm {d} t,\quad \int _{g(t)=a}^{g(t)=b}f(t)\,\mathrm {d} t.$
By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing, but is more compact horizontally, especially when longer expressions are used in the limits:
$\int \limits _{0}^{T}f(t)\,\mathrm {d} t,\quad \int \limits _{\!\!\!\!\!g(t)=a\!\!\!\!\!}^{\!\!\!\!\!g(t)=b\!\!\!\!\!}f(t)\,\mathrm {d} t.$
See also
• Capital sigma notation
• Capital pi notation
Notes
1. Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675).
2. Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017.
3. Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017
4. Stillwell, John (1989). Mathematics and its History. Springer. p. 110.
5. "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
6. "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.
7. "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021.
References
• Stewart, James (2003). "Integrals". Single Variable Calculus: Early Transcendentals (5th ed.). Belmont, CA: Brooks/Cole. p. 381. ISBN 0-534-39330-6.
• Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (1999), "Russian Typographical Traditions in Mathematical Literature" (PDF), Russian Typographical Traditions in Mathematical Literature, EuroTeX'99 Proceedings
External links
• Fileformat.info
Infinitesimals
History
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• Leibniz's notation
• Integral symbol
• Criticism of nonstandard analysis
• The Analyst
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• Cavalieri's principle
Related branches
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Mathematicians
• Gottfried Wilhelm Leibniz
• Abraham Robinson
• Pierre de Fermat
• Augustin-Louis Cauchy
• Leonhard Euler
Textbooks
• Analyse des Infiniment Petits
• Elementary Calculus
• Cours d'Analyse
| Wikipedia |
Nimber
In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition and ordinal multiplication.
Not to be confused with Number.
Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.
The nimber addition and multiplication operations are associative and commutative. Each nimber is its own negative. In particular for some pairs of ordinals, their nimber sum is smaller than either addend.[1] The minimum excludant operation is applied to sets of nimbers.
Uses
Nim
Main article: Nim
Nim is a game in which two players take turns removing objects from distinct heaps. As moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric, Nim is an impartial game. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap. The goal of the game is to be the player who removes the last object. The nimber of a heap is simply the number of objects in that heap. Using nim addition, one can calculate the nimber of the game as a whole. The winning strategy is to force the nimber of the game to 0 for the opponent's turn.[2]
Cram
Main article: Cram (game)
Cram is a game often played on a rectangular board in which players take turns placing dominoes either horizontally or vertically until no more dominoes can be placed. The first player that cannot make a move loses. As the possible moves for both players are the same, it is an impartial game and can have a nimber value. For example, any board that is an even size by an even size will have a nimber of 0. Any board that is even by odd will have a non-zero nimber. Any 2×n board will have a nimber of 0 for all even n and a nimber of 1 for all odd n.
Northcott's game
In Northcott's game, pegs for each player are placed along a column with a finite number of spaces. Each turn each player must move the piece up or down the column, but may not move past the other player's piece. Several columns are stacked together to add complexity. The player that can no longer make any moves loses. Unlike many other nimber related games, the number of spaces between the two tokens on each row are the sizes of the Nim heaps. If your opponent increases the number of spaces between two tokens, just decrease it on your next move. Else, play the game of Nim and make the Nim-sum of the number of spaces between the tokens on each row be 0.[3]
Hackenbush
Hackenbush is a game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line. Players take turns removing line segments. An impartial game version, thereby a game able to be analyzed using nimbers, can be found by removing distinction from the lines, allowing either player to cut any branch. Any segments reliant on the newly removed segment in order to connect to the ground line are removed as well. In this way, each connection to the ground can be considered a nim heap with a nimber value. Additionally, all the separate connections to the ground line can also be summed for a nimber of the game state.
Addition
Nimber addition (also known as nim-addition) can be used to calculate the size of a single nim heap equivalent to a collection of nim heaps. It is defined recursively by
α ⊕ β = mex({α′ ⊕ β : α' < α} ∪ {α ⊕ β′ : β′ < β}),
where the minimum excludant mex(S) of a set S of ordinals is defined to be the smallest ordinal that is not an element of S.
For finite ordinals, the nim-sum is easily evaluated on a computer by taking the bitwise exclusive or (XOR, denoted by ⊕) of the corresponding numbers. For example, the nim-sum of 7 and 14 can be found by writing 7 as 111 and 14 as 1110; the ones place adds to 1; the twos place adds to 2, which we replace with 0; the fours place adds to 2, which we replace with 0; the eights place adds to 1. So the nim-sum is written in binary as 1001, or in decimal as 9.
This property of addition follows from the fact that both mex and XOR yield a winning strategy for Nim and there can be only one such strategy; or it can be shown directly by induction: Let α and β be two finite ordinals, and assume that the nim-sum of all pairs with one of them reduced is already defined. The only number whose XOR with α is α ⊕ β is β, and vice versa; thus α ⊕ β is excluded. On the other hand, for any ordinal γ < α ⊕ β, XORing ξ ≔ α ⊕ β ⊕ γ with all of α, β and γ must lead to a reduction for one of them (since the leading 1 in ξ must be present in at least one of the three); since ξ ⊕ γ = α ⊕ β > γ, we must have α > ξ ⊕ α = β ⊕ γ or β > ξ ⊕ β = α ⊕ γ; thus γ is included as (β ⊕ γ) ⊕ β or as α ⊕ (α ⊕ γ), and hence α ⊕ β is the minimum excluded ordinal.
Nimber addition is associative and commutative, with 0 as the additive identity element. Moreover, a nimber is its own additive inverse.[4] It follows that α ⊕ β = 0 if and only if α = β.
Multiplication
Nimber multiplication (nim-multiplication) is defined recursively by
α β = mex({α′ β ⊕ α β′ ⊕ α' β′ : α′ < α, β′ < β}).
Nimber multiplication is associative and commutative, with the ordinal 1 as the multiplicative identity element. Moreover, nimber multiplication distributes over nimber addition.[4]
Thus, except for the fact that nimbers form a proper class and not a set, the class of nimbers forms a ring. In fact, it even determines an algebraically closed field of characteristic 2, with the nimber multiplicative inverse of a nonzero ordinal α given by
α−1 = mex(S),
where S is the smallest set of ordinals (nimbers) such that
1. 0 is an element of S;
2. if 0 < α′ < α and β′ is an element of S, then (1 + (α′ − α) β′) / α′−1 is also an element of S.
For all natural numbers n, the set of nimbers less than 22n form the Galois field GF(22n) of order 22n. Therefore, the set of finite nimbers is isomorphic to the direct limit as n → ∞ of the fields GF(22n). This subfield is not algebraically closed, since no field GF(2k) with k not a power of 2 is contained in any of those fields, and therefore not in their direct limit; for instance the polynomial x3 + x + 1, which has a root in GF(23), does not have a root in the set of finite nimbers.
Just as in the case of nimber addition, there is a means of computing the nimber product of finite ordinals. This is determined by the rules that
1. The nimber product of a Fermat 2-power (numbers of the form 22n) with a smaller number is equal to their ordinary product;
2. The nimber square of a Fermat 2-power x is equal to 3x/2 as evaluated under the ordinary multiplication of natural numbers.
The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal ωωω, where ω is the smallest infinite ordinal. It follows that as a nimber, ωωω is transcendental over the field.[5]
Addition and multiplication tables
The following tables exhibit addition and multiplication among the first 16 nimbers.
This subset is closed under both operations, since 16 is of the form 22n. (If you prefer simple text tables, they are here.)
See also
• Surreal number
Notes
1. Advances in computer games : 14th International Conference, ACG 2015, Leiden, the Netherlands, July 1-3, 2015, Revised selected papers. Herik, Jaap van den,, Plaat, Aske,, Kosters, Walter. Cham. 2015-12-24. ISBN 978-3319279923. OCLC 933627646.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
2. Anany., Levitin (2012). Introduction to the design & analysis of algorithms (3rd ed.). Boston: Pearson. ISBN 9780132316811. OCLC 743298766.
3. "Theory of Impartial Games" (PDF). Feb 3, 2009.
4. Brown, Ezra; Guy, Richard K. (2021). "2.5 Nim arithmetic and Nim algebra". The Unity of Combinatorics. Vol. 36 of The Carus Mathematical Monographs (reprint ed.). American Mathematical Society. p. 35. ISBN 978-1-4704-6509-4.
5. Conway 1976, p. 61.
References
• Conway, John Horton (1976). On Numbers and Games. Academic Press Inc. (London) Ltd.
• Lenstra, H. W. (1978). Nim multiplication. Report IHES/M/78/211. Institut des hautes études scientifiques. hdl:1887/2125.
• Schleicher, Dierk; Stoll, Michael (2004). "An Introduction to Conway's Games and Numbers". arXiv:math.DO/0410026. which discusses games, surreal numbers, and nimbers.
| Wikipedia |
Smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (X, x0) and (Y, y0) is the quotient of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x in X and y in Y. The smash product is itself a pointed space, with basepoint being the equivalence class of (x0, y0). The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous).
For the smash product in the theory of Hopf algebras, see Hopf smash product.
One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum $X\vee Y=(X\amalg Y)\;/{\sim }$. In particular, {x0} × Y in X × Y is identified with Y in $X\vee Y$, ditto for X × {y0} and X. In $X\vee Y$, subspaces X and Y intersect in the single point $x_{0}\sim y_{0}$. The smash product is then the quotient
$X\wedge Y=(X\times Y)/(X\vee Y).$
The smash product shows up in homotopy theory, a branch of algebraic topology. In homotopy theory, one often works with a different category of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complexes is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
Examples
• The smash product of any pointed space X with a 0-sphere (a discrete space with two points) is homeomorphic to X.
• The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere.
• More generally, the smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm+n.
• The smash product of a space X with a circle is homeomorphic to the reduced suspension of X:
$\Sigma X\cong X\wedge S^{1}.$
• The k-fold iterated reduced suspension of X is homeomorphic to the smash product of X and a k-sphere
$\Sigma ^{k}X\cong X\wedge S^{k}.$
• In domain theory, taking the product of two domains (so that the product is strict on its arguments).
As a symmetric monoidal product
For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g., that of compactly generated spaces), there are natural (basepoint preserving) homeomorphisms
${\begin{aligned}X\wedge Y&\cong Y\wedge X,\\(X\wedge Y)\wedge Z&\cong X\wedge (Y\wedge Z).\end{aligned}}$
However, for the naive category of pointed spaces, this fails, as shown by the counterexample $X=Y=\mathbb {Q} $ and $Z=\mathbb {N} $ found by Dieter Puppe.[1] A proof due to Kathleen Lewis that Puppe's counterexample is indeed a counterexample can be found in the book of Johann Sigurdsson and J. Peter May.[2]
These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product in an appropriate category of pointed spaces.
Adjoint relationship
Adjoint functors make the analogy between the tensor product and the smash product more precise. In the category of R-modules over a commutative ring R, the tensor functor $(-\otimes _{R}A)$ is left adjoint to the internal Hom functor $\mathrm {Hom} (A,-)$, so that
$\mathrm {Hom} (X\otimes A,Y)\cong \mathrm {Hom} (X,\mathrm {Hom} (A,Y)).$
In the category of pointed spaces, the smash product plays the role of the tensor product in this formula: if $A,X$ are compact Hausdorff then we have an adjunction
$\mathrm {Maps_{*}} (X\wedge A,Y)\cong \mathrm {Maps_{*}} (X,\mathrm {Maps_{*}} (A,Y))$
where $\operatorname {Maps_{*}} $ denotes continuous maps that send basepoint to basepoint, and $\mathrm {Maps_{*}} (A,Y)$ carries the compact-open topology.[3]
In particular, taking $A$ to be the unit circle $S^{1}$, we see that the reduced suspension functor $\Sigma $ is left adjoint to the loop space functor $\Omega $:
$\mathrm {Maps_{*}} (\Sigma X,Y)\cong \mathrm {Maps_{*}} (X,\Omega Y).$
Notes
1. Puppe, Dieter (1958). "Homotopiemengen und ihre induzierten Abbildungen. I.". Mathematische Zeitschrift. 69: 299–344. doi:10.1007/BF01187411. MR 0100265. S2CID 121402726. (p. 336)
2. May, J. Peter; Sigurdsson, Johann (2006). Parametrized Homotopy Theory. Mathematical Surveys and Monographs. Vol. 132. Providence, RI: American Mathematical Society. section 1.5. ISBN 978-0-8218-3922-5. MR 2271789.
3. "Algebraic Topology", Maunder, Theorem 6.2.38c
References
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.
| Wikipedia |
Hexagon
In geometry, a hexagon (from Greek ἕξ, hex, meaning "six", and γωνία, gonía, meaning "corner, angle") is a six-sided polygon.[1] The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
For the crystal system, see Hexagonal crystal family.
Regular hexagon
A regular hexagon
TypeRegular polygon
Edges and vertices6
Schläfli symbol{6}, t{3}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D6), order 2×6
Internal angle (degrees)120°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular hexagon
A regular hexagon has Schläfli symbol {6}[2] and can also be constructed as a truncated equilateral triangle, t{3}, which alternates two types of edges.
A step-by-step animation of the construction of a regular hexagon using compass and straightedge, given by Euclid's Elements, Book IV, Proposition 15: this is possible as 6 $=$ 2 × 3, a product of a power of two and distinct Fermat primes.
When the side length AB is given, drawing a circular arc from point A and point B gives the intersection M, the center of the circumscribed circle. Transfer the line segment AB four times on the circumscribed circle and connect the corner points.
A regular hexagon is defined as a hexagon that is both equilateral and equiangular. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle).
The common length of the sides equals the radius of the circumscribed circle or circumcircle, which equals ${\tfrac {2}{\sqrt {3}}}$ times the apothem (radius of the inscribed circle). All internal angles are 120 degrees. A regular hexagon has six rotational symmetries (rotational symmetry of order six) and six reflection symmetries (six lines of symmetry), making up the dihedral group D6. The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral.
Parameters
The maximal diameter (which corresponds to the long diagonal of the hexagon), D, is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor:
${\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t$ and, similarly, $d={\frac {\sqrt {3}}{2}}D.$
The area of a regular hexagon
${\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}$
For any regular polygon, the area can also be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, and p${}=6R=4r{\sqrt {3}}$, so
${\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}$
The regular hexagon fills the fraction ${\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270$ of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD.
It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 between parallel sides.
Point in plane
For an arbitrary point in the plane of a regular hexagon with circumradius $R$, whose distances to the centroid of the regular hexagon and its six vertices are $L$ and $d_{i}$ respectively, we have[3]
$d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),$
$d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),$
$d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).$
If $d_{i}$ are the distances from the vertices of a regular hexagon to any point on its circumcircle, then [3]
$\left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.$
Symmetry
Example hexagons by symmetry
r12
regular
i4
d6
isotoxal
g6
directed
p6
isogonal
d2
g2
general
parallelogon
p2
g3
a1
The regular hexagon has D6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order.[4] r12 is full symmetry, and a1 is no symmetry. p6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can seen as directed edges.
Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations.
p6m (*632) cmm (2*22) p2 (2222) p31m (3*3) pmg (22*) pg (××)
r12
i4
g2
d2
d2
p2
a1
Dih6 Dih2 Z2 Dih1 Z1
A2 and G2 groups
A2 group roots
G2 group roots
The 6 roots of the simple Lie group A2, represented by a Dynkin diagram , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
The 12 roots of the Exceptional Lie group G2, represented by a Dynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
Dissection
6-cube projection 12 rhomb dissection
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2m(m − 1) parallelograms.[5] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube, with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids.
Dissection of hexagons into three rhombs and parallelograms
2D Rhombs Parallelograms
Regular {6} Hexagonal parallelogons
3D Square faces Rectangular faces
Cube Rectangular cuboid
Related polygons and tilings
A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of the regular hexagonal tiling, {6,3}, with three hexagonal faces around each vertex.
A regular hexagon can also be created as a truncated equilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.
A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle, {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling.
A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling.
Regular
{6}
Truncated
t{3} = {6}
Hypertruncated triangles Stellated
Star figure 2{3}
Truncated
t{6} = {12}
Alternated
h{6} = {3}
Crossed
hexagon
A concave hexagon A self-intersecting hexagon (star polygon) Extended
Central {6} in {12}
A skew hexagon, within cube Dissected {6} projection
octahedron
Complete graph
Self-crossing hexagons
There are six self-crossing hexagons with the vertex arrangement of the regular hexagon:
Self-intersecting hexagons with regular vertices
Dih2 Dih1 Dih3
Figure-eight
Center-flip
Unicursal
Fish-tail
Double-tail
Triple-tail
Hexagonal structures
From bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression.
Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation.
Hexagonal prism tessellations
Form Hexagonal tiling Hexagonal prismatic honeycomb
Regular
Parallelogonal
Tesselations by hexagons
Main article: Hexagonal tiling
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane.
Hexagon inscribed in a conic section
Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
Cyclic hexagon
The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point.
If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf.[6]
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent.[7]
If a hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.[8]: p. 179
Hexagon tangential to a conic section
Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.
In a hexagon that is tangential to a circle and that has consecutive sides a, b, c, d, e, and f,[9]
$a+c+e=b+d+f.$
Equilateral triangles on the sides of an arbitrary hexagon
If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle.[10]: Thm. 1
Skew hexagon
A skew hexagon is a skew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes.
A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D3d, [2+,6] symmetry, order 12.
The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Skew hexagons on 3-fold axes
Cube
Octahedron
Petrie polygons
The regular skew hexagon is the Petrie polygon for these higher dimensional regular, uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections:
4D 5D
3-3 duoprism
3-3 duopyramid
5-simplex
Convex equilateral hexagon
A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a, there exists[11]: p.184, #286.3 a principal diagonal d1 such that
${\frac {d_{1}}{a}}\leq 2$
and a principal diagonal d2 such that
${\frac {d_{2}}{a}}>{\sqrt {3}}.$
Polyhedra with hexagons
There is no Platonic solid made of only regular hexagons, because the hexagons tessellate, not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron, truncated octahedron, truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron. These hexagons can be considered truncated triangles, with Coxeter diagrams of the form and .
Hexagons in Archimedean solids
Tetrahedral Octahedral Icosahedral
truncated tetrahedron
truncated octahedron
truncated cuboctahedron
truncated icosahedron
truncated icosidodecahedron
There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0):
Hexagons in Goldberg polyhedra
Tetrahedral Octahedral Icosahedral
Chamfered tetrahedron
Chamfered cube
Chamfered dodecahedron
There are also 9 Johnson solids with regular hexagons:
Johnson solids with hexagons
triangular cupola
elongated triangular cupola
gyroelongated triangular cupola
augmented hexagonal prism
parabiaugmented hexagonal prism
metabiaugmented hexagonal prism
triaugmented hexagonal prism
augmented truncated tetrahedron
triangular hebesphenorotunda
Truncated triakis tetrahedron
Prismoids with hexagons
Hexagonal prism
Hexagonal antiprism
Hexagonal pyramid
Tilings with regular hexagons
Regular 1-uniform
{6,3}
r{6,3}
rr{6,3}
tr{6,3}
2-uniform tilings
Gallery of natural and artificial hexagons
• The ideal crystalline structure of graphene is a hexagonal grid.
• Assembled E-ELT mirror segments
• A beehive honeycomb
• The scutes of a turtle's carapace
• Saturn's hexagon, a hexagonal cloud pattern around the north pole of the planet
• Micrograph of a snowflake
• Benzene, the simplest aromatic compound with hexagonal shape.
• Hexagonal order of bubbles in a foam.
• Crystal structure of a molecular hexagon composed of hexagonal aromatic rings.
• Naturally formed basalt columns from Giant's Causeway in Northern Ireland; large masses must cool slowly to form a polygonal fracture pattern
• An aerial view of Fort Jefferson in Dry Tortugas National Park
• The James Webb Space Telescope mirror is composed of 18 hexagonal segments.
• In French, l'Hexagone refers to Metropolitan France for its vaguely hexagonal shape.
• Hexagonal Hanksite crystal, one of many hexagonal crystal system minerals
• Hexagonal barn
• The Hexagon, a hexagonal theatre in Reading, Berkshire
• Władysław Gliński's hexagonal chess
• Pavilion in the Taiwan Botanical Gardens
• Hexagonal window
See also
• 24-cell: a four-dimensional figure which, like the hexagon, has orthoplex facets, is self-dual and tessellates Euclidean space
• Hexagonal crystal system
• Hexagonal number
• Hexagonal tiling: a regular tiling of hexagons in a plane
• Hexagram: six-sided star within a regular hexagon
• Unicursal hexagram: single path, six-sided star, within a hexagon
• Honeycomb conjecture
• Havannah: abstract board game played on a six-sided hexagonal grid
References
1. Cube picture
2. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, archived from the original on 2016-01-02, retrieved 2015-11-06.
3. Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340. doi:10.26713/cma.v11i3.1420 (inactive 1 August 2023).{{cite journal}}: CS1 maint: DOI inactive as of August 2023 (link)
4. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
5. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
6. Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40.
7. Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17.
8. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960).
9. Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", Archived 2012-05-11 at the Wayback Machine, Accessed 2012-04-17.
10. Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12.
11. Inequalities proposed in "Crux Mathematicorum", Archived 2017-08-30 at the Wayback Machine.
External links
Look up hexagon in Wiktionary, the free dictionary.
• Weisstein, Eric W. "Hexagon". MathWorld.
• Definition and properties of a hexagon with interactive animation and construction with compass and straightedge.
• An Introduction to Hexagonal Geometry on Hexnet a website devoted to hexagon mathematics.
• Hexagons are the Bestagons on YouTube – an animated internet video about hexagons by CGP Grey.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Soroban
The soroban (算盤, そろばん, counting tray) is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century.[1][nb 1] Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electronic calculators.
Construction
The soroban is composed of an odd number of columns or rods, each having beads: one separate bead having a value of five, called go-dama (五玉, ごだま, "five-bead") and four beads each having a value of one, called ichi-dama (一玉, いちだま, "one-bead"). Each set of beads of each rod is divided by a bar known as a reckoning bar. The number and size of beads in each rod make a standard-sized 13-rod soroban much less bulky than a standard-sized suanpan of similar expressive power.
The number of rods in a soroban is always odd and never fewer than seven. Basic models usually have thirteen rods, but the number of rods on practical or standard models often increases to 21, 23, 27 or even 31, thus allowing calculation of more digits or representations of several different numbers at the same time. Each rod represents a digit, and a larger number of rods allows the representation of more digits, either in singular form or during operations.
The beads and rods are made of a variety of different materials. Most soroban made in Japan are made of wood and have wood, metal, rattan, or bamboo rods for the beads to slide on. The beads themselves are usually biconal (shaped like a double-cone). They are normally made of wood, although the beads of some soroban, especially those made outside Japan, can be marble, stone, or even plastic. The cost of a soroban is commensurate with the materials used in its construction.
One unique feature that sets the soroban apart from its Chinese cousin is a dot marking every third rod in a soroban. These are unit rods and any one of them is designated to denote the last digit of the whole number part of the calculation answer. Any number that is represented on rods to the right of this designated rod is part of the decimal part of the answer, unless the number is part of a division or multiplication calculation. Unit rods to the left of the designated one also aid in place value by denoting the groups in the number (such as thousands, millions, etc.). Suanpan usually do not have this feature.
Usage
Representation of numbers
The soroban uses a bi-quinary coded decimal system, where each of the rods can represent a single digit from 0 to 9. By moving beads towards the reckoning bar, they are put in the "on" position; i.e., they assume value. For the "five bead" this means it is moved downwards, while "one beads" are moved upwards. In this manner, all digits from 0 to 9 can be represented by different configurations of beads, as shown below:
Representation of digits 0 - 9 on the soroban
0123456789
These digits can subsequently be used to represent multiple-digit numbers. This is done in the same way as in Western, decimal notation: the rightmost digit represents units, the one to the left of it represents tens, etc. The number 8036, for instance, is represented by the following configuration:
8036
The soroban user is free to choose which rod is used for the units; typically this will be one of the rods marked with a dot (see the 6 in the example above). Any digits to the right of the units represent decimals: tenths, hundredths, etc. In order to change 8036 into 80.36, for instance, the user places the digits in such a way that the 0 falls on a rod marked with a dot:
80.36
Methods of operation
The methods of addition and subtraction on a soroban are basically the same as the equivalent operations on a suanpan, with basic addition and subtraction making use of a complementary number to add or subtract ten in carrying over.
There are many methods to perform both multiplication and division on a soroban, especially Chinese methods that came with the importation of the suanpan. The authority in Japan on the soroban, the Japan Abacus Committee, has recommended so-called standard methods for both multiplication and division which require only the use of the multiplication table. These methods were chosen for efficiency and speed in calculation.
Because the soroban developed through a reduction in the number of beads from seven, to six, and then to the present five, these methods can be used on the suanpan as well as on soroban produced before the 1930s, which have five "one" beads and one "five" bead.
Modern use
The Japanese abacus has been taught in school for over 500 years, deeply rooted in the value of learning the fundamentals as a form of art.[3] However, the introduction of the West during the Meiji period and then again after World War II has gradually altered the Japanese education system. Now, the strive is for speed and turning out deliverables rather than understanding the subtle intricacies of the concepts behind the product. Calculators have since replaced sorobans, and elementary schools are no longer required to teach students how to use the soroban, though some do so by choice. The growing popularity of calculators within the context of Japanese modernization has driven the study of soroban from public schools to private after school classrooms. Where once it was an institutionally required subject in school for children grades 2 to 6, current laws have made keeping this art form and perspective on math practiced amongst the younger generations more lenient.[4] Today, it shifted from a given to a game where one can take The Japanese Chamber of Commerce and Industry's examination in order to obtain a certificate and license.[5]
There are six levels of mastery, starting from sixth-grade (very skilled) all the way up to first-grade (for those who have completely mastered the use of the soroban). Those obtaining at least a third-grade certificate/license are qualified to work in public corporations.
The soroban is still taught in some primary schools as a way to visualize and grapple with mathematical concepts. The practice of soroban includes the teacher reciting a string of numbers (addition, subtraction, multiplication, and division) in a song-like manner where at the end, the answer is given by the teacher. This helps train the ability to follow the tempo given by the teacher while remaining calm and accurate. In this way, it reflects on a fundamental aspect of Japanese culture of practicing meditative repetition in every aspect of life.[3] Primary school students often bring two soroban to class, one with the modern configuration and the one having the older configuration of one heavenly bead and five earth beads.
Shortly after the beginning of one's soroban studies, drills to enhance mental calculation, known as anzan (暗算, "blind calculation") in Japanese are incorporated. Students are asked to solve problems mentally by visualizing the soroban and working out the solution by moving the beads theoretically in one's mind. The mastery of anzan is one reason why, despite the access to handheld calculators, some parents still send their children to private tutors to learn the soroban.
The soroban is also the basis for two kinds of abaci developed for the use of blind people. One is the toggle-type abacus wherein flip switches are used instead of beads. The second is the Cranmer abacus which has circular beads, longer rods, and a leather backcover so the beads do not slide around when in use.
Brief history
The soroban's physical resemblance is derived from the suanpan but the number of beads is identical to the Roman abacus, which had four beads below and one at the top.
Most historians on the soroban agree that it has its roots on the suanpan's importation to Japan via the Korean peninsula around the 14th century.[1][nb 1] When the suanpan first became native to Japan as the soroban (with its beads modified for ease of use), it had two heavenly beads and five earth beads. But the soroban was not widely used until the 17th century, although it was in use by Japanese merchants since its introduction.[6] Once the soroban became popularly known, several Japanese mathematicians, including Seki Kōwa, studied it extensively. These studies became evident on the improvements on the soroban itself and the operations used on it.
In the construction of the soroban itself, the number of beads had begun to decrease. In around 1850, one heavenly bead was removed from the suanpan configuration of two heavenly beads and five earth beads. This new Japanese configuration existed concurrently with the suanpan until the start of the Meiji era, after which the suanpan fell completely out of use. In 1891, Irie Garyū further removed one earth bead, forming the modern configuration of one heavenly bead and four earth beads.[7] This configuration was later reintroduced in 1930 and became popular in the 1940s.
Also, when the suanpan was imported to Japan, it came along with its division table. The method of using the table was called kyūkihō (九帰法, "nine returning method") in Japanese, while the table itself was called the hassan (八算, "eight calculation"). The division table used along with the suanpan was more popular because of the original hexadecimal configuration of Japanese currency . But because using the division table was complicated and it should be remembered along with the multiplication table, it soon fell out in 1935 (soon after the soroban's present form was reintroduced in 1930), with a so-called standard method replacing the use of the division table. This standard method of division, recommended today by the Japan Abacus Committee, is in fact an old method which used counting rods, first suggested by mathematician Momokawa Chubei in 1645,[8] and therefore had to compete with the division table during the latter's heyday
Comparison with the electric calculator
On November 12, 1946, a contest was held in Tokyo between the Japanese soroban, used by Kiyoshi Matsuzaki, and an electric calculator, operated by US Army Private Thomas Nathan Wood. The basis for scoring in the contest was speed and accuracy of results in all four basic arithmetic operations and a problem which combines all four. The soroban won 4 to 1, with the electric calculator prevailing in multiplication.[9]
About the event, the Nippon Times newspaper reported that "Civilization ... tottered" that day, while the Stars and Stripes newspaper described the soroban's "decisive" victory as an event in which "the machine age took a step backward....".
The breakdown of results is as follows:
• Five additions problems for each heat, each problem consisting of 50 three- to six-digit numbers. The soroban won in two successive heats.
• Five subtraction problems for each heat, each problem having six- to eight-digit minuends and subtrahends. The soroban won in the first and third heats; the second heat was a no contest.
• Five multiplication problems, each problem having five- to 12-digit factors. The calculator won in the first and third heats; the soroban won on the second.
• Five division problems, each problem having five- to 12-digit dividends and divisors. The soroban won in the first and third heats; the calculator won on the second.
• A composite problem which the soroban answered correctly and won on this round. It consisted of:
• An addition problem involving 30 six-digit numbers
• Three subtraction problems, each with two six-digit numbers
• Three multiplication problems, each with two figures containing a total of five to twelve digits
• Three division problems, each with two figures containing a total of five to twelve digits
Even with the improvement of technology involving calculators, this event has yet to be replicated officially.
See also
• Abacus
• Suanpan
• Chisanbop
• Pental system
• Bi-quinary coded decimal
Notes
1. Some sources give a date of introduction of around 1600.[2]
Footnotes
Wikimedia Commons has media related to Soroban.
1. Gullberg 1997, p. 169 harvnb error: no target: CITEREFGullberg1997 (help)
2. Fernandes 2013
3. Suzuki, Daisetz T. (1959). Zen and the Japanese Culture. Princeton University Press.
4. "Soroban in Education and Modern Japanese Society". History of Soroban. Retrieved 21 November 2018.
5. Kojima, Takashi (1954). The Japanese Abacus: its Use and Theory. Tokyo: Charles E. Tuttle. ISBN 0-8048-0278-5.
6. "そろばんの歴史 ー 西欧、中国、そして日本へ", "トモエそろばん", Retrieved 2017-10-19.
7. Frédéric, Louis (2002). Japan encyclopedia. Translated by Roth, Käthe. Harvard University Press. pp. 303, 903. ISBN 9780674017535.
8. Smith, David Eugene; Mikami, Yoshio (1914). "Chapter III: The Development of the Soroban.". A History of Japanese Mathematics. The Open Court Publishing. pp. 43–44. digital copy Archived 2010-12-03 at the Wayback Machine
9. Stoddard, Edward (1994). Speed Mathematics Simplified. Dover. p. 12.
References
• Kojima, Takashi (1963). Advanced Abacus: Japanese Theory and Practice. Tokyo: Charles E. Tuttle.
• Soroban. Japan: The Japan Chamber of Commerce and Industry. 1989.
• Bernazzani, David (March 2, 2005). Soroban Abacus Handbook (PDF) (Rev 1.05 ed.).
• Fernandes, Luis (2013). "The Abacus: A Brief History". ee.ryerson.ca. Archived from the original on March 3, 2000. Retrieved July 31, 2014.
• Heffelfinger, Totton; Flom, Gary (2004). Abacus: Mystery of the Bead.
• Knott, Cargill Gilston (1886). "The Abacus, in Its Historic and Scientific Aspects" (PDF). The Transactions of the Asiatic Society of Japan. xiv: 18–72.
External links
• Japanese Soroban Association (in English)
| Wikipedia |
Tangram
The tangram (Chinese: 七巧板; pinyin: qīqiǎobǎn; lit. 'seven boards of skill') is a dissection puzzle consisting of seven flat polygons, called tans, which are put together to form shapes. The objective is to replicate a pattern (given only an outline) generally found in a puzzle book using all seven pieces without overlap. Alternatively the tans can be used to create original minimalist designs that are either appreciated for their inherent aesthetic merits or as the basis for challenging others to replicate its outline. It is reputed to have been invented in China sometime around the late 18th century and then carried over to America and Europe by trading ships shortly after.[1] It became very popular in Europe for a time, and then again during World War I. It is one of the most widely recognized dissection puzzles in the world and has been used for various purposes including amusement, art, and education. [2][3][4]
Etymology
The origin of the English word 'tangram' is unclear. One conjecture holds that it is a compound of the Greek element '-gram' derived from γράμμα ('written character, letter, that which is drawn') with the 'tan-' element being variously conjectured to be Chinese t'an 'to extend' or Cantonese t'ang 'Chinese'.[5] Alternatively, the word may be derivative of the archaic English 'tangram' meaning "an odd, intricately contrived thing".[6]
In either case, the first known use of the word is believed to be found in the 1848 book Geometrical Puzzle for the Young by mathematician and future Harvard University president Thomas Hill.[7] Hill likely coined the term in the same work, and vigorously promoted the word in numerous articles advocating for the puzzle's use in education, and in 1864 the word received official recognition in the English language when it was included in Noah Webster's American Dictionary.[8]
History
Origins
Despite its relatively recent emergence in the West, there is a much older tradition of dissection amusements in China which likely played a role in its inspiration. In particular, the modular banquet tables of the Song dynasty bear an uncanny resemblance to the playing pieces of the Tangram and there were books dedicated to arranging them together to form pleasing patterns.[9]
Several Chinese sources broadly report a well-known Song dynasty polymath Huang Bosi 黄伯思 who developed a form of entertainment for his dinner guests based on creative arrangements of six small tables called 宴几 or 燕几（feast tables or swallow tables respectively). One diagram shows these as oblong rectangles, and other reports suggest a seventh table being added later, perhaps by a later inventor.
According to Western sources, however, the tangram's historical Chinese inventor is unknown except through the pen name Yang-cho-chu-shih (Dim-witted (?) recluse, recluse = 处士). It is believed that the puzzle was originally introduced in a book titled Ch'i chi'iao t'u which was already being reported as lost in 1815 by Shan-chiao in his book New Figures of the Tangram. Nevertheless, it is generally reputed that the puzzle's origins would have been around 20 years earlier than this.[10]
The prominent third-century mathematician Liu Hui made use of construction proofs in his works and some bear a striking resemblance to the subsequently developed Banquet tables which in turn seem to anticipate the Tangram. While there is no reason to suspect that tangrams were used in the proof of the Pythagorean theorem, as is sometimes reported, it is likely that this style of geometric reasoning went on to exert an influence on Chinese cultural life that lead directly to the puzzle.[11]
The early years of attempting to date the Tangram were confused by the popular but fraudulently written history by famed puzzle maker Samuel Loyd in his 1908 The Eighth Book Of Tan. This work contains many whimsical features that aroused both interest and suspicion amongst contemporary scholars who attempted to verify the account. By 1910 it was clear that it was a hoax. A letter dated from this year from the Oxford Dictionary editor Sir James Murray on behalf of a number of Chinese scholars to the prominent puzzlist Henry Dudeney reads "The result has been to show that the man Tan, the god Tan, and the Book of Tan are entirely unknown to Chinese literature, history or tradition."[6] Along with its many strange details The Eighth Book of Tan's date of creation for the puzzle of 4000 years in antiquity had to be regarded as entirely baseless and false.
Reaching the Western world (1815–1820s)
The earliest extant tangram was given to the Philadelphia shipping magnate and congressman Francis Waln in 1802 but it was not until over a decade later that Western audiences, at large, would be exposed to the puzzle.[1] In 1815, American Captain M. Donnaldson was given a pair of author Sang-Hsia-koi's books on the subject (one problem and one solution book) when his ship, Trader docked there. They were then brought with the ship to Philadelphia, in February 1816. The first tangram book to be published in America was based on the pair brought by Donnaldson.[12]
The puzzle eventually reached England, where it became very fashionable. The craze quickly spread to other European countries. This was mostly due to a pair of British tangram books, The Fashionable Chinese Puzzle, and the accompanying solution book, Key.[13] Soon, tangram sets were being exported in great number from China, made of various materials, from glass, to wood, to tortoise shell.[14]
Many of these unusual and exquisite tangram sets made their way to Denmark. Danish interest in tangrams skyrocketed around 1818, when two books on the puzzle were published, to much enthusiasm.[15] The first of these was Mandarinen (About the Chinese Game). This was written by a student at Copenhagen University, which was a non-fictional work about the history and popularity of tangrams. The second, Det nye chinesiske Gaadespil (The new Chinese Puzzle Game), consisted of 339 puzzles copied from The Eighth Book of Tan, as well as one original.[15]
One contributing factor in the popularity of the game in Europe was that although the Catholic Church forbade many forms of recreation on the sabbath, they made no objection to puzzle games such as the tangram.[16]
Second craze in Germany (1891–1920s)
Tangrams were first introduced to the German public by industrialist Friedrich Adolf Richter around 1891.[17] The sets were made out of stone or false earthenware,[18] and marketed under the name "The Anchor Puzzle".[17]
More internationally, the First World War saw a great resurgence of interest in tangrams, on the homefront and trenches of both sides. During this time, it occasionally went under the name of "The Sphinx" an alternative title for the "Anchor Puzzle" sets.[19][20]
Paradoxes
A tangram paradox is a dissection fallacy: Two figures composed with the same set of pieces, one of which seems to be a proper subset of the other.[21] One famous paradox is that of the two monks, attributed to Henry Dudeney, which consists of two similar shapes, one with and the other missing a foot.[22] In reality, the area of the foot is compensated for in the second figure by a subtly larger body.
The two-monks paradox – two similar shapes but one missing a foot:
The Magic Dice Cup tangram paradox – from Sam Loyd's book The 8th Book of Tan (1903).[23] Each of these cups was composed using the same seven geometric shapes. But the first cup is whole, and the others contain vacancies of different sizes. (Notice that the one on the left is slightly shorter than the other two. The one in the middle is ever-so-slightly wider than the one on the right, and the one on the left is narrower still.)[24]
Clipped square tangram paradox – from Loyd's book The Eighth Book of Tan (1903):[23]
The seventh and eighth figures represent the mysterious square, built with seven pieces: then with a corner clipped off, and still the same seven pieces employed.[25]
Number of configurations
Over 6500 different tangram problems have been created from 19th century texts alone, and the current number is ever-growing.[26] Fu Traing Wang and Chuan-Chih Hsiung proved in 1942 that there are only thirteen convex tangram configurations (config segment drawn between any two points on the configuration's edge always pass through the configuration's interior, i.e., configurations with no recesses in the outline).[27][28]
Pieces
Choosing a unit of measurement so that the seven pieces can be assembled to form a square of side one unit and having area one square unit, the seven pieces are:[29]
• 2 large right triangles (hypotenuse 1, sides √2/2, area 1/4)
• 1 medium right triangle (hypotenuse √2/2, sides 1/2, area 1/8)
• 2 small right triangles (hypotenuse 1/2, sides √2/4, area 1/16)
• 1 square (sides √2/4, area 1/8)
• 1 parallelogram (sides of 1/2 and √2/4, height of 1/4, area 1/8)
Of these seven pieces, the parallelogram is unique in that it has no reflection symmetry but only rotational symmetry, and so its mirror image can be obtained only by flipping it over. Thus, it is the only piece that may need to be flipped when forming certain shapes.
See also
• Tangram (video game)
• Egg of Columbus (tangram puzzle)
• Mathematical puzzle
• Ostomachion
• Tiling puzzle
References
1. Slocum (2003), p. 21.
2. Campillo-Robles, Jose M.; Alonso, Ibon; Gondra, Ane; Gondra, Nerea (1 September 2022). "Calculation and measurement of center of mass: An all-in-one activity using Tangram puzzles". American Journal of Physics. 90 (9): 652. Bibcode:2022AmJPh..90..652C. doi:10.1119/5.0061884. ISSN 0002-9505. S2CID 251917733.
3. Slocum (2001), p. 9.
4. Forbrush, William Byron (1914). Manual of Play. Jacobs. p. 315. Retrieved 2010-10-13.
5. Oxford English Dictionary, 1910, s.v.
6. Slocum (2003), p. 23.
7. Hill, Thomas (1848). Puzzles to teach geometry : in seventeen cards numbered from the first to the seventeenth inclusive. Boston : Wm. Crosby & H.P. Nichols.
8. Slocum (2003), p. 25.
9. Slocum (2003), p. 16.
10. Slocum (2003), pp. 16–19.
11. Slocum (2003), p. 15.
12. Slocum (2003), p. 30.
13. Slocum (2003), p. 31.
14. Slocum (2003), p. 49.
15. Slocum (2003), pp. 99–100.
16. Slocum (2003), p. 51.
17. "Tangram the incredible timeless 'Chinese' puzzle". www.archimedes-lab.org.
18. Treasury Decisions Under customs and other laws, Volume 25. United States Department Of The Treasury. 1890–1926. p. 1421. Retrieved 2010-09-16.
19. Wyatt (26 April 2006). "Tangram – The Chinese Puzzle". h2g2. BBC. Archived from the original on 2011-10-02. Retrieved 2010-10-03.
20. Braman, Arlette (2002). Kids Around The World Play!. John Wiley and Sons. p. 10. ISBN 978-0-471-40984-7. Retrieved 2010-09-05.
21. Tangram Paradox, by Barile, Margherita, From MathWorld – A Wolfram Web Resource, created by Eric W. Weisstein.
22. Dudeney, H. (1958). Amusements in Mathematics. New York: Dover Publications.
23. The 8th Book of Tan by Sam Loyd. 1903 – via Tangram Channel.
24. "The Magic Dice Cup". 2 April 2011.
25. Loyd, Sam (1968). The eighth book of Tan – 700 Tangrams by Sam Loyd with an introduction and solutions by Peter Van Note. New York: Dover Publications. p. 25.
26. Slocum 2001, p. 37.
27. Fu Traing Wang; Chuan-Chih Hsiung (November 1942). "A Theorem on the Tangram". The American Mathematical Monthly. 49 (9): 596–599. doi:10.2307/2303340. JSTOR 2303340.
28. Read, Ronald C. (1965). Tangrams : 330 Puzzles. New York: Dover Publications. p. 53. ISBN 0-486-21483-4.
29. Brooks, David J. (1 December 2018). "How to Make a Classic Tangram Puzzle". Boys' Life magazine. Retrieved 2020-03-10.
Sources
• Slocum, Jerry (2001). The Tao of Tangram. Barnes & Noble. ISBN 978-1-4351-0156-2.
• Slocum, Jerry (2003). The Tangram Book. Sterling. ISBN 978-1-4027-0413-0.
Further reading
• Anno, Mitsumasa. Anno's Math Games (three volumes). New York: Philomel Books, 1987. ISBN 0-399-21151-9 (v. 1), ISBN 0-698-11672-0 (v. 2), ISBN 0-399-22274-X (v. 3).
• Botermans, Jack, et al. The World of Games: Their Origins and History, How to Play Them, and How to Make Them (translation of Wereld vol spelletjes). New York: Facts on File, 1989. ISBN 0-8160-2184-8.
• Dudeney, H. E. Amusements in Mathematics. New York: Dover Publications, 1958.
• Gardner, Martin. "Mathematical Games—on the Fanciful History and the Creative Challenges of the Puzzle Game of Tangrams", Scientific American Aug. 1974, p. 98–103.
• Gardner, Martin. "More on Tangrams", Scientific American Sep. 1974, p. 187–191.
• Gardner, Martin. The 2nd Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster, 1961. ISBN 0-671-24559-7.
• Loyd, Sam. Sam Loyd's Book of Tangram Puzzles (The 8th Book of Tan Part I). Mineola, New York: Dover Publications, 1968.
• Slocum, Jerry, et al. Puzzles of Old and New: How to Make and Solve Them. De Meern, Netherlands: Plenary Publications International (Europe); Amsterdam, Netherlands: ADM International; Seattle: Distributed by University of Washington Press, 1986. ISBN 0-295-96350-6.
External links
Wikimedia Commons has media related to Tangrams.
• Past & Future: The Roots of Tangram and Its Developments
• Turning Your Set of Tangram Into A Magic Math Puzzle by puzzle designer G. Sarcone
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Shinichi Mochizuki
Shinichi Mochizuki (望月 新一, Mochizuki Shin'ichi, born March 29, 1969) is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. Mochizuki has also worked in Hodge–Arakelov theory and p-adic Teichmüller theory. Mochizuki developed inter-universal Teichmüller theory,[2][3][4][5] which has attracted attention from non-mathematicians due to claims it provides a resolution of the abc conjecture.[6]
Shinichi Mochizuki
Born (1969-03-29) March 29, 1969[1]
Tokyo, Japan[1]
NationalityJapanese
Alma materPrinceton University
Known forAnabelian geometry
Inter-universal Teichmüller theory
AwardsJSPS Prize, Japan Academy Medal[1]
Scientific career
FieldsMathematics
InstitutionsKyoto University
Doctoral advisorGerd Faltings
Biography
Early life
Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki.[7] When he was five years old, Shinichi Mochizuki and his family left Japan to live in the United States. His father was Fellow of the Center for International Affairs and Center for Middle Eastern Studies at Harvard University (1974–76).[8] Mochizuki attended Phillips Exeter Academy and graduated in 1985.[9]
Mochizuki entered Princeton University as an undergraduate student at the age of 16 and graduated as salutatorian with an A.B. in mathematics in 1988.[9] He completed his senior thesis, titled "Curves and their deformations," under the supervision of Gerd Faltings.[10]
He remained at Princeton for graduate studies and received his Ph.D. in mathematics in 1992 after completing his doctoral dissertation, titled "The geometry of the compactification of the Hurwitz scheme," also under the supervision of Faltings.[11]
After his graduate studies, Mochizuki spent two years at Harvard University and then in 1994 moved back to Japan to join the Research Institute for Mathematical Sciences (RIMS) at Kyoto University in 1992, and was promoted to professor in 2002.[1][12]
Career
Mochizuki proved Grothendieck's conjecture on anabelian geometry in 1996. He was an invited speaker at the International Congress of Mathematicians in 1998.[13] In 2000–2008 he discovered several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta theory for line bundles over tempered covers of the Tate curve.
On August 30, 2012 Mochizuki released four preprints, whose total size was about 500 pages, that developped inter-universal Teichmüller theory and applied it in an attempt to prove several very famous problems in Diophantine geometry.[14] These include the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. In September 2018, Mochizuki posted a report on his work by Peter Scholze and Jakob Stix asserting that the third preprint contains an irreparable flaw; he also posted several documents containing his rebuttal of their criticism.[15] The majority of number theorists have found Mochizuki's preprints very difficult to follow and have not accepted the conjectures as settled, although there are a few prominent exceptions, including Go Yamashita, Ivan Fesenko, and Yuichiro Hoshi, who vouch for the work and have written expositions of the theory.[16][17]
On April 3, 2020, two Japanese mathematicians, Masaki Kashiwara and Akio Tamagawa, announced that Mochizuki's claimed proof of the abc conjecture would be published in Publications of the Research Institute for Mathematical Sciences, a journal of which Mochizuki is chief editor.[18] The announcement was received with skepticism by Kiran Kedlaya and Edward Frenkel, as well as being described by Nature as "unlikely to move many researchers over to Mochizuki's camp".[18] The special issue containing Mochizuki's articles was published on March 5, 2021.[2][3][4][5]
Publications
• Mochizuki, Shinichi (1997), "A Version of the Grothendieck Conjecture for p-adic Local Fields" (PDF), International Journal of Mathematics, Singapore: World Scientific Pub. Co., 8 (3): 499–506, CiteSeerX 10.1.1.161.7778, doi:10.1142/S0129167X97000251, ISSN 0129-167X
• Mochizuki, Shinichi (1998), "The intrinsic Hodge theory of p-adic hyperbolic curves, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica: 187–196, ISSN 1431-0635, MR 1648069
• Mochizuki, Shinichi (1999), Foundations of p-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1190-0, MR 1700772
Inter-universal Teichmüller theory
• Mochizuki, Shinichi (2011), "Inter-universal Teichmüller Theory: A Progress Report" (PDF), Development of Galois–Teichmüller Theory and Anabelian Geometry, The 3rd Mathematical Society of Japan, Seasonal Institute.
• Mochizuki, Shinichi (2012a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF).
• Mochizuki, Shinichi (2012b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF).
• Mochizuki, Shinichi (2012c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF).
• Mochizuki, Shinichi (2012d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF).
References
1. Mochizuki, Shinichi. "Curriculum Vitae" (PDF). Retrieved 14 September 2012.
2. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory I: Construction of Hodge Theaters" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 3–207. doi:10.4171/PRIMS/57-1-1. S2CID 233829305.
3. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 209–401. doi:10.4171/PRIMS/57-1-2. S2CID 233794971.
4. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory III: Canonical Splittings of the Log-Theta-Lattice" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 403–626. doi:10.4171/PRIMS/57-1-3. S2CID 233777314.
5. Mochizuki, Shinichi (2021). "Inter-universal Teichmüller Theory IV: Log-Volume Computations and Set-Theoretic Foundations" (PDF). Publications of the Research Institute for Mathematical Sciences. 57 (1–2): 627–723. doi:10.4171/PRIMS/57-1-4. S2CID 3135393.
6. Crowell 2017.
7. Leah P. (Edelman) Rauch Philly.com on Mar. 6, 2005
8. MOCHIZUKI, Kiichi Dr. National Association of Japan-America Societies, Inc.
9. "Seniors address commencement crowd". Princeton Weekly Bulletin. Vol. 77. 20 June 1988. p. 4. Archived from the original on 3 April 2013.{{cite news}}: CS1 maint: bot: original URL status unknown (link)
10. Mochizuki, Shinichi (1988). Curves and their deformations. Princeton, NJ: Department of Mathematics.
11. Mochizuki, Shinichi (1992). The geometry of the compactification of the Hurwitz scheme.
12. Castelvecchi 2015.
13. "International Congress of Mathematicians 1998". Archived from the original on 2015-12-19.
14. Inter-universal Teichmüller theory IV: log-volume computations and set-theoretic foundations, Shinichi Mochizuki, August 2012
15. Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
16. Fesenko, Ivan (2016), "Fukugen", Inference: International Review of Science, 2 (3), doi:10.37282/991819.16.25
17. Roberts, David Michael (2019), "A crisis of identification", Inference: International Review of Science, 4 (3), doi:10.37282/991819.19.2, S2CID 232514600
18. Castelvecchi, Davide (April 3, 2020). "Mathematical proof that rocked number theory will be published". Nature. 580 (7802): 177. Bibcode:2020Natur.580..177C. doi:10.1038/d41586-020-00998-2. PMID 32246118. S2CID 214786566. Retrieved April 4, 2020.
Sources
• Castelvecchi, Davide (7 October 2015), "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof", Nature, 526 (7572): 178–181, Bibcode:2015Natur.526..178C, doi:10.1038/526178a, PMID 26450038
• Crowell, Rachel (19 September 2017). "On a summary of Shinichi Mochizuki's proof for the abc conjecture". American Mathematical Society.
• Ishikura, Tetsuya (16 December 2017). "Mathematician in Kyoto cracks formidable brainteaser". The Asahi Shimbun.
External links
Wikiquote has quotations related to Shinichi Mochizuki
• Shinichi Mochizuki at the Mathematics Genealogy Project
• Personal website
• Papers of Shinichi Mochizuki
• A brief introduction to inter-universal geometry
• On inter-universal Teichmüller theory of Shinichi Mochizuki, colloquium talk by Ivan Fesenko
• Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki by Ivan Fesenko
• Introduction to inter-universal Teichmüller theory (in Japanese), a survey by Yuichiro Hoshi
• RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory, March 2015, Kyoto*
• CMI workshop on IUT theory of Shinichi Mochizuki, December 2015, Oxford*
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Tally marks
Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system.
They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.
Early history
Counting aids other than body parts appear in the Upper Paleolithic. The oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age.
The so-called Wolf bone is a prehistoric artifact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia, led by Karl Absolon. Dated to the Aurignacian, approximately 30,000 years ago, the bone is marked with 55 marks which may be tally marks. The head of an ivory Venus figurine was excavated close to the bone.[1]
The Ishango bone, found in the Ishango region of the present-day Democratic Republic of Congo, is dated to over 20,000 years old. Upon discovery, it was thought to portray a series of prime numbers. In the book How Mathematics Happened: The First 50,000 Years, Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[2] Alexander Marshack examined the Ishango bone microscopically, and concluded that it may represent a six-month lunar calendar.[3]
Clustering
Tally marks are typically clustered in groups of five for legibility. The cluster size 5 has the advantages of (a) easy conversion into decimal for higher arithmetic operations and (b) avoiding error, as humans can far more easily correctly identify a cluster of 5 than one of 10.
• Tally marks representing (from left to right) the numbers 1, 2, 3, 4 and 5 that was used in most of Europe, the Anglosphere, and Southern Africa. In some variants, the diagonal/horizontal slash is used on its own when five or more units are added at once.
• Cultures using Chinese characters tally by forming the character 正,[lower-alpha 1] which consists of five strokes.[4][5]
• Tally marks used in France, Portugal, Spain, and their former colonies, including Latin America. 1 to 5 and so on. These are most commonly used for registering scores in card games, like Truco.
• In the dot and line (or dot-dash) tally, dots represent counts from 1 to 4, lines 5 to 8, and diagonal lines 9 and 10. This method is commonly used in forestry and related fields.[6]
Writing systems
Part of a series on
Numeral systems
Place-value notation
Hindu-Arabic numerals
• Western Arabic
• Eastern Arabic
• Bengali
• Devanagari
• Gujarati
• Gurmukhi
• Odia
• Sinhala
• Tamil
• Malayalam
• Telugu
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East Asian systems
Contemporary
• Chinese
• Suzhou
• Hokkien
• Japanese
• Korean
• Vietnamese
Historic
• Counting rods
• Tangut
Other systems
• History
Ancient
• Babylonian
Post-classical
• Cistercian
• Mayan
• Muisca
• Pentadic
• Quipu
• Rumi
Contemporary
• Cherokee
• Kaktovik (Iñupiaq)
By radix/base
Common radices/bases
• 2
• 3
• 4
• 5
• 6
• 8
• 10
• 12
• 16
• 20
• 60
• (table)
Non-standard radices/bases
• Bijective (1)
• Signed-digit (balanced ternary)
• Mixed (factorial)
• Negative
• Complex (2i)
• Non-integer (φ)
• Asymmetric
Sign-value notation
Non-alphabetic
• Aegean
• Attic
• Aztec
• Brahmi
• Chuvash
• Egyptian
• Etruscan
• Kharosthi
• Prehistoric counting
• Proto-cuneiform
• Roman
• Tally marks
Alphabetic
• Abjad
• Armenian
• Alphasyllabic
• Akṣarapallī
• Āryabhaṭa
• Kaṭapayādi
• Coptic
• Cyrillic
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• Georgian
• Glagolitic
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List of numeral systems
Roman numerals, the Brahmi and Chinese numerals for one through three (一 二 三), and rod numerals were derived from tally marks, as possibly was the ogham script.[7]
Base 1 arithmetic notation system is a unary positional system similar to tally marks. It is rarely used as a practical base for counting due to its difficult readability.
The numbers 1, 2, 3, 4, 5, 6 ... would be represented in this system as[8]
0, 00, 000, 0000, 00000, 000000 ...
Base 1 notation is widely used in type numbers of flour; the higher number represents a higher grind.
Unicode
In 2015, Ken Lunde and Daisuke Miura submitted a proposal to encode various systems of tally marks in the Unicode Standard.[9] However, the box tally and dot-and-dash tally characters were not accepted for encoding, and only the five ideographic tally marks (正 scheme) and two Western tally digits were added to the Unicode Standard in the Counting Rod Numerals block in Unicode version 11.0 (June 2018). Only the tally marks for the numbers 1 and 5 are encoded, and tally marks for the numbers 2, 3 and 4 are intended to be composed from sequences of tally mark 1 at the font level.
Counting Rod Numerals[1][2]
Official Unicode Consortium code chart (PDF)
0123456789ABCDEF
U+1D36x 𝍠 𝍡 𝍢 𝍣 𝍤 𝍥 𝍦 𝍧 𝍨 𝍩 𝍪 𝍫 𝍬 𝍭 𝍮 𝍯
U+1D37x 𝍰 𝍱 𝍲 𝍳 𝍴 𝍵 𝍶 𝍷 𝍸
Notes
1.^ As of Unicode version 15.0
2.^ Grey areas indicate non-assigned code points
See also
• History of writing ancient numbers
• Abacus
• Australian Aboriginal enumeration
• Carpenters' marks
• Cherty i rezy
• Chuvash numerals
• Counting rods
• Finger counting
• Hangman (game)
• History of communication
• History of mathematics
• Lebombo bone
• List of international common standards
• Paleolithic tally sticks
• Prehistoric numerals
• Quipu
• Roman numerals
• Tally stick
Wikimedia Commons has media related to Tally marks.
Notes
1. This character was apparently chosen purely due to appropriateness of the physical process of writing it using the conventional stroke-order system—i.e., the physical movements of the strokes have a distinct alternation right-down-right-down-right working down the character, but the semantics of the character have no particular relation to the concept of "5" (neither in the character etymology nor the word etymology, which in languages using Chinese characters are two originally-separate-but-historically-complexly-interacting things). By contrast, the character for "five", 五, which looks like it also has 5 distinct lines, has only 4 strokes when written using conventional stroke-order.)
References
• Graham Flegg, Numbers: their history and meaning, Courier Dover Publications, 2002 ISBN 978-0-486-42165-0, pp. 41-42.
1. Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.
2. Marshack, Alexander (1991): The Roots of Civilization, Colonial Hill, Mount Kisco, NY.
3. Hsieh, Hui-Kuang (1981) "Chinese tally mark", The American Statistician, 35 (3), p. 174, doi:10.2307/2683999
4. Ken Lunde, Daisuke Miura, L2/16-046: Proposal to encode five ideographic tally marks, 2016
5. Schenck, Carl A. (1898) Forest mensuration. The University Press. (Note: The linked reference appears to actually be "Bulletin of the Ohio Agricultural Experiment Station", Number 302, August 1916)
6. Macalister, R. A. S., Corpus Inscriptionum Insularum Celticarum Vol. I and II, Dublin: Stationery Office (1945).
7. Hext, Jan (1990), Programming Structures: Machines and programs, Programming Structures, vol. 1, Prentice Hall, p. 33, ISBN 9780724809400.
8. Lunde, Ken; Miura, Daisuke (30 November 2015). "Proposal to encode tally marks" (PDF). Unicode Consortium.
| Wikipedia |
Chinese numerals
Chinese numerals are words and characters used to denote numbers in Chinese.
Part of a series on
Numeral systems
Place-value notation
Hindu-Arabic numerals
• Western Arabic
• Eastern Arabic
• Bengali
• Devanagari
• Gujarati
• Gurmukhi
• Odia
• Sinhala
• Tamil
• Malayalam
• Telugu
• Kannada
• Dzongkha
• Tibetan
• Balinese
• Burmese
• Javanese
• Khmer
• Lao
• Mongolian
• Sundanese
• Thai
East Asian systems
Contemporary
• Chinese
• Suzhou
• Hokkien
• Japanese
• Korean
• Vietnamese
Historic
• Counting rods
• Tangut
Other systems
• History
Ancient
• Babylonian
Post-classical
• Cistercian
• Mayan
• Muisca
• Pentadic
• Quipu
• Rumi
Contemporary
• Cherokee
• Kaktovik (Iñupiaq)
By radix/base
Common radices/bases
• 2
• 3
• 4
• 5
• 6
• 8
• 10
• 12
• 16
• 20
• 60
• (table)
Non-standard radices/bases
• Bijective (1)
• Signed-digit (balanced ternary)
• Mixed (factorial)
• Negative
• Complex (2i)
• Non-integer (φ)
• Asymmetric
Sign-value notation
Non-alphabetic
• Aegean
• Attic
• Aztec
• Brahmi
• Chuvash
• Egyptian
• Etruscan
• Kharosthi
• Prehistoric counting
• Proto-cuneiform
• Roman
• Tally marks
Alphabetic
• Abjad
• Armenian
• Alphasyllabic
• Akṣarapallī
• Āryabhaṭa
• Kaṭapayādi
• Coptic
• Cyrillic
• Geʽez
• Georgian
• Glagolitic
• Greek
• Hebrew
List of numeral systems
Today, speakers of Chinese languages use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These may be shared with other languages of the Chinese cultural sphere such as Korean, Japanese, and Vietnamese. Most people and institutions in China primarily use the Arabic or mixed Arabic-Chinese systems for convenience, with traditional Chinese numerals used in finance, mainly for writing amounts on cheques, banknotes, some ceremonial occasions, some boxes, and on commercials.
The other indigenous system consists of the Suzhou numerals, or huama, a positional system, the only surviving form of the rod numerals. These were once used by Chinese mathematicians, and later by merchants in Chinese markets, such as those in Hong Kong until the 1990s, but were gradually supplanted by Arabic numerals.
Characters used to represent numbers
The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals. Similar to spelling-out numbers in English (e.g., "one thousand nine hundred forty-five"), it is not an independent system per se. Since it reflects spoken language, it does not use the positional system as in Arabic numerals, in the same way that spelling out numbers in English does not.
Standard numbers
There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, ten thousands and hundred millions. There are two sets of characters for Chinese numerals: one for everyday writing, known as xiǎoxiě (traditional Chinese: 小寫; simplified Chinese: 小写; lit. 'small writing'), and one for use in commercial, accounting or financial contexts, known as dàxiě (traditional Chinese: 大寫; simplified Chinese: 大写; lit. 'big writing'). The latter arose because the characters used for writing numerals are geometrically simple, so simply using those numerals cannot prevent forgeries in the same way spelling numbers out in English would.[1] A forger could easily change the everyday characters 三十 (30) to 五千 (5000) just by adding a few strokes. That would not be possible when writing using the financial characters 參拾 (30) and 伍仟 (5000). They are also referred to as "banker's numerals", "anti-fraud numerals", or "banker's anti-fraud numerals". For the same reason, rod numerals were never used in commercial records.
T denotes Traditional Chinese characters, while S denotes Simplified Chinese characters.
Financial Normal Value Pīnyīn
(Mandarin)
Jyutping
(Cantonese)
Pe̍h-ōe-jī
(Hokkien)
Wugniu
(Shanghainese)
Notes
Character (T)Character (S)Character (T)Character (S)
零零〇0 língling4khòng/lênglin Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero. The original Chinese character is 空 or 〇, 零 is referred as remainder something less than 1 yet not nil [說文] referred. The traditional 零 is more often used in schools. In Unicode, 〇 is treated as a Chinese symbol or punctuation, rather than a Chinese ideograph.
壹一1 yījat1it/chi̍tiq Also 弌 (obsolete financial), can be easily manipulated into 弍 (two) or 弎 (three).
貳贰二2 èrji6jī/nn̄ggni/er/lian Also 弍 (obsolete financial), can be easily manipulated into 弌 (one) or 弎 (three).
Also 兩 (T) or 两 (S), see Characters with regional usage section.
參参三3 sānsaam1sam/saⁿsé Also 弎 (obsolete financial), which can be easily manipulated into 弌 (one) or 弍 (two).
肆四4 sìsei3sù/sìsy Also 䦉 (obsolete financial)[nb 1]
伍五5 wǔng5ngó͘/gō͘ng
陸陆六6 liùluk6lio̍k/la̍kloq
柒七7 qīcat1chhitchiq
捌八8 bābaat3pat/pehpaq
玖九9 jiǔgau2kiú/káucieu
拾十10 shísap6si̍p/cha̍pzeq Although some people use 什 as financial, it is not ideal because it can be easily manipulated into 伍 (five) or 仟 (thousand).
佰百100 bǎibaak3pek/pahpaq
仟千1,000 qiāncin1chhian/chhengchi
萬萬万104 wànmaan6bānve Chinese numbers group by ten-thousands; see Reading and transcribing numbers below.
億億亿105/108 yìjik1eki For variant meanings and words for higher values, see Large numbers below and ja:大字 (数字).
Characters with regional usage
Financial Normal Value Pinyin (Mandarin) Standard alternative Notes
空 0 kòng 零 Historically, the use of 空 for "zero" predates 零. This is now archaic in most varieties of Chinese, but it is still used in Southern Min.
洞 0 dòng 零 Literally means "a hole" and is analogous to the shape of "0" and "〇", it is used to unambiguously pronounce "#0" in radio communication. [2][3]
幺 1 yāo 一 Literally means "the smallest", it is used to unambiguously pronounce "#1" in radio communication. [2][3] This usage is not observed in Cantonese except for 十三幺 (a special winning hand) in Mahjong.
蜀 1 shǔ 一 In most Min varieties, there are two words meaning "one". For example, in Hokkien, chi̍t is used before a classifier: "one person" is chi̍t ê lâng, not it ê lâng. In written Hokkien, 一 is often used for both chi̍t and it, but some authors differentiate, writing 蜀 for chi̍t and 一 for it.
兩(T) or
两(S)
2 liǎng 二 Used instead of 二 before a classifier. For example, "two people" is "两个人", not "二个人". However, in some lects, such as Shanghainese, 兩 is the generic term used for two in most contexts, such as "四十兩" and not "四十二". It appears where "a pair of" would in English, but 两 is always used in such cases. It is also used for numbers, with usage varying from dialect to dialect, even person to person. For example, "2222" can be read as "二千二百二十二", "兩千二百二十二" or even "兩千兩百二十二" in Mandarin. It is used to unambiguously pronounce "#2" in radio communication. [2][3]
倆(T) or
俩(S)
2 liǎ 兩 In regional dialects of Northeastern Mandarin, 倆 represents a "lazy" pronunciation of 兩 within the local dialect. It can be used as an alternative for 兩个 "two of" (e.g. 我们倆 Wǒmen liǎ, "the two of us", as opposed to 我们兩个 Wǒmen liǎng gè). A measure word (such as 个) never follows after 倆.
仨 3 sā 三 In regional dialects of Northeastern Mandarin, 仨 represents a "lazy" pronunciation of three within the local dialect. It can be used as a general number to represent "three" (e.g.第仨号 dì sā hào, "number three"; 星期仨 xīngqīsā, "Wednesday"), or as an alternative for 三个 "three of" (e.g. 我们仨 Wǒmen sā, "the three of us", as opposed to 我们三个 Wǒmen sān gè). Regardless of usage, a measure word (such as 个) never follows after 仨.
拐 7 guǎi 七 Literally means "a turn" or "a walking stick" and is analogous to the shape of "7" and "七", it is used to unambiguously pronounce "#7" in radio communication. [2][3]
勾 9 gōu 九 Literally means "a hook" and is analogous to the shape of "9", it is used to unambiguously pronounce "#9" in radio communication. [2][3]
呀 10 yà 十 In spoken Cantonese, 呀 (aa6) can be used in place of 十 when it is used in the middle of a number, preceded by a multiplier and followed by a ones digit, e.g. 六呀三, 63; it is not used by itself to mean 10. This usage is not observed in Mandarin.
念 廿 20 niàn 二十 A contraction of 二十. The written form is still used to refer to dates, especially Chinese calendar dates.
Spoken form is still used in various dialects of Chinese. See Reading and transcribing numbers section below.
In spoken Cantonese, 廿 (jaa6) can be used in place of 二十 when followed by another digit such as in numbers 21-29 (e.g. 廿三, 23), a measure word (e.g. 廿個), a noun, or in a phrase like 廿幾 ("twenty-something"); it is not used by itself to mean 20.
卄 is a rare variant.
卅 30 sà 三十 A contraction of 三十. The written form is still used to abbreviate date references in Chinese. For example, May 30 Movement (五卅運動).
Spoken form is still used in various dialects of Chinese.
In spoken Cantonese, 卅 (saa1) can be used in place of 三十 when followed by another digit such as in numbers 31–39, a measure word (e.g. 卅個), a noun, or in phrases like 卅幾 ("thirty-something"); it is not used by itself to mean 30. When spoken 卅 is pronounced as 卅呀 (saa1 aa6). Thus 卅一 (31), is pronounced as saa1 aa6 jat1.
卌 40 xì 四十 A contraction of 四十. Found in historical writings written in Classical Chinese.
Spoken form is still used in various dialects of Chinese, albeit very rare. See Reading and transcribing numbers section below.
In spoken Cantonese 卌 (sei3) can be used in place of 四十 when followed by another digit such as in numbers 41–49, a measure word (e.g. 卌個), a noun, or in phrases like 卌幾 ("forty-something"); it is not used by itself to mean 40. When spoken, 卌 is pronounced as 卌呀 (sei3 aa6). Thus 卌一 (41), is pronounced as sei3 aa6 jat1.
皕 200 bì 二百 Very rarely used; one example is in the name of a library in Huzhou, 皕宋樓 (Bìsòng Lóu).
Large numbers
For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with unique names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan, Wujing suanshu (Arithmetic in Five Classics). In modern Chinese only the second system is used, in which the same ancient names are used, but each represents a number 10,000 (myriad, 萬 wàn) times the previous:
Character (T) 萬 億 兆 京 垓 秭 穰 溝 澗 正 載 Factor of increase
Character (S) 万 亿 兆 京 垓 秭 穰 沟 涧 正 载
Pinyin wàn yì zhào jīng gāi zǐ ráng gōu jiàn zhèng zǎi
Jyutping maan6 jik1 siu6 ging1 goi1 zi2 joeng4 kau1 gaan3 zing3 zoi2
Hokkien POJ bān ek tiāu keng kai chí jiông ko͘ kàn chèng cháiⁿ
Shanghainese ve i zau cín ké tsy gnian kéu ké tsen tse
Alternative 經/经 𥝱 壤
Rank 1 2 3 4 5 6 7 8 9 10 11 =n
"short scale"
(下數)
104 105 106 107 108 109 1010 1011 1012 1013 1014 =10n+3
Each numeral is 10 (十 shí) times the previous.
"myriad scale"
(萬進, current usage)
104 108 1012 1016 1020 1024 1028 1032 1036 1040 1044 =104n
Each numeral is 10,000 (萬 (T) or 万 (S) wàn) times the previous.
"mid-scale"
(中數)
104 108 1016 1024 1032 1040 1048 1056 1064 1072 1080 =108(n-1)
Starting with 亿, each numeral is 108 (萬乘以萬 (T) or 万乘以万 (S) wàn chéng yǐ wàn, 10000 times 10000) times the previous.
"long scale"
(上數)
104 108 1016 1032 1064 10128 10256 10512 101024 102048 104096 =102n+1
Each numeral is the square of the previous. This is similar to the -yllion system.
In practice, this situation does not lead to ambiguity, with the exception of 兆 (zhào), which means 1012 according to the system in common usage throughout the Chinese communities as well as in Japan and Korea, but has also been used for 106 in recent years (especially in mainland China for megabyte). To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, but uses 万亿 (wànyì) or 太 (tài, as the translation for tera) instead. Partly due to this, combinations of 万 and 亿 are often used instead of the larger units of the traditional system as well, for example 亿亿 (yìyì) instead of 京. The ROC government in Taiwan uses 兆 (zhào) to mean 1012 in official documents.
Large numbers from Buddhism
Numerals beyond 載 zǎi come from Buddhist texts in Sanskrit, but are mostly found in ancient texts. Some of the following words are still being used today, but may have transferred meanings.
Character (T) Character (S) Pinyin Jyutping Hokkien POJ Shanghainese Value Notes
極 极 jí gik1 ke̍k jiq5 1048 Literally means "Extreme".
恆河沙 恒河沙 héng hé shā hang4 ho4 sa1 hêng-hô-soa ghen3-wu-so 1052 Literally means "Sands of the Ganges"; a metaphor used in a number of Buddhist texts referring to the grains of sand in the Ganges River.
阿僧祇 ā sēng qí aa1 zang1 kei4 a-seng-kî a1-sen-ji 1056 From Sanskrit Asaṃkhyeya असंख्येय, meaning "incalculable, innumerable, infinite".
那由他 nà yóu tā naa5 jau4 taa1 ná-iû-thaⁿ na1-yeu-tha 1060 From Sanskrit nayuta नियुत, meaning "myriad".
不可思議 不可思议 bùkě sīyì bat1 ho2 si1 ji3 put-khó-su-gī peq4-khu sy1-gni 1064 Literally translated as "unfathomable". This word is commonly used in Chinese as a chengyu, meaning "unimaginable", instead of its original meaning of the number 1064.
無量大數 无量大数 wú liàng dà shù mou4 loeng6 daai6 sou3 bû-liōng tāi-siàu m3-lian du3-su 1068 "无量" literally translated as "without measure", and can mean 1068. This word is also commonly used in Chinese as a commendatory term, means "no upper limit". E.g.: 前途无量 lit. front journey no limit, which means "a great future". "大数" literally translated as "a large number; the great number", and can mean 1072.
Small numbers
The following are characters used to denote small order of magnitude in Chinese historically. With the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse.
Character(s) (T) Character(s) (S) Pinyin Value Notes
漠 mò 10−12 (Ancient Chinese)
皮 corresponds to the SI prefix pico-.
渺 miǎo 10−11 (Ancient Chinese)
埃 āi 10−10 (Ancient Chinese)
塵 尘 chén 10−9 Literally, "Dust"
奈 (T) or 纳 (S) corresponds to the SI prefix nano-.
沙 shā 10−8 Literally, "Sand"
纖 纤 xiān 10−7 Literally, "Fiber"
微 wēi 10−6 still in use, corresponds to the SI prefix micro-.
忽 hū 10−5 (Ancient Chinese)
絲 丝 sī 10−4 also 秒.
Literally, "Thread"
毫 háo 10−3 also 毛.
still in use, corresponds to the SI prefix milli-.
厘 lí 10−2 also 釐.
still in use, corresponds to the SI prefix centi-.
分 fēn 10−1 still in use, corresponds to the SI prefix deci-.
Small numbers from Buddhism
Character(s) (T) Character(s) (S) Pinyin Value Notes
涅槃寂靜 涅槃寂静 niè pán jì jìng 10−24 Literally, "Nirvana's Tranquility"
攸 (T) or 幺 (S) corresponds to the SI prefix yocto-.
阿摩羅 阿摩罗 ā mó luó 10−23 (Ancient Chinese, from Sanskrit अमल amala)
阿頼耶 阿赖耶 ā lài yē 10−22 (Ancient Chinese, from Sanskrit आलय ālaya)
清靜 清净 qīng jìng 10−21 Literally, "Quiet"
介 (T) or 仄 (S) corresponds to the SI prefix zepto-.
虛空 虚空 xū kōng 10−20 Literally, "Void"
六德 liù dé 10−19 (Ancient Chinese)
剎那 刹那 chà nà 10−18 Literally, "Brevity", from Sanskrit क्षण ksaṇa
阿 corresponds to the SI prefix atto-.
彈指 弹指 tán zhǐ 10−17 Literally, "Flick of a finger". Still commonly used in the phrase "弹指一瞬间" (A very short time)
瞬息 shùn xī 10−16 Literally, "Moment of Breath". Still commonly used in Chengyu "瞬息万变" (Many things changed in a very short time)
須臾 须臾 xū yú 10−15 (Ancient Chinese, rarely used in Modern Chinese as "a very short time")
飛 (T) or 飞 (S) corresponds to the SI prefix femto-.
逡巡 qūn xún 10−14 (Ancient Chinese)
模糊 mó hu 10−13 Literally, "Blurred"
SI prefixes
In the People's Republic of China, the early translation for the SI prefixes in 1981 was different from those used today. The larger (兆, 京, 垓, 秭, 穰) and smaller Chinese numerals (微, 纖, 沙, 塵, 渺) were defined as translation for the SI prefixes as mega, giga, tera, peta, exa, micro, nano, pico, femto, atto, resulting in the creation of yet more values for each numeral.[4]
The Republic of China (Taiwan) defined 百萬 as the translation for mega and 兆 as the translation for tera. This translation is widely used in official documents, academic communities, informational industries, etc. However, the civil broadcasting industries sometimes use 兆赫 to represent "megahertz".
Today, the governments of both China and Taiwan use phonetic transliterations for the SI prefixes. However, the governments have each chosen different Chinese characters for certain prefixes. The following table lists the two different standards together with the early translation.
SI Prefixes
Value Symbol English Early translation PRC standard ROC standard
1024Yyotta- 尧 yáo 佑 yòu
1021Zzetta- 泽 zé 皆 jiē
1018Eexa- 穰[4] ráng艾 ài 艾 ài
1015Ppeta- 秭[4] zǐ拍 pāi 拍 pāi
1012Ttera- 垓[4] gāi太 tài 兆 zhào
109Ggiga- 京[4] jīng吉 jí 吉 jí
106Mmega- 兆[4] zhào兆 zhào 百萬 bǎiwàn
103kkilo- 千 qiān千 qiān 千 qiān
102hhecto- 百 bǎi百 bǎi百 bǎi
101dadeca- 十 shí十 shí 十 shí
100(base)one 一 yī 一 yī
10−1ddeci- 分 fēn分 fēn 分 fēn
10−2ccenti- 厘 lí厘 lí 厘 lí
10−3mmilli- 毫 háo毫 háo 毫 háo
10−6µmicro- 微[4] wēi 微 wēi 微 wēi
10−9nnano- 纖[4] xiān 纳 nà 奈 nài
10−12ppico- 沙[4] shā皮 pí 皮 pí
10−15ffemto- 塵[4] chén飞 fēi 飛 fēi
10−18aatto- 渺[4] miǎo 阿 à 阿 à
10−21zzepto- 仄 zè 介 jiè
10−24yyocto- 幺 yāo 攸 yōu
Reading and transcribing numbers
Whole numbers
Multiple-digit numbers are constructed using a multiplicative principle; first the digit itself (from 1 to 9), then the place (such as 10 or 100); then the next digit.
In Mandarin, the multiplier 兩 (liǎng) is often used rather than 二 (èr) for all numbers 200 and greater with the "2" numeral (although as noted earlier this varies from dialect to dialect and person to person). Use of both 兩 (liǎng) or 二 (èr) are acceptable for the number 200. When writing in the Cantonese dialect, 二 (yi6) is used to represent the "2" numeral for all numbers. In the southern Min dialect of Chaozhou (Teochew), 兩 (no6) is used to represent the "2" numeral in all numbers from 200 onwards. Thus:
Number Structure Characters
Mandarin Cantonese Chaozhou Shanghainese
60[6] [10]六十六十六十六十
20[2] [10] or [20]二十二十 or 廿二十廿
200[2] (èr or liǎng) [100]二百 or 兩百二百 or 兩百兩百兩百
2000[2] (èr or liǎng) [1000]二千 or 兩千二千 or 兩千兩千兩千
45[4] [10] [5]四十五四十五 or 卌五四十五四十五
2,362[2] [1000] [3] [100] [6] [10] [2]兩千三百六十二二千三百六十二兩千三百六十二兩千三百六十二
For the numbers 11 through 19, the leading "one" (一; yī) is usually omitted. In some dialects, like Shanghainese, when there are only two significant digits in the number, the leading "one" and the trailing zeroes are omitted. Sometimes, the one before "ten" in the middle of a number, such as 213, is omitted. Thus:
Number Strict Putonghua Colloquial or dialect usage
Structure Characters Structure Characters
14[10] [4]十四
12000[1] [10000] [2] [1000]一萬兩千[1] [10000] [2]一萬二 or 萬二
114[1] [100] [1] [10] [4]一百一十四[1] [100] [10] [4]一百十四
1158[1] [1000] [1] [100] [5] [10] [8]一千一百五十八See note 1 below
Notes:
1. Nothing is ever omitted in large and more complicated numbers such as this.
In certain older texts like the Protestant Bible or in poetic usage, numbers such as 114 may be written as [100] [10] [4] (百十四).
Outside of Taiwan, digits are sometimes grouped by myriads instead of thousands. Hence it is more convenient to think of numbers here as in groups of four, thus 1,234,567,890 is regrouped here as 12,3456,7890. Larger than a myriad, each number is therefore four zeroes longer than the one before it, thus 10000 × wàn (萬) = yì (億). If one of the numbers is between 10 and 19, the leading "one" is omitted as per the above point. Hence (numbers in parentheses indicate that the number has been written as one number rather than expanded):
Number Structure Taiwan Mainland China
12,345,678,902,345
(12,3456,7890,2345)
(12) [1,0000,0000,0000] (3456) [1,0000,0000] (7890) [1,0000] (2345)十二兆三千四百五十六億七千八百九十萬兩千三百四十五十二兆三千四百五十六亿七千八百九十万二千三百四十五
In Taiwan, pure Arabic numerals are officially always and only grouped by thousands.[5] Unofficially, they are often not grouped, particularly for numbers below 100,000. Mixed Arabic-Chinese numerals are often used in order to denote myriads. This is used both officially and unofficially, and come in a variety of styles:
Number Structure Mixed numerals
12,345,000(1234) [1,0000] (5) [1,000]1,234萬5千[6]
123,450,000 (1) [1,0000,0000] (2345) [1,0000] 1億2345萬[7]
12,345 (1) [1,0000] (2345) 1萬2345[8]
Interior zeroes before the unit position (as in 1002) must be spelt explicitly. The reason for this is that trailing zeroes (as in 1200) are often omitted as shorthand, so ambiguity occurs. One zero is sufficient to resolve the ambiguity. Where the zero is before a digit other than the units digit, the explicit zero is not ambiguous and is therefore optional, but preferred. Thus:
Number Structure Characters
205[2] [100] [0] [5]二百零五
100,004
(10,0004)
[10] [10,000] [0] [4]十萬零四
10,050,026
(1005,0026)
(1005) [10,000] (026) or
(1005) [10,000] (26)
一千零五萬零二十六 or
一千零五萬二十六
Fractional values
To construct a fraction, the denominator is written first, followed by 分; fēn; 'parts', then the literary possessive particle 之; zhī; 'of this', and lastly the numerator. This is the opposite of how fractions are read in English, which is numerator first. Each half of the fraction is written the same as a whole number. For example, to express "two thirds", the structure "three parts of-this two" is used. Mixed numbers are written with the whole-number part first, followed by 又; yòu; 'and', then the fractional part.
Fraction Structure
2⁄3
三
sān
3
分
fēn
parts
之
zhī
of this
二
èr
2
三 分 之 二
sān fēn zhī èr
3 parts {of this} 2
15⁄32
三
sān
3
十
shí
10
二
èr
2
分
fēn
parts
之
zhī
of this
十
shí
10
五
wǔ
5
三 十 二 分 之 十 五
sān shí èr fēn zhī shí wǔ
3 10 2 parts {of this} 10 5
1⁄3000
三
sān
3
千
qiān
1000
分
fēn
parts
之
zhī
of this
一
yī
1
三 千 分 之 一
sān qiān fēn zhī yī
3 1000 parts {of this} 1
3 5⁄6
三
sān
3
又
yòu
and
六
liù
6
分
fēn
parts
之
zhī
of this
五
wǔ
5
三 又 六 分 之 五
sān yòu liù fēn zhī wǔ
3 and 6 parts {of this} 5
Percentages are constructed similarly, using 百; bǎi; '100' as the denominator. (The number 100 is typically expressed as 一百; yībǎi; 'one hundred', like the English "one hundred". However, for percentages, 百 is used on its own.)
Percentage Structure
25%
百
bǎi
100
分
fēn
parts
之
zhī
of this
二
èr
2
十
shí
10
五
wǔ
5
百 分 之 二 十 五
bǎi fēn zhī èr shí wǔ
100 parts {of this} 2 10 5
110%
百
bǎi
100
分
fēn
parts
之
zhī
of this
一
yī
1
百
bǎi
100
一
yī
1
十
shí
10
百 分 之 一 百 一 十
bǎi fēn zhī yī bǎi yī shí
100 parts {of this} 1 100 1 10
Because percentages and other fractions are formulated the same, Chinese are more likely than not to express 10%, 20% etc. as "parts of 10" (or 1/10, 2/10, etc. i.e. 十分之一; shí fēnzhī yī, 十分之二; shí fēnzhī èr, etc.) rather than "parts of 100" (or 10/100, 20/100, etc. i.e. 百分之十; bǎi fēnzhī shí, 百分之二十; bǎi fēnzhī èrshí, etc.)
In Taiwan, the most common formation of percentages in the spoken language is the number per hundred followed by the word 趴; pā, a contraction of the Japanese パーセント; pāsento, itself taken from the English "percent". Thus 25% is 二十五趴; èrshíwǔ pā.[nb 2]
Decimal numbers are constructed by first writing the whole number part, then inserting a point (simplified Chinese: 点; traditional Chinese: 點; pinyin: diǎn), and finally the fractional part. The fractional part is expressed using only the numbers for 0 to 9, similarly to English.
Decimal expression Structure
16.98
十
shí
10
六
liù
6
點
diǎn
point
九
jiǔ
9
八
bā
8
十 六 點 九 八
shí liù diǎn jiǔ bā
10 6 point 9 8
12345.6789
一
yī
1
萬
wàn
10000
兩
liǎng
2
千
qiān
1000
三
sān
3
百
bǎi
100
四
sì
4
十
shí
10
五
wǔ
5
點
diǎn
point
六
liù
6
七
qī
7
八
bā
8
九
jiǔ
9
一 萬 兩 千 三 百 四 十 五 點 六 七 八 九
yī wàn liǎng qiān sān bǎi sì shí wǔ diǎn liù qī bā jiǔ
1 10000 2 1000 3 100 4 10 5 point 6 7 8 9
75.4025
七
七
qī
7
十
十
shí
10
五
五
wǔ
5
點
點
diǎn
point
四
四
sì
4
〇
零
líng
0
二
二
èr
2
五
五
wǔ
5
七 十 五 點 四 〇 二 五
七 十 五 點 四 零 二 五
qī shí wǔ diǎn sì líng èr wǔ
7 10 5 point 4 0 2 5
0.1
零
líng
0
點
diǎn
point
一
yī
1
零 點 一
líng diǎn yī
0 point 1
半; bàn; 'half' functions as a number and therefore requires a measure word. For example: 半杯水; bàn bēi shuǐ; 'half a glass of water'.
Ordinal numbers
Ordinal numbers are formed by adding 第; dì ("sequence") before the number.
Ordinal Structure
1st
第
dì
sequence
一
yī
1
第 一
dì yī
sequence 1
2nd
第
dì
sequence
二
èr
2
第 二
dì èr
sequence 2
82nd
第
dì
sequence
八
bā
8
十
shí
10
二
èr
2
第 八 十 二
dì bā shí èr
sequence 8 10 2
The Heavenly Stems are a traditional Chinese ordinal system.
Negative numbers
Negative numbers are formed by adding fù (负; 負) before the number.
Number Structure
−1158
負
fù
negative
一
yī
1
千
qiān
1000
一
yī
1
百
bǎi
100
五
wǔ
5
十
shí
10
八
bā
8
負 一 千 一 百 五 十 八
fù yī qiān yī bǎi wǔ shí bā
negative 1 1000 1 100 5 10 8
−3 5/6
負
fù
negative
三
sān
3
又
yòu
and
六
liù
6
分
fēn
parts
之
zhī
of this
五
wǔ
5
負 三 又 六 分 之 五
fù sān yòu liù fēn zhī wǔ
negative 3 and 6 parts {of this} 5
−75.4025
負
fù
negative
七
qī
7
十
shí
10
五
wǔ
5
點
diǎn
point
四
sì
4
零
líng
0
二
èr
2
五
wǔ
5
負 七 十 五 點 四 零 二 五
fù qī shí wǔ diǎn sì líng èr wǔ
negative 7 10 5 point 4 0 2 5
Usage
Chinese grammar requires the use of classifiers (measure words) when a numeral is used together with a noun to express a quantity. For example, "three people" is expressed as 三个人; 三個人; sān ge rén , "three (ge particle) person", where 个/個 ge is a classifier. There exist many different classifiers, for use with different sets of nouns, although 个/個 is the most common, and may be used informally in place of other classifiers.
Chinese uses cardinal numbers in certain situations in which English would use ordinals. For example, 三楼/三樓; sān lóu (literally "three story/storey") means "third floor" ("second floor" in British § Numbering). Likewise, 二十一世纪/二十一世紀; èrshí yī shìjì (literally "twenty-one century") is used for "21st century".[9]
Numbers of years are commonly spoken as a sequence of digits, as in 二零零一; èr líng líng yī ("two zero zero one") for the year 2001.[10] Names of months and days (in the Western system) are also expressed using numbers: 一月; yīyuè ("one month") for January, etc.; and 星期一; xīngqīyī ("week one") for Monday, etc. There is only one exception: Sunday is 星期日; xīngqīrì, or informally 星期天; xīngqītiān, both literally "week day". When meaning "week", "星期" xīngqī and "禮拜; 礼拜" lǐbài are interchangeable. "禮拜天" lǐbàitiān or "禮拜日" lǐbàirì means "day of worship". Chinese Catholics call Sunday "主日" zhǔrì, "Lord's day".[11]
Full dates are usually written in the format 2001年1月20日 for January 20, 2001 (using 年; nián "year", 月; yuè "month", and 日; rì "day") – all the numbers are read as cardinals, not ordinals, with no leading zeroes, and the year is read as a sequence of digits. For brevity the nián, yuè and rì may be dropped to give a date composed of just numbers. For example "6-4" in Chinese is "six-four", short for "month six, day four" i.e. June Fourth, a common Chinese shorthand for the 1989 Tiananmen Square protests (because of the violence that occurred on June 4). For another example 67, in Chinese is sixty seven, short for year nineteen sixty seven, a common Chinese shorthand for the Hong Kong 1967 leftist riots.
Counting rod and Suzhou numerals
In the same way that Roman numerals were standard in ancient and medieval Europe for mathematics and commerce, the Chinese formerly used the rod numerals, which is a positional system. The Suzhou numerals (simplified Chinese: 苏州花码; traditional Chinese: 蘇州花碼; pinyin: Sūzhōu huāmǎ) system is a variation of the Southern Song rod numerals. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices.
Hand gestures
There is a common method of using of one hand to signify the numbers one to ten. While the five digits on one hand can easily express the numbers one to five, six to ten have special signs that can be used in commerce or day-to-day communication.
Historical use of numerals in China
Most Chinese numerals of later periods were descendants of the Shang dynasty oracle numerals of the 14th century BC. The oracle bone script numerals were found on tortoise shell and animal bones. In early civilizations, the Shang were able to express any numbers, however large, with only nine symbols and a counting board though it was still not positional .[13]
Some of the bronze script numerals such as 1, 2, 3, 4, 10, 11, 12, and 13 became part of the system of rod numerals.
In this system, horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that "one is vertical, ten is horizontal".[14]
七 一 八 二 四
7 1 8 2 4
The counting rod numerals system has place value and decimal numerals for computation, and was used widely by Chinese merchants, mathematicians and astronomers from the Han dynasty to the 16th century.
In 690 AD, Empress Wǔ promulgated Zetian characters, one of which was "〇". The word is now used as a synonym for the number zero.[nb 3]
Alexander Wylie, Christian missionary to China, in 1853 already refuted the notion that "the Chinese numbers were written in words at length", and stated that in ancient China, calculation was carried out by means of counting rods, and "the written character is evidently a rude presentation of these". After being introduced to the rod numerals, he said "Having thus obtained a simple but effective system of figures, we find the Chinese in actual use of a method of notation depending on the theory of local value [i.e. place-value], several centuries before such theory was understood in Europe, and while yet the science of numbers had scarcely dawned among the Arabs."[15]
During the Ming and Qing dynasties (after Arabic numerals were introduced into China), some Chinese mathematicians used Chinese numeral characters as positional system digits. After the Qing period, both the Chinese numeral characters and the Suzhou numerals were replaced by Arabic numerals in mathematical writings.
Cultural influences
Traditional Chinese numeric characters are also used in Japan and Korea and were used in Vietnam before the 20th century. In vertical text (that is, read top to bottom), using characters for numbers is the norm, while in horizontal text, Arabic numerals are most common. Chinese numeric characters are also used in much the same formal or decorative fashion that Roman numerals are in Western cultures. Chinese numerals may appear together with Arabic numbers on the same sign or document.
See also
Wikimedia Commons has media related to Chinese numerals.
• Chinese number gestures
• Numbers in Chinese culture
• Chinese units of measurement
• Chinese classifier
• Chinese grammar
• Japanese numerals
• Korean numerals
• Vietnamese numerals
• Celestial stem
• List of numbers in Sinitic languages
Notes
1. Variant Chinese character of 肆, with a 镸 radical next to a 四 character. Not all browsers may be able to display this character, which forms a part of the Unicode CJK Unified Ideographs Extension A group.
2. This usage can also be found in written sources, such as in the headline of this article (while the text uses "%") and throughout this article.
3. The code for the lowercase 〇 (IDEOGRAPHIC NUMBER ZERO) is U+3007, not to be confused with the O mark (CIRCLE).
References
1. 大寫數字『 Archived 2011-07-22 at the Wayback Machine
2. Li, Suming (18 March 2016). Qiao, Meng (ed.). ""军语"里的那些秘密 武警少将亲自为您揭开" [Secrets in the "Military Lingo", Reveled by PAP General]. People's Armed Police. Retrieved 2021-06-18.
3. 飛航管理程序 [Air Traffic Management Procedures] (14 ed.). 30 November 2015.
4. (in Chinese) 1981 Gazette of the State Council of the People's Republic of China Archived 2012-01-11 at the Wayback Machine, No. 365 Archived 2014-11-04 at the Wayback Machine, page 575, Table 7: SI prefixes
5. 中華民國統計資訊網（專業人士）. 中華民國統計資訊網 (in Chinese). Archived from the original on 5 August 2016. Retrieved 31 July 2016.
6. 中華民國統計資訊網（專業人士） (in Chinese). 中華民國統計資訊網. Archived from the original on 28 August 2016. Retrieved 31 July 2016.
7. "石化氣爆 高市府代位求償訴訟中". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Archived from the original on 1 August 2016. Retrieved 31 July 2016.
8. "陳子豪雙響砲 兄弟連2天轟猿動紫趴". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Archived from the original on 31 July 2016. Retrieved 31 July 2016.
9. Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 12.
10. Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 13.
11. Days of the Week in Chinese: Three Different Words for 'Week' http://www.cjvlang.com/Dow/dowchin.html Archived 2016-03-06 at the Wayback Machine
12. The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, Table 20, p. 6, Cambridge University Press ISBN 0-521-23582-0
13. The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, p5, Cambridge University Press ISBN 0-521-23582-0
14. Chinese Wikisource Archived 2012-02-22 at the Wayback Machine 孫子算經: 先識其位，一從十橫，百立千僵，千十相望，萬百相當。
15. Alexander Wylie, Jottings on the Sciences of the Chinese, North Chinese Herald, 1853, Shanghai
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| Wikipedia |
Toshikazu Sunada
Toshikazu Sunada (砂田 利一, Sunada Toshikazu, born September 7, 1948) is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recognition of achievement over the course of an academic career. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University (1988–1991), at the University of Tokyo (1991–1993), and at Tohoku University (1993–2003). Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean (2013–2017). Since 2019, he is President of Mathematics Education Society of Japan.
Toshikazu Sunada
Born1948 (age 74–75)
Tokyo, Japan
Alma materTokyo Institute of Technology
Awards
• 1987 Iyanaga Award of Mathematical Society of Japan
• 2013 Publication Prize of Mathematical Society of Japan
• 2017 Hiroshi Fujiwara Prize for Mathematical Sciences
• 2018 Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology)
• 2019 the 1st Kodaira Kunihiko Prize
Scientific career
FieldsMathematics (Spectral geometry and discrete geometric analysis)
InstitutionsNagoya University
Tokyo University
Tohoku University
Meiji University
Main work
Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, discrete geometric analysis, and mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory, and is considered to be a breakthrough in the problem proposed by Mark Kac in "Can one hear the shape of a drum?" (see Hearing the shape of a drum). Sunada's idea was taken up by Carolyn S. Gordon, David Webb, and Scott A. Wolpert when they constructed a counterexample for Kac's problem. For this work, Sunada was awarded the Iyanaga Prize of the Mathematical Society of Japan (MSJ) in 1987. He was also awarded Publication Prize of MSJ in 2013, the Hiroshi Fujiwara Prize for Mathematical Sciences in 2017, the Prize for Science and Technology (the Commendation for Science and Technology by the Minister of Education, Culture, Sports, Science and Technology) in 2018, and the 1st Kodaira Kunihiko Prize in 2019.
In a joint work with Atsushi Katsuda, Sunada also established a geometric analogue of Dirichlet's theorem on arithmetic progressions in the context of dynamical systems (1988). One can see, in this work as well as the one above, how the concepts and ideas in totally different fields (geometry, dynamical systems, and number theory) are put together to formulate problems and to produce new results.
His study of discrete geometric analysis includes a graph-theoretic interpretation of Ihara zeta functions, a discrete analogue of periodic magnetic Schrödinger operators as well as the large time asymptotic behaviors of random walk on crystal lattices. The study of random walk led him to the discovery of a "mathematical twin" of the diamond crystal out of an infinite universe of hypothetical crystals (2005). He named it the K4 crystal due to its mathematical relevance (see the linked article). What was noticed by him is that the K4 crystal has the "strong isotropy property", meaning that for any two vertices x and y of the crystal net, and for any ordering of the edges adjacent to x and any ordering of the edges adjacent to y, there is a net-preserving congruence taking x to y and each x-edge to the similarly ordered y-edge. This property is shared only by the diamond crystal (the strong isotropy should not be confused with the edge-transitivity or the notion of symmetric graph; for instance, the primitive cubic lattice is a symmetric graph, but not strongly isotropic). The K4 crystal and the diamond crystal as networks in space are examples of “standard realizations”, the notion introduced by Sunada and Motoko Kotani as a graph-theoretic version of Albanese maps (Abel-Jacobi maps) in algebraic geometry.
For his work, see also Isospectral, Reinhardt domain, Ihara zeta function, Ramanujan graph, quantum ergodicity, quantum walk.
Selected publications by Sunada
• T. Sunada, Holomorphic equivalence problem for bounded Reinhardt domains, Mathematische Annalen 235 (1978), 111–128
• T. Sunada, Rigidity of certain harmonic mappings, Inventiones Mathematicae 51 (1979), 297–307
• J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings into algebraic varieties, American Journal of Mathematics 104 (1982), 887–900
• T. Sunada, Riemannian coverings and isospectral manifolds, Annals of Mathematics 121 (1985), 169–186
• T. Sunada, L-functions and some applications, Lecture Notes in Mathematics 1201 (1986), Springer-Verlag, 266–284
• A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, American Journal of Mathematics 110(1988), 145–156
• T. Sunada, Unitary representations of fundamental groups and the spectrum of twisted Laplacians, Topology 28 (1989), 125–132
• A. Katsuda and T. Sunada, Closed orbits in homology classes, Publications Mathématiques de l'IHÉS 71 (1990), 5–32
• M. Nishio and T. Sunada, Trace formulae in spectral geometry, Proc. ICM-90 Kyoto, Springer-Verlag, Tokyo, (1991), 577–585
• T. Sunada, Quantum ergodicity, Trend in Mathematics, Birkhauser Verlag, Basel, 1997, 175–196
• M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Communications in Mathematical Physics 209 (2000), 633–670
• M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary Mathematics 338 (2003), 271–305
• T. Sunada, Crystals that nature might miss creating, Notices of the American Mathematical Society 55 (2008), 208–215
• T. Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics (ed. by P. Exner, J. P. Keating, P. Kuchment, T. Sunada, A. Teplyaev), 77 (2008), 51–86
• K. Shiga and T. Sunada, A Mathematical Gift, III, American Mathematical Society
• T. Sunada, Lecture on topological crystallography, Japan Journal of Mathematics 7 (2012), 1–39
• T. Sunada, Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer, 2013, ISBN 978-4-431-54176-9 (print) ISBN 978-4-431-54177-6 (online)
• T. Sunada, Generalized Riemann sums, in From Riemann to Differential Geometry and Relativity, Editors: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada, Springer (2017), 457–479
• T. Sunada, Topics on mathematical crystallography, Proceedings of the symposium Groups, graphs and random walks, London Mathematical Society Lecture Note Series 436, Cambridge University Press, 2017, 473–513
• T. Sunada, From Euclid to Riemann and beyond, in Geometry in History, Editors: S. G. Dani, Athanase Papadopoulos, Springer (2019), 213–304
References
• Atsushi Katsuda and Polly Wee Sy,, An overview of Sunada's work
• Meiji U. Homepage (Mathematics Department)
• David Bradley, , Diamond's chiral chemical cousin
• M. Itoh et al., New metallic carbon crystal, Phys. Rev. Lett. 102, 055703 (2009)
• Diamond twin, Meiji U. Homepage
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| Wikipedia |
Shisanji Hokari
Dr. Shisanji Hokari (穂刈 四三二, Hokari Shisanji, 28 March 1908 – 2 January 2004) was a Japanese mathematician. He was admitted to the American Mathematical Society in 1966.[1] He was a professor emeritus of Tokyo Metropolitan University and the president of Josai University.
Year Age Milestone
1926 18 Enrolled to Tokyo University of Science.
1928 20 Graduated from Tokyo University of Science.
1931 23 Enrolled to Hokkaido University.
1934 26 Graduated from Hokkaido University.
1939 31 Lecturer at Hokkaido University.
1940 32 Received a doctorate. Assistant Professor at Hokkaido University.
1949 41 Professor at Tokyo Metropolitan University.
1971 63 Professor at Josai University. Dean of Faculty of Science.
1971 63 Professor Emeritus at Tokyo Metropolitan University.
1977 69 Honorary Member of Japan Society of Mathematical Education.
1978 70 President of Josai University.
1980 72 President Emeritus of Josai University.
1982 74 Professor Emeritus at Josai University.
1987 79 Received 3rd Class Order of the Rising Sun.
References
1. Green, John W.; Sherman, Seymour. The annual meeting in Chicago. Bull. Amer. Math. Soc. 72 (1966), no. 3, p. 476 http://projecteuclid.org/euclid.bams/1183527951.
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1
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of unit length is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer.[1] It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0.
← 0 1 2 →
−1 0 1 2 3 4 5 6 7 8 9 →
• List of numbers
• Integers
← 0 10 20 30 40 50 60 70 80 90 →
Cardinalone
Ordinal1st
(first)
Numeral systemunary
Factorization∅
Divisors1
Greek numeralΑ´
Roman numeralI, i
Greek prefixmono-/haplo-
Latin prefixuni-
Binary12
Ternary13
Senary16
Octal18
Duodecimal112
Hexadecimal116
Greek numeralα'
Arabic, Kurdish, Persian, Sindhi, Urdu١
Assamese & Bengali১
Chinese numeral一/弌/壹
Devanāgarī१
Ge'ez፩
GeorgianႠ/ⴀ/ა(Ani)
Hebrewא
Japanese numeral一/壱
Kannada೧
Khmer១
Malayalam൧
Meitei꯱
Thai๑
Tamil௧
Telugu೧
Counting rod𝍠
The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is the smallest possible difference between two distinct natural numbers.
The unique mathematical properties of the number have led to its unique uses in other fields, ranging from science to sports. It commonly denotes the first, leading, or top thing in a group.
As a word
One is most commonly a determiner used with singular countable nouns, as in one day at a time.[2] One is also a pronoun used to refer to an unspecified person or to people in general as in one should take care of oneself.[3] Finally, one is a noun when it refers to the number one as in one plus one is two and when it is used as a pro form, as in the green one is nice or those ones look good.
Etymology
One comes from the English word an,[4] which comes from the Proto-Germanic root *ainaz.[4] The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-.[4]
Compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish en, Dutch een, German eins and Old Norse einn.
Compare the Proto-Indo-European root *oi-no- (which means "one, single")[4] to Greek oinos (which means "ace" on dice),[4] Latin unus (one),[4] Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin and Breton un (one).[4]
As a number
One, sometimes referred to as unity,[5][1] is the first non-zero natural number. It is thus the integer after zero.
Any number multiplied by one remains that number, as one is the identity for multiplication. As a result, 1 is its own factorial, its own square and square root, its own cube and cube root, and so on. One is also the result of the empty product, as any number multiplied by one is itself. It is also the only natural number that is neither composite nor prime with respect to division, but is instead considered a unit (meaning of ring theory).
As a digit
Main article: History of the Hindu–Arabic numeral system
The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Brahmic script of ancient India, where it was a simple vertical line. It was transmitted to Europe via the Maghreb and Andalusia during the Middle Ages, through scholarly works written in Arabic.
In some countries, the serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph used for seven in other countries. In styles in which the digit 1 is written with a long upstroke, the digit 7 is often written with a horizontal stroke through the vertical line, to disambiguate them. Styles that do not use the long upstroke on digit 1 usually do not use the horizontal stroke through the vertical of the digit 7 either.
While the shape of the character for the digit 1 has an ascender in most modern typefaces, in typefaces with text figures, the glyph usually is of x-height, as, for example, in .
Many older typewriters lack a separate key for 1, using the lowercase letter l or uppercase I instead. It is possible to find cases when the uppercase J is used, though it may be for decorative purposes. In some typefaces, different glyphs are used for I and 1, but the numeral 1 resembles a small caps version of I, with parallel serifs at top and bottom, with the capital I being full-height.
Mathematics
Definitions
Mathematically, 1 is:
• in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers;
• more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.
Formalizations of the natural numbers have their own representations of 1. In the Peano axioms, 1 is the successor of 0. In Principia Mathematica, it is defined as the set of all singletons (sets with one element), and in the Von Neumann cardinal assignment of natural numbers, it is defined as the set {0}.
In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is also traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields.
By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). The term unit matrix is sometimes used to mean a matrix composed entirely of 1s.
By definition, 1 is the probability of an event that is absolutely or almost certain to occur.
In category theory, 1 is sometimes used to denote the terminal object of a category.
In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be approximately 1.08366, but was proven to equal exactly 1 in 1899.
Properties
Tallying is often referred to as "base 1", since only one mark – the tally itself – is needed. This is more formally referred to as a unary numeral system. Unlike base 2 or base 10, this is not a positional notation.
Since the base 1 exponential function (1x) always equals 1, its inverse does not exist (which would be called the logarithm base 1 if it did exist).
In many mathematical and engineering problems, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Likewise, vectors are often normalized into unit vectors (i.e., vectors of magnitude one), because these often have more desirable properties. Functions, too, are often normalized by the condition that they have integral one, maximum value one, or square integral one, depending on the application.
Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must be equal to 1.
There are two ways to write the real number 1 as a recurring decimal: as 1.000..., and as 0.999.... 1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few.
1 is also the first and second number in the Fibonacci sequence (0 being the zeroth) and is the first number in many other mathematical sequences.
The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all.
1 is the most common leading digit in many sets of data, a consequence of Benford's law.
1 is the only known Tamagawa number for a simply connected algebraic group over a number field.
The generating function that has all coefficients equal to 1 is a geometric series, given by
${\frac {1}{1-x}}=1+x+x^{2}+x^{3}+\ldots $
The zeroth metallic mean is 1, with the golden section equal to the continued fraction [1;1,1,...], and the infinitely nested square root $\scriptstyle {\sqrt {1+{\sqrt {{\text{ }}1+\cdots {\text{ }}}}}}.$
The series of unit fractions that most rapidly converge to 1 are the reciprocals of Sylvester's sequence, which generate the infinite Egyptian fraction
$1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}+\cdots $
Primality
Main article: Prime number § Primality of one
1 is by convention neither a prime number nor a composite number, but a unit (meaning of ring theory) like −1 and, in the Gaussian integers, i and −i.
The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units. For example, 4 = 22, but if units are included, is also equal to, say, (−1)6 × 123 × 22, among infinitely many similar "factorizations".
1 appears to meet the naïve definition of a prime number, being evenly divisible only by 1 and itself (also 1). As such, some mathematicians considered it a prime number as late as the middle of the 20th century, but mathematical consensus has generally and since then universally been to exclude it for a variety of reasons (such as complicating the fundamental theorem of arithmetic and other theorems related to prime numbers).
1 is the only positive integer divisible by exactly one positive integer, whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers.
Table of basic calculations
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
1 × x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 ÷ x 1 0.5 0.3 0.25 0.2 0.16 0.142857 0.125 0.1 0.1 0.09 0.083 0.076923 0.0714285 0.06
x ÷ 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
In technology
• The resin identification code used in recycling to identify polyethylene terephthalate.[6]
• The ITU country code for the North American Numbering Plan area, which includes the United States, Canada, and parts of the Caribbean.
• A binary code is a sequence of 1 and 0 that is used in computers for representing any kind of data.
• In many physical devices, 1 represents the value for "on", which means that electricity is flowing.[7][8]
• The numerical value of true in many programming languages.
• 1 is the ASCII code of "Start of Header".
In science
• Dimensionless quantities are also known as quantities of dimension one.
• 1 is the atomic number of hydrogen.
• +1 is the electric charge of positrons and protons.
• Group 1 of the periodic table consists of the alkali metals.
• Period 1 of the periodic table consists of just two elements, hydrogen and helium.
• The dwarf planet Ceres has the minor-planet designation 1 Ceres because it was the first asteroid to be discovered.
• The Roman numeral I often stands for the first-discovered satellite of a planet or minor planet (such as Neptune I, a.k.a. Triton). For some earlier discoveries, the Roman numerals originally reflected the increasing distance from the primary instead.
In philosophy
In the philosophy of Plotinus (and that of other neoplatonists), The One is the ultimate reality and source of all existence.[9] Philo of Alexandria (20 BC – AD 50) regarded the number one as God's number, and the basis for all numbers ("De Allegoriis Legum," ii.12 [i.66]).
The Neopythagorean philosopher Nicomachus of Gerasa affirmed that one is not a number, but the source of number. He also believed the number two is the embodiment of the origin of otherness. His number theory was recovered by Boethius in his Latin translation of Nicomachus's treatise Introduction to Arithmetic.[10]
In sports
In many professional sports, the number 1 is assigned to the player who is first or leading in some respect, or otherwise important; the number is printed on his or her sports uniform or equipment. This is the pitcher in baseball, the goalkeeper in association football (soccer), the starting fullback in most of rugby league, the starting loosehead prop in rugby union and the previous year's world champion in Formula One. 1 may be the lowest possible player number, like in the American–Canadian National Hockey League (NHL) since the 1990s or in American football.
In other fields
• Number One is Royal Navy informal usage for the chief executive officer of a ship, the captain's deputy responsible for discipline and all normal operation of a ship and its crew.
• 1 is the value of an ace in many playing card games, such as cribbage.
• List of highways numbered 1
• List of public transport routes numbered 1
• 1 is often used to denote the Gregorian calendar month of January.
• 1 CE, the first year of the Common Era
• 01, the former dialling code for Greater London (now 020)
• For Pythagorean numerology (a pseudoscience), the number 1 is the number that means beginning, new beginnings, new cycles, it is a unique and absolute number.
• PRS One, a German paraglider design
• In some countries, a street address of "1" is considered prestigious and developers will attempt to obtain such an address for a building, to the point of lobbying for a street or portion of a street to be renamed, even if this makes the address less useful for wayfinding. The construction of a new street to serve the development may also provide the possibility of a "1" address. An example of such an address is the Apple Campus, located at 1 Infinite Loop, Cupertino, California.
See also
• −1
• +1 (disambiguation)
• List of mathematical constants
• One (word)
• Root of unity
• List of highways numbered 1
References
Wikimedia Commons has media related to:
1 (number) (category)
1. Weisstein, Eric W. "1". mathworld.wolfram.com. Archived from the original on 2020-07-26. Retrieved 2020-08-10.
2. Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 117. ISBN 978-1-316-51464-1. OCLC 1255524478.
3. Huddleston, Rodney D.; Pullum, Geoffrey K.; Reynolds, Brett (2022). A student's introduction to English grammar (2nd ed.). Cambridge, United Kingdom: Cambridge University Press. p. 140. ISBN 978-1-316-51464-1. OCLC 1255524478.
4. "Online Etymology Dictionary". etymonline.com. Douglas Harper. Archived from the original on 2013-12-30. Retrieved 2013-12-30.
5. Skoog, Douglas. Principles of Instrumental Analysis. Brooks/Cole, 2007, p. 758.
6. "Plastic Packaging Resins" (PDF). American Chemistry Council. Archived from the original (PDF) on 2011-07-21.
7. Woodford, Chris (2006), Digital Technology, Evans Brothers, p. 9, ISBN 978-0-237-52725-9, retrieved 2016-03-24
8. Godbole, Achyut S. (1 September 2002), Data Comms & Networks, Tata McGraw-Hill Education, p. 34, ISBN 978-1-259-08223-8
9. Olson, Roger (2017). The Essentials of Christian Thought: Seeing Reality through the Biblical Story. Zondervan Academic. ISBN 9780310521563.{{cite book}}: CS1 maint: location missing publisher (link)
10. British Society for the History of Science (July 1, 1977). "From Abacus to Algorism: Theory and Practice in Medieval Arithmetic". The British Journal for the History of Science. Cambridge University PRess. 10 (2): Abstract. doi:10.1017/S0007087400015375. S2CID 145065082. Archived from the original on May 16, 2021. Retrieved May 16, 2021.
External links
Wikiquote has quotations related to 1 (number).
• The Number 1
• The Positive Integer 1
• Prime curiosities: 1
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| Wikipedia |
Percent sign
The percent sign % (sometimes per cent sign in British English) is the symbol used to indicate a percentage, a number or ratio as a fraction of 100. Related signs include the permille (per thousand) sign ‰ and the permyriad (per ten thousand) sign ‱ (also known as a basis point), which indicate that a number is divided by one thousand or ten thousand, respectively. Higher proportions use parts-per notation.
"%" redirects here. For the mod sign as used in modular arithmetic, see Modular arithmetic.
%
Percent sign
In UnicodeU+0025 % PERCENT SIGN (%)
Different from
Different fromU+2052 ⁒ COMMERCIAL MINUS SIGN
U+00F7 ÷ DIVISION SIGN
Related
See alsoU+2030 ‰ PER MILLE SIGN
U+2031 ‱ PER TEN THOUSAND SIGN (Basis point)
Correct style
Form and spacing
English style guides prescribe writing the percent sign following the number without any space between (e.g. 50%).[sources 1] However, the International System of Units and ISO 31-0 standard prescribe a space between the number and percent sign,[8][9][10] in line with the general practice of using a non-breaking space between a numerical value and its corresponding unit of measurement.
Other languages have other rules for spacing in front of the percent sign:
• In Czech and in Slovak, the percent sign is spaced with a non-breaking space if the number is used as a noun.[11] In Czech, no space is inserted if the number is used as an adjective (e.g. “a 50% increase”),[12] whereas Slovak uses a non-breaking space in this case as well.[13]
• In Finnish, the percent sign is always spaced, and a case suffix can be attached to it using the colon (e.g. 50 %:n kasvu 'an increase of 50%').[14]
• In French, the percent sign must be spaced with a non-breaking space.[15][16]
• According to the Real Academia Española, in Spanish, the percent sign should be spaced now, despite the fact that it is not the linguistic norm.[17] Despite that, in North American Spanish (Mexico and the US), several style guides and institutions either recommend the percent sign be written following the number without any space between or do so in their own publications in accordance with common usage in that region.[18][19]
• In Russian, the percent sign is rarely spaced, contrary to the guidelines of the GOST 8.417-2002 state standard.
• In Chinese, the percent sign is almost never spaced, probably because Chinese does not use spaces to separate characters or words at all.
• According to the Swedish Language Council, the percent sign should be preceded by a space in Swedish, as all other units.
• In German, the space is prescribed by the regulatory body in the national standard DIN 5008.
• In Turkish and some other Turkic languages, the percent sign precedes rather than follows the number, without an intervening space.
• In Persian texts, the percent sign may either precede or follow the number, in either case without a space.
• In Arabic, the percent sign follows the number; as Arabic is written from right to left, this means that the percent sign is to the left of the number, usually without a space.
• In Hebrew, the percent sign is written to the right of the number, just as in English, without an intervening space. This is because numbers in Hebrew (which otherwise is written from right to left) are written from left to right, as in English.
• In Dutch, the official rule (NBN Z 01-002) is to place a space between the number and the sign (e.g. "een stijging van 50 %"), but most of the time, the space is missing (e.g. "een stijging van 50%").[20]
Usage in text
It is often recommended that the percent sign only be used in tables and other places with space restrictions. In running text, it should be spelled out as percent or per cent (often in newspapers). For example, not "Sales increased by 24% over 2006" but "Sales increased by 24 percent over 2006".[21][22][23]
Evolution
Prior to 1425, there is no known evidence of a special symbol being used for percentage. The Italian term per cento, "for a hundred", was used as well as several different abbreviations (e.g. "per 100", "p 100", "p cento", etc.). Examples of this can be seen in the 1339 arithmetic text (author unknown) depicted below.[24] The letter p with its descender crossed by a horizontal or diagonal strike (ꝑ in Unicode) conventionally stood for per, por, par, or pur in Medieval and Renaissance palaeography.[25]
At some point, a scribe used the abbreviation "pc" with a tiny loop or circle (depicting the ending -o used in Italian ordinals, as in primo, secondo, etc.; it is analogous to the English "-th" as in "25th"). This appears in some additional pages of a 1425 text which were probably added around 1435.[26] This is shown below (source, Rara Arithmetica p. 440).
The "pc" with a loop eventually evolved into a horizontal fraction sign by 1650 (see below for an example in a 1684 text[27]) and thereafter lost the "per".[28]
In 1925, D. E. Smith wrote, "The solidus form () is modern."[29]
Usage
Unicode
The Unicode code points are:
• U+0025 % PERCENT SIGN (HTML %, %[30]),
• U+FF05 ％ FULLWIDTH PERCENT SIGN see fullwidth forms
• U+FE6A ﹪ SMALL PERCENT SIGN see Small Form Variants
• U+066A ٪ ARABIC PERCENT SIGN, which has the circles replaced by square dots set on edge, the shape of the digit 0 in Eastern Arabic numerals.
ASCII
The ASCII code for the percent character is 37, or 0x25 in hexadecimal.
In computers
Names for the percent sign include percent sign (in ITU-T), mod, grapes (in hacker jargon) , and the humorous double-oh-seven (in INTERCAL).
In computing, the percent character is also used for the modulo operation in programming languages that derive their syntax from the C programming language, which in turn acquired this usage from the earlier B.[31]
In the textual representation of URIs, a % immediately followed by a 2-digit hexadecimal number denotes an octet specifying (part of) a character that might otherwise not be allowed in URIs (see percent-encoding).
In SQL, the percent sign is a wildcard character in "LIKE" expressions, for example SELECT * FROM table WHERE fullname LIKE 'Lisa %' will fetch all records whose names start with "Lisa ".
In TeX (and therefore also in LaTeX) and PostScript, and in GNU Octave and MATLAB,[32] a % denotes a line comment.
In BASIC, Visual Basic, ASP, and VBA a trailing % after a variable name marks it as an integer.
In ASP, the percent sign can be used to indicate the start and end of the ASP code <%...... %>
In Perl % is the sigil for hashes.
In many programming languages' string formatting operations (performed by functions such as printf and scanf), the percent sign denotes parts of the template string that will be replaced with arguments. (See printf format string.) In Python and Ruby the percent sign is also used as the string formatting operator.[33][34][35]
In the command processors COMMAND.COM (DOS) and CMD.EXE (OS/2 and Windows), %1, %2,... stand for the first, second,... parameters of a batch file. %0 stands for the specification of the batch file itself as typed on the command line. The % sign is also used similarly in the FOR command. %VAR1% represents the value of an environment variable named VAR1. Thus: set PATH=c:\;%PATH% sets a new value for PATH, that being the old value preceded by "c:\;". Because these uses give the percent sign special meaning, the sequence %% (two percent signs) is used to represent a literal percent sign, so that: set PATH=c:\;%%PATH%% would set PATH to the literal value "c:\;%PATH%".
In the C Shell, % is part of the default command prompt.
In linguistics
In linguistics, the percent sign is prepended to an example string to show that it is judged well-formed by some speakers and ill-formed by others. This may be due to differences in dialect or even individual idiolects. This is similar to the asterisk to mark ill-formed or reconstructed strings, the question mark to mark strings where well-formedness is unclear, and the number sign to mark strings that are syntactically well-formed but semantically nonsensical.
See also
• Metacharacter
• U+2030 ‰ PER MILLE SIGN,
• U+2031 ‱ PER TEN THOUSAND SIGN (also known as basis point)
Reference notes
1. [1][2][3][4][5][6][7]
Notes
1. "Guardian and Observer style guide: P". The Guardian. 30 April 2021. Archived from the original on 28 December 2019. Retrieved 16 March 2023.
2. "The Chicago Manual of Style". University of Chicago Press. 2003. Archived from the original on 5 January 2009. Retrieved 5 January 2007.
3. Publication Manual of the American Psychological Association. 1994. Washington, DC: American Psychological Association, p. 114.
4. Merriam-Webster's Manual for Writers and Editors. 1998. Springfield, MA: Merriam-Webster, p. 128.
5. Jenkins, Jana et al. 2011. The IBM Style Guide: Conventions for Writers and Editors. Boston, MA: Pearson Education, p. 162.
6. Covey, Stephen R. FranklinCovey Style Guide: For Business and Technical Communication. Salt Lake City, UT: FranklinCovey, p. 287.
7. Dodd, Janet S. 1997. The ACS Style Guide: A Manual for Authors and Editors. Washington, DC: American Chemical Society, p. 264.
8. "SI Brochure". International Bureau of Weights and Measures. 2006. Archived from the original on 21 March 2019. Retrieved 5 May 2016.
9. "The International System of Units" (PDF). International Bureau of Weights and Measures. 2006. Archived (PDF) from the original on 13 March 2020. Retrieved 6 August 2007.
10. "Quantities and units – Part 0: General principles". International Organization for Standardization. 22 December 1999. Archived from the original on 29 May 2007. Retrieved 5 January 2007.
11. "Internetová jazyková příručka". Ústav pro jazyk český Akademie věd ČR. 2014. Archived from the original on 14 February 2015. Retrieved 24 November 2014.
12. "Jazyková poradna ÚJČ AV ČR: FAQ". Ústav pro jazyk český Akademie věd ČR. 2002. Archived from the original on 19 April 2002. Retrieved 16 March 2009.
13. "Jazyková poradňa". Petit Press, a.s. 2013. Archived from the original on 21 February 2009. Retrieved 26 October 2019.
14. "Kielikello 2/2006". kotus.fi. Kotimaisten kielten keskus. 2006. Archived from the original on 1 May 2015. Retrieved 30 June 2015.
15. Guide des principales règles typographiques (PDF). Université Joseph-Fourier. Archived (PDF) from the original on 3 March 2016. Retrieved 8 June 2022.
16. André, Jacques. Petites leçons de typographie (PDF). Rennes: Institut de recherche en informatique et systèmes aléatoires. p. 34. Archived (PDF) from the original on 20 January 2011. Retrieved 8 June 2022.
17. "El % se escribe separado de la cifra a la que acompaña". fundeu.es. Fundeu. 2012. Archived from the original on 24 November 2021. Retrieved 24 November 2021.
18. "Normas particulares de estilo". colmex.mx. Colegio de México. 2020. Archived from the original on 10 May 2022. Retrieved 5 April 2022.
19. "¿En un texto, es correcto usar el signo de porcentaje o tiene que escribirse por ciento?". academia.org.mx. Academia Mexicana de la Lengua. 2020. Archived from the original on 6 February 2023. Retrieved 5 April 2022.
20. "procentteken (spatie)". www.vlaanderen.be (in Dutch). Archived from the original on 27 November 2021. Retrieved 27 November 2021.
21. American Economic Review: Style Guide Archived 2007-12-25 at the Wayback Machine
22. "UNC Pharmacy style guide". Archived from the original on 12 June 2007. Retrieved 16 October 2007.
23. "University of Colorado style guide". Archived from the original on 2 November 2007. Retrieved 16 October 2007.
24. Smith 1898, p. 437
25. Letter p. Archived 18 April 2009 at the Wayback Machine / Cappelli, Adriano: Lexicon Abbreviaturarum Archived 8 May 2015 at the Wayback Machine. 2. verb. Aufl. Leipzig 1928. Wörterbuch der Abkürzungen: P. pages 256–257
26. Smith 1898, pp. 439-440
27. Smith 1898, p. 441
28. Smith 1898, p. 440
29. Smith 1925, Vol. 2, p. 250 in Dover reprint of 1958, ISBN 0-486-20430-8
30. HTML5 is the only version of HTML that has a named entity for the percent sign, see https://www.w3.org/TR/html4/sgml/entities.html Archived 1 April 2018 at the Wayback Machine ("The following sections present the complete lists of character entity references.") and https://www.w3.org/TR/2014/CR-html5-20140731/syntax.html#named-character-references Archived 5 August 2017 at the Wayback Machine ("percnt;").
31. Thompson, Ken (1996). "Users' Reference to B". Archived from the original on 6 July 2006.
32. "2.7.1 Single Line Comments". GNU. Archived from the original on 20 July 2018. Retrieved 20 July 2018.
33. "Python 2 – String Formatting Operations". Archived from the original on 4 November 2015. Retrieved 28 October 2015.
34. "Python 3 – printf-style String Formatting". Archived from the original on 14 June 2020. Retrieved 28 October 2015.
35. "Ruby – String#%". Archived from the original on 8 October 2011. Retrieved 28 October 2015.
References
• Smith, D. E. (1898), Rara Arithmetica: a catalogue of the arithmetics written before MDCI, with description of those in the library of George Arthur Plimpton of New York, Boston: Ginn
• Smith, D. E. (1925), History of Mathematics, Boston: Ginn
Common punctuation marks and other typographical marks or symbols
• space
• , comma
• : colon
• ; semicolon
• ‐ hyphen
• ’ ' apostrophe
• ′ ″ ‴ prime
• . full stop
• & ampersand
• @ at sign
• ^ caret
• / slash
• \ backslash
• … ellipsis
• * asterisk
• ⁂ asterism
• * * * dinkus
• - hyphen-minus
• ‒ – — dash
• = ⸗ double hyphen
• ? question mark
• ! exclamation mark
• ‽ interrobang
• ¡ ¿ inverted ! and ?
• ⸮ irony punctuation
• # number sign
• № numero sign
• º ª ordinal indicator
• % percent sign
• ‰ per mille
• ‱ basis point
• ° degree symbol
• ⌀ diameter sign
• + − plus and minus signs
• × multiplication sign
• ÷ division sign
• ~ tilde
• ± plus–minus sign
• ∓ minus-plus sign
• _ underscore
• ⁀ tie
• | ¦ ‖ vertical bar
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• † ‡ dagger
• ❧ hedera/floral heart
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| Wikipedia |
Material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol $\rightarrow $ is interpreted as material implication, a formula $P\rightarrow Q$ is true unless $P$ is true and $Q$ is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.
Not to be confused with Material inference or Material implication (rule of inference).
Material conditional
IMPLY
Definition$x\rightarrow y$
Truth table$(1011)$
Logic gate
Normal forms
Disjunctive${\overline {x}}+y$
Conjunctive${\overline {x}}+y$
Zhegalkin polynomial$1\oplus x\oplus xy$
Post's lattices
0-preservingno
1-preservingyes
Monotoneno
Affineno
Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.
Notation
In logic and related fields, the material conditional is customarily notated with an infix operator $\to $.[1] The material conditional is also notated using the infixes $\supset $ and $\Rightarrow $. In the prefixed Polish notation, conditionals are notated as $Cpq$. In a conditional formula $p\to q$, the subformula $p$ is referred to as the antecedent and $q$ is termed the consequent of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula $(p\to q)\to (r\to s)$.
History
In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition “If $A$ then $B$” as $A$ Ɔ $B$ with the symbol Ɔ, which is the opposite of C.[2] He also expressed the proposition $A\supset B$ as $A$ Ɔ $B$.[lower-alpha 1][3][4] Hilbert expressed the proposition “If A then B” as $A\to B$ in 1918.[1] Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition “If A then B” as $A\supset B$. Following Russell, Gentzen expressed the proposition “If A then B” as $A\supset B$. Heyting expressed the proposition “If A then B” as $A\supset B$ at first but later came to express it as $A\to B$ with a right-pointing arrow. Bourbaki expressed the proposition “If A then B” as $A\Rightarrow B$ in 1954.[5]
Definitions
Semantics
From a semantic perspective, material implication is the binary truth functional operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in a truth table such as the one below.
Truth table
The truth table of p → q:
$p$$q$$p$ → $q$
TrueTrueTrue
TrueFalseFalse
FalseTrueTrue
FalseFalseTrue
The 3rd and 4th logical cases of this truth table, where the antecedent p is false and p → q is true, are called vacuous truths.
Deductive definition
Material implication can also be characterized deductively in terms of the following rules of inference.
1. Modus ponens
2. Conditional proof
3. Classical contraposition
4. Classical reductio ad absurdum
Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic, which rejects proofs by contraposition as valid rules of inference, (p → q) ⇒ ¬p ∨ q is not a propositional theorem, but the material conditional is used to define negation.
Formal properties
When disjunction, conjunction and negation are classical, material implication validates the following equivalences:
• Contraposition: $P\to Q\equiv \neg Q\to \neg P$
• Import-Export: $P\to (Q\to R)\equiv (P\land Q)\to R$
• Negated conditionals: $\neg (P\to Q)\equiv P\land \neg Q$
• Or-and-if: $P\to Q\equiv \neg P\lor Q$
• Commutativity of antecedents: ${\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}$
• Distributivity: ${\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}$
Similarly, on classical interpretations of the other connectives, material implication validates the following entailments:
• Antecedent strengthening: $P\to Q\models (P\land R)\to Q$
• Vacuous conditional: $\neg P\models P\to Q$
• Transitivity: $(P\to Q)\land (Q\to R)\models P\to R$
• Simplification of disjunctive antecedents: $(P\lor Q)\to R\models (P\to R)\land (Q\to R)$
Tautologies involving material implication include:
• Reflexivity: $\models P\to P$
• Totality: $\models (P\to Q)\lor (Q\to P)$
• Conditional excluded middle: $\models (P\to Q)\lor (P\to \neg Q)$
Discrepancies with natural language
Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the paradoxes of material implication.[6] In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, counterfactual conditionals would all be vacuously true on such an account.[7]
In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals denote material implication but end up conveying additional information when they interact with conversational norms such as Grice's maxims.[6][8] Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals.[8] In particular, such work has often rejected the assumption that natural-language conditionals are truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q.[6] Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.[8][6][9]
Similar discrepancies have been observed by psychologists studying conditional reasoning. For instance, the notorious Wason selection task study, less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to confirm to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.[10][11][12]
See also
• Boolean domain
• Boolean function
• Boolean logic
• Conditional quantifier
• Implicational propositional calculus
• Laws of Form
• Logical graph
• Logical equivalence
• Material implication (rule of inference)
• Peirce's law
• Propositional calculus
• Sole sufficient operator
Conditionals
• Counterfactual conditional
• Indicative conditional
• Corresponding conditional
• Strict conditional
Notes
1. Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.
References
1. Hilbert, D. (1918). Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.).
2. Jean van Heijenoort, ed. (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Harvard University Press. pp. 84–87. ISBN 0-674-32449-8.
3. Michael Nahas (25 Apr 2022). "English Translation of "Arithmetices Principia, Nova Methodo Exposita"" (PDF). GitHub. p. VI. Retrieved 2022-08-10.
4. Mauro ALLEGRANZA (2015-02-13). "elementary set theory - Is there any connection between the symbol $\supset $ when it means implication and its meaning as superset?". Mathematics Stack Exchange. Stack Exchange Inc. Answer. Retrieved 2022-08-10.
5. Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14.
6. Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
7. Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
8. Gillies, Thony (2017). "Conditionals" (PDF). In Hale, B.; Wright, C.; Miller, A. (eds.). A Companion to the Philosophy of Language. Wiley Blackwell. pp. 401–436. doi:10.1002/9781118972090.ch17. ISBN 9781118972090.
9. von Fintel, Kai (2011). "Conditionals" (PDF). In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.). Semantics: An international handbook of meaning. de Gruyter Mouton. pp. 1515–1538. doi:10.1515/9783110255072.1515. hdl:1721.1/95781. ISBN 978-3-11-018523-2.
10. Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection". Psychological Review. 101 (4): 608–631. CiteSeerX 10.1.1.174.4085. doi:10.1037/0033-295X.101.4.608.
11. Stenning, K.; van Lambalgen, M. (2004). "A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning". Cognitive Science. 28 (4): 481–530. CiteSeerX 10.1.1.13.1854. doi:10.1016/j.cogsci.2004.02.002.
12. von Sydow, M. (2006). Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules. Göttingen: Göttingen University Press. doi:10.53846/goediss-161. S2CID 246924881.
Further reading
• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
• Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
• Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
• Stalnaker, Robert, "Indicative Conditionals", Philosophia, 5 (1975): 269–286.
External links
• Media related to Material conditional at Wikimedia Commons
• Edgington, Dorothy. "Conditionals". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
Common logical connectives
• Tautology/True $\top $
• Alternative denial (NAND gate) $\uparrow $
• Converse implication $\leftarrow $
• Implication (IMPLY gate) $\rightarrow $
• Disjunction (OR gate) $\lor $
• Negation (NOT gate) $\neg $
• Exclusive or (XOR gate) $\not \leftrightarrow $
• Biconditional (XNOR gate) $\leftrightarrow $
• Statement (Digital buffer)
• Joint denial (NOR gate) $\downarrow $
• Nonimplication (NIMPLY gate) $\nrightarrow $
• Converse nonimplication $\nleftarrow $
• Conjunction (AND gate) $\land $
• Contradiction/False $\bot $
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| Wikipedia |
Power set
In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself.[1] In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.[2] The powerset of S is variously denoted as P(S), 𝒫(S), P(S), $\mathbb {P} (S)$, $\wp (S)$, or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.[1]
Power set
The elements of the power set of {x, y, z} ordered with respect to inclusion.
TypeSet operation
FieldSet theory
StatementThe power set is the set that contains all subsets of a given set.
Symbolic statement$x\in P(S)\iff x\subseteq S$
Any subset of P(S) is called a family of sets over S.
Example
If S is the set {x, y, z}, then all the subsets of S are
• {} (also denoted $\varnothing $ or $\emptyset $, the empty set or the null set)
• {x}
• {y}
• {z}
• {x, y}
• {x, z}
• {y, z}
• {x, y, z}
and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.[3]
Properties
If S is a finite set with the cardinality |S| = n (i.e., the number of all elements in the set S is n), then the number of all the subsets of S is |P(S)| = 2n. This fact as well as the reason of the notation 2S denoting the power set P(S) are demonstrated in the below.
An indicator function or a characteristic function of a subset A of a set S with the cardinality |S| = n is a function from S to the two elements set {0, 1}, denoted as IA: S → {0, 1}, and it indicates whether an element of S belongs to A or not; If x in S belongs to A, then IA(x) = 1, and 0 otherwise. Each subset A of S is identified by or equivalent to the indicator function IA, and {0,1}S as the set of all the functions from S to {0,1} consists of all the indicator functions of all the subsets of S. In other words, {0,1}S is equivalent or bijective to the power set P(S). Since each element in S corresponds to either 0 or 1 under any function in {0,1}S, the number of all the functions in {0,1}S is 2n. Since the number 2 can be defined as {0,1} (see, for example, von Neumann ordinals), the P(S) is also denoted as 2S. Obviously |2S| = 2|S| holds. Generally speaking, XY is the set of all functions from Y to X and |XY| = |X||Y|.
Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see Cardinality of the continuum).
The power set of a set S, together with the operations of union, intersection and complement, can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras, this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem).
The power set of a set S forms an abelian group when it is considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse), and a commutative monoid when considered with the operation of intersection. It can hence be shown, by proving the distributive laws, that the power set considered together with both of these operations forms a Boolean ring.
Representing subsets as functions
In set theory, XY is the notation representing the set of all functions from Y to X. As "2" can be defined as {0,1} (see, for example, von Neumann ordinals), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. As shown above, 2S and the power set of S, P(S), are considered identical set-theoretically.
This equivalence can be applied to the example above, in which S = {x, y, z}, to get the isomorphism with the binary representations of numbers from 0 to 2n − 1, with n being the number of elements in the set S or |S| = n. First, the enumerated set { (x, 1), (y, 2), (z, 3) } is defined in which the number in each ordered pair represents the position of the paired element of S in a sequence of binary digits such as {x, y} = 011(2); x of S is located at the first from the right of this sequence and y is at the second from the right, and 1 in the sequence means the element of S corresponding to the position of it in the sequence exists in the subset of S for the sequence while 0 means it does not.
For the whole power set of S, we get:
Subset Sequence
of binary digits
Binary
interpretation
Decimal
equivalent
{ }0, 0, 0000(2)0(10)
{ x }0, 0, 1001(2)1(10)
{ y }0, 1, 0010(2)2(10)
{ x, y }0, 1, 1011(2)3(10)
{ z }1, 0, 0100(2)4(10)
{ x, z }1, 0, 1101(2)5(10)
{ y, z }1, 1, 0110(2)6(10)
{ x, y, z }1, 1, 1111(2)7(10)
Such a injective mapping from P(S) to integers is arbitrary, so this representation of all the subsets of S is not unique, but the sort order of the enumerated set does not change its cardinality. (E.g., { (y, 1), (z, 2), (x, 3) } can be used to construct another injective mapping from P(S) to the integers without changing the number of one-to-one correspondences.)
However, such finite binary representation is only possible if S can be enumerated. (In this example, x, y, and z are enumerated with 1, 2, and 3 respectively as the position of binary digit sequences.) The enumeration is possible even if S has an infinite cardinality (i.e., the number of elements in S is infinite), such as the set of integers or rationals, but not possible for example if S is the set of real numbers, in which case we cannot enumerate all irrational numbers.
Relation to binomial theorem
The binomial theorem is closely related to the power set. A k–elements combination from some set is another name for a k–elements subset, so the number of combinations, denoted as C(n, k) (also called binomial coefficient) is a number of subsets with k elements in a set with n elements; in other words it's the number of sets with k elements which are elements of the power set of a set with n elements.
For example, the power set of a set with three elements, has:
• C(3, 0) = 1 subset with 0 elements (the empty subset),
• C(3, 1) = 3 subsets with 1 element (the singleton subsets),
• C(3, 2) = 3 subsets with 2 elements (the complements of the singleton subsets),
• C(3, 3) = 1 subset with 3 elements (the original set itself).
Using this relationship, we can compute $ \left|2^{S}\right|$ using the formula:
$\left|2^{S}\right|=\sum _{k=0}^{|S|}{\binom {|S|}{k}}$
Therefore, one can deduce the following identity, assuming $ |S|=n$:
$\left|2^{S}\right|=2^{n}=\sum _{k=0}^{n}{\binom {n}{k}}$
Recursive definition
If $S$ is a finite set, then a recursive definition of $P(S)$ proceeds as follows:
• If $S=\{\}$, then $P(S)=\{\,\{\}\,\}$.
• Otherwise, let $e\in S$ and $T=S\setminus \{e\}$; then $P(S)=P(T)\cup \{t\cup \{e\}:t\in P(T)\}$.
In words:
• The power set of the empty set is a singleton whose only element is the empty set.
• For a non-empty set $S$, let $e$ be any element of the set and $T$ its relative complement; then the power set of $S$ is a union of a power set of $T$ and a power set of $T$ whose each element is expanded with the $e$ element.
Subsets of limited cardinality
The set of subsets of S of cardinality less than or equal to κ is sometimes denoted by Pκ(S) or [S]κ, and the set of subsets with cardinality strictly less than κ is sometimes denoted P< κ(S) or [S]<κ. Similarly, the set of non-empty subsets of S might be denoted by P≥ 1(S) or P+(S).
Power object
A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra.
The power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard, subalgebras behave analogously to subsets.
However, there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way.
Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set HG of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore, the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph ΩG, called the power object of G.
What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object YX, in topos theory Y is required to be Ω.
Functors and quantifiers
In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.[4]
See also
• Cantor's theorem
• Family of sets
• Field of sets
• Combination
References
1. Weisstein, Eric W. "Power Set". mathworld.wolfram.com. Retrieved 2020-09-05.
2. Devlin 1979, p. 50
3. Puntambekar 2007, pp. 1–2
4. Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58
Bibliography
• Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN 0-387-90441-7. Zbl 0407.04003.
• Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.
• Puntambekar, A. A. (2007). Theory Of Automata And Formal Languages. Technical Publications. ISBN 978-81-8431-193-8.
External links
Look up power set in Wiktionary, the free dictionary.
• Power set at PlanetMath.
• Power set at the nLab
• Power object at the nLab
• Power set Algorithm in C++
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| Wikipedia |
Rhombus
In plane Euclidean geometry, a rhombus (PL: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after the French sweet[1] – also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle.
Rhombus
A rhombus in two different orientations
Typequadrilateral, trapezoid, parallelogram, kite
Edges and vertices4
Schläfli symbol{ } + { }
{2α}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D2), [2], (*22), order 4
Area$K={\frac {p\cdot q}{2}}$ (half the product of the diagonals)
Propertiesconvex, isotoxal
Dual polygonrectangle
Every rhombus is simple (non-self-intersecting), and is a special case of a parallelogram and a kite. A rhombus with right angles is a square.[2]
Etymology
The word "rhombus" comes from Ancient Greek: ῥόμβος, romanized: rhombos, meaning something that spins,[3] which derives from the verb ῥέμβω, romanized: rhémbō, meaning "to turn round and round."[4] The word was used both by Euclid and Archimedes, who used the term "solid rhombus" for a bicone, two right circular cones sharing a common base.[5]
The surface we refer to as rhombus today is a cross section of the bicone on a plane through the apexes of the two cones.
Characterizations
A simple (non-self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]
• a parallelogram in which a diagonal bisects an interior angle
• a parallelogram in which at least two consecutive sides are equal in length
• a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
• a quadrilateral with four sides of equal length (by definition)
• a quadrilateral in which the diagonals are perpendicular and bisect each other
• a quadrilateral in which each diagonal bisects two opposite interior angles
• a quadrilateral ABCD possessing a point P in its plane such that the four triangles ABP, BCP, CDP, and DAP are all congruent[8]
• a quadrilateral ABCD in which the incircles in triangles ABC, BCD, CDA and DAB have a common point[9]
Basic properties
Every rhombus has two diagonals connecting pairs of opposite vertices, and two pairs of parallel sides. Using congruent triangles, one can prove that the rhombus is symmetric across each of these diagonals. It follows that any rhombus has the following properties:
• Opposite angles of a rhombus have equal measure.
• The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral.
• Its diagonals bisect opposite angles.
The first property implies that every rhombus is a parallelogram. A rhombus therefore has all of the properties of a parallelogram: for example, opposite sides are parallel; adjacent angles are supplementary; the two diagonals bisect one another; any line through the midpoint bisects the area; and the sum of the squares of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q, in every rhombus
$\displaystyle 4a^{2}=p^{2}+q^{2}.$
Not every parallelogram is a rhombus, though any parallelogram with perpendicular diagonals (the second property) is a rhombus. In general, any quadrilateral with perpendicular diagonals, one of which is a line of symmetry, is a kite. Every rhombus is a kite, and any quadrilateral that is both a kite and parallelogram is a rhombus.
A rhombus is a tangential quadrilateral.[10] That is, it has an inscribed circle that is tangent to all four sides.
Diagonals
The length of the diagonals p = AC and q = BD can be expressed in terms of the rhombus side a and one vertex angle α as
$p=a{\sqrt {2+2\cos {\alpha }}}$
and
$q=a{\sqrt {2-2\cos {\alpha }}}.$
These formulas are a direct consequence of the law of cosines.
Inradius
The inradius (the radius of a circle inscribed in the rhombus), denoted by r, can be expressed in terms of the diagonals p and q as[10]
$r={\frac {p\cdot q}{2{\sqrt {p^{2}+q^{2}}}}},$
or in terms of the side length a and any vertex angle α or β as
$r={\frac {a\sin \alpha }{2}}={\frac {a\sin \beta }{2}}.$
Area
As for all parallelograms, the area K of a rhombus is the product of its base and its height (h). The base is simply any side length a:
$K=a\cdot h.$
The area can also be expressed as the base squared times the sine of any angle:
$K=a^{2}\cdot \sin \alpha =a^{2}\cdot \sin \beta ,$
or in terms of the height and a vertex angle:
$K={\frac {h^{2}}{\sin \alpha }},$
or as half the product of the diagonals p, q:
$K={\frac {p\cdot q}{2}},$
or as the semiperimeter times the radius of the circle inscribed in the rhombus (inradius):
$K=2a\cdot r.$
Another way, in common with parallelograms, is to consider two adjacent sides as vectors, forming a bivector, so the area is the magnitude of the bivector (the magnitude of the vector product of the two vectors), which is the determinant of the two vectors' Cartesian coordinates: K = x1y2 – x2y1.[11]
Dual properties
The dual polygon of a rhombus is a rectangle:[12]
• A rhombus has all sides equal, while a rectangle has all angles equal.
• A rhombus has opposite angles equal, while a rectangle has opposite sides equal.
• A rhombus has an inscribed circle, while a rectangle has a circumcircle.
• A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides.
• The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
• The figure formed by joining the midpoints of the sides of a rhombus is a rectangle, and vice versa.
Cartesian equation
The sides of a rhombus centered at the origin, with diagonals each falling on an axis, consist of all points (x, y) satisfying
$\left|{\frac {x}{a}}\right|\!+\left|{\frac {y}{b}}\right|\!=1.$
The vertices are at $(\pm a,0)$ and $(0,\pm b).$ This is a special case of the superellipse, with exponent 1.
Other properties
• One of the five 2D lattice types is the rhombic lattice, also called centered rectangular lattice.
• Identical rhombi can tile the 2D plane in three different ways, including, for the 60° rhombus, the rhombille tiling.
As topological square tilings As 30-60 degree rhombille tiling
• Three-dimensional analogues of a rhombus include the bipyramid and the bicone as a surface of revolution.
As the faces of a polyhedron
Convex polyhedra with rhombi include the infinite set of rhombic zonohedrons, which can be seen as projective envelopes of hypercubes.
• A rhombohedron (also called a rhombic hexahedron) is a three-dimensional figure like a cuboid (also called a rectangular parallelepiped), except that its 3 pairs of parallel faces are up to 3 types of rhombi instead of rectangles.
• The rhombic dodecahedron is a convex polyhedron with 12 congruent rhombi as its faces.
• The rhombic triacontahedron is a convex polyhedron with 30 golden rhombi (rhombi whose diagonals are in the golden ratio) as its faces.
• The great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron with 30 intersecting rhombic faces.
• The rhombic hexecontahedron is a stellation of the rhombic triacontahedron. It is nonconvex with 60 golden rhombic faces with icosahedral symmetry.
• The rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces, with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim ones.
• The rhombic icosahedron is a polyhedron composed of 20 rhombic faces, of which three, four, or five meet at each vertex. It has 10 faces on the polar axis with 10 faces following the equator.
Example polyhedra with all rhombic faces
Isohedral Isohedral golden rhombi 2-isohedral 3-isohedral
Trigonal trapezohedron Rhombic dodecahedron Rhombic triacontahedron Rhombic icosahedron Rhombic enneacontahedron Rhombohedron
See also
• Merkel-Raute
• Rhombus of Michaelis, in human anatomy
• Rhomboid, either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle
• Rhombic antenna
• Rhombic Chess
• Flag of the Department of North Santander of Colombia, containing four stars in the shape of a rhombus
• Superellipse (includes a rhombus with rounded corners)
References
1. Alsina, Claudi; Nelsen, Roger B. (31 December 2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. ISBN 9781614442165.
2. Note: Euclid's original definition and some English dictionaries' definition of rhombus excludes squares, but modern mathematicians prefer the inclusive definition. See, e.g., De Villiers, Michael (February 1994). "The role and function of a hierarchical classification of quadrilaterals". For the Learning of Mathematics. 14 (1): 11–18. JSTOR 40248098.
3. ῥόμβος Archived 2013-11-08 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
4. ρέμβω Archived 2013-11-08 at the Wayback Machine, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
5. "The Origin of Rhombus". Archived from the original on 2015-04-02. Retrieved 2005-01-25.
6. Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition Archived 2020-02-26 at the Wayback Machine", Information Age Publishing, 2008, pp. 55-56.
7. Owen Byer, Felix Lazebnik and Deirdre Smeltzer, Methods for Euclidean Geometry Archived 2019-09-01 at the Wayback Machine, Mathematical Association of America, 2010, p. 53.
8. Paris Pamfilos (2016), "A Characterization of the Rhombus", Forum Geometricorum 16, pp. 331–336, Archived 2016-10-23 at the Wayback Machine
9. "IMOmath, "26-th Brazilian Mathematical Olympiad 2004"" (PDF). Archived (PDF) from the original on 2016-10-18. Retrieved 2020-01-06.
10. Weisstein, Eric W. "Rhombus". MathWorld.
11. WildLinAlg episode 4 Archived 2017-02-05 at the Wayback Machine, Norman J Wildberger, Univ. of New South Wales, 2010, lecture via youtube
12. de Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
External links
Look up rhombus in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Rhombi.
• Parallelogram and Rhombus - Animated course (Construction, Circumference, Area)
• Rhombus definition, Math Open Reference with interactive applet.
• Rhombus area, Math Open Reference - shows three different ways to compute the area of a rhombus, with interactive applet
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
Authority control: National
• Germany
| Wikipedia |
Less-than sign
The less-than sign is a mathematical symbol that denotes an inequality between two values. The widely adopted form of two equal-length strokes connecting in an acute angle at the left, <, has been found in documents dated as far back as the 1560s. In mathematical writing, the less-than sign is typically placed between two values being compared and signifies that the first number is less than the second number. Examples of typical usage include 1⁄2 < 1 and −2 < 0.
<
Less-than sign
In UnicodeU+003C < LESS-THAN SIGN (<, <)
Different from
Different fromU+2329 〈 LEFT-POINTING ANGLE BRACKET
Related
See alsoU+003E > GREATER-THAN SIGN
U+2264 ≤ LESS-THAN OR EQUAL TO
U+2A7D ⩽ LESS-THAN OR SLANTED EQUAL TO used e.g. in Poland
U+226E ≮ NOT LESS-THAN
U+226A ≪ MUCH LESS-THAN
Since the development of computer programming languages, the less-than sign and the greater-than sign have been repurposed for a range of uses and operations.
Computing
The less-than sign, <, is an original ASCII character (hex 3C, decimal 60).
The less-than sign may be used for an approximation of the opening angle bracket, ⟨. ASCII does not have angle brackets but are standard in Unicode (U+2329 〈 LEFT-POINTING ANGLE BRACKET). The latter is expected in formal texts.
Programming
In BASIC, Lisp-family languages, and C-family languages (including Java and C++), comparison operator < means "less than".
In Coldfusion, operator .lt. means "less than".
In Fortran, operator .LT. means "less than"; later versions allow <.
Shell scripts
In Bourne shell (and many other shells), operator -lt means "less than". Less-than sign is used to redirect input from a file. Less-than plus ampersand (<&) is used to redirect from a file descriptor.
Double less-than sign
The double less-than sign, <<, may be used for an approximation of the much-less-than sign (≪) or of the opening guillemet («). ASCII does not encode either of these signs, though they are both included in Unicode.
In Bash, Perl, and Ruby, operator <<EOF (where "EOF" is an arbitrary string, but commonly "EOF" denoting "end of file") is used to denote the beginning of a here document.
In C and C++, operator << represents a binary left shift.
In the C++ Standard Library, operator <<, when applied on an output stream, acts as insertion operator and performs an output operation on the stream.
In Ruby, operator << acts as append operator when used between an array and the value to be appended.
In XPath the << operator returns true if the left operand precedes the right operand in document order; otherwise it returns false.[1]
Triple less-than sign
In PHP, operator <<<OUTPUT is used to denote the beginning of a heredoc statement (where OUTPUT is an arbitrary named variable.)
In Bash, <<<word is used as a "here string", where word is expanded and supplied to the command on its standard input, similar to a heredoc.
Less-than sign with equals sign
The less-than sign with the equals sign, <=, may be used for an approximation of the less-than-or-equal-to sign, ≤. ASCII does not have a less-than-or-equal-to sign, but Unicode defines it at code point U+2264.
In BASIC, Lisp-family languages, and C-family languages (including Java and C++), operator <= means "less than or equal to". In Sinclair BASIC it is encoded as a single-byte code point token.
In Prolog, =< means "less than or equal to" (as distinct from the arrow <=).
In Fortran, operators .LE. and <= both mean "less than or equal to".
In Bourne shell and Windows PowerShell, the operator -le means "less than or equal to".
Less-than sign with hyphen-minus
In the R programming language, the less-than sign is used in conjunction with a hyphen-minus to create an arrow (<-), this can be used as the left assignment operator.
Spaceship operator
Less-than sign is used in the spaceship operator.
HTML
In HTML (and SGML and XML), the less-than sign is used at the beginning of tags. The less-than sign may be included with <. The less-than-or-equal-to sign, ≤, may be included with ≤.
Unicode
Unicode provides various Less Than Symbol:[2]
SymbolNameCode Point
⍃Apl Functional Symbol Quad Less ThanU+2343
⧀Circled Less ThanU+29C0
⦖Double Right Arc Less Than BracketU+2996
⋜Equal To Or Less ThanU+22DC
⦓Left Arc Less Than BracketU+2993
⥷Leftwards Arrow Through Less ThanU+2977
⥶Less Than Above Leftwards ArrowU+2976
≨Less Than But Not Equal ToU+2268
⋦Less Than But Not Equivalent ToU+22E6
≤Less Than Or Equal ToU+2264
≲Less Than Or Equivalent ToU+2272
≦Less Than Over Equal ToU+2266
<Less Than SignU+003C
⩹Less Than With Circle InsideU+2A79
⋖Less Than With DotU+22D6
≪Much Less ThanU+226A
≰Neither Less Than Nor Equal ToU+2270
≴Neither Less Than Nor Equivalent ToU+2274
≮Not Less ThanU+226E
⋘Very Much Less ThanU+22D8
Mathematics
In an inequality, the less-than sign and greater-than sign always "point" to the smaller number. Put another way, the "jaws" (the wider section of the symbol) always direct to the larger number.
See also
• Inequality (mathematics)
• Greater-than sign
• Relational operator
• Much-less-than sign
References
1. "XML Path Language (XPath) 2.0 (Second Edition)". www.w3.org. W3C. 14 December 2010. Archived from the original on 7 October 2022. Retrieved 29 October 2019.
2. "Less than symbol". Archived from the original on 2023-05-16. Retrieved 2023-06-06.
| Wikipedia |
'Mamphono Khaketla
'Mamphono Khaketla (born 5 March 1960) is a Lesotho mathematician and senator who served as Minister of Finance from March 2015 to June 2017.
'Mamphono Khaketla
Personal details
Born (1960-03-05) March 5, 1960
Maseru, Lesotho
CitizenshipLesotho
NationalityMosotho
Parent(s)Bennett Makalo
'Masechele Caroline Ntseliseng Khaketla
EducationNational University of Lesotho 1980
Alma materUniversity of Wisconsin 1991
OccupationMathematician
Early life and education
Khaketla was born in Maseru on 5 March 1960 to Bennett Makalo and Caroline Ntseliseng ’Masechele Khaketla.[1] Her father was a novelist, journalist, politician and former minister, as well as the major shareholder of Mohlabani Property Company, and left her a sizeable estate.[2][3] Her mother was a teacher and author, one of the first women published in Lesotho.[4][5]
Khaketla did her primary and secondary schooling Maseru, before receiving a Bachelor of Education from the National University of Lesotho in 1980.[4] She has a master's degree in education and a PhD in mathematics education from the University of Wisconsin (1991).[4] Her thesis was titled "An analysis of the Lesotho Junior Certificate Mathematics Examination and its impact on instructions".[1]
Career
Khaketla was a lecturer in mathematics at the National Teacher Training College from 1981 until 1995 and became assistant director of the college.[4] She worked at the Institute of Development Management in Lesotho and Botswana from 1996 until 2001 before becoming the director of the Centre for Accounting Studies.
Khaketla was appointed as a senator by Prime Minister Pakalitha Mosisili in 2002 and served as Minister of Communications, Science and Technology from 2002 until 2004.[4] At the 2007 election, she lost her seat but was elected to the National Assembly as one of the Lesotho Congress for Democracy members on a party list for proportional representation submitted by the National Independence Party.[4] She served as Minister of Education and Training from 2007 until 2012. In 2011, Khaketla was one of seven women ministers in the Cabinet, alongside: Mannete Ramali, Maphoka Motoboli, Mathabiso Lepono, Mphu Keneileo Ramatlapeng, Mpeo Mahase-Moiloa and Pontso Suzan Matumelo Sekatle.[6] On 30 March 2015 she was appointed Minister of Finance.
In November 2015, she presided over the 102nd session of the African, Caribbean and Pacific Group of States Council of Ministers.[7]
In July 2016, Khaketla was accused of soliciting a bribe for a major government contract in a case that was before the courts. She denied the allegation.[8][9]
Publications
• Romberg, Thomas A.; Wilson, Linda; Khaketla, Mamphono (1989). "An examination of six standard mathematics tests for grade eight". National Center for Research in Mathematical Sciences Education. {{cite journal}}: Cite journal requires |journal= (help)
• Romberg, T. A.; Wilson, L.; Khaketla, M. (1989). "The alignment of six standardized tests with the NCTM standards". Washington, DC: National Summit on Mathematics Assessment convened by the Mathematical Sciences Education Board. {{cite journal}}: Cite journal requires |journal= (help)
• Romberg, Thomas A.; Wilson, Linda; Khaketla, 'Mamphono; Chavarria, Silvia (1992). "Curriculum and Test Alignment". In Thomas A. Romberg (ed.). Mathematics Assessment and Evaluation: Imperatives for Mathematics Educators. SUNY Press. pp. 61–74. ISBN 9780791409008.
References
1. African Doctorates in Mathematics: A Catalogue. Lulu.com. 2007. p. 147.
2. "Minister wins back father's estate". Lesotho Times. 29 September 2011. Retrieved 16 May 2017.
3. "Khaketla estate wrangle rages on". Sunday Express. 10 December 2010. Retrieved 16 May 2017.
4. Rosenberg, Scott; Wesifelder, Richard F. (2013). Historical Dictionary of Lesotho. Scarecrow Press. pp. 204–205. ISBN 9780810879829.
5. Sheldon, Kathleen (2016). Historical Dictionary of Women in Sub-Saharan Africa. Rowman & Littlefield. p. 146. ISBN 9781442262935.
6. Political handbook of the world 2012. Lansford, Tom. Los Angeles: Sage. 2012. pp. 832. ISBN 978-1-4522-3434-2. OCLC 794595888.{{cite book}}: CS1 maint: others (link)
7. "Statement by the President of the 102nd session of the ACP Council of Ministers, Hon. Dr. Mamphono Khaketla, Minister of Finance of Lesotho". Brussels: ACP. 24 November 2015. Retrieved 16 May 2017.
8. Mohloboli, Keiso (14 August 2016). "Lesotho's great car 'con'". City Press. Retrieved 16 May 2017.
9. Ntsukunyane, Lekhetho (28 July 2016). "DC Youths blast Minister". Lesotho Times. Retrieved 16 May 2017.
External links
• O'Connor, John J.; Robertson, Edmund F., "'Mamphono Khaketla", MacTutor History of Mathematics Archive, University of St Andrews
• Government Profile
| Wikipedia |
(2+1)-dimensional topological gravity
In two spatial and one time dimensions, general relativity turns out to have no propagating gravitational degrees of freedom. In fact, it can be shown that in a vacuum, spacetime will always be locally flat (or de Sitter or anti-de Sitter depending upon the cosmological constant). This makes (2+1)-dimensional topological gravity (2+1D topological gravity) a topological theory with no gravitational local degrees of freedom.
Physicists became interested in the relation between Chern–Simons theory and gravity during the 1980s.[1] During this period, Edward Witten[2] argued that 2+1D topological gravity is equivalent to a Chern–Simons theory with the gauge group $SO(2,2)$ for a negative cosmological constant, and $SO(3,1)$ for a positive one. This theory can be exactly solved, making it a toy model for quantum gravity. The Killing form involves the Hodge dual.
Witten later changed his mind,[3] and argued that nonperturbatively 2+1D topological gravity differs from Chern–Simons because the functional measure is only over nonsingular vielbeins. He suggested the CFT dual is a monster conformal field theory, and computed the entropy of BTZ black holes.
References
1. Achúcarro, A.; Townsend, P. (1986). "A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories". Phys. Lett. B180 (1–2): 89. Bibcode:1986PhLB..180...89A. doi:10.1016/0370-2693(86)90140-1.
2. Witten, Edward (19 Dec 1988). "(2+1)-Dimensional Gravity as an Exactly Soluble System". Nuclear Physics B. 311 (1): 46–78. Bibcode:1988NuPhB.311...46W. doi:10.1016/0550-3213(88)90143-5. hdl:10338.dmlcz/143077.url=http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/Witten2.pdf
3. Witten, Edward (22 June 2007). "Three-Dimensional Gravity Revisited". arXiv:0706.3359 [hep-th].
Quantum gravity
Central concepts
• AdS/CFT correspondence
• Ryu–Takayanagi conjecture
• Causal patch
• Gravitational anomaly
• Graviton
• Holographic principle
• IR/UV mixing
• Planck units
• Quantum foam
• Trans-Planckian problem
• Weinberg–Witten theorem
• Faddeev–Popov ghost
• Batalin-Vilkovisky formalism
• CA-duality
Toy models
• 2+1D topological gravity
• CGHS model
• Jackiw–Teitelboim gravity
• Liouville gravity
• RST model
• Topological quantum field theory
Quantum field theory
in curved spacetime
• Bunch–Davies vacuum
• Hawking radiation
• Semiclassical gravity
• Unruh effect
Black holes
• Black hole complementarity
• Black hole information paradox
• Black-hole thermodynamics
• Bekenstein bound
• Bousso's holographic bound
• Cosmic censorship hypothesis
• ER = EPR
• Firewall (physics)
• Gravitational singularity
Approaches
String theory
• Bosonic string theory
• M-theory
• Supergravity
• Superstring theory
Canonical quantum gravity
• Loop quantum gravity
• Wheeler–DeWitt equation
Euclidean quantum gravity
• Hartle–Hawking state
Others
• Causal dynamical triangulation
• Causal sets
• Noncommutative geometry
• Spin foam
• Group field theory
• Superfluid vacuum theory
• Twistor theory
• Dual graviton
Applications
• Quantum cosmology
• Eternal inflation
• Multiverse
• FRW/CFT duality
• See also: Template:Quantum mechanics topics
Theories of gravitation
Standard
Newtonian gravity (NG)
• Newton's law of universal gravitation
• Gauss's law for gravity
• Poisson's equation for gravity
• History of gravitational theory
General relativity (GR)
• Introduction
• History
• Mathematics
• Exact solutions
• Resources
• Tests
• Post-Newtonian formalism
• Linearized gravity
• ADM formalism
• Gibbons–Hawking–York boundary term
Alternatives to
general relativity
Paradigms
• Classical theories of gravitation
• Quantum gravity
• Theory of everything
Classical
• Poincaré gauge theory
• Einstein–Cartan
• Teleparallelism
• Bimetric theories
• Gauge theory gravity
• Composite gravity
• f(R) gravity
• Infinite derivative gravity
• Massive gravity
• Modified Newtonian dynamics, MOND
• AQUAL
• Tensor–vector–scalar
• Nonsymmetric gravitation
• Scalar–tensor theories
• Brans–Dicke
• Scalar–tensor–vector
• Conformal gravity
• Scalar theories
• Nordström
• Whitehead
• Geometrodynamics
• Induced gravity
• Degenerate Higher-Order Scalar-Tensor theories
Quantum-mechanical
• Euclidean quantum gravity
• Canonical quantum gravity
• Wheeler–DeWitt equation
• Loop quantum gravity
• Spin foam
• Causal dynamical triangulation
• Asymptotic safety in quantum gravity
• Causal sets
• DGP model
• Rainbow gravity theory
Unified-field-theoric
• Kaluza–Klein theory
• Supergravity
Unified-field-theoric and
quantum-mechanical
• Noncommutative geometry
• Semiclassical gravity
• Superfluid vacuum theory
• Logarithmic BEC vacuum
• String theory
• M-theory
• F-theory
• Heterotic string theory
• Type I string theory
• Type 0 string theory
• Bosonic string theory
• Type II string theory
• Little string theory
• Twistor theory
• Twistor string theory
Generalisations /
extensions of GR
• Liouville gravity
• Lovelock theory
• (2+1)-dimensional topological gravity
• Gauss–Bonnet gravity
• Jackiw–Teitelboim gravity
Pre-Newtonian
theories and
toy models
• Aristotelian physics
• CGHS model
• RST model
• Mechanical explanations
• Fatio–Le Sage
• Entropic gravity
• Gravitational interaction of antimatter
• Physics in the medieval Islamic world
• Theory of impetus
Related topics
• Graviton
| Wikipedia |
(2,3,7) triangle group
In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84(g − 1), of its automorphism group.
A note on terminology – the "(2,3,7) triangle group" most often refers, not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2.
Torsion-free normal subgroups of the (2,3,7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.
Constructions
Hyperbolic construction
To construct the triangle group, start with a hyperbolic triangle with angles π/2, π/3, and π/7. This triangle, the smallest hyperbolic Schwarz triangle, tiles the plane by reflections in its sides. Consider then the group generated by reflections in the sides of the triangle, which (since the triangle tiles) is a non-Euclidean crystallographic group (a discrete subgroup of hyperbolic isometries) with this triangle for fundamental domain; the associated tiling is the order-3 bisected heptagonal tiling. The (2,3,7) triangle group is defined as the index 2 subgroups consisting of the orientation-preserving isometries, which is a Fuchsian group (orientation-preserving NEC group).
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
Group presentation
It has a presentation in terms of a pair of generators, g2, g3, modulo the following relations:
$g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=1.$
Geometrically, these correspond to rotations by ${\frac {2\pi }{2}},{\frac {2\pi }{3}}$, and ${\frac {2\pi }{7}}$ about the vertices of the Schwarz triangle.
Quaternion algebra
The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center ±1.
Let η = 2cos(2π/7). Then from the identity
$(2-\eta )^{3}=7(\eta -1)^{2}.$
we see that Q(η) is a totally real cubic extension of Q. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i2 = j2 = η, ij = −ji. One chooses a suitable Hurwitz quaternion order ${\mathcal {Q}}_{\mathrm {Hur} }$ in the quaternion algebra. Here the order ${\mathcal {Q}}_{\mathrm {Hur} }$ is generated by elements
$g_{2}={\tfrac {1}{\eta }}ij$
$g_{3}={\tfrac {1}{2}}(1+(\eta ^{2}-2)j+(3-\eta ^{2})ij).$
In fact, the order is a free Z[η]-module over the basis $1,g_{2},g_{3},g_{2}g_{3}$. Here the generators satisfy the relations
$g_{2}^{2}=g_{3}^{3}=(g_{2}g_{3})^{7}=-1,\,$
which descend to the appropriate relations in the triangle group, after quotienting by the center.
Relation to SL(2,R)
Extending the scalars from Q(η) to R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the algebra M(2,R) of real 2 by 2 matrices. Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group in SL(2,R), specifically as a quotient of the modular group. This can be visualized by the associated tilings, as depicted at right: the (2,3,7) tiling on the Poincaré disc is a quotient of the modular tiling on the upper half-plane.
However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements (and hence also translation lengths of hyperbolic elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups) can be calculated by means of the reduced trace in the quaternion algebra, and the formula
$\operatorname {tr} (\gamma )=2\cosh(\ell _{\gamma }/2).$
References
1. Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp
Further reading
• Elkies, N.D. (1998). "Shimura curve computations". In Buhler, J.P (ed.). Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science. Vol. 1423. Springer. pp. 1–47. arXiv:math.NT/0005160. doi:10.1007/BFb0054850. ISBN 978-3-540-69113-6.
• Katz, M.; Schaps, M.; Vishne, U. (2007). "Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups". J. Differential Geom. 76 (3): 399–422. arXiv:math.DG/0505007.
| Wikipedia |
(G,X)-manifold
In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds. This notion is often used with G being a Lie group and X a homogeneous space for G. Foundational examples are hyperbolic manifolds and affine manifolds.
Definition and examples
Formal definition
Let $X$ be a connected differential manifold and $G$ be a subgroup of the group of diffeomorphisms of $X$ which act analytically in the following sense:
if $g_{1},g_{2}\in G$ and there is a nonempty open subset $U\subset X$ such that $g_{1},g_{2}$ are equal when restricted to $U$ then $g_{1}=g_{2}$
(this definition is inspired by the analytic continuation property of analytic diffeomorphisms on an analytic manifold).
A $(G,X)$-structure on a topological space $M$ is a manifold structure on $M$ whose atlas' charts has values in $X$ and transition maps belong to $G$. This means that there exists:
• a covering of $M$ by open sets $U_{i},i\in I$ (i.e. $M=\bigcup _{i\in I}U_{i}$);
• open embeddings $\varphi _{i}:U_{i}\to X$ called charts;
such that every transition map $\varphi _{i}\circ \varphi _{j}^{-1}:\varphi _{j}(U_{i}\cap U_{j})\to \varphi _{i}(U_{i}\cap U_{j})$ is the restriction of a diffeomorphism in $G$.
Two such structures $(U_{i},\varphi _{i}),(V_{j},\psi _{j})$ are equivalent when they are contained in a maximal one, equivalently when their union is also a $(G,X)$ structure (i.e. the maps $\varphi _{i}\circ \psi _{j}^{-1}$ and $\psi _{j}\circ \varphi _{i}^{-1}$ are restrictions of diffeomorphisms in $G$).
Riemannian examples
If $G$ is a Lie group and $X$ a Riemannian manifold with a faithful action of $G$ by isometries then the action is analytic. Usually one takes $G$ to be the full isometry group of $X$. Then the category of $(G,X)$ manifolds is equivalent to the category of Riemannian manifolds which are locally isometric to $X$ (i.e. every point has a neighbourhood isometric to an open subset of $X$).
Often the examples of $X$ are homogeneous under $G$, for example one can take $X=G$ with a left-invariant metric. A particularly simple example is $X=\mathbb {R} ^{n}$ and $G$ the group of euclidean isometries. Then a $(G,X)$ manifold is simply a flat manifold.
A particularly interesting example is when $X$ is a Riemannian symmetric space, for example hyperbolic space. The simplest such example is the hyperbolic plane, whose isometry group is isomorphic to $G=\mathrm {PGL} _{2}(\mathbb {R} )$.
Pseudo-Riemannian examples
When $X$ is Minkowski space and $G$ the Lorentz group the notion of a $(G,X)$-structure is the same as that of a flat Lorentzian manifold.
Other examples
When $X$ is the affine space and $G$ the group of affine transformations then one gets the notion of an affine manifold.
When $X=\mathbb {P} ^{n}(\mathbb {R} )$ is the n-dimensional real projective space and $G=\mathrm {PGL} _{n+1}(\mathbb {R} )$ one gets the notion of a projective structure.[1]
Developing map and completeness
Developing map
Let $M$ be a $(G,X)$-manifold which is connected (as a topological space). The developing map is a map from the universal cover ${\tilde {M}}$ to $X$ which is only well-defined up to composition by an element of $G$.
A developing map is defined as follows:[2] fix $p\in {\tilde {M}}$ and let $q\in {\tilde {M}}$ be any other point, $\gamma $ a path from $p$ to $q$, and $\varphi :U\to X$ (where $U$ is a small enough neighbourhood of $p$) a map obtained by composing a chart of $M$ with the projection ${\tilde {M}}\to M$. We may use analytic continuation along $\gamma $ to extend $\varphi $ so that its domain includes $q$. Since ${\tilde {M}}$ is simply connected the value of $\varphi (q)$ thus obtained does not depend on the original choice of $\gamma $, and we call the (well-defined) map $\varphi :{\tilde {M}}\to X$ :{\tilde {M}}\to X} a developing map for the $(G,X)$-structure. It depends on the choice of base point and chart, but only up to composition by an element of $G$.
Monodromy
Given a developing map $\varphi $, the monodromy or holonomy[3] of a $(G,X)$-structure is the unique morphism $h:\pi _{1}(M)\to G$ which satisfies
$\forall \gamma \in \pi _{1}(M),p\in {\tilde {M}}:\varphi (\gamma \cdot p)=h(\gamma )\cdot \varphi (p)$.
It depends on the choice of a developing map but only up to an inner automorphism of $G$.
Complete (G,X)-structures
A $(G,X)$ structure is said to be complete if it has a developing map which is also a covering map (this does not depend on the choice of developing map since they differ by a diffeomorphism). For example, if $X$ is simply connected the structure is complete if and only if the developing map is a diffeomorphism.
Riemannian (G,X)-structures
If $X$ is a Riemannian manifold and $G$ its full group of isometry, then a $(G,X)$-structure is complete if and only if the underlying Riemannian manifold is geodesically complete (equivalently metrically complete). In particular, in this case if the underlying space of a $(G,X)$-manifold is compact then the latter is automatically complete.
In the case where $X$ is the hyperbolic plane the developing map is the same map as given by the Uniformisation Theorem.
Other cases
In general compactness of the space does not imply completeness of a $(G,X)$-structure. For example, an affine structure on the torus is complete if and only if the monodromy map has its image inside the translations. But there are many affine tori which do not satisfy this condition, for example any quadrilateral with its opposite sides glued by an affine map yields an affine structure on the torus, which is complete if and only if the quadrilateral is a parallelogram.
Interesting examples of complete, noncompact affine manifolds are given by the Margulis spacetimes.
(G,X)-structures as connections
Main article: Ehresmann connection
In the work of Charles Ehresmann $(G,X)$-structures on a manifold $M$ are viewed as flat Ehresmann connections on fiber bundles with fiber $X$ over $M$, whose monodromy maps lie in $G$.
Notes
1. Dumas, David (2009). "Complex projective structures". In Papadopoulos, Athanase (ed.). Handbook of Teichmüller theory, Volume II. European MAth. soc.
2. Thurston 1997, Chapter 3.4.
3. Thurston 1997, p. 141.
References
• Thurston, William (1997). Three-dimensional geometry and topology. Vol. 1. Princeton University Press.
| Wikipedia |
(a, b)-decomposition
In graph theory, the (a, b)-decomposition of an undirected graph is a partition of its edges into a + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree b. If this graph is also a forest, then we call this a F(a, b)-decomposition.
A graph with arboricity a is (a, 0)-decomposable. Every (a, 0)-decomposition or (a, 1)-decomposition is a F(a, 0)-decomposition or a F(a, 1)-decomposition respectively.
Graph classes
• Every planar graph is F(2, 4)-decomposable.[1]
• Every planar graph $G$ with girth at least $g$ is
• F(2, 0)-decomposable if $g\geq 4$.[2]
• (1, 4)-decomposable if $g\geq 5$.[3]
• F(1, 2)-decomposable if $g\geq 6$.[4]
• F(1, 1)-decomposable if $g\geq 8$,[5] or if every cycle of $G$ is either a triangle or a cycle with at least 8 edges not belonging to a triangle.[6]
• (1, 5)-decomposable if $G$ has no 4-cycles.[7]
• Every outerplanar graph is F(2, 0)-decomposable[2] and (1, 3)-decomposable.[8]
Notes
1. Gonçalves (2009), conjectured by Balogh et al. (2005). Improving results by Nash-Williams (1964) then Balogh et al. (2005).
2. Implied by Nash-Williams (1964).
3. He et al. (2002)
4. Implied by Montassier et al. (2012), improving results by He et al. (2002), then Kleitman (2008).
5. Independently proved by Wang & Zhang (2011) and implied by Montassier et al. (2012), improving results by He et al. (2002) for girth 11, then Bassa et al. (2010) for girth 10 and Borodin et al. (2008a) for girth 9.
6. Borodin et al. (2009b), even if not explicitly stated.
7. Borodin et al. (2009a), improving results by He et al. (2002), then Borodin et al. (2008b).
8. Proved without explicit reference by Guan & Zhu (1999).
References (chronological order)
• Nash-Williams, Crispin St. John Alvah (1964). "Decomposition of finite graphs into forests". Journal of the London Mathematical Society. 39 (1): 12. doi:10.1112/jlms/s1-39.1.12. MR 0161333.
• Guan, D. J.; Zhu, Xuding (1999). "Game chromatic number of outerplanar graphs". Journal of Graph Theory. 30 (1): 67–70. doi:10.1002/(sici)1097-0118(199901)30:1<67::aid-jgt7>3.0.co;2-m.
• He, Wenjie; Hou, Xiaoling; Lih, Ko-Wei; Shao, Jiating; Wang, Weifan; Zhu, Xuding (2002). "Edge-partitions of planar graphs and their game coloring numbers". Journal of Graph Theory. 41 (4): 307–311. doi:10.1002/jgt.10069. S2CID 20929383.
• Balogh, József; Kochol, Martin; Pluhár, András; Yu, Xingxing (2005). "Covering planar graphs with forests". Journal of Combinatorial Theory, Series B. 94 (1): 147–158. doi:10.1016/j.ejc.2007.06.020.
• Borodin, Oleg V.; Kostochka, Alexandr V.; Sheikh, Naeem N.; Yu, Gexin (2008). "Decomposing a planar graph with girth 9 into a forest and a matching". European Journal of Combinatorics. 29 (5): 1235–1241. doi:10.1016/j.ejc.2007.06.020.
• Borodin, Oleg V.; Kostochka, Alexandr V.; Sheikh, Naeem N.; Yu, Gexin (2008). "M-Degrees of Quadrangle-Free Planar Graphs" (PDF). Journal of Graph Theory. 60 (1): 80–85. CiteSeerX 10.1.1.224.8397. doi:10.1002/jgt.20346. S2CID 7486622.
• Kleitman, Daniel J. (2008). "Partitioning the Edges of a Girth 6 Planar Graph into those of a Forest and those of a Set of Disjoint Paths and Cycles". Manuscript.
• Gonçalves, Daniel (2009). "Covering planar graphs with forests, one having bounded maximum degree". Journal of Combinatorial Theory, Series B. 99 (2): 314–322. doi:10.1016/j.jctb.2008.07.004.
• Borodin, Oleg V.; Ivanova, Anna O.; Kostochka, Alexandr V.; Sheikh, Naeem N. (2009). "Decompositions of Quadrangle-Free Planar Graphs" (PDF). Discussiones Mathematicae Graph Theory. 29: 87–99. CiteSeerX 10.1.1.224.8787. doi:10.7151/dmgt.1434.
• Borodin, Oleg V.; Ivanova, Anna O.; Kostochka, Alexandr V.; Sheikh, Naeem N. (2009). "Planar graphs decomposable into a forest and a matching". Discrete Mathematics. 309 (1): 277–279. doi:10.1016/j.disc.2007.12.104.
• Bassa, A.; Burns, J.; Campbell, J.; Deshpande, A.; Farley, J.; Halsey, L.; Ho, S.-Y.; Kleitman, D.; Michalakis, S.; Persson, P.-O.; Pylyavskyy, P.; Rademacher, L.; Riehl, A.; Rios, M.; Samuel, J.; Tenner, B.E.; Vijayasarathy, A.; Zhao, L. (2010). "Partitioning a Planar Graph of Girth 10 into a Forest and a Matching". European Journal of Combinatorics. 124 (3): 213–228. doi:10.1111/j.1467-9590.2009.00468.x. S2CID 120663098.
• Wang, Yingqian; Zhang, Qijun (2011). "Decomposing a planar graph with girth at least 8 into a forest and a matching". Discrete Mathematics. 311 (10–11): 844–849. doi:10.1016/j.disc.2011.01.019.
• Montassier, Mickaël; Ossona de Mendez, Patrice; André, Raspaud; Zhu, Xuding (2012). "Decomposing a graph into forests". Journal of Combinatorial Theory, Series B. 102 (1): 38–52. doi:10.1016/j.jctb.2011.04.001.
| Wikipedia |
(g,K)-module
In mathematics, more specifically in the representation theory of reductive Lie groups, a $({\mathfrak {g}},K)$-module is an algebraic object, first introduced by Harish-Chandra,[1] used to deal with continuous infinite-dimensional representations using algebraic techniques. Harish-Chandra showed that the study of irreducible unitary representations of a real reductive Lie group, G, could be reduced to the study of irreducible $({\mathfrak {g}},K)$-modules, where ${\mathfrak {g}}$ is the Lie algebra of G and K is a maximal compact subgroup of G.[2]
Definition
Let G be a real Lie group. Let ${\mathfrak {g}}$ be its Lie algebra, and K a maximal compact subgroup with Lie algebra ${\mathfrak {k}}$. A $({\mathfrak {g}},K)$-module is defined as follows:[3] it is a vector space V that is both a Lie algebra representation of ${\mathfrak {g}}$ and a group representation of K (without regard to the topology of K) satisfying the following three conditions
1. for any v ∈ V, k ∈ K, and X ∈ ${\mathfrak {g}}$
$k\cdot (X\cdot v)=(\operatorname {Ad} (k)X)\cdot (k\cdot v)$
2. for any v ∈ V, Kv spans a finite-dimensional subspace of V on which the action of K is continuous
3. for any v ∈ V and Y ∈ ${\mathfrak {k}}$
$\left.\left({\frac {d}{dt}}\exp(tY)\cdot v\right)\right|_{t=0}=Y\cdot v.$
In the above, the dot, $\cdot $, denotes both the action of ${\mathfrak {g}}$ on V and that of K. The notation Ad(k) denotes the adjoint action of G on ${\mathfrak {g}}$, and Kv is the set of vectors $k\cdot v$ as k varies over all of K.
The first condition can be understood as follows: if G is the general linear group GL(n, R), then ${\mathfrak {g}}$ is the algebra of all n by n matrices, and the adjoint action of k on X is kXk−1; condition 1 can then be read as
$kXv=kXk^{-1}kv=\left(kXk^{-1}\right)kv.$
In other words, it is a compatibility requirement among the actions of K on V, ${\mathfrak {g}}$ on V, and K on ${\mathfrak {g}}$. The third condition is also a compatibility condition, this time between the action of ${\mathfrak {k}}$ on V viewed as a sub-Lie algebra of ${\mathfrak {g}}$ and its action viewed as the differential of the action of K on V.
Notes
1. Page 73 of Wallach 1988
2. Page 12 of Doran & Varadarajan 2000
3. This is James Lepowsky's more general definition, as given in section 3.3.1 of Wallach 1988
References
• Doran, Robert S.; Varadarajan, V. S., eds. (2000), The mathematical legacy of Harish-Chandra, Proceedings of Symposia in Pure Mathematics, vol. 68, AMS, ISBN 978-0-8218-1197-9, MR 1767886
• Wallach, Nolan R. (1988), Real reductive groups I, Pure and Applied Mathematics, vol. 132, Academic Press, ISBN 978-0-12-732960-4, MR 0929683
| Wikipedia |
Riffle shuffle permutation
In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of $n$ items that can be obtained by a single riffle shuffle, in which a sorted deck of $n$ cards is cut into two packets and then the two packets are interleaved (e.g. by moving cards one at a time from the bottom of one or the other of the packets to the top of the sorted deck). Beginning with an ordered set (1 rising sequence), mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences.[1] The permutations with 1 rising sequence are the identity permutations.
As a special case of this, a $(p,q)$-shuffle, for numbers $p$ and $q$ with $p+q=n$, is a riffle in which the first packet has $p$ cards and the second packet has $q$ cards.[2]
Combinatorial enumeration
Since a $(p,q)$-shuffle is completely determined by how its first $p$ elements are mapped, the number of $(p,q)$-shuffles is
${\binom {p+q}{p}}.$
However, the number of distinct riffles is not quite the sum of this formula over all choices of $p$ and $q$ adding to $n$ (which would be $2^{n}$), because the identity permutation can be represented in multiple ways as a $(p,q)$-shuffle for different values of $p$ and $q$. Instead, the number of distinct riffle shuffle permutations of a deck of $n$ cards, for $n=1,2,3,\dots $, is
1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, ... (sequence A000325 in the OEIS)
More generally, the formula for this number is $2^{n}-n$; for instance, there are 4503599627370444 riffle shuffle permutations of a 52-card deck.
The number of permutations that are both a riffle shuffle permutation and the inverse permutation of a riffle shuffle is[3]
${\binom {n+1}{3}}+1.$
For $n=1,2,3,\dots $, this is
1, 2, 5, 11, 21, 36, 57, 85, 121, 166, 221, ... (sequence A050407 in the OEIS)
and for $n=52$ there are exactly 23427 invertible shuffles.
Random distribution
The Gilbert–Shannon–Reeds model describes a random probability distribution on riffle shuffles that is a good match for observed human shuffles.[4] In this model, the identity permutation has probability $(n+1)/2^{n}$ of being generated, and all other riffle permutations have equal probability $1/2^{n}$ of being generated. Based on their analysis of this model, mathematicians have recommended that a deck of 52 cards be given seven riffles in order to thoroughly randomize it.[5]
Permutation patterns
A pattern in a permutation is a smaller permutation formed from a subsequence of some $k$ values in the permutation by reducing these values to the range from 1 to $k$ while preserving their order. Several important families of permutations can be characterized by a finite set of forbidden patterns, and this is true also of the riffle shuffle permutations: they are exactly the permutations that do not have 321, 2143, and 2413 as patterns.[3] Thus, for instance, they are a subclass of the vexillary permutations, which have 2143 as their only minimal forbidden pattern.[6]
Perfect shuffles
A perfect shuffle is a riffle in which the deck is split into two equal-sized packets, and in which the interleaving between these two packets strictly alternates between the two. There are two types of perfect shuffle, an in shuffle and an out shuffle, both of which can be performed consistently by some well-trained people. When a deck is repeatedly shuffled using these permutations, it remains much less random than with typical riffle shuffles, and it will return to its initial state after only a small number of perfect shuffles. In particular, a deck of 52 playing cards will be returned to its original ordering after 52 in shuffles or 8 out shuffles. This fact forms the basis of several magic tricks.[7]
Algebra
Riffle shuffles may be used to define the shuffle algebra. This is a Hopf algebra where the basis is a set of words, and the product is the shuffle product denoted by the sha symbol ш, the sum of all riffle shuffles of two words.
In exterior algebra, the wedge product of a $p$-form and a $q$-form can be defined as a sum over $(p,q)$-shuffles.[2]
See also
• Gilbreath permutations, the permutations formed by reversing one of the two packets of cards before riffling them
References
1. Aldous, David; Diaconis, Persi (1986), "Shuffling cards and stopping times" (PDF), The American Mathematical Monthly, 93 (5): 333–348, doi:10.2307/2323590, JSTOR 2323590, MR 0841111
2. Weibel, Charles (1994). An Introduction to Homological Algebra, p. 181. Cambridge University Press, Cambridge.
3. Atkinson, M. D. (1999), "Restricted permutations", Discrete Mathematics, 195 (1–3): 27–38, doi:10.1016/S0012-365X(98)00162-9, MR 1663866.
4. Diaconis, Persi (1988), Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 11, Hayward, CA: Institute of Mathematical Statistics, ISBN 0-940600-14-5, MR 0964069.
5. Kolata, Gina (January 9, 1990), "In Shuffling Cards, 7 Is Winning Number", New York Times.
6. Claesson, Anders (2004), Permutation patterns, continued fractions, and a group determined by an ordered set, Ph.D. thesis, Department of Mathematics, Chalmers University of Technology, CiteSeerX 10.1.1.103.2001.
7. Diaconis, Persi; Graham, R. L.; Kantor, William M. (1983), "The mathematics of perfect shuffles", Advances in Applied Mathematics, 4 (2): 175–196, CiteSeerX 10.1.1.77.7769, doi:10.1016/0196-8858(83)90009-X, MR 0700845.
| Wikipedia |
Addition
Addition (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.[2] The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 plus 2 is equal to 5").
Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
Addition has several important properties. It is commutative, meaning that the order of the operands does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 1 is the same as counting (see Successor function). Addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
Performing addition is one of the simplest numerical tasks to do. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months, and even some members of other animal species. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.
Arithmetic operations
Addition (+)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{sum}}$
Subtraction (−)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{difference}}$
Multiplication (×)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{product}}$
Division (÷)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,$ $\scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.$
Exponentiation (^)
$\scriptstyle {\text{base}}^{\text{exponent}}\,=\,$ $\scriptstyle {\text{power}}$
nth root (√)
$\scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,$ $\scriptstyle {\text{root}}$
Logarithm (log)
$\scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,$ $\scriptstyle {\text{logarithm}}$
Notation and terminology
Addition is written using the plus sign "+" between the terms;[3] that is, in infix notation. The result is expressed with an equals sign. For example,
$1+2=3$ ("one plus two equals three")
$5+4+2=11$ (see "associativity" below)
$3+3+3+3=12$ (see "multiplication" below)
There are also situations where addition is "understood", even though no symbol appears:
• A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.[4] For example,
$3{\frac {1}{2}}=3+{\frac {1}{2}}=3.5.$
This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.[5]
The sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example,
$\sum _{k=1}^{5}k^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=55.$
Terms
The numbers or the objects to be added in general addition are collectively referred to as the terms,[6] the addends[7][8][9] or the summands;[10] this terminology carries over to the summation of multiple terms. This is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend.[7][8][9] In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the commutative property of addition, "augend" is rarely used, and both terms are generally called addends.[11]
All of the above terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb addere, which is in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to give"; thus to add is to give to.[11] Using the gerundive suffix -nd results in "addend", "thing to be added".[lower-alpha 1] Likewise from augere "to increase", one gets "augend", "thing to be increased".
"Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.[13] Addere and summare date back at least to Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.[14]
The plus sign "+" (Unicode:U+002B; ASCII: +) is an abbreviation of the Latin word et, meaning "and".[15] It appears in mathematical works dating back to at least 1489.[16]
Interpretations
Addition is used to model many physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.
Combining sets
Possibly the most basic interpretation of addition lies in combining sets:
• When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.
This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see § Natural numbers below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.[17]
One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods.[18] Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.
Extending a length
A second interpretation of addition comes from extending an initial length by a given length:
• When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.[19]
The sum a + b can be interpreted as a binary operation that combines a and b, in an algebraic sense, or it can be interpreted as the addition of b more units to a. Under the latter interpretation, the parts of a sum a + b play asymmetric roles, and the operation a + b is viewed as applying the unary operation +b to a.[20] Instead of calling both a and b addends, it is more appropriate to call a the augend in this case, since a plays a passive role. The unary view is also useful when discussing subtraction, because each unary addition operation has an inverse unary subtraction operation, and vice versa.
Properties
Commutativity
Addition is commutative, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if a and b are any two numbers, then
a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other binary operations are commutative, such as multiplication, but many others are not, such as subtraction and division.
Associativity
Addition is associative, which means that when three or more numbers are added together, the order of operations does not change the result.
As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)? Given that addition is associative, the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that (a + b) + c = a + (b + c). For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3).
When addition is used together with other operations, the order of operations becomes important. In the standard order of operations, addition is a lower priority than exponentiation, nth roots, multiplication and division, but is given equal priority to subtraction.[21]
Identity element
Adding zero to any number, does not change the number; this means that zero is the identity element for addition, and is also known as the additive identity. In symbols, for every a, one has
a + 0 = 0 + a = a.
This law was first identified in Brahmagupta's Brahmasphutasiddhanta in 628 AD, although he wrote it as three separate laws, depending on whether a is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later Indian mathematicians refined the concept; around the year 830, Mahavira wrote, "zero becomes the same as what is added to it", corresponding to the unary statement 0 + a = a. In the 12th century, Bhaskara wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement a + 0 = a.[22]
Successor
Within the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a.[23] For instance, 3 is the successor of 2 and 7 is the successor of 6. Because of this succession, the value of a + b can also be seen as the bth successor of a, making addition iterated succession. For example, 6 + 2 is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd successor of 6.
Units
To numerically add physical quantities with units, they must be expressed with common units .[24] For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.[25]
Performing addition
Innate ability
Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation: infants look longer at situations that are unexpected.[26] A seminal experiment by Karen Wynn in 1992 involving Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies.[27] Another 1992 experiment with older toddlers, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.[28]
Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaque and cottontop tamarin monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the Arabic numerals 0 through 4, one chimpanzee was able to compute the sum of two numerals without further training.[29] More recently, Asian elephants have demonstrated an ability to perform basic arithmetic.[30]
Childhood learning
Typically, children first master counting. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, five" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers.[31] Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, five." Eventually children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that 6 + 6 = 12 and then reason that 6 + 7 is one more, or 13.[32] Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.[33]
Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.[34] However, throughout the world, addition is taught by the end of the first year of elementary school.[35]
Table
Children are often presented with the addition table of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.
+ 0123456789
0 0123456789
1 12345678910
2 234567891011
3 3456789101112
4 45678910111213
5 567891011121314
6 6789101112131415
7 78910111213141516
8 891011121314151617
9 9101112131415161718
Decimal system
The prerequisite to addition in the decimal system is the fluent recall or derivation of the 100 single-digit "addition facts". One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most people, more efficient:[36]
• Commutative property: Mentioned above, using the pattern a + b = b + a reduces the number of "addition facts" from 100 to 55.
• One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition.[36]
• Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; word problems may help rationalize the "exception" of zero.[36]
• Doubles: Adding a number to itself is related to counting by two and to multiplication. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.[36]
• Near-doubles: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact 6 + 6 = 12 by adding one more, or from 7 + 7 = 14 but subtracting one.[36]
• Five and ten: Sums of the form 5 + x and 10 + x are usually memorized early and can be used for deriving other facts. For example, 6 + 7 = 13 can be derived from 5 + 7 = 12 by adding one more.[36]
• Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14.[36]
As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.[33]
Carry
Main article: Carry (arithmetic)
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is "carried" into the next column. For example, in the addition 27 + 59
¹
27
+ 59
————
86
7 + 9 = 16, and the digit 1 is the carry.[lower-alpha 2] An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods.
Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum.[37] This decision was criticized,[38] which is why some states and counties did not support this experiment.
Decimal fractions
Decimal fractions can be added by a simple modification of the above process.[39] One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.
As an example, 45.1 + 4.34 can be solved as follows:
4 5 . 1 0
+ 0 4 . 3 4
————————————
4 9 . 4 4
Scientific notation
Main article: Scientific notation § Basic operations
In scientific notation, numbers are written in the form $x=a\times 10^{b}$, where $a$ is the significand and $10^{b}$ is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.
For example:
$2.34\times 10^{-5}+5.67\times 10^{-6}=2.34\times 10^{-5}+0.567\times 10^{-5}=2.907\times 10^{-5}$
Non-decimal
Main article: Binary addition
Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary.[40] Adding two single-digit binary numbers is relatively simple, using a form of carrying:
0 + 0 → 0
0 + 1 → 1
1 + 0 → 1
1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21))
Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101))
7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101))
This is known as carrying.[41] When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
1 1 1 1 1 (carried digits)
0 1 1 0 1
+ 1 0 1 1 1
—————————————
1 0 0 1 0 0 = 36
In this example, two numerals are being added together: 011012 (1310) and 101112 (2310). The top row shows the carry bits used. Starting in the rightmost column, 1 + 1 = 102. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 1 + 0 + 1 = 102 again; the 1 is carried, and 0 is written at the bottom. The third column: 1 + 1 + 1 = 112. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 1001002 (3610).
Computers
Analog computers work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an averaging lever. If the addends are the rotation speeds of two shafts, they can be added with a differential. A hydraulic adder can add the pressures in two chambers by exploiting Newton's second law to balance forces on an assembly of pistons. The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but a better design exploits an operational amplifier.[42]
Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the carry mechanism, is an important limitation to overall performance.
The abacus, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer.[43]
Blaise Pascal invented the mechanical calculator in 1642;[44] it was the first operational adding machine. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century[45] and the earliest automatic, digital computer. Pascal's calculator was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the Pascal's calculator's complement, which required as many steps as an addition. Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically.
Adders execute integer addition in electronic digital computers, usually using binary arithmetic. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing 999 + 1, but one bypasses the group of 9s and skips to the answer.[46]
In practice, computational addition may be achieved via XOR and AND bitwise logical operations in conjunction with bitshift operations as shown in the pseudocode below. Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of full adder circuits which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all floating-point operations as well as such basic tasks as address generation during memory access and fetching instructions during branching. To increase speed, modern designs calculate digits in parallel; these schemes go by such names as carry select, carry lookahead, and the Ling pseudocarry. Many implementations are, in fact, hybrids of these last three designs.[47][48] Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both numbers vanish.[49] In modern times, the ADD instruction of a microprocessor often replaces the augend with the sum but preserves the addend.[50] In a high-level programming language, evaluating a + b does not change either a or b; if the goal is to replace a with the sum this must be explicitly requested, typically with the statement a = a + b. Some languages such as C or C++ allow this to be abbreviated as a += b.
// Iterative algorithm
int add(int x, int y) {
int carry = 0;
while (y != 0) {
carry = AND(x, y); // Logical AND
x = XOR(x, y); // Logical XOR
y = carry << 1; // left bitshift carry by one
}
return x;
}
// Recursive algorithm
int add(int x, int y) {
return x if (y == 0) else add(XOR(x, y), AND(x, y) << 1);
}
On a computer, if the result of an addition is too large to store, an arithmetic overflow occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of program errors. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests.[51] The Year 2000 problem was a series of bugs where overflow errors occurred due to use of a 2-digit format for years.[52]
Addition of numbers
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the natural numbers. In set theory, addition is then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers.[53] (In mathematics education,[54] positive fractions are added before negative numbers are even considered; this is also the historical route.[55])
Natural numbers
Further information: Natural number
There are two popular ways to define the sum of two natural numbers a and b. If one defines natural numbers to be the cardinalities of finite sets, (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:
• Let N(S) be the cardinality of a set S. Take two disjoint sets A and B, with N(A) = a and N(B) = b. Then a + b is defined as $N(A\cup B)$.[56]
Here, A ∪ B is the union of A and B. An alternate version of this definition allows A and B to possibly overlap and then takes their disjoint union, a mechanism that allows common elements to be separated out and therefore counted twice.
The other popular definition is recursive:
• Let n+ be the successor of n, that is the number following n in the natural numbers, so 0+=1, 1+=2. Define a + 0 = a. Define the general sum recursively by a + (b+) = (a + b)+. Hence 1 + 1 = 1 + 0+ = (1 + 0)+ = 1+ = 2.[57]
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the recursion theorem on the partially ordered set N2.[58] On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers a to be temporarily "fixed", applies recursion on b to define a function "a +", and pastes these unary operations for all a together to form the full binary operation.[59]
This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades.[60] He proved the associative and commutative properties, among others, through mathematical induction.
Integers
Further information: Integer
The simplest conception of an integer is that it consists of an absolute value (which is a natural number) and a sign (generally either positive or negative). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:
• For an integer n, let |n| be its absolute value. Let a and b be integers. If either a or b is zero, treat it as an identity. If a and b are both positive, define a + b = |a| + |b|. If a and b are both negative, define a + b = −(|a| + |b|). If a and b have different signs, define a + b to be the difference between |a| and |b|, with the sign of the term whose absolute value is larger.[61] As an example, −6 + 4 = −2; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.
Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such differences, a – b and c – d are equal if and only if a + d = b + c. So, one can define formally the integers as the equivalence classes of ordered pairs of natural numbers under the equivalence relation
(a, b) ~ (c, d) if and only if a + d = b + c.
The equivalence class of (a, b) contains either (a – b, 0) if a ≥ b, or (0, b – a) otherwise. If n is a natural number, one can denote +n the equivalence class of (n, 0), and by –n the equivalence class of (0, n). This allows identifying the natural number n with the equivalence class +n.
Addition of ordered pairs is done component-wise:
$(a,b)+(c,d)=(a+c,b+d).$
A straightforward computation shows that the equivalence class of the result depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers.[62] Another straightforward computation shows that this addition is the same as the above case definition.
This way of defining integers as equivalence classes of pairs of natural numbers, can be used to embed into a group any commutative semigroup with cancellation property. Here, the semigroup is formed by the natural numbers and the group is the additive group of integers. The rational numbers are constructed similarly, by taking as semigroup the nonzero integers with multiplication.
This construction has been also generalized under the name of Grothendieck group to the case of any commutative semigroup. Without the cancellation property the semigroup homomorphism from the semigroup into the group may be non-injective. Originally, the Grothendieck group was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an abelian category, with the direct sum as semigroup operation.
Rational numbers (fractions)
Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler definition involves only integer addition and multiplication:
• Define ${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.$
As an example, the sum ${\frac {3}{4}}+{\frac {1}{8}}={\frac {3\times 8+4\times 1}{4\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}$.
Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same: ${\frac {a}{c}}+{\frac {b}{c}}={\frac {a+b}{c}}$, so ${\frac {1}{4}}+{\frac {2}{4}}={\frac {1+2}{4}}={\frac {3}{4}}$.[63]
The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic.[64] For a more rigorous and general discussion, see field of fractions.
Real numbers
Further information: Construction of the real numbers
A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a Dedekind cut of rationals: a non-empty set of rationals that is closed downward and has no greatest element. The sum of real numbers a and b is defined element by element:
• Define $a+b=\{q+r\mid q\in a,r\in b\}.$[65]
This definition was first published, in a slightly modified form, by Richard Dedekind in 1872.[66] The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.[67]
Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers.[68] Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a Cauchy sequence of rationals, lim an. Addition is defined term by term:
• Define $\lim _{n}a_{n}+\lim _{n}b_{n}=\lim _{n}(a_{n}+b_{n}).$[69]
This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different.[70] One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions.[71]
Complex numbers
Complex numbers are added by adding the real and imaginary parts of the summands.[72][73] That is to say:
$(a+bi)+(c+di)=(a+c)+(b+d)i.$
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.
Generalizations
There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in set theory and category theory.
Vectors
Main article: Vector addition
In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as a vector from the origin in the Euclidean plane to the point (a,b) in the plane. The sum of two vectors is obtained by adding their individual coordinates:
$(a,b)+(c,d)=(a+c,b+d).$
This addition operation is central to classical mechanics, in which velocities, accelerations and forces are all represented by vectors.[74]
Matrices
Main article: Matrix addition
Matrix addition is defined for two matrices of the same dimensions. The sum of two m × n (pronounced "m by n") matrices A and B, denoted by A + B, is again an m × n matrix computed by adding corresponding elements:[75][76]
${\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}}$
For example:
${\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}$
Modular arithmetic
Main article: Modular arithmetic
In modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function. A similar "wrap around" operation arises in geometry, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori.
General theory
The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups.
Set theory and category theory
A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most addition operations, addition of ordinal numbers is not commutative.[77] Addition of cardinal numbers, however, is a commutative operation closely related to the disjoint union operation.
In category theory, disjoint union is seen as a particular case of the coproduct operation,[78] and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as direct sum and wedge sum, are named to evoke their connection with addition.
Related operations
Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in elementary arithmetic.
Arithmetic
Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x and subtracting x are inverse functions.
Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be described as a set that is closed under subtraction.[79]
Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number.
In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:[80]
$e^{a+b}=e^{a}e^{b}.$
This identity allows multiplication to be carried out by consulting a table of logarithms and computing addition by hand; it also enables multiplication on a slide rule. The formula is still a good first-order approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated Lie algebra.[81]
There are even more generalizations of multiplication than addition.[82] In general, multiplication operations always distribute over addition; this requirement is formalized in the definition of a ring. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; by expanding the product (1 + 1)(a + b) in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.[83]
Division is an arithmetic operation remotely related to addition. Since a/b = a(b−1), division is right distributive over addition: (a + b) / c = a/c + b/c.[84] However, division is not left distributive over addition; 1 / (2 + 2) is not the same as 1/2 + 1/2.
Ordering
The maximum operation "max (a, b)" is a binary operation similar to addition. In fact, if two nonnegative numbers a and b are of different orders of magnitude, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating Taylor series. However, it presents a perpetual difficulty in numerical analysis, essentially since "max" is not invertible. If b is much greater than a, then a straightforward calculation of (a + b) − b can accumulate an unacceptable round-off error, perhaps even returning zero. See also Loss of significance.
The approximation becomes exact in a kind of infinite limit; if either a or b is an infinite cardinal number, their cardinal sum is exactly equal to the greater of the two.[86] Accordingly, there is no subtraction operation for infinite cardinals.[87]
Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
$a+\max(b,c)=\max(a+b,a+c).$
For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is negative infinity.[88] Some authors prefer to replace addition with minimization; then the additive identity is positive infinity.[89]
Tying these observations together, tropical addition is approximately related to regular addition through the logarithm:
$\log(a+b)\approx \max(\log a,\log b),$
which becomes more accurate as the base of the logarithm increases.[90] The approximation can be made exact by extracting a constant h, named by analogy with Planck's constant from quantum mechanics,[91] and taking the "classical limit" as h tends to zero:
$\max(a,b)=\lim _{h\to 0}h\log(e^{a/h}+e^{b/h}).$
In this sense, the maximum operation is a dequantized version of addition.[92]
Other ways to add
Incrementation, also known as the successor operation, is the addition of 1 to a number.
Summation describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero.[93] An infinite summation is a delicate procedure known as a series.[94]
Counting a finite set is equivalent to summing 1 over the set.
Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.
Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.[95]
Convolution is used to add two independent random variables defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication.[96] In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.
See also
• Lunar arithmetic
• Mental arithmetic
• Parallel addition (mathematics)
• Verbal arithmetic (also known as cryptarithms), puzzles involving addition
Notes
1. "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".
2. Some authors think that "carry" may be inappropriate for education; Van de Walle (p. 211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.
Footnotes
1. From Enderton (p. 138): "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks."
2. Lewis, Rhys (1974). "Arithmetic". First-Year Technician Mathematics. Palgrave, London: The MacMillan Press Ltd. p. 1. doi:10.1007/978-1-349-02405-6_1. ISBN 978-1-349-02405-6.
3. "Addition". www.mathsisfun.com. Retrieved 2020-08-25.
4. Devine et al. p. 263
5. Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press, 2014. p. 161
6. Department of the Army (1961) Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics . Section 5.1
7. Shmerko, V.P.; Yanushkevich [Ânuškevič], Svetlana N. [Svitlana N.]; Lyshevski, S.E. (2009). Computer arithmetics for nanoelectronics. CRC Press. p. 80.
8. Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, NY: John Wiley & Sons. ISBN 0-471-76180-X. and Schmid, Hermann (1983) [1974]. Decimal Computation (reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company. ISBN 978-0-89874-318-0.
9. Weisstein, Eric W. "Addition". mathworld.wolfram.com. Retrieved 2020-08-25.
10. Hosch, W.L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38
11. Schwartzman p. 19
12. Karpinski pp. 56–57, reproduced on p. 104
13. Schwartzman (p. 212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (p. 103) writes that Leonard of Pisa "introduces the novelty of writing the sum above the addends"; it is unclear whether Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe.
14. Karpinski pp. 150–153
15. Cajori, Florian (1928). "Origin and meanings of the signs + and -". A History of Mathematical Notations, Vol. 1. The Open Court Company, Publishers.
16. "plus". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
17. See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".
18. Adding it up (p. 73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature."
19. Mosley, F (2001). Using number lines with 5–8 year olds. Nelson Thornes. p. 8
20. Li, Y., & Lappan, G. (2014). Mathematics curriculum in school education. Springer. p. 204
21. Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "2.4.1.1.". In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B.G. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120. ISBN 978-3-87144-492-0.
22. Kaplan pp. 69–71
23. Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7
24. R. Fierro (2012) Mathematics for Elementary School Teachers. Cengage Learning. Sec 2.3
25. Moebs, William; et al. (2022). "1.4 Dimensional Analysis". University Physics Volume 1. OpenStax. ISBN 978-1-947172-20-3.
26. Wynn p. 5
27. Wynn p. 15
28. Wynn p. 17
29. Wynn p. 19
30. Randerson, James (21 August 2008). "Elephants have a head for figures". The Guardian. Archived from the original on 2 April 2015. Retrieved 29 March 2015.
31. F. Smith p. 130
32. Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson, Susan (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. ISBN 978-0-325-00137-1.
33. Henry, Valerie J.; Brown, Richard S. (2008). "First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard". Journal for Research in Mathematics Education. 39 (2): 153–183. doi:10.2307/30034895. JSTOR 30034895.
34. Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1-8. Chicago
35. Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". American Educator, 26(2), 1–18.
36. Fosnot and Dolk p. 99
37. "Vertical addition and subtraction strategy". primarylearning.org. Retrieved April 20, 2022.
38. "Reviews of TERC: Investigations in Number, Data, and Space". nychold.com. Retrieved April 20, 2022.
39. Rebecca Wingard-Nelson (2014) Decimals and Fractions: It's Easy Enslow Publishers, Inc.
40. Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155
41. P.E. Bates Bothman (1837) The common school arithmetic. Henry Benton. p. 31
42. Truitt and Rogers pp. 1;44–49 and pp. 2;77–78
43. Ifrah, Georges (2001). The Universal History of Computing: From the Abacus to the Quantum Computer. New York: John Wiley & Sons, Inc. ISBN 978-0-471-39671-0. p. 11
44. Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963)
45. See Competing designs in Pascal's calculator article
46. Flynn and Overman pp. 2, 8
47. Flynn and Overman pp. 1–9
48. Yeo, Sang-Soo, et al., eds. Algorithms and Architectures for Parallel Processing: 10th International Conference, ICA3PP 2010, Busan, Korea, May 21–23, 2010. Proceedings. Vol. 1. Springer, 2010. p. 194
49. Karpinski pp. 102–103
50. The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz and Hill p. 679; for ADD in 68k see p. 767.
51. Joshua Bloch, "Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken" Archived 2016-04-01 at the Wayback Machine. Official Google Research Blog, June 2, 2006.
52. Neumann, Peter G. (2 February 1987). "The Risks Digest Volume 4: Issue 45". The Risks Digest. 4 (45). Archived from the original on 2014-12-28. Retrieved 2015-03-30.
53. Enderton chapters 4 and 5, for example, follow this development.
54. According to a survey of the nations with highest TIMSS mathematics test scores; see Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(2), p. 4.
55. Baez (p. 37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"
56. Begle p. 49, Johnson p. 120, Devine et al. p. 75
57. Enderton p. 79
58. For a version that applies to any poset with the descending chain condition, see Bergman p. 100.
59. Enderton (p. 79) observes, "But we want one binary operation +, not all these little one-place functions."
60. Ferreirós p. 223
61. K. Smith p. 234, Sparks and Rees p. 66
62. Enderton p. 92
63. Schyrlet Cameron, and Carolyn Craig (2013)Adding and Subtracting Fractions, Grades 5–8 Mark Twain, Inc.
64. The verifications are carried out in Enderton p. 104 and sketched for a general field of fractions over a commutative ring in Dummit and Foote p. 263.
65. Enderton p. 114
66. Ferreirós p. 135; see section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Machine.
67. The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton p. 117 for details.
68. Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of." Lecture Notes in Computer Science (1995).
69. Textbook constructions are usually not so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, drawn-out development of addition with Cauchy sequences.
70. Ferreirós p. 128
71. Burrill p. 140
72. Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 978-0-387-90328-6
73. Joshi, Kapil D (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6
74. Gbur, p. 1
75. Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.
76. Riley, K.F.; Hobson, M.P.; Bence, S.J. (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
77. Cheng, pp. 124–132
78. Riehl, p. 100
79. The set still must be nonempty. Dummit and Foote (p. 48) discuss this criterion written multiplicatively.
80. Rudin p. 178
81. Lee p. 526, Proposition 20.9
82. Linderholm (p. 49) observes, "By multiplication, properly speaking, a mathematician may mean practically anything. By addition he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."
83. Dummit and Foote p. 224. For this argument to work, one still must assume that addition is a group operation and that multiplication has an identity.
84. For an example of left and right distributivity, see Loday, especially p. 15.
85. Compare Viro Figure 1 (p. 2)
86. Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the Axiom of Choice.
87. Enderton p. 164
88. Mikhalkin p. 1
89. Akian et al. p. 4
90. Mikhalkin p. 2
91. Litvinov et al. p. 3
92. Viro p. 4
93. Martin p. 49
94. Stewart p. 8
95. Rieffel and Polak, p. 16
96. Gbur, p. 300
References
History
• Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.
• Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
• Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. ISBN 978-0-88385-511-9.
• Williams, Michael (1985). A History of Computing Technology. Prentice-Hall. ISBN 978-0-13-389917-7.
Elementary mathematics
• Sparks, F.; Rees C. (1979). A Survey of Basic Mathematics. McGraw-Hill. ISBN 978-0-07-059902-4.
Education
• Begle, Edward (1975). The Mathematics of the Elementary School. McGraw-Hill. ISBN 978-0-07-004325-1.
• California State Board of Education mathematics content standards Adopted December 1997, accessed December 2005.
• Devine, D.; Olson, J.; Olson, M. (1991). Elementary Mathematics for Teachers (2e ed.). Wiley. ISBN 978-0-471-85947-5.
• National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822. ISBN 978-0-309-06995-3.
• Van de Walle, John (2004). Elementary and Middle School Mathematics: Teaching developmentally (5e ed.). Pearson. ISBN 978-0-205-38689-5.
Cognitive science
• Fosnot, Catherine T.; Dolk, Maarten (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction. Heinemann. ISBN 978-0-325-00353-5.
• Wynn, Karen (1998). "Numerical competence in infants". The Development of Mathematical Skills. Taylor & Francis. ISBN 0-86377-816-X.
Mathematical exposition
• Bogomolny, Alexander (1996). "Addition". Interactive Mathematics Miscellany and Puzzles (cut-the-knot.org). Archived from the original on April 26, 2006. Retrieved 3 February 2006.
• Cheng, Eugenia (2017). Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Basic Books. ISBN 978-1-541-64413-7.
• Dunham, William (1994). The Mathematical Universe. Wiley. ISBN 978-0-471-53656-7.
• Johnson, Paul (1975). From Sticks and Stones: Personal Adventures in Mathematics. Science Research Associates. ISBN 978-0-574-19115-1.
• Linderholm, Carl (1971). Mathematics Made Difficult. Wolfe. ISBN 978-0-7234-0415-6.
• Smith, Frank (2002). The Glass Wall: Why Mathematics Can Seem Difficult. Teachers College Press. ISBN 978-0-8077-4242-6.
• Smith, Karl (1980). The Nature of Modern Mathematics (3rd ed.). Wadsworth. ISBN 978-0-8185-0352-8.
Advanced mathematics
• Bergman, George (2005). An Invitation to General Algebra and Universal Constructions (2.3 ed.). General Printing. ISBN 978-0-9655211-4-7.
• Burrill, Claude (1967). Foundations of Real Numbers. McGraw-Hill. LCC QA248.B95.
• Dummit, D.; Foote, R. (1999). Abstract Algebra (2 ed.). Wiley. ISBN 978-0-471-36857-1.
• Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.
• Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
• Lee, John (2003). Introduction to Smooth Manifolds. Springer. ISBN 978-0-387-95448-6.
• Martin, John (2003). Introduction to Languages and the Theory of Computation (3 ed.). McGraw-Hill. ISBN 978-0-07-232200-2.
• Riehl, Emily (2016). Category Theory in Context. Dover. ISBN 978-0-486-80903-8.
• Rudin, Walter (1976). Principles of Mathematical Analysis (3 ed.). McGraw-Hill. ISBN 978-0-07-054235-8.
• Stewart, James (1999). Calculus: Early Transcendentals (4 ed.). Brooks/Cole. ISBN 978-0-534-36298-0.
Mathematical research
• Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem". INRIA Reports. arXiv:math.SP/0402090. Bibcode:2004math......2090A.
• Baez, J.; Dolan, J. (2001). Mathematics Unlimited – 2001 and Beyond. From Finite Sets to Feynman Diagrams. p. 29. arXiv:math.QA/0004133. ISBN 3-540-66913-2.
• Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999). Idempotent mathematics and interval analysis. Reliable Computing, Kluwer.
• Loday, Jean-Louis (2002). "Arithmetree". Journal of Algebra. 258: 275. arXiv:math/0112034. doi:10.1016/S0021-8693(02)00510-0.
• Mikhalkin, Grigory (2006). Sanz-Solé, Marta (ed.). Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Tropical Geometry and its Applications. Zürich: European Mathematical Society. pp. 827–852. arXiv:math.AG/0601041. ISBN 978-3-03719-022-7. Zbl 1103.14034.
• Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.). European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Paper. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. arXiv:math/0005163. Bibcode:2000math......5163V. ISBN 978-3-7643-6417-5. Zbl 1024.14026.
Computing
• Flynn, M.; Oberman, S. (2001). Advanced Computer Arithmetic Design. Wiley. ISBN 978-0-471-41209-0.
• Horowitz, P.; Hill, W. (2001). The Art of Electronics (2 ed.). Cambridge UP. ISBN 978-0-521-37095-0.
• Jackson, Albert (1960). Analog Computation. McGraw-Hill. LCC QA76.4 J3.
• Rieffel, Eleanor G.; Polak, Wolfgang H. (4 March 2011). Quantum Computing: A Gentle Introduction. MIT Press. ISBN 978-0-262-01506-6.
• Truitt, T.; Rogers, A. (1960). Basics of Analog Computers. John F. Rider. LCC QA76.4 T7.
• Marguin, Jean (1994). Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942 (in French). Hermann. ISBN 978-2-7056-6166-3.
• Taton, René (1963). Le Calcul Mécanique. Que Sais-Je ? n° 367 (in French). Presses universitaires de France. pp. 20–28.
Further reading
• Baroody, Arthur; Tiilikainen, Sirpa (2003). The Development of Arithmetic Concepts and Skills. Two perspectives on addition development. Routledge. p. 75. ISBN 0-8058-3155-X.
• Davison, David M.; Landau, Marsha S.; McCracken, Leah; Thompson, Linda (1999). Mathematics: Explorations & Applications (TE ed.). Prentice Hall. ISBN 978-0-13-435817-8.
• Bunt, Lucas N.H.; Jones, Phillip S.; Bedient, Jack D. (1976). The Historical roots of Elementary Mathematics. Prentice-Hall. ISBN 978-0-13-389015-0.
• Poonen, Bjorn (2010). "Addition". Girls' Angle Bulletin. 3 (3–5). ISSN 2151-5743.
• Weaver, J. Fred (1982). "Addition and Subtraction: A Cognitive Perspective". Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction. Taylor & Francis. p. 60. ISBN 0-89859-171-6.
Elementary arithmetic
+
Addition
(+)
−
Subtraction
(−)
×
Multiplication
(× or ·)
÷
Division
(÷ or ∕)
Hyperoperations
Primary
• Successor (0)
• Addition (1)
• Multiplication (2)
• Exponentiation (3)
• Tetration (4)
• Pentation (5)
Inverse for left argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Root extraction (3)
• Super-root (4)
Inverse for right argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Logarithm (3)
• Super-logarithm (4)
Related articles
• Ackermann function
• Conway chained arrow notation
• Grzegorczyk hierarchy
• Knuth's up-arrow notation
• Steinhaus–Moser notation
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| Wikipedia |
Plus and minus signs
The plus sign + and the minus sign − are mathematical symbols used to represent the notions of positive and negative, respectively. In addition, + represents the operation of addition, which results in a sum, while − represents subtraction, resulting in a difference.[1] Their use has been extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning "more" and "less", respectively.
For the ± symbol, see Plus-minus sign.
+ −
Plus and minus signs
In UnicodeU+002B + PLUS SIGN (+)
U+2212 − MINUS SIGN (−)
Different from
Different fromU+002D - HYPHEN-MINUS
U+2010 ‐ HYPHEN
(many) - Dash
Related
See alsoU+00B1 ± PLUS-MINUS SIGN
U+2213 ∓ MINUS-OR-PLUS SIGN
U+2052 ⁒ COMMERCIAL MINUS SIGN
History
Though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembled a pair of legs walking in the direction in which the text was written (Egyptian could be written either from right to left or left to right), with the reverse sign indicating subtraction:[2]
or
Nicole Oresme's manuscripts from the 14th century show what may be one of the earliest uses of + as a sign for plus.[3]
In early 15th century Europe, the letters "P" and "M" were generally used.[4][5] The symbols (P with overline, p̄, for più (more), i.e., plus, and M with overline, m̄, for meno (less), i.e., minus) appeared for the first time in Luca Pacioli's mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494.[6]
The + sign is a simplification of the Latin: et (comparable to the evolution of the ampersand &).[7] The − may be derived from a tilde written over ⟨m⟩ when used to indicate subtraction; or it may come from a shorthand version of the letter ⟨m⟩ itself.[8]
In his 1489 treatise, Johannes Widmann referred to the symbols − and + as minus and mer (Modern German mehr; "more"): "[...] was − ist das ist minus [...] und das + das ist mer das zu addirst"[9][10][11] They weren't used for addition and subtraction in the treatise, but were used to indicate surplus and deficit; usage in the modern sense is attested in a 1518 book by Henricus Grammateus.[12][13]
Robert Recorde, the designer of the equals sign, introduced plus and minus to Britain in 1557 in The Whetstone of Witte:[14] "There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made − and betokeneth lesse."
Plus sign
The plus sign, +, is a binary operator that indicates addition, as in 2 + 3 = 5. It can also serve as a unary operator that leaves its operand unchanged (+x means the same as x). This notation may be used when it is desired to emphasize the positiveness of a number, especially in contrast with the negative numbers (+5 versus −5).
The plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures, such as vector spaces and matrix rings, have some operation which is called, or is equivalent to, addition. It is though conventional to use the plus sign to only denote commutative operations.[15]
The symbol is also used in chemistry and physics. For more, see § Other uses.
Minus sign
The minus sign, −, has three main uses in mathematics:[16]
1. The subtraction operator: a binary operator to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the inverse of addition.[1]
2. The function whose value for any real or complex argument is the additive inverse of that argument. For example, if x = 3, then −x = −3, but if x = −3, then −x = +3. Similarly, −(−x) = x.
3. A prefix of a numeric constant. When it is placed immediately before an unsigned numeral, the combination names a negative number, the additive inverse of the positive number that the numeral would otherwise name. In this usage, '−5' names a number the same way 'semicircle' names a geometric figure, with the caveat that 'semi' does not have a separate use as a function name.
In many contexts, it does not matter whether the second or the third of these usages is intended: −5 is the same number. When it is important to distinguish them, a raised minus sign ¯ is sometimes used for negative constants, as in elementary education, the programming language APL, and some early graphing calculators.[lower-alpha 1]
All three uses can be referred to as "minus" in everyday speech, though the binary operator is sometimes read as "take away".[17] In American English nowadays, −5 (for example) is generally referred to as "negative five" though speakers born before 1950 often refer to it as "minus five". (Temperatures tend to follow the older usage; −5° is generally called "minus five degrees".)[18] Further, a few textbooks in the United States encourage −x to be read as "the opposite of x" or "the additive inverse of x"—to avoid giving the impression that −x is necessarily negative (since x itself may already be negative).[19]
In mathematics and most programming languages, the rules for the order of operations mean that −52 is equal to −25: Exponentiation binds more strongly than the unary minus, which binds more strongly than multiplication or division. However, in some programming languages (Microsoft Excel in particular), unary operators bind strongest, so in those cases −5^2 is 25, but 0−5^2 is −25.[20]
Similar to the plus sign, the minus sign is also used in chemistry and physics. For more, see § Other uses below.
Use in elementary education
Some elementary teachers use raised minus signs before numbers to disambiguate them from the operation of subtraction. The same convention is also used in some computer languages. For example, subtracting −5 from 3 might be read as "positive three take away negative 5", and be shown as
3 − −5 becomes 3 + 5 = 8,
or even as
+3 − −5 becomes +3 + +5 = +8.
Use as a qualifier
In grading systems (such as examination marks), the plus sign indicates a grade one level higher and the minus sign a grade lower. For example, B− ("B minus") is one grade lower than B. In some occasions, this is extended to two plus or minus signs (e.g., A++ being two grades higher than A).
Positive and negative are sometimes abbreviated as +ve and −ve.[21]
Mathematics
In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → +$\infty $ as x → 0+ but 1/x → −$\infty $ as x → 0−.
Blood
Blood types are often qualified with a plus or minus to indicate the presence or absence of the Rh factor. For example, A+ means type A blood with the Rh factor present, while B− means type B blood with the Rh factor absent.
Music
In music, augmented chords are symbolized with a plus sign, although this practice is not universal (as there are other methods for spelling those chords). For example, "C+" is read "C augmented chord". Sometimes the plus is written as a superscript.
Uses in computing
As well as the normal mathematical usage, plus and minus signs may be used for a number of other purposes in computing.
Plus and minus signs are often used in tree view on a computer screen—to show if a folder is collapsed or not.
In some programming languages, concatenation of strings is written "a" + "b", and results in "ab".
In most programming languages, subtraction and negation are indicated with the ASCII hyphen-minus character, -. In APL a raised minus sign (Unicode U+00AF) is used to denote a negative number, as in ¯3. While in J a negative number is denoted by an underscore, as in _5.
In C and some other computer programming languages, two plus signs indicate the increment operator and two minus signs a decrement; the position of the operator before or after the variable indicates whether the new or old value is read from it. For example, if x equals 6, then y = x++ increments x to 7 but sets y to 6, whereas y = ++x would set both x and y to 7. By extension, ++ is sometimes used in computing terminology to signify an improvement, as in the name of the language C++.
In regular expressions, + is often used to indicate "1 or more" in a pattern to be matched. For example, x+ means "one or more of the letter x".
There is no concept of negative zero in mathematics, but in computing −0 may have a separate representation from zero. In the IEEE floating-point standard, 1 / −0 is negative infinity ($-\infty $) whereas 1 / 0 is positive infinity ($\infty $).
+ is also used to denote added lines in diff output in the context format or the unified format.
Other uses
In physics, the use of plus and minus signs for different electrical charges was introduced by Georg Christoph Lichtenberg.
In chemistry, superscripted plus and minus signs are used to indicate an ion with a positive or negative charge of 1 (e.g., NH+
4
). If the charge is greater than 1, a number indicating the charge is written before the sign (as in SO2−
4
). The minus sign is also used, in place of an en dash, for a single covalent bond between two atoms as in the skeletal formula.
A plus sign prefixed to a telephone number is used to indicate the form used for International Direct Dialing.[22] Its precise usage varies by technology and national standards. In the International Phonetic Alphabet, subscripted plus and minus signs are used as diacritics to indicate advanced or retracted articulations of speech sounds.
The minus sign is also used as tone letter in the orthographies of Dan, Krumen, Karaboro, Mwan, Wan, Yaouré, Wè, Nyabwa and Godié.[23] The Unicode character used for the tone letter (U+02D7) is different from the mathematical minus sign.
The plus sign sometimes represents /ɨ/ in the orthography of Huichol.[24]
In the algebraic notation used to record games of chess, the plus sign + is used to denote a move that puts the opponent into check, while a double plus ++ is sometimes used to denote double check. Combinations of the plus and minus signs are used to evaluate a move (+/−, +/=, =/+, −/+).
In linguistics, a superscript plus + sometimes replaces the asterisk, which denotes unattested linguistic reconstruction.
In botanical names, a plus sign denotes graft-chimaera.
Character codes
- + −
hyphen-minus, plus, minus signs compared
ReadCharacterUnicodeASCIIin URLHTML notations
Plus+U+002B43dec, 2Bhex%2B+, +
Minus−U+2212%E2%88%92− − −
Hyphen-minus-U+002D45dec, 2Dhex%2D-
Small Hyphen-minus﹣U+FE63%EF%B9%A3﹣ ﹣
Full-width Plus＋U+FF0B%EF%BC%8B＋ ＋
Full-width Hyphen-minus－U+FF0D%EF%BC%8D－ －
The hyphen-minus sign, -, is the original ASCII version of the minus sign, which doubles as a hyphen. It is usually shorter in length than the plus sign and often at a different height to the plus-sign's cross bar. It can be used as a substitute for the true minus sign when the character set is limited to ASCII. Most programming languages and other computer readable languages do this, since ASCII is generally available as a subset of most character encodings, while U+2212 is a Unicode feature only. Also several other software programs usable for calculations do not accept the U+2212 minus. For example, pasting =3−2 into Excel or 3−2= into the Windows calculator won't work.
As the true minus is not available on most keyboard layouts, typographers sometimes use the very similar en dash, U+2013, to represent the minus sign although it is "not preferred" in mathematical typesetting.[25] Ways of producing the en dash are available on most computers; see Dash § Typing the characters.
Alternative minus signs
There is a commercial minus sign, ⁒, which is used in Germany and Scandinavia. The symbol ÷ is used to denote subtraction in Scandinavia.[26]
Alternative plus sign
See also: Up tack
A Jewish tradition that dates from at least the 19th century is to write plus using the symbol ﬩.[27] This practice was adopted into Israeli schools and is still commonplace today in elementary schools (including secular schools) but in fewer secondary schools.[28] It is also used occasionally in books by religious authors, but most books for adults use the international symbol +. The reason for this practice is that it avoids the writing of a symbol + that looks like a Christian cross.[27][28] Unicode has this symbol at position U+FB29 ﬩ HEBREW LETTER ALTERNATIVE PLUS SIGN.[29]
See also
• En dash, a dash that looks similar to the subtraction symbol but is used for different purposes
• Plus–minus sign ±
• Glossary of mathematical symbols
• ⊕ (disambiguation)
References and footnotes
1. at least the early Texas Instruments models, including the TI-81 and TI-82
1. Weisstein, Eric W. "Subtraction". mathworld.wolfram.com. Archived from the original on 2020-09-14. Retrieved 2020-08-26.
2. Karpinski, Louis C. (1917). "Algebraical Developments Among the Egyptians and Babylonians". The American Mathematical Monthly. 24 (6): 257–265. doi:10.2307/2973180. JSTOR 2973180. MR 1518824.
3. The birth of symbols – Zdena Lustigova, Faculty of Mathematics and Physics Charles University, Prague Archived 2013-07-08 at archive.today
4. Ley, Willy (April 1965). "Symbolically Speaking". For Your Information. Galaxy Science Fiction. pp. 57–67.
5. Stallings, Lynn (May 2000). "A brief history of algebraic notation". School Science and Mathematics. 100 (5): 230–235. doi:10.1111/j.1949-8594.2000.tb17262.x. Retrieved 13 April 2009.
6. Sangster, Alan; Stoner, Greg; McCarthy, Patricia (2008). "The market for Luca Pacioli's Summa Arithmetica" (PDF). Accounting Historians Journal. 35 (1): 111–134 [p. 115]. doi:10.2308/0148-4184.35.1.111. Archived (PDF) from the original on 2018-01-26. Retrieved 2012-04-29.
7. Cajori, Florian (1928). "Origin and meanings of the signs + and -". A History of Mathematical Notations, Vol. 1. The Open Court Company, Publishers.
8. Wright, D. Franklin; New, Bill D. (2000). Intermediate Algebra (4th ed.). Thomson Learning. p. 1. The minus sign or bar, — , is thought to be derived from the habit of early scribes of using a bar to represent the letter m
9. Widmann, Johannes (1489). "Behe[n]de vnd hubsche Rechenung auff allen kauffmanschafft". Leipzig : Konrad Kachelofen. p. 176. Archived from the original on 2022-05-03. Retrieved 2022-05-03.
10. Widmann, Johannes (1508). "Behend vnd hüpsch Rechnung vff allen Kauffmanschafften". Kolophon: Gedruck zů Pfhortzheim von Thoman Anßhelm. p. 122. Archived from the original on 2022-05-03. Retrieved 2022-05-03.
11. "plus". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
12. Smith, D.E. (1951). History of Mathematics. Vol. 1. Courier Dover Publications. pp. 258, 330. ISBN 0486204308.
13. "Earliest Uses of Symbols of Operation". Archived from the original on 2022-04-29. Retrieved 2022-05-03.
14. Cajori, Florian (2007), A History of Mathematical Notations, Cosimo, p. 164, ISBN 9781602066847.
15. Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 52. ISBN 0-201-52821-5.
16. Henri Picciotto (1990). The Algebra Lab. Creative Publications. p. 9. ISBN 978-0-88488-964-9.
17. "Subtraction". www.mathsisfun.com. Archived from the original on 2020-08-12. Retrieved 2020-08-26.
18. Schwartzman, Steven (1994). The words of mathematics. The Mathematical Association of America. p. 136. ISBN 9780883855119.
19. Wheeler, Ruric E. (2001). Modern Mathematics (11 ed.). p. 171.
20. "Microsoft Office Excel Calculation operators and precedence". Archived from the original on 2009-08-11. Retrieved 2009-07-29.
21. Castledine, George; Close, Ann (2009). Oxford Handbook of Adult Nursing. Oxford University Press. p. xvii. ISBN 9780191039676..
22. "Recommendation E.123: Notation for national and international telephone numbers, e-mail addresses and Web addresses". International Telecommunication Union. 2001. Archived from the original on 2021-05-05. Retrieved 2021-03-18.
23. Hartell, Rhonda L., ed. (1993), The Alphabets of Africa. Dakar: UNESCO and SIL.
24. Biglow, Brad Morris (2001). Ethno-Nationalist Politics and Cultural Preservation: Education and Bordered Identities Among the Wixaritari (Huichol) of Tateikita, Jalisco, Mexico (PDF) (PhD). University of Florida. p. 284. Archived (PDF) from the original on 2021-06-02. Retrieved 2021-05-29.
25. "The Unicode Standard, Version 13.0, Chapter 6.2" (PDF). 2020. General Punctuation § Dashes and Hyphens. Archived (PDF) from the original on 2021-01-22. Retrieved 2020-12-29.
26. "6. Writing Systems and Punctuation". The Unicode Standard: Version 10.0 – Core Specification (PDF). Unicode Consortium. June 2017. p. 280, Obelus. Archived (PDF) from the original on 2021-10-04. Retrieved 2022-04-11.
27. Kaufmann Kohler (1901–1906). "Cross". In Cyrus Adler; et al. (eds.). Jewish Encyclopedia. Archived from the original on 2017-01-06. Retrieved 2017-02-12.
28. Christian-Jewish Dialogue: Theological Foundations By Peter von der Osten-Sacken (1986 – Fortress Press) Archived 2023-04-08 at the Wayback Machine ISBN 0-8006-0771-6 "In Israel the plus sign used in mathematics is represented by a horizontal stroke with a vertical hook instead of the sign otherwise used all over the world, because the latter is reminiscent of a cross." (Page 96)
29. Unicode U+FB29 reference page Archived 2009-01-26 at the Wayback Machine This form of the plus sign is also used on the control buttons at individual seats on board the El Al Israel Airlines aircraft.
External links
• The dictionary definition of plus sign at Wiktionary
• The dictionary definition of minus sign at Wiktionary
Common punctuation marks and other typographical marks or symbols
• space
• , comma
• : colon
• ; semicolon
• ‐ hyphen
• ’ ' apostrophe
• ′ ″ ‴ prime
• . full stop
• & ampersand
• @ at sign
• ^ caret
• / slash
• \ backslash
• … ellipsis
• * asterisk
• ⁂ asterism
• * * * dinkus
• - hyphen-minus
• ‒ – — dash
• = ⸗ double hyphen
• ? question mark
• ! exclamation mark
• ‽ interrobang
• ¡ ¿ inverted ! and ?
• ⸮ irony punctuation
• # number sign
• № numero sign
• º ª ordinal indicator
• % percent sign
• ‰ per mille
• ‱ basis point
• ° degree symbol
• ⌀ diameter sign
• + − plus and minus signs
• × multiplication sign
• ÷ division sign
• ~ tilde
• ± plus–minus sign
• ∓ minus-plus sign
• _ underscore
• ⁀ tie
• | ¦ ‖ vertical bar
• • bullet
• · interpunct
• © copyright symbol
• © copyleft
• ℗ sound recording copyright
• ® registered trademark
• SM service mark symbol
• TM trademark symbol
• ‘ ’ “ ” ' ' " " quotation mark
• ‹ › « » guillemet
• ( ) [ ] { } ⟨ ⟩ bracket
• ” 〃 ditto mark
• † ‡ dagger
• ❧ hedera/floral heart
• ☞ manicule
• ◊ lozenge
• ¶ ⸿ pilcrow (paragraph mark)
• ※ reference mark
• § section mark
• Version of this table as a sortable list
• Currency symbols
• Diacritics (accents)
• Logic symbols
• Math symbols
• Whitespace
• Chinese punctuation
• Hebrew punctuation
• Japanese punctuation
• Korean punctuation
| Wikipedia |
+ h.c.
+ h.c. is an abbreviation for "plus the Hermitian conjugate"; it means is that there are additional terms which are the Hermitian conjugates of all of the preceding terms, and is a convenient shorthand to omit half the terms actually present.[1][2]
Context and use
The notation convention "+ h.c." is common in physics in the context of writing out formulas for Lagrangians and Hamiltonians, which conventionally are both required to be Hermitian operators.[2]
The expression
$A+B+C+~{\text{h.c.}}~$
means
$A+B+C+A^{\dagger }+B^{\dagger }+C^{\dagger }~.$[2]
The mathematics of quantum mechanics is based on complex numbers, whereas almost all observations (measurements) are only real numbers. Adding its own conjugate to an operator guarantees that the combination is Hermitian, which in turn guarantees that the combined operator's eigenvalues will be real numbers, suitable for predicting values of observations / measurements.[1]
Dagger and asterisk notation
In the expressions above, $A^{\dagger }$ is used as the symbol for the Hermitian conjugate (also called the conjugate transpose) of $A$, defined as applying both the complex conjugate and the transpose transformations to the operator $A$, in any order.
The dagger ($\dagger $) is an old notation in mathematics, but is still widespread in quantum-mechanics. In mathematics (particularly linear algebra) the Hermitian conjugate of $A$ is commonly written as $A^{\ast }$, but in quantum mechanics the asterisk ($\ast $) notation is sometimes used for the complex conjugate only, and not the combined conjugate transpose (Hermitian conjugate).
References
1. Özkan, Tristan; Lin, Huey-Wen (29 May 2019). "Quantum3: Learning QCD through Intuitive Play". Proceedings of the 36th Annual International Symposium on Lattice Field Theory (LATTICE2018). Proceedings of Science. Vol. 334. p. 326. arXiv:1901.00022. doi:10.22323/1.334.0326. OCLC 1082145757. S2CID 119383350.
2. "Meaning of h.c. in Lagrangians (& elsewhere?)". Physics Forums. 5 December 2010. Retrieved 23 May 2018.
| Wikipedia |
Plus–minus sign
The plus–minus sign, ±, is a mathematical symbol with multiple meanings:
• In mathematics, it generally indicates a choice of exactly two possible values, one of which is obtained through addition and the other through subtraction.[1]
• In experimental sciences, the sign commonly indicates the confidence interval or uncertainty bounding a range of possible errors in a measurement, often the standard deviation or standard error.[2] The sign may also represent an inclusive range of values that a reading might have.
• In medicine, it means "with or without".[3][4]
• In engineering, the sign indicates the tolerance, which is the range of values that are considered to be acceptable, safe, or which comply with some standard or with a contract.
• In botany, it is used in morphological descriptions to notate "more or less".
• In chemistry, the sign is used to indicate a racemic mixture.
• In chess, the sign indicates a clear advantage for the white player; the complementary minus-plus sign, ∓, indicates the same advantage for the black player.[5]
• In electronics, this sign may indicate a dual voltage power supply, such as ±5 volts means +5 volts and −5 volts, when used with audio circuits and operational amplifiers.
• In linguistics, it may indicate a distinctive feature, such as [±voiced].[6]
• In philosophy, the symbol ± or ∓ can be used to indicate a yinyang concept. Although Yin(-) and Yang(+) are in opposition, they coordinate and help each other in a unity. Yin and Yang are interdependent and coexist as two sides of the same concept.
±
Plus–minus sign
In UnicodeU+00B1 ± PLUS-MINUS SIGN (±, ±, ±)
Related
See alsoU+2213 ∓ MINUS-OR-PLUS SIGN (∓, ∓, ∓)
History
A version of the sign, including also the French word ou ("or"), was used in its mathematical meaning by Albert Girard in 1626, and the sign in its modern form was used as early as 1631, in William Oughtred's Clavis Mathematicae.[7]
Usage
In mathematics
In mathematical formulas, the ± symbol may be used to indicate a symbol that may be replaced by either the plus and minus signs, + or −, allowing the formula to represent two values or two equations.[8]
If x2 = 9, one may give the solution as x = ±3. This indicates that the equation has two solutions: x = +3 and x = −3. A common use of this notation is found in the quadratic formula
$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$
which describes the two solutions to the quadratic equation ax2+bx+c = 0.
Similarly, the trigonometric identity
$\sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)$
can be interpreted as a shorthand for two equations: one with + on both sides of the equation, and one with − on both sides.
The minus–plus sign, ∓, is generally used in conjunction with the ± sign, in such expressions as x ± y ∓ z, which can be interpreted as meaning x + y − z or x − y + z (but not x + y + z or x − y − z). The ∓ always has the opposite sign to ±.
The above expression can be rewritten as x ± (y − z) to avoid use of ∓, but cases such as the trigonometric identity are most neatly written using the "∓" sign:
$\cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)$
which represents the two equations:
${\begin{aligned}\cos(A+B)&=\cos(A)\cos(B)-\sin(A)\sin(B)\\\cos(A-B)&=\cos(A)\cos(B)+\sin(A)\sin(B)\end{aligned}}$
Another example is the conjugate of the perfect squares
$x^{3}\pm y^{3}=(x\pm y)\left((x\mp y\right)^{2}\pm xy)$
which represents the two equations:
$x^{3}+y^{3}=(x+y)\left((x-y\right)^{2}+xy)$
$x^{3}-y^{3}=(x-y)\left((x+y\right)^{2}-xy)$
A related usage is found in this presentation of the formula for the Taylor series of the sine function:
$\sin \left(x\right)=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \pm {\frac {1}{(2n+1)!}}x^{2n+1}+\cdots ~.$
Here, the plus-or-minus sign indicates that the term may be added or subtracted depending on whether n is odd or even; a rule which can be deduced from the first few terms. A more rigorous presentation would multiply each term by a factor of (−1)n, which gives +1 when n is even, and −1 when n is odd. In older texts one occasionally finds (−)n, which means the same.
When the standard presumption that the plus-or-minus signs all take on the same value of +1 or all −1 is not true, then the line of text that immediately follows the equation must contain a brief description of the actual connection, if any, most often of the form "where the ‘±’ signs are independent" or similar. If a brief, simple description is not possible, the equation must be re-written to provide clarity; e.g. by introducing variables such as s1, s2, ... and specifying a value of +1 or −1 separately for each, or some appropriate relation, like $s_{3}=s_{1}\cdot (s_{2})^{n}\,,$ or similar.
In statistics
The use of ± for an approximation is most commonly encountered in presenting the numerical value of a quantity, together with its tolerance or its statistical margin of error.[2] For example, 5.7 ± 0.2 may be anywhere in the range from 5.5 to 5.9 inclusive. In scientific usage, it sometimes refers to a probability of being within the stated interval, usually corresponding to either 1 or 2 standard deviations (a probability of 68.3% or 95.4% in a normal distribution).
Operations involving uncertain values should always try to preserve the uncertainty, in order to avoid propagation of error. If $~n=a\pm b\;,$ any operation of the form $~m=f(n)~$ must return a value of the form $~m=c\pm d~$, where c is $\,f(n)\,$ and d is range updated using interval arithmetic.
A percentage may also be used to indicate the error margin. For example, 230 ±10% V refers to a voltage within 10% (or 23 V) of either side of 230 V (from 207 V to 253 V inclusive). Separate values for the upper and lower bounds may also be used. For example, to indicate that a value is most likely 5.7, but may be as high as 5.9 or as low as 5.6, one may write 5.7+0.2
−0.1
.
In chess
The symbols ± and ∓ are used in chess notation to denote an advantage for white and black, respectively. However, the more common chess notation would be to only use + and –.[5] If several different symbols are used together, then the symbols + and − denote a clearer advantage than ± and ∓. When finer evaluation is desired, three pairs of symbols are used: ⩲ and ⩱ for only a slight advantage; ± and ∓ for a significant advantage; and +– and –+ for a potentially winning advantage, in each case for white or black respectively.[9]
Encodings
• In Unicode: U+00B1 ± PLUS-MINUS SIGN
• In ISO 8859-1, -7, -8, -9, -13, -15, and -16, the plus–minus symbol is code 0xB1hex. This location was copied to Unicode.
• The symbol also has a HTML entity representations of ±, ±, and ±.
• The rarer minus–plus sign is not generally found in legacy encodings, but is available in Unicode as U+2213 ∓ MINUS-OR-PLUS SIGN so can be used in HTML using ∓ or ∓.
• In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively.
• Although these characters may also be produced using underlining or overlining + symbol ( + or + ), this is deprecated or discouraged because the formatting may be stripped at a later date, changing the meaning. It also makes the meaning less accessible to blind users with screen readers.
Typing
• Windows: Alt+241 or Alt+0177 (numbers typed on the numeric keypad).
• Macintosh: ⌥ Option+⇧ Shift+= (equal sign on the non-numeric keypad).
• Unix-like systems: Compose,+,- or ⇧ Shift+Ctrl+u B1space (second works on Chromebook)
• In the Vim text editor (in Insert mode): Ctrl+k +- or Ctrl+v 177 or Ctrl+v x B1 or Ctrl+v u 00B1
• AutoCAD shortcut string: %%p
Similar characters
Look up 士, 土, or 干 in Wiktionary, the free dictionary.
The plus–minus sign resembles the Chinese characters 土 (Radical 32) and 士 (Radical 33), whereas the minus–plus sign resembles 干 (Radical 51).
See also
• ≈ (approximately equal to)
• Engineering tolerance
• Plus and minus signs
• Sign (mathematics)
• Table of mathematical symbols
References
1. Weisstein, Eric W. "Plus or Minus". mathworld.wolfram.com. Retrieved 2020-08-28.
2. Brown, George W. (1982). "Standard deviation, standard error: Which 'standard' should we use?". American Journal of Diseases of Children. 136 (10): 937–941. doi:10.1001/archpedi.1982.03970460067015. PMID 7124681.
3. Naess, I. A.; Christiansen, S. C.; Romundstad, P.; Cannegieter, S. C.; Rosendaal, F. R.; Hammerstrøm, J. (2007). "Incidence and mortality of venous thrombosis: a population-based study". Journal of Thrombosis and Haemostasis. 5 (4): 692–699. doi:10.1111/j.1538-7836.2007.02450.x. ISSN 1538-7933. PMID 17367492.
4. Heit, J. A.; Silverstein, M. D.; Mohr, D. N.; Petterson, T. M.; O'Fallon, W. M.; Melton, L. J. (1999-03-08). "Predictors of survival after deep vein thrombosis and pulmonary embolism: a population-based, cohort study". Archives of Internal Medicine. 159 (5): 445–453. doi:10.1001/archinte.159.5.445. ISSN 0003-9926. PMID 10074952.
5. Eade, James (2005), Chess For Dummies (2nd ed.), John Wiley & Sons, p. 272, ISBN 9780471774334.
6. Hornsby, David. Linguistics, A Complete Introduction. p. 99. ISBN 9781444180336.
7. Cajori, Florian (1928), A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics, Open Court, p. 245.
8. "Definition of PLUS/MINUS SIGN". www.merriam-webster.com. Retrieved 2020-08-28.
9. For details, see Chess annotation symbols#Positions.
Common punctuation marks and other typographical marks or symbols
• space
• , comma
• : colon
• ; semicolon
• ‐ hyphen
• ’ ' apostrophe
• ′ ″ ‴ prime
• . full stop
• & ampersand
• @ at sign
• ^ caret
• / slash
• \ backslash
• … ellipsis
• * asterisk
• ⁂ asterism
• * * * dinkus
• - hyphen-minus
• ‒ – — dash
• = ⸗ double hyphen
• ? question mark
• ! exclamation mark
• ‽ interrobang
• ¡ ¿ inverted ! and ?
• ⸮ irony punctuation
• # number sign
• № numero sign
• º ª ordinal indicator
• % percent sign
• ‰ per mille
• ‱ basis point
• ° degree symbol
• ⌀ diameter sign
• + − plus and minus signs
• × multiplication sign
• ÷ division sign
• ~ tilde
• ± plus–minus sign
• ∓ minus-plus sign
• _ underscore
• ⁀ tie
• | ¦ ‖ vertical bar
• • bullet
• · interpunct
• © copyright symbol
• © copyleft
• ℗ sound recording copyright
• ® registered trademark
• SM service mark symbol
• TM trademark symbol
• ‘ ’ “ ” ' ' " " quotation mark
• ‹ › « » guillemet
• ( ) [ ] { } ⟨ ⟩ bracket
• ” 〃 ditto mark
• † ‡ dagger
• ❧ hedera/floral heart
• ☞ manicule
• ◊ lozenge
• ¶ ⸿ pilcrow (paragraph mark)
• ※ reference mark
• § section mark
• Version of this table as a sortable list
• Currency symbols
• Diacritics (accents)
• Logic symbols
• Math symbols
• Whitespace
• Chinese punctuation
• Hebrew punctuation
• Japanese punctuation
• Korean punctuation
| Wikipedia |
0,1-simple lattice
In lattice theory, a bounded lattice L is called a 0,1-simple lattice if nonconstant lattice homomorphisms of L preserve the identity of its top and bottom elements. That is, if L is 0,1-simple and ƒ is a function from L to some other lattice that preserves joins and meets and does not map every element of L to a single element of the image, then it must be the case that ƒ−1(ƒ(0)) = {0} and ƒ−1(ƒ(1)) = {1}.
For instance, let Ln be a lattice with n atoms a1, a2, ..., an, top and bottom elements 1 and 0, and no other elements. Then for n ≥ 3, Ln is 0,1-simple. However, for n = 2, the function ƒ that maps 0 and a1 to 0 and that maps a2 and 1 to 1 is a homomorphism, showing that L2 is not 0,1-simple.
External links
• Matt Insall. "0,1-Simple Lattice". MathWorld.
| Wikipedia |
Zero-based numbering
Zero-based numbering is a way of numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday non-mathematical or non-programming circumstances. Under zero-based numbering, the initial element is sometimes termed the zeroth element,[1] rather than the first element; zeroth is a coined ordinal number corresponding to the number zero. In some cases, an object or value that does not (originally) belong to a given sequence, but which could be naturally placed before its initial element, may be termed the zeroth element. There is not wide agreement regarding the correctness of using zero as an ordinal (nor regarding the use of the term zeroth), as it creates ambiguity for all subsequent elements of the sequence when lacking context.
Numbering sequences starting at 0 is quite common in mathematics notation, in particular in combinatorics, though programming languages for mathematics usually index from 1.[2][3][4] In computer science, array indices usually start at 0 in modern programming languages, so computer programmers might use zeroth in situations where others might use first, and so forth. In some mathematical contexts, zero-based numbering can be used without confusion, when ordinal forms have well established meaning with an obvious candidate to come before first; for instance, a zeroth derivative of a function is the function itself, obtained by differentiating zero times. Such usage corresponds to naming an element not properly belonging to the sequence but preceding it: the zeroth derivative is not really a derivative at all. However, just as the first derivative precedes the second derivative, so also does the zeroth derivative (or the original function itself) precede the first derivative.
Computer programming
Origin
Martin Richards, creator of the BCPL language (a precursor of C), designed arrays initiating at 0 as the natural position to start accessing the array contents in the language, since the value of a pointer p used as an address accesses the position p + 0 in memory.[5][6] BCPL was first compiled for the IBM 7094; the language introduced no run-time indirection lookups, so the indirection optimization provided by these arrays was done at compile time.[6] The optimization was nevertheless important.[6][7]
In 1982 Edsger W. Dijkstra in his pertinent note Why numbering should start at zero[8] argued that arrays subscripts should start at zero as the latter being the most natural number. Discussing possible designs of array ranges by enclosing them in a chained inequality, combining sharp and standard inequalities to four possibilities, demonstrating that to his conviction zero-based arrays are best represented by non-overlapping index ranges, which start at zero, alluding to open, half-open and closed intervals as with the real numbers. Dijkstra's criteria for preferring this convention are in detail that it represents empty sequences in a more natural way (a ≤ i < a ?) than closed "intervals" (a ≤ i ≤ (a − 1) ?), and that with half-open "intervals" of naturals, the length of a sub-sequence equals the upper minus the lower bound (a ≤ i < b gives (b − a) possible values for i, with a, b, i all integers).
Usage in programming languages
This usage follows from design choices embedded in many influential programming languages, including C, Java, and Lisp. In these three, sequence types (C arrays, Java arrays and lists, and Lisp lists and vectors) are indexed beginning with the zero subscript. Particularly in C, where arrays are closely tied to pointer arithmetic, this makes for a simpler implementation: the subscript refers to an offset from the starting position of an array, so the first element has an offset of zero.
Referencing memory by an address and an offset is represented directly in computer hardware on virtually all computer architectures, so this design detail in C makes compilation easier, at the cost of some human factors. In this context using "zeroth" as an ordinal is not strictly correct, but a widespread habit in this profession. Other programming languages, such as Fortran or COBOL, have array subscripts starting with one, because they were meant as high-level programming languages, and as such they had to have a correspondence to the usual ordinal numbers which predate the invention of the zero by a long time.
Pascal allows the range of an array to be of any ordinal type (including enumerated types). APL allows setting the index origin to 0 or 1 during runtime programatically.[9][10] Some recent languages, such as Lua and Visual Basic, have adopted the same convention for the same reason.
Zero is the lowest unsigned integer value, one of the most fundamental types in programming and hardware design. In computer science, zero is thus often used as the base case for many kinds of numerical recursion. Proofs and other sorts of mathematical reasoning in computer science often begin with zero. For these reasons, in computer science it is not unusual to number from zero rather than one.
In recent years this trait has also been observed among many pure mathematicians, where many constructions are defined to be numbered from 0.
If an array is used to represent a cycle, it is convenient to obtain the index with a modulo function, which can result in zero.
Numerical properties
With zero-based numbering, a range can be expressed as the half-open interval, [0, n), as opposed to the closed interval, [1, n]. Empty ranges, which often occur in algorithms, are tricky to express with a closed interval without resorting to obtuse conventions like [1, 0]. Because of this property, zero-based indexing potentially reduces off-by-one and fencepost errors.[8] On the other hand, the repeat count n is calculated in advance, making the use of counting from 0 to n − 1 (inclusive) less intuitive. Some authors prefer one-based indexing, as it corresponds more closely to how entities are indexed in other contexts.[11]
Another property of this convention is in the use of modular arithmetic as implemented in modern computers. Usually, the modulo function maps any integer modulo N to one of the numbers 0, 1, 2, ..., N − 1, where N ≥ 1. Because of this, many formulas in algorithms (such as that for calculating hash table indices) can be elegantly expressed in code using the modulo operation when array indices start at zero.
Pointer operations can also be expressed more elegantly on a zero-based index due to the underlying address/offset logic mentioned above. To illustrate, suppose a is the memory address of the first element of an array, and i is the index of the desired element. To compute the address of the desired element, if the index numbers count from 1, the desired address is computed by this expression:
$a+s\times (i-1),$
where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes
$a+s\times i.$
This simpler expression is more efficient to compute at run time.
However, a language wishing to index arrays from 1 could adopt the convention that every array address is represented by a′ = a – s; that is, rather than using the address of the first array element, such a language would use the address of a fictitious element located immediately before the first actual element. The indexing expression for a 1-based index would then be
$a'+s\times i.$
Hence, the efficiency benefit at run time of zero-based indexing is not inherent, but is an artifact of the decision to represent an array with the address of its first element rather than the address of the fictitious zeroth element. However, the address of that fictitious element could very well be the address of some other item in memory not related to the array.
Superficially, the fictitious element doesn't scale well to multidimensional arrays. Indexing multidimensional arrays from zero makes a naive (contiguous) conversion to a linear address space (systematically varying one index after the other) look simpler than when indexing from one. For instance, when mapping the three-dimensional array A[P][N][M] to a linear array L[M⋅N⋅P], both with M ⋅ N ⋅ P elements, the index r in the linear array to access a specific element with L[r] = A[z][y][x] in zero-based indexing, i.e. [0 ≤ x < P], [0 ≤ y < N], [0 ≤ z < M], and [0 ≤ r < M ⋅ N ⋅ P], is calculated by
$r=z\cdot M\cdot N+y\cdot M+x.$
Organizing all arrays with 1-based indices ([1 ≤ x′ ≤ P], [1 ≤ y′ ≤ N], [1 ≤ z′ ≤ M], [1 ≤ r′ ≤ M ⋅ N ⋅ P]), and assuming an analogous arrangement of the elements, gives
$r'=(z'-1)\cdot M\cdot N+(y'-1)\cdot M+(x'-0)$
to access the same element, which arguably looks more complicated. Of course, r′ = r + 1, since [z = z′ – 1], [y = y′ – 1], and [x = x′ – 1]. A simple and everyday-life example is positional notation, which the invention of the zero made possible. In positional notation, tens, hundreds, thousands and all other digits start with zero, only units start at one.[12]
• Zero-based indices
x
y
012..$x=x'-1$..89
0 0001020809
1 1011121819
2 2021222829
..
$y=y'-1$ $y\cdot M+x$
..
8 8081828889
9 9091929899
The table content represents the index r.
•
• One-based indices
x'
y'
123..$x'=x+1$..910
1 0102030910
2 1112131920
3 2122232930
..
$y'=y+1$ $(y'-1)\cdot M+x'$
..
9 8182838990
10 91929399100
The table content represents the index r′.
This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". Therefore, an analogy from the ordinal numbers to the quantity of objects numbered appears; the highest index of n objects will be n − 1, and it refers to the nth element. For this reason, the first element is sometimes referred to as the zeroth element, in an attempt to avoid confusion.
Science
In mathematics, many sequences of numbers or of polynomials are indexed by nonnegative integers, for example, the Bernoulli numbers and the Bell numbers.
In both mechanics and statistics, the zeroth moment is defined, representing total mass in the case of physical density, or total probability, i.e. one, for a probability distribution.
The zeroth law of thermodynamics was formulated after the first, second, and third laws, but considered more fundamental, thus its name.
In biology, an organism is said to have zero-order intentionality if it shows "no intention of anything at all". This would include a situation where the organism's genetically predetermined phenotype results in a fitness benefit to itself, because it did not "intend" to express its genes.[13] In the similar sense, a computer may be considered from this perspective a zero-order intentional entity, as it does not "intend" to express the code of the programs it runs.[14]
In biological or medical experiments, initial measurements made before any experimental time has passed are said to be on the 0 day of the experiment.
In genomics, both 0-based and 1-based systems are used for genome coordinates.
Patient zero (or index case) is the initial patient in the population sample of an epidemiological investigation.
Other fields
The year zero does not exist in the widely used Gregorian calendar or in its predecessor, the Julian calendar. Under those systems, the year 1 BC is followed by AD 1. However, there is a year zero in astronomical year numbering (where it coincides with the Julian year 1 BC) and in ISO 8601:2004 (where it coincides with the Gregorian year 1 BC), as well as in all Buddhist and Hindu calendars.
In many countries, the ground floor in buildings is considered as floor number 0 rather than as the "1st floor", the naming convention usually found in the United States of America. This makes a consistent set with underground floors marked with negative numbers.
While the ordinal of 0 mostly finds use in communities directly connected to mathematics, physics, and computer science, there are also instances in classical music. The composer Anton Bruckner regarded his early Symphony in D minor to be unworthy of including in the canon of his works, and he wrote gilt nicht ("doesn't count") on the score and a circle with a crossbar, intending it to mean "invalid". But posthumously, this work came to be known as Symphony No. 0 in D minor, even though it was actually written after Symphony No. 1 in C minor. There is an even earlier Symphony in F minor of Bruckner's, which is sometimes called No. 00. The Russian composer Alfred Schnittke also wrote a Symphony No. 0.
In some universities, including Oxford and Cambridge, "week 0" or occasionally "noughth week" refers to the week before the first week of lectures in a term. In Australia, some universities refer to this as "O week", which serves as a pun on "orientation week". As a parallel, the introductory weeks at university educations in Sweden are generally called nollning (zeroing).
The United States Air Force starts basic training each Wednesday, and the first week (of eight) is considered to begin with the following Sunday. The four days before that Sunday are often referred to as "zero week".
24-hour clocks and the international standard ISO 8601 use 0 to denote the first (zeroth) hour of the day, consistent with using the 0 to denote the first (zeroth) minute of the hour and the first (zeroth) second of the minute. Also, the 12-hour clocks used in Japan use 0 to denote the hour immediately after midnight and noon in contrast to 12 used elsewhere, in order to avoid confusion whether 12 a.m. and 12 p.m. represent noon or midnight.
King's Cross station in London, Edinburgh Haymarket, and stations in Uppsala, Yonago, Stockport and Cardiff have a Platform 0.
Robert Crumb's drawings for the first issue of Zap Comix were stolen, so he drew a whole new issue, which was published as issue 1. Later he re-inked his photocopies of the stolen artwork and published it as issue 0.
The Brussels ring road in Belgium is numbered R0. It was built after the ring road around Antwerp, but Brussels (being the capital city) was deemed deserving of a more basic number. Similarly the (unfinished) orbital motorway around Budapest in Hungary is called M0.
Zero is sometimes used in street addresses, especially in schemes where even numbers are one side of the street and odd numbers on the other. A case in point is Christ Church on Harvard Square, whose address is 0 Garden Street.
Formerly in Formula One, when a defending world champion did not compete in the following season, the number 1 was not assigned to any driver, but one driver of the world champion team would carry the number 0, and the other, number 2. This did happen both in 1993 and 1994 with Damon Hill carrying the number 0 in both seasons, as defending champion Nigel Mansell quit after 1992, and defending champion Alain Prost quit after 1993. However, in 2014 the series moved to drivers carrying career-long personalised numbers, instead of team-allocated numbers, other than the defending champion still having the option to carry number 1. Therefore 0 is no longer used in this scenario. It is not clear if it is available as a driver's chosen number, or whether they must be between 2 and 99, but it has not been used to date under this system.
Some team sports allow 0 to be chosen as a player's uniform number (in addition to the typical range of 1-99). The NFL voted to allow this from 2023 onwards.
A chronological prequel of a series may be numbered as 0, such as Ring 0: Birthday or Zork Zero.
The Swiss Federal Railways number certain classes of rolling stock from zero, for example, Re 460 000 to 118.
In the realm of fiction, Isaac Asimov eventually added a Zeroth Law to his Three Laws of Robotics, essentially making them four laws.
A standard roulette wheel contains the number 0 as well as 1-36. It appears in green, so is classed as neither a “red” nor “black” number for betting purposes. The card game Uno has number cards running from 0 to 9 along with special cards, within each coloured suit.
See also
• Zeroth-order approximation
• Off-by-one error
References
Citations
1. M. Seed, Graham (1965). An Introduction to Object-Oriented Programming in C++ with Applications in Computer Graphics (2nd ed.). British Library: Springer. p. 391. ISBN 1852334509. Retrieved 11 February 2020.
2. Steve Eddins and Loren Shure. "Matrix Indexing in MATLAB". Retrieved 23 February 2021.
3. "How to : Get Elements of Lists". Wolfram. Retrieved 23 February 2021.
4. "Indexing Arrays, Matrices, and Vectors". Maplesoft. Retrieved 23 February 2021.
5. Martin Richards (1967). The BCPL Reference Manual (PDF). Massachusetts Institute of Technology. p. 11.
6. Mike Hoye. "Citation Needed". Retrieved 28 January 2014.
7. Tom Van Vleck (1995). "The IBM 7094 and CTSS". Retrieved 28 January 2014.
8. Dijkstra, Edsger Wybe (May 2, 2008). "Why numbering should start at zero (EWD 831)". E. W. Dijkstra Archive. University of Texas at Austin. Retrieved 2011-03-16.
9. Brown, Jim (December 1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. S2CID 40187000.
10. Hui, Roger. "Is Index Origin 0 a Hindrance?". jsoftware.com. JSoftware. Retrieved 19 January 2015.
11. Programming Microsoft® Visual C# 2005 by Donis Marshall.
12. Sal Khan. Math 1st Grade / Place Value / Number grid. Khan Academy. Retrieved July 28, 2018. Youtube title: Number grid / Counting / Early Math / Khan Academy.
13. Byrne, Richard W. "The Thinking Ape: Evolutionary Origins of Intelligence". Retrieved 2010-05-18.
14. Dunbar, Robin. "The Human Story – A new history of mankind's Evolution". Retrieved 2010-05-18.
Sources
• This article incorporates material taken from zeroth at the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.
| Wikipedia |
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces.
Connected and disconnected subspaces of R²
From top to bottom: red space A, pink space B, yellow space C and orange space D are all connected spaces, whereas green space E (made of subsets E1, E2, E3, and E4) is disconnected. Furthermore, A and B are also simply connected (genus 0), while C and D are not: C has genus 1 and D has genus 4.
A subset of a topological space $X$ is a connected set if it is a connected space when viewed as a subspace of $X$.
Some related but stronger conditions are path connected, simply connected, and $n$-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
Formal definition
A topological space $X$ is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, $X$ is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
For a topological space $X$ the following conditions are equivalent:
1. $X$ is connected, that is, it cannot be divided into two disjoint non-empty open sets.
2. The only subsets of $X$ which are both open and closed (clopen sets) are $X$ and the empty set.
3. The only subsets of $X$ with empty boundary are $X$ and the empty set.
4. $X$ cannot be written as the union of two non-empty separated sets (sets for which each is disjoint from the other's closure).
5. All continuous functions from $X$ to $\{0,1\}$ are constant, where $\{0,1\}$ is the two-point space endowed with the discrete topology.
Historically this modern formulation of the notion of connectedness (in terms of no partition of $X$ into two separated sets) first appeared (independently) with N.J. Lennes, Frigyes Riesz, and Felix Hausdorff at the beginning of the 20th century. See [1] for details.
Connected components
Given some point $x$ in a topological space $X,$ the union of any collection of connected subsets such that each contains $x$ will once again be a connected subset. The connected component of a point $x$ in $X$ is the union of all connected subsets of $X$ that contain $x;$ it is the unique largest (with respect to $\subseteq $) connected subset of $X$ that contains $x.$ The maximal connected subsets (ordered by inclusion $\subseteq $) of a non-empty topological space are called the connected components of the space. The components of any topological space $X$ form a partition of $X$: they are disjoint, non-empty and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the rational numbers are the one-point sets (singletons), which are not open. Proof: Any two distinct rational numbers $q_{1}<q_{2}$ are in different components. Take an irrational number $q_{1}<r<q_{2},$ and then set $A=\{q\in \mathbb {Q} :q<r\}$ and $B=\{q\in \mathbb {Q} :q>r\}.$ Then $(A,B)$ is a separation of $\mathbb {Q} ,$ and $q_{1}\in A,q_{2}\in B$. Thus each component is a one-point set.
Let $\Gamma _{x}$ be the connected component of $x$ in a topological space $X,$ and $\Gamma _{x}'$ be the intersection of all clopen sets containing $x$ (called quasi-component of $x.$) Then $\Gamma _{x}\subset \Gamma '_{x}$ where the equality holds if $X$ is compact Hausdorff or locally connected. [2]
Disconnected spaces
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space $X$ is called totally separated if, for any two distinct elements $x$ and $y$ of $X$, there exist disjoint open sets $U$ containing $x$ and $V$ containing $y$ such that $X$ is the union of $U$ and $V$. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers $\mathbb {Q} $, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even Hausdorff, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff.
Examples
• The closed interval $[0,2)$ in the standard subspace topology is connected; although it can, for example, be written as the union of $[0,1)$ and $[1,2),$ the second set is not open in the chosen topology of $[0,2).$
• The union of $[0,1)$ and $(1,2]$ is disconnected; both of these intervals are open in the standard topological space $[0,1)\cup (1,2].$
• $(0,1)\cup \{3\}$ is disconnected.
• A convex subset of $\mathbb {R} ^{n}$ is connected; it is actually simply connected.
• A Euclidean plane excluding the origin, $(0,0),$ is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
• A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
• $\mathbb {R} $, the space of real numbers with the usual topology, is connected.
• The Sorgenfrey line is disconnected.[3]
• If even a single point is removed from $\mathbb {R} $, the remainder is disconnected. However, if even a countable infinity of points are removed from $\mathbb {R} ^{n}$, where $n\geq 2,$ the remainder is connected. If $n\geq 3$, then $\mathbb {R} ^{n}$ remains simply connected after removal of countably many points.
• Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field (such as $\mathbb {R} $ or $\mathbb {C} $), is simply connected.
• Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space.[4]
• On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space.
• The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
• If a space $X$ is homotopy equivalent to a connected space, then $X$ is itself connected.
• The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.
• The general linear group $\operatorname {GL} (n,\mathbb {R} )$ (that is, the group of $n$-by-$n$ real, invertible matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, $\operatorname {GL} (n,\mathbb {C} )$ is connected. More generally, the set of invertible bounded operators on a complex Hilbert space is connected.
• The spectra of commutative local ring and integral domains are connected. More generally, the following are equivalent[5]
1. The spectrum of a commutative ring $\mathbb {R} $ is connected
2. Every finitely generated projective module over $\mathbb {R} $ has constant rank.
3. $\mathbb {R} $ has no idempotent $\neq 0,1$ (i.e., $\mathbb {R} $ is not a product of two rings in a nontrivial way).
An example of a space that is not connected is a plane with an infinite line deleted from it. Other examples of disconnected spaces (that is, spaces which are not connected) include the plane with an annulus removed, as well as the union of two disjoint closed disks, where all examples of this paragraph bear the subspace topology induced by two-dimensional Euclidean space.
Path connectedness
A path-connected space is a stronger notion of connectedness, requiring the structure of a path. A path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval $[0,1]$ to $X$ with $f(0)=x$ and $f(1)=y$. A path-component of $X$ is an equivalence class of $X$ under the equivalence relation which makes $x$ equivalent to $y$ if there is a path from $x$ to $y$. The space $X$ is said to be path-connected (or pathwise connected or $\mathbf {0} $-connected) if there is exactly one path-component, i.e. if there is a path joining any two points in $X$. Again, many authors exclude the empty space (by this definition, however, the empty space is not path-connected because it has zero path-components; there is a unique equivalence relation on the empty set which has zero equivalence classes).
Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended long line $L^{*}$ and the topologist's sine curve.
Subsets of the real line $\mathbb {R} $ are connected if and only if they are path-connected; these subsets are the intervals of $\mathbb {R} $. Also, open subsets of $\mathbb {R} ^{n}$ or $\mathbb {C} ^{n}$ are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for finite topological spaces.
Arc connectedness
A space $X$ is said to be arc-connected or arcwise connected if any two topologically distinguishable points can be joined by an arc, which is an embedding $f:[0,1]\to X$. An arc-component of $X$ is a maximal arc-connected subset of $X$; or equivalently an equivalence class of the equivalence relation of whether two points can be joined by an arc or by a path whose points are topologically indistinguishable.
Every Hausdorff space that is path-connected is also arc-connected; more generally this is true for a $\Delta $-Hausdorff space, which is a space where each image of a path is closed. An example of a space which is path-connected but not arc-connected is given by the line with two origins; its two copies of $0$ can be connected by a path but not by an arc.
Intuition for path-connected spaces does not readily transfer to arc-connected spaces. Let $X$ be the line with two origins. The following are facts whose analogues hold for path-connected spaces, but do not hold for arc-connected spaces:
• Continuous image of arc-connected space may not be arc-connected: for example, a quotient map from an arc-connected space to its quotient with countably many (at least 2) topologically distinguishable points cannot be arc-connected due to too small cardinality.
• Arc-components may not be disjoint. For example, $X$ has two overlapping arc-components.
• Arc-connected product space may not be a product of arc-connected spaces. For example, $X\times \mathbb {R} $ is arc-connected, but $X$ is not.
• Arc-components of a product space may not be products of arc-components of the marginal spaces. For example, $X\times \mathbb {R} $ has a single arc-component, but $X$ has two arc-components.
• If arc-connected subsets have a non-empty intersection, then their union may not be arc-connected. For example, the arc-components of $X$ intersect, but their union is not arc-connected.
Local connectedness
Main article: Locally connected space
A topological space is said to be locally connected at a point $x$ if every neighbourhood of $x$ contains a connected open neighbourhood. It is locally connected if it has a base of connected sets. It can be shown that a space $X$ is locally connected if and only if every component of every open set of $X$ is open.
Similarly, a topological space is said to be locally path-connected if it has a base of path-connected sets. An open subset of a locally path-connected space is connected if and only if it is path-connected. This generalizes the earlier statement about $\mathbb {R} ^{n}$ and $\mathbb {C} ^{n}$, each of which is locally path-connected. More generally, any topological manifold is locally path-connected.
Locally connected does not imply connected, nor does locally path-connected imply path connected. A simple example of a locally connected (and locally path-connected) space that is not connected (or path-connected) is the union of two separated intervals in $\mathbb {R} $, such as $(0,1)\cup (2,3)$.
A classical example of a connected space that is not locally connected is the so called topologist's sine curve, defined as $T=\{(0,0)\}\cup \left\{\left(x,\sin \left({\tfrac {1}{x}}\right)\right):x\in (0,1]\right\}$, with the Euclidean topology induced by inclusion in $\mathbb {R} ^{2}$.
Set operations
The intersection of connected sets is not necessarily connected.
The union of connected sets is not necessarily connected, as can be seen by considering $X=(0,1)\cup (1,2)$.
Each ellipse is a connected set, but the union is not connected, since it can be partitioned to two disjoint open sets $U$ and $V$.
This means that, if the union $X$ is disconnected, then the collection $\{X_{i}\}$ can be partitioned to two sub-collections, such that the unions of the sub-collections are disjoint and open in $X$ (see picture). This implies that in several cases, a union of connected sets is necessarily connected. In particular:
1. If the common intersection of all sets is not empty ($ \bigcap X_{i}\neq \emptyset $), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected.
2. If the intersection of each pair of sets is not empty ($\forall i,j:X_{i}\cap X_{j}\neq \emptyset $) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected.
3. If the sets can be ordered as a "linked chain", i.e. indexed by integer indices and $\forall i:X_{i}\cap X_{i+1}\neq \emptyset $, then again their union must be connected.
4. If the sets are pairwise-disjoint and the quotient space $X/\{X_{i}\}$ is connected, then X must be connected. Otherwise, if $U\cup V$ is a separation of X then $q(U)\cup q(V)$ is a separation of the quotient space (since $q(U),q(V)$ are disjoint and open in the quotient space).[6]
The set difference of connected sets is not necessarily connected. However, if $X\supseteq Y$ and their difference $X\setminus Y$ is disconnected (and thus can be written as a union of two open sets $X_{1}$ and $X_{2}$), then the union of $Y$ with each such component is connected (i.e. $Y\cup X_{i}$ is connected for all $i$).
Proof[7]
By contradiction, suppose $Y\cup X_{1}$ is not connected. So it can be written as the union of two disjoint open sets, e.g. $Y\cup X_{1}=Z_{1}\cup Z_{2}$. Because $Y$ is connected, it must be entirely contained in one of these components, say $Z_{1}$, and thus $Z_{2}$ is contained in $X_{1}$. Now we know that:
$X=\left(Y\cup X_{1}\right)\cup X_{2}=\left(Z_{1}\cup Z_{2}\right)\cup X_{2}=\left(Z_{1}\cup X_{2}\right)\cup \left(Z_{2}\cap X_{1}\right)$
The two sets in the last union are disjoint and open in $X$, so there is a separation of $X$, contradicting the fact that $X$ is connected.
Theorems
• Main theorem of connectedness: Let $X$ and $Y$ be topological spaces and let $f:X\rightarrow Y$ be a continuous function. If $X$ is (path-)connected then the image $f(X)$ is (path-)connected. This result can be considered a generalization of the intermediate value theorem.
• Every path-connected space is connected.
• Every locally path-connected space is locally connected.
• A locally path-connected space is path-connected if and only if it is connected.
• The closure of a connected subset is connected. Furthermore, any subset between a connected subset and its closure is connected.
• The connected components are always closed (but in general not open)
• The connected components of a locally connected space are also open.
• The connected components of a space are disjoint unions of the path-connected components (which in general are neither open nor closed).
• Every quotient of a connected (resp. locally connected, path-connected, locally path-connected) space is connected (resp. locally connected, path-connected, locally path-connected).
• Every product of a family of connected (resp. path-connected) spaces is connected (resp. path-connected).
• Every open subset of a locally connected (resp. locally path-connected) space is locally connected (resp. locally path-connected).
• Every manifold is locally path-connected.
• Arc-wise connected space is path connected, but path-wise connected space may not be arc-wise connected
• Continuous image of arc-wise connected set is arc-wise connected.
Graphs
Graphs have path connected subsets, namely those subsets for which every pair of points has a path of edges joining them. But it is not always possible to find a topology on the set of points which induces the same connected sets. The 5-cycle graph (and any $n$-cycle with $n>3$ odd) is one such example.
As a consequence, a notion of connectedness can be formulated independently of the topology on a space. To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets (Muscat & Buhagiar 2006). Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs.
However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
Stronger forms of connectedness
There are stronger forms of connectedness for topological spaces, for instance:
• If there exist no two disjoint non-empty open sets in a topological space $X$, $X$ must be connected, and thus hyperconnected spaces are also connected.
• Since a simply connected space is, by definition, also required to be path connected, any simply connected space is also connected. If the "path connectedness" requirement is dropped from the definition of simple connectivity, a simply connected space does not need to be connected.
• Yet stronger versions of connectivity include the notion of a contractible space. Every contractible space is path connected and thus also connected.
In general, any path connected space must be connected but there exist connected spaces that are not path connected. The deleted comb space furnishes such an example, as does the above-mentioned topologist's sine curve.
See also
• Connected component (graph theory) – Maximal subgraph whose vertices can reach each otherPages displaying short descriptions of redirect targets
• Connectedness locus
• Domain (mathematical analysis) – Connected open subset of a topological space
• Extremally disconnected space – Topological space in which the closure of every open set is open
• Locally connected space – Property of topological spaces
• n-connected
• Uniformly connected space – Type of uniform space
• Pixel connectivity
References
1. Wilder, R.L. (1978). "Evolution of the Topological Concept of "Connected"". American Mathematical Monthly. 85 (9): 720–726. doi:10.2307/2321676. JSTOR 2321676.
2. "General topology - Components of the set of rational numbers".
3. Stephen Willard (1970). General Topology. Dover. p. 191. ISBN 0-486-43479-6.
4. George F. Simmons (1968). Introduction to Topology and Modern Analysis. McGraw Hill Book Company. p. 144. ISBN 0-89874-551-9.
5. Charles Weibel, The K-book: An introduction to algebraic K-theory
6. Brandsma, Henno (February 13, 2013). "How to prove this result involving the quotient maps and connectedness?". Stack Exchange.
7. Marek (February 13, 2013). "How to prove this result about connectedness?". Stack Exchange.
Further reading
• Munkres, James R. (2000). Topology, Second Edition. Prentice Hall. ISBN 0-13-181629-2.
• Weisstein, Eric W. "Connected Set". MathWorld.
• V. I. Malykhin (2001) [1994], "Connected space", Encyclopedia of Mathematics, EMS Press
• Muscat, J; Buhagiar, D (2006). "Connective Spaces" (PDF). Mem. Fac. Sci. Eng. Shimane Univ., Series B: Math. Sc. 39: 1–13. Archived from the original (PDF) on 2016-03-04. Retrieved 2010-05-17..
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Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive identity is the identity element in an additive group. It corresponds to the element 0 such that for all x in the group, 0 + x = x + 0 = x. Some examples of additive identity include:
• The zero vector under vector addition: the vector of length 0 and whose components are all 0. Often denoted as $\mathbf {0} $ or ${\vec {0}}$.[1][2]
• The zero function or zero map defined by z(x) = 0, under pointwise addition (f + g)(x) = f(x) + g(x)
• The empty set under set union
• An empty sum or empty coproduct
• An initial object in a category (an empty coproduct, and so an identity under coproducts)
Absorbing elements
An absorbing element in a multiplicative semigroup or semiring generalises the property 0 ⋅ x = 0. Examples include:
• The empty set, which is an absorbing element under Cartesian product of sets, since { } × S = { }
• The zero function or zero map defined by z(x) = 0 under pointwise multiplication (f ⋅ g)(x) = f(x) ⋅ g(x)
Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a field or ring, which is both the additive identity and the multiplicative absorbing element, and whose principal ideal is the smallest ideal.
Zero objects
A zero object in a category is both an initial and terminal object (and so an identity under both coproducts and products). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:
• The trivial group, containing only the identity (a zero object in the category of groups)
• The zero module, containing only the identity (a zero object in the category of modules over a ring)
Zero morphisms
A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0XY : X → Y is the zero morphism among morphisms from X to Y, and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0XY = 0XB and 0XY ∘ f = 0AY.
If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0XY : X → Y. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.
Least elements
A least element in a partially ordered set or lattice may sometimes be called a zero element, and written either as 0 or ⊥.
Zero module
In mathematics, the zero module is the module consisting of only the additive identity for the module's addition function. In the integers, this identity is zero, which gives the name zero module. That the zero module is in fact a module is simple to show; it is closed under addition and multiplication trivially.
Zero ideal
In mathematics, the zero ideal in a ring $R$ is the ideal $\{0\}$ consisting of only the additive identity (or zero element). The fact that this is an ideal follows directly from the definition.
Zero matrix
Main article: Zero matrix
In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. It is alternately denoted by the symbol $O$. Some examples of zero matrices are
$0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\ $
The set of m × n matrices with entries in a ring K forms a module $K_{m,n}$. The zero matrix $0_{K_{m,n}}$ in $K_{m,n}$ is the matrix with all entries equal to $0_{K}$, where $0_{K}$ is the additive identity in K.
$0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}$
The zero matrix is the additive identity in $K_{m,n}$. That is, for all $A\in K_{m,n}$:
$0_{K_{m,n}}+A=A+0_{K_{m,n}}=A$
There is exactly one zero matrix of any given size m × n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.
The zero matrix also represents the linear transformation which sends all vectors to the zero vector.
Zero tensor
In mathematics, the zero tensor is a tensor, of any order, all of whose components are zero. The zero tensor of order 1 is sometimes known as the zero vector.
Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Adding the zero tensor is equivalent to the identity operation.
See also
• Null semigroup
• Zero divisor
• Zero object
• Zero of a function
• Zero — non-mathematical uses
References
1. Weisstein, Eric W. "Zero Vector". mathworld.wolfram.com. Retrieved 2020-08-12.
2. "Definition of ZERO VECTOR". www.merriam-webster.com. Retrieved 2020-08-12.
| Wikipedia |
0/1-polytope
A 0/1-polytope is a convex polytope generated by the convex hull of a subset of d coordinates value 0 or 1, {0,1}d. The full domain is the unit hypercube with cut planes passing through these coordinates.[1] A d-polytope requires at least d+1 vertices, and can't be all in the same hyperplanes.
n-simplex polytopes for example can be generated (n+1) vertices, using the origin, and one vertex along each primary axis, (1,0....), etc.
References
1. Branko Grunbaum, Convex Polytopes, 2003. 4.9 Additional notes and comments, p.69a
| Wikipedia |
Zero to the power of zero
Zero to the power of zero, denoted by 00, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 00 = 1. In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression.
Discrete exponents
Many widely used formulas involving natural-number exponents require 00 to be defined as 1. For example, the following three interpretations of b0 make just as much sense for b = 0 as they do for positive integers b:
• The interpretation of b0 as an empty product assigns it the value 1.
• The combinatorial interpretation of b0 is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple.
• The set-theoretic interpretation of b0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function.[1]
All three of these specialize to give 00 = 1.
Polynomials and power series
When evaluating polynomials, it is convenient to define 00 as 1. A (real) polynomial is an expression of the form a0x0 + ⋅⋅⋅ + anxn, where x is an indeterminate, and the coefficients ai are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring R[x]. The multiplicative identity of R[x] is the polynomial x0; that is, x0 times any polynomial p(x) is just p(x).[2] Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism evr : R[x] → R such that evr(x) = r. Because evr is unital, evr(x0) = 1. That is, r0 = 1 for each real number r, including 0. The same argument applies with R replaced by any ring.[3]
Defining 00 = 1 is necessary for many polynomial identities. For example, the binomial theorem (1 + x)n = Σn
k=0
(n
k
) xk
holds for x = 0 only if 00 = 1.[4]
Similarly, rings of power series require x0 to be defined as 1 for all specializations of x. For example, identities like 1/1−x = Σ∞
n=0
xn
and ex = Σ∞
n=0
xn/n!
hold for x = 0 only if 00 = 1.[5]
In order for the polynomial x0 to define a continuous function R → R, one must define 00 = 1.
In calculus, the power rule d/dxxn = nxn−1 is valid for n = 1 at x = 0 only if 00 = 1.
Continuous exponents
Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.[6] The expression 00 is an indeterminate form: Given real-valued functions f(t) and g(t) approaching 0 (as t approaches a real number or ±∞) with f(t) > 0, the limit of f(t)g(t) can be any non-negative real number or +∞, or it can diverge, depending on f and g. For example, each limit below involves a function f(t)g(t) with f(t), g(t) → 0 as t → 0+ (a one-sided limit), but their values are different:
$\lim _{t\to 0^{+}}{t}^{t}=1,$
$\lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{t}=0,$
$\lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{-t}=+\infty ,$
$\lim _{t\to 0^{+}}\left(e^{-1/t}\right)^{at}=e^{-a}.$
Thus, the two-variable function xy, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on {(x, y) : x > 0} ∪ {(0, 0)}, no matter how one chooses to define 00.[7]
On the other hand, if f and g are analytic functions on an open neighborhood of a number c, then f(t)g(t) → 1 as t approaches c from any side on which f is positive.[8] This and more general results can be obtained by studying the limiting behavior of the function ln(f(t)g(t)) = g(t) ln f(t).[9][10]
Complex exponents
In the complex domain, the function zw may be defined for nonzero z by choosing a branch of log z and defining zw as ew log z. This does not define 0w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.[11][12][13]
History
As a value
In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1[14] and explicitly mentioned that 00 = 1.[15] An annotation attributed[16] to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis[17] offered the "justification"
$0^{0}=(a-a)^{n-n}={\frac {(a-a)^{n}}{(a-a)^{n}}}=1$
as well as another more involved justification. In the 1830s, Libri[18][16] published several further arguments attempting to justify the claim 00 = 1, though these were far from convincing, even by standards of rigor at the time.[19]
As a limiting form
Euler, when setting 00 = 1, mentioned that consequently the values of the function 0x take a "huge jump", from ∞ for x < 0, to 1 at x = 0, to 0 for x > 0.[14] In 1814, Pfaff used a squeeze theorem argument to prove that xx → 1 as x → 0+.[8]
On the other hand, in 1821 Cauchy[20] explained why the limit of xy as positive numbers x and y approach 0 while being constrained by some fixed relation could be made to assume any value between 0 and ∞ by choosing the relation appropriately. He deduced that the limit of the full two-variable function xy without a specified constraint is "indeterminate". With this justification, he listed 00 along with expressions like 0/0 in a table of indeterminate forms.
Apparently unaware of Cauchy's work, Möbius[8] in 1834, building on Pfaff's argument, claimed incorrectly that f(x)g(x) → 1 whenever f(x),g(x) → 0 as x approaches a number c (presumably f is assumed positive away from c). Möbius reduced to the case c = 0, but then made the mistake of assuming that each of f and g could be expressed in the form Pxn for some continuous function P not vanishing at 0 and some nonnegative integer n, which is true for analytic functions, but not in general. An anonymous commentator pointed out the unjustified step;[21] then another commentator who signed his name simply as "S" provided the explicit counterexamples (e−1/x)x → e−1 and (e−1/x)2x → e−2 as x → 0+ and expressed the situation by writing that "00 can have many different values".[21]
Current situation
• Some authors define 00 as 1 because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness. If we refrain from defining 00, then certain assertions become unnecessarily awkward. ... The consensus is to use the definition 00 = 1, although there are textbooks that refrain from defining 00."[22] Knuth (1992) contends more strongly that 00 "has to be 1"; he draws a distinction between the value 00, which should equal 1, and the limiting form 00 (an abbreviation for a limit of f(t)g(t) where f(t), g(t) → 0), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."[19]
• Other authors leave 00 undefined because 00 is an indeterminate form: f(t), g(t) → 0 does not imply f(t)g(t) → 1.[23][24]
There do not seem to be any authors assigning 00 a specific value other than 1.[22]
Treatment on computers
IEEE floating-point standard
The IEEE 754-2008 floating-point standard is used in the design of most floating-point libraries. It recommends a number of operations for computing a power:[25]
• pown (whose exponent is an integer) treats 00 as 1; see § Discrete exponents.
• pow (whose intent is to return a non-NaN result when the exponent is an integer, like pown) treats 00 as 1.
• powr treats 00 as NaN (Not-a-Number) due to the indeterminate form; see § Continuous exponents.
The pow variant is inspired by the pow function from C99, mainly for compatibility.[26] It is useful mostly for languages with a single power function. The pown and powr variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above).[27]
Programming languages
The C and C++ standards do not specify the result of 00 (a domain error may occur). But for C, as of C99, if the normative annex F is supported, the result for real floating-point types is required to be 1 because there are significant applications for which this value is more useful than NaN[28] (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annex G is supported. The Java standard,[29] the .NET Framework method System.Math.Pow,[30] Julia, and Python[31][32] also treat 00 as 1. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua[33] and Perl's ** operator[34] (where it is explicitly mentioned that the result of 0**0 is platform-dependent).
Mathematical and scientific software
APL, R,[35] Stata, SageMath,[36] Matlab, Magma, GAP, Singular, PARI/GP,[37] and GNU Octave evaluate x0 to 1. Mathematica[38] and Macsyma simplify x0 to 1 even if no constraints are placed on x; however, if 00 is entered directly, it is treated as an error or indeterminate. SageMath does not simplify 0x. Maple, Mathematica[38] and PARI/GP[37][39] further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a 1 of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error.
See also
• 0/0
References
1. Bourbaki, Nicolas (2004). "III.§3.5". Elements of Mathematics, Theory of Sets. Springer-Verlag.
2. Bourbaki, Nicolas (1970). "§III.2 No. 9". Algèbre. Springer. L'unique monôme de degré 0 est l'élément unité de A[(Xi)i∈I]; on l'identifie souvent à l'élément unité 1 de A
3. Bourbaki, Nicolas (1970). "§IV.1 No. 3". Algèbre. Springer.
4. Graham, Ronald; Knuth, Donald; Patashnik, Oren (1989-01-05). "Binomial coefficients". Concrete Mathematics (1st ed.). Addison-Wesley Longman Publishing Co. p. 162. ISBN 0-201-14236-8. Some textbooks leave the quantity 00 undefined, because the functions x0 and 0x have different limiting values when x decreases to 0. But this is a mistake. We must define x0 = 1, for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0x is quite unimportant.
5. Vaughn, Herbert E. (1970). "The expression 00". The Mathematics Teacher. 63: 111–112.
6. Malik, S. C.; Arora, Savita (1992). Mathematical Analysis. New York, USA: Wiley. p. 223. ISBN 978-81-224-0323-7. In general the limit of φ(x)/ψ(x) when x = a in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞, 0 × ∞, ∞ − ∞, 00, 1∞ and ∞0.
7. Paige, L. J. (March 1954). "A note on indeterminate forms". American Mathematical Monthly. 61 (3): 189–190. doi:10.2307/2307224. JSTOR 2307224.
8. Möbius, A. F. (1834). "Beweis der Gleichung 00 = 1, nach J. F. Pfaff" [Proof of the equation 00 = 1, according to J. F. Pfaff]. Journal für die reine und angewandte Mathematik (in German). 1834 (12): 134–136. doi:10.1515/crll.1834.12.134. S2CID 199547186.
9. Baxley, John V.; Hayashi, Elmer K. (June 1978). "Indeterminate Forms of Exponential Type". The American Mathematical Monthly. 85 (6): 484–486. doi:10.2307/2320074. JSTOR 2320074. Retrieved 2021-11-23.
10. Xiao, Jinsen; He, Jianxun (December 2017). "On Indeterminate Forms of Exponential Type". Mathematics Magazine. 90 (5): 371–374. doi:10.4169/math.mag.90.5.371. JSTOR 10.4169/math.mag.90.5.371. S2CID 125602000. Retrieved 2021-11-23.
11. Carrier, George F.; Krook, Max; Pearson, Carl E. (2005). Functions of a Complex Variable: Theory and Technique. p. 15. ISBN 0-89871-595-4. Since log(0) does not exist, 0z is undefined. For Re(z) > 0, we define it arbitrarily as 0.
12. Gonzalez, Mario (1991). Classical Complex Analysis. Chapman & Hall. p. 56. ISBN 0-8247-8415-4. For z = 0, w ≠ 0, we define 0w = 0, while 00 is not defined.
13. Meyerson, Mark D. (June 1996). "The xx Spindle". Mathematics Magazine. Vol. 69, no. 3. pp. 198–206. doi:10.1080/0025570X.1996.11996428. ... Let's start at x = 0. Here xx is undefined.
14. Euler, Leonhard (1988). "Chapter 6, §97". Introduction to analysis of the infinite, Book 1. Translated by Blanton, J. D. Springer. p. 75. ISBN 978-0-387-96824-7.
15. Euler, Leonhard (1988). "Chapter 6, §99". Introduction to analysis of the infinite, Book 1. Translated by Blanton, J. D. Springer. p. 76. ISBN 978-0-387-96824-7.
16. Libri, Guillaume (1833). "Mémoire sur les fonctions discontinues". Journal für die reine und angewandte Mathematik (in French). 1833 (10): 303–316. doi:10.1515/crll.1833.10.303. S2CID 121610886.
17. Euler, Leonhard (1787). Institutiones calculi differentialis, Vol. 2. Ticini. ISBN 978-0-387-96824-7.
18. Libri, Guillaume (1830). "Note sur les valeurs de la fonction 00x". Journal für die reine und angewandte Mathematik (in French). 1830 (6): 67–72. doi:10.1515/crll.1830.6.67. S2CID 121706970.
19. Knuth, Donald E. (1992). "Two Notes on Notation". The American Mathematical Monthly. 99 (5): 403–422. arXiv:math/9205211. Bibcode:1992math......5211K. doi:10.1080/00029890.1992.11995869.
20. Cauchy, Augustin-Louis (1821), Cours d'Analyse de l'École Royale Polytechnique, Oeuvres Complètes: 2 (in French), vol. 3, pp. 65–69
21. Anonymous (1834). "Bemerkungen zu dem Aufsatze überschrieben "Beweis der Gleichung 00 = 1, nach J. F. Pfaff"" [Remarks on the essay "Proof of the equation 00 = 1, according to J. F. Pfaff"]. Journal für die reine und angewandte Mathematik (in German). 1834 (12): 292–294. doi:10.1515/crll.1834.12.292.
22. Benson, Donald C. (1999). Written at New York, USA. The Moment of Proof: Mathematical Epiphanies. Oxford, UK: Oxford University Press. p. 29. ISBN 978-0-19-511721-9.
23. Edwards; Penney (1994). Calculus (4th ed.). Prentice-Hall. p. 466.
24. Keedy; Bittinger; Smith (1982). Algebra Two. Addison-Wesley. p. 32.
25. Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. p. 216. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668. ISBN 978-0-8176-4705-6 (online), ISBN 0-8176-4704-X (print)
26. "More transcendental questions". grouper.ieee.org. Archived from the original on 2017-11-14. Retrieved 2019-05-27. (NB. Beginning of the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)
27. "Re: A vague specification". grouper.ieee.org. Archived from the original on 2017-11-14. Retrieved 2019-05-27. (NB. Suggestion of variants in the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)
28. Rationale for International Standard—Programming Languages—C (PDF) (Report). Revision 5.10. April 2003. p. 182.
29. "Math (Java Platform SE 8) pow". Oracle.
30. ".NET Framework Class Library Math.Pow Method". Microsoft.
31. "Built-in Types — Python 3.8.1 documentation". Retrieved 2020-01-25. Python defines pow(0, 0) and 0 ** 0 to be 1, as is common for programming languages.
32. "math — Mathematical functions — Python 3.8.1 documentation". Retrieved 2020-01-25. Exceptional cases follow Annex 'F' of the C99 standard as far as possible. In particular, pow(1.0, x) and pow(x, 0.0) always return 1.0, even when x is a zero or a NaN.
33. "Lua 5.3 Reference Manual". Retrieved 2019-05-27.
34. "perlop – Exponentiation". Retrieved 2019-05-27.
35. The R Core Team (2023-06-11). "R: A Language and Environment for Statistical Computing – Reference Index" (PDF). Version 4.3.0. p. 25. Retrieved 2019-11-22. 1 ^ y and y ^ 0 are 1, always.
36. The Sage Development Team (2020). "Sage 9.2 Reference Manual: Standard Commutative Rings. Elements of the ring Z of integers". Retrieved 2021-01-21. For consistency with Python and MPFR, 0^0 is defined to be 1 in Sage.
37. "pari.git / commitdiff – 10- x ^ t_FRAC: return an exact result if possible; e.g. 4^(1/2) is now 2". Retrieved 2018-09-10.
38. "Wolfram Language & System Documentation: Power". Wolfram. Retrieved 2018-08-02.
39. The PARI Group (2018). "Users' Guide to PARI/GP (version 2.11.0)" (PDF). pp. 10, 122. Retrieved 2018-09-04. There is also the exponentiation operator ^, when the exponent is of type integer; otherwise, it is considered as a transcendental function. ... If the exponent n is an integer, then exact operations are performed using binary (left-shift) powering techniques. ... If the exponent n is not an integer, powering is treated as the transcendental function exp(n log x).
External links
• sci.math FAQ: What is 00?
• What does 00 (zero to the zeroth power) equal? on AskAMathematician.com
| Wikipedia |
Rectified 7-simplexes
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
7-simplex
Rectified 7-simplex
Birectified 7-simplex
Trirectified 7-simplex
Orthogonal projections in A7 Coxeter plane
There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.
Rectified 7-simplex
Rectified 7-simplex
Typeuniform 7-polytope
Coxeter symbol051
Schläfli symbolr{36} = {35,1}
or $\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}$
Coxeter diagrams
Or
6-faces16
5-faces84
4-faces224
Cells350
Faces336
Edges168
Vertices28
Vertex figure6-simplex prism
Petrie polygonOctagon
Coxeter groupA7, [36], order 40320
Propertiesconvex
The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
7
.
Alternate names
• Rectified octaexon (Acronym: roc) (Jonathan Bowers)
Coordinates
The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Birectified 7-simplex
Birectified 7-simplex
Typeuniform 7-polytope
Coxeter symbol042
Schläfli symbol2r{3,3,3,3,3,3} = {34,2}
or $\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}$
Coxeter diagrams
Or
6-faces16:
8 r{35}
8 2r{35}
5-faces112:
28 {34}
56 r{34}
28 2r{34}
4-faces392:
168 {33}
(56+168) r{33}
Cells770:
(420+70) {3,3}
280 {3,4}
Faces840:
(280+560) {3}
Edges420
Vertices56
Vertex figure{3}x{3,3,3}
Coxeter groupA7, [36], order 40320
Propertiesconvex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
7
. It is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
• Birectified octaexon (Acronym: broc) (Jonathan Bowers)
Coordinates
The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Trirectified 7-simplex
Trirectified 7-simplex
Typeuniform 7-polytope
Coxeter symbol033
Schläfli symbol3r{36} = {33,3}
or $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$
Coxeter diagrams
Or
6-faces16 2r{35}
5-faces112
4-faces448
Cells980
Faces1120
Edges560
Vertices70
Vertex figure{3,3}x{3,3}
Coxeter groupA7×2, [[36]], order 80640
Propertiesconvex, isotopic
The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
7
.
This polytope is the vertex figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names
• Hexadecaexon (Acronym: he) (Jonathan Bowers)
Coordinates
The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.
The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).
Images
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]
Related polytopes
Isotopic uniform truncated simplices
Dim. 2 3 4 5 6 7 8
Name
Coxeter
Hexagon
=
t{3} = {6}
Octahedron
=
r{3,3} = {31,1} = {3,4}
$\left\{{\begin{array}{l}3\\3\end{array}}\right\}$
Decachoron
2t{33}
Dodecateron
2r{34} = {32,2}
$\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$
Tetradecapeton
3t{35}
Hexadecaexon
3r{36} = {33,3}
$\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$
Octadecazetton
4t{37}
Images
Vertex figure ( )∨( )
{ }×{ }
{ }∨{ }
{3}×{3}
{3}∨{3}
{3,3}×{3,3}
{3,3}∨{3,3}
Facets {3} t{3,3} r{3,3,3} 2t{3,3,3,3} 2r{3,3,3,3,3} 3t{3,3,3,3,3,3}
As
intersecting
dual
simplexes
∩
∩
∩
∩
∩ ∩ ∩
Related polytopes
These polytopes are three of 71 uniform 7-polytopes with A7 symmetry.
A7 polytopes
t0
t1
t2
t3
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3
t0,4
t1,4
t2,4
t0,5
t1,5
t0,6
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4
t1,3,4
t2,3,4
t0,1,5
t0,2,5
t1,2,5
t0,3,5
t1,3,5
t0,4,5
t0,1,6
t0,2,6
t0,3,6
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,2,3,4
t1,2,3,4
t0,1,2,5
t0,1,3,5
t0,2,3,5
t1,2,3,5
t0,1,4,5
t0,2,4,5
t1,2,4,5
t0,3,4,5
t0,1,2,6
t0,1,3,6
t0,2,3,6
t0,1,4,6
t0,2,4,6
t0,1,5,6
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,3,4,5
t0,2,3,4,5
t1,2,3,4,5
t0,1,2,3,6
t0,1,2,4,6
t0,1,3,4,6
t0,2,3,4,6
t0,1,2,5,6
t0,1,3,5,6
t0,1,2,3,4,5
t0,1,2,3,4,6
t0,1,2,3,5,6
t0,1,2,4,5,6
t0,1,2,3,4,5,6
See also
• List of A7 polytopes
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Rectified 8-simplexes
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
8-simplex
Rectified 8-simplex
Birectified 8-simplex
Trirectified 8-simplex
Orthogonal projections in A8 Coxeter plane
There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.
Rectified 8-simplex
Rectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol061
Schläfli symbolt1{37}
r{37} = {36,1}
or $\left\{{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams
or
7-faces18
6-faces108
5-faces336
4-faces630
Cells756
Faces588
Edges252
Vertices36
Vertex figure7-simplex prism, {}×{3,3,3,3,3}
Petrie polygonenneagon
Coxeter groupA8, [37], order 362880
Propertiesconvex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
8
. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as .
Coordinates
The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Birectified 8-simplex
Birectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol052
Schläfli symbolt2{37}
2r{37} = {35,2} or
$\left\{{\begin{array}{l}3,3,3,3,3\\3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams
or
7-faces18
6-faces144
5-faces588
4-faces1386
Cells2016
Faces1764
Edges756
Vertices84
Vertex figure{3}×{3,3,3,3}
Coxeter groupA8, [37], order 362880
Propertiesconvex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
8
. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as .
The birectified 8-simplex is the vertex figure of the 152 honeycomb.
Coordinates
The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Trirectified 8-simplex
Trirectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol043
Schläfli symbolt3{37}
3r{37} = {34,3} or
$\left\{{\begin{array}{l}3,3,3,3\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagrams
or
7-faces9 + 9
6-faces36 + 72 + 36
5-faces84 + 252 + 252 + 84
4-faces126 + 504 + 756 + 504
Cells630 + 1260 + 1260
Faces1260 + 1680
Edges1260
Vertices126
Vertex figure{3,3}×{3,3,3}
Petrie polygonenneagon
Coxeter groupA7, [37], order 362880
Propertiesconvex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
8
. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as .
Coordinates
The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Related polytopes
This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.
It is also one of 135 uniform 8-polytopes with A8 symmetry.
A8 polytopes
t0
t1
t2
t3
t01
t02
t12
t03
t13
t23
t04
t14
t24
t34
t05
t15
t25
t06
t16
t07
t012
t013
t023
t123
t014
t024
t124
t034
t134
t234
t015
t025
t125
t035
t135
t235
t045
t145
t016
t026
t126
t036
t136
t046
t056
t017
t027
t037
t0123
t0124
t0134
t0234
t1234
t0125
t0135
t0235
t1235
t0145
t0245
t1245
t0345
t1345
t2345
t0126
t0136
t0236
t1236
t0146
t0246
t1246
t0346
t1346
t0156
t0256
t1256
t0356
t0456
t0127
t0137
t0237
t0147
t0247
t0347
t0157
t0257
t0167
t01234
t01235
t01245
t01345
t02345
t12345
t01236
t01246
t01346
t02346
t12346
t01256
t01356
t02356
t12356
t01456
t02456
t03456
t01237
t01247
t01347
t02347
t01257
t01357
t02357
t01457
t01267
t01367
t012345
t012346
t012356
t012456
t013456
t023456
t123456
t012347
t012357
t012457
t013457
t023457
t012367
t012467
t013467
t012567
t0123456
t0123457
t0123467
t0123567
t01234567
Notes
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "8D Uniform polytopes (polyzetta)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene
External links
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Indeterminate form
In calculus and other branches of mathematical analysis, when the limit of the sum, difference, product, quotient or power of two functions is taken, it may often be possible to simply add, subtract, multiply, divide or exponentiate the corresponding limits of these two functions respectively. However, there are occasions where it is unclear what the sum, difference, product or power of these two limits ought to be. For example, it is unclear what the following expressions ought to evaluate to:[1]
${\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,~0^{0},~1^{\infty },{\text{ and }}\infty ^{0}.$
These seven expressions are known as indeterminate forms. More specifically, such expressions are obtained by naively applying the algebraic limit theorem to evaluate the limit of the corresponding arithmetic operation of two functions, yet there are examples of pairs of functions that after being operated on converge to 0, converge to another finite value, diverge to infinity or just diverge. This inability to decide what the limit ought to be explains why these forms are regarded as indeterminate. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity).[1] The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.
The most common example of an indeterminate form is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by $0/0$. For example, as $x$ approaches $0~$, the ratios $x/x^{3}$, $x/x$, and $x^{2}/x$ go to $\infty $, $1$, and $0~$ respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is $0/0$, which is indeterminate. In this sense, $0/0$ can take on the values $0~$, $1$, or $\infty $, by appropriate choices of functions to put in the numerator and denominator. A pair of functions for which the limit is any particular given value may in fact be found. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, $x\sin(1/x)/x$.
So the fact that two functions $f(x)$ and $g(x)$ converge to $0~$ as $x$ approaches some limit point $c$ is insufficient to determinate the limit
$\lim _{x\to c}{\frac {f(x)}{g(x)}}.$
An expression that arises by ways other than applying the algebraic limit theorem may have the same form of an indeterminate form. However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits. For example, $0/0$ which arises from substituting $0~$ for $x$ in the equation $f(x)=|x|/(|x-1|-1)$ is not an indeterminate form since this expression is not made in the determination of a limit (it is in fact undefined as division by zero). Another example is the expression $0^{0}$. Whether this expression is left undefined, or is defined to equal $1$, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that $0^{\infty }$ and other expressions involving infinity are not indeterminate forms.
Some examples and non-examples
Indeterminate form 0/0
"0/0" redirects here. For the symbol, see Percent sign.
• Fig. 1: y = x/x
• Fig. 2: y = x2/x
• Fig. 3: y = sin x/x
• Fig. 4: y = x − 49/√x − 7 (for x = 49)
• Fig. 5: y = ax/x where a = 2
• Fig. 6: y = x/x3
The indeterminate form $0/0$ is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.
As mentioned above,
$\lim _{x\to 0}{\frac {x}{x}}=1,\qquad $ (see fig. 1)
while
$\lim _{x\to 0}{\frac {x^{2}}{x}}=0,\qquad $ (see fig. 2)
This is enough to show that $0/0$ is an indeterminate form. Other examples with this indeterminate form include
$\lim _{x\to 0}{\frac {\sin(x)}{x}}=1,\qquad $ (see fig. 3)
and
$\lim _{x\to 49}{\frac {x-49}{{\sqrt {x}}\,-7}}=14,\qquad $ (see fig. 4)
Direct substitution of the number that $x$ approaches into any of these expressions shows that these are examples correspond to the indeterminate form $0/0$, but these limits can assume many different values. Any desired value $a$ can be obtained for this indeterminate form as follows:
$\lim _{x\to 0}{\frac {ax}{x}}=a.\qquad $ (see fig. 5)
The value $\infty $ can also be obtained (in the sense of divergence to infinity):
$\lim _{x\to 0}{\frac {x}{x^{3}}}=\infty .\qquad $ (see fig. 6)
Indeterminate form 00
Main article: Zero to the power of zero
• Fig. 7: y = x0
• Fig. 8: y = 0x
The following limits illustrate that the expression $0^{0}$ is an indeterminate form:
$\lim _{x\to 0^{+}}x^{0}=1,\qquad $ (see fig. 7)
$\lim _{x\to 0^{+}}0^{x}=0.\qquad $ (see fig. 8)
Thus, in general, knowing that $\textstyle \lim _{x\to c}f(x)\;=\;0$ and $\textstyle \lim _{x\to c}g(x)\;=\;0$ is not sufficient to evaluate the limit
$\lim _{x\to c}f(x)^{g(x)}.$
If the functions $f$ and $g$ are analytic at $c$, and $f$ is positive for $x$ sufficiently close (but not equal) to $c$, then the limit of $f(x)^{g(x)}$ will be $1$.[2] Otherwise, use the transformation in the table below to evaluate the limit.
Expressions that are not indeterminate forms
The expression $1/0$ is not commonly regarded as an indeterminate form, because if the limit of $f/g$ exists then there is no ambiguity as to its value, as it always diverges. Specifically, if $f$ approaches $1$ and $g$ approaches $0~$, then $f$ and $g$ may be chosen so that:
1. $f/g$ approaches $+\infty $
2. $f/g$ approaches $-\infty $
3. The limit fails to exist.
In each case the absolute value $|f/g|$ approaches $+\infty $, and so the quotient $f/g$ must diverge, in the sense of the extended real numbers (in the framework of the projectively extended real line, the limit is the unsigned infinity $\infty $ in all three cases[3]). Similarly, any expression of the form $a/0$ with $a\neq 0$ (including $a=+\infty $ and $a=-\infty $) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.
The expression $0^{\infty }$ is not an indeterminate form. The expression $0^{+\infty }$ obtained from considering $\lim _{x\to c}f(x)^{g(x)}$ gives the limit $0~$, provided that $f(x)$ remains nonnegative as $x$ approaches $c$. The expression $0^{-\infty }$ is similarly equivalent to $1/0$; if $f(x)>0$ as $x$ approaches $c$, the limit comes out as $+\infty $.
To see why, let $L=\lim _{x\to c}f(x)^{g(x)},$ where $\lim _{x\to c}{f(x)}=0,$ and $\lim _{x\to c}{g(x)}=\infty .$ By taking the natural logarithm of both sides and using $\lim _{x\to c}\ln {f(x)}=-\infty ,$ we get that $\ln L=\lim _{x\to c}({g(x)}\times \ln {f(x)})=\infty \times {-\infty }=-\infty ,$ which means that $L={e}^{-\infty }=0.$
Evaluating indeterminate forms
The adjective indeterminate does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.
Equivalent infinitesimal
When two variables $\alpha $ and $\beta $ converge to zero at the same limit point and $\textstyle \lim {\frac {\beta }{\alpha }}=1$, they are called equivalent infinitesimal (equiv. $\alpha \sim \beta $).
Moreover, if variables $\alpha '$ and $\beta '$ are such that $\alpha \sim \alpha '$ and $\beta \sim \beta '$, then:
$\lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta '}{\alpha '}}$
Here is a brief proof:
Suppose there are two equivalent infinitesimals $\alpha \sim \alpha '$ and $\beta \sim \beta '$.
$\lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta \beta '\alpha '}{\beta '\alpha '\alpha }}=\lim {\frac {\beta }{\beta '}}\lim {\frac {\alpha '}{\alpha }}\lim {\frac {\beta '}{\alpha '}}=\lim {\frac {\beta '}{\alpha '}}$
For the evaluation of the indeterminate form $0/0$, one can make use of the following facts about equivalent infinitesimals (e.g., $x\sim \sin x$ if x becomes closer to zero):[4]
$x\sim \sin x,$
$x\sim \arcsin x,$
$x\sim \sinh x,$
$x\sim \tan x,$
$x\sim \arctan x,$
$x\sim \ln(1+x),$
$1-\cos x\sim {\frac {x^{2}}{2}},$
$\cosh x-1\sim {\frac {x^{2}}{2}},$
$a^{x}-1\sim x\ln a,$
$e^{x}-1\sim x,$
$(1+x)^{a}-1\sim ax.$
For example:
${\begin{aligned}\lim _{x\to 0}{\frac {1}{x^{3}}}\left[\left({\frac {2+\cos x}{3}}\right)^{x}-1\right]&=\lim _{x\to 0}{\frac {e^{x\ln {\frac {2+\cos x}{3}}}-1}{x^{3}}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln {\frac {2+\cos x}{3}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln \left({\frac {\cos x-1}{3}}+1\right)\\&=\lim _{x\to 0}{\frac {\cos x-1}{3x^{2}}}\\&=\lim _{x\to 0}-{\frac {x^{2}}{6x^{2}}}\\&=-{\frac {1}{6}}\end{aligned}}$
In the 2nd equality, $e^{y}-1\sim y$ where $y=x\ln {2+\cos x \over 3}$ as y become closer to 0 is used, and $y\sim \ln {(1+y)}$ where $y={{\cos x-1} \over 3}$ is used in the 4th equality, and $1-\cos x\sim {x^{2} \over 2}$ is used in the 5th equality.
L'Hôpital's rule
Main article: L'Hôpital's rule
L'Hôpital's rule is a general method for evaluating the indeterminate forms $0/0$ and $\infty /\infty $. This rule states that (under appropriate conditions)
$\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}},$
where $f'$ and $g'$ are the derivatives of $f$ and $g$. (Note that this rule does not apply to expressions $\infty /0$, $1/0$, and so on, as these expressions are not indeterminate forms.) These derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.
L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 00:
$\ln \lim _{x\to c}f(x)^{g(x)}=\lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}.$
The right-hand side is of the form $\infty /\infty $, so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved $f$ and $g$ may (or may not) be as long as $f$ is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.)
Although L'Hôpital's rule applies to both $0/0$ and $\infty /\infty $, one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms by transforming $f/g$ to $(1/g)/(1/f)$.
List of indeterminate forms
The following table lists the most common indeterminate forms and the transformations for applying l'Hôpital's rule.
Indeterminate form Conditions Transformation to $0/0$ Transformation to $\infty /\infty $
$0$/$0$ $\lim _{x\to c}f(x)=0,\ \lim _{x\to c}g(x)=0\!$
—
$\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {1/g(x)}{1/f(x)}}\!$
$\infty $/$\infty $ $\lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=\infty \!$ $\lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {1/g(x)}{1/f(x)}}\!$
—
$0\cdot \infty $ $\lim _{x\to c}f(x)=0,\ \lim _{x\to c}g(x)=\infty \!$ $\lim _{x\to c}f(x)g(x)=\lim _{x\to c}{\frac {f(x)}{1/g(x)}}\!$ $\lim _{x\to c}f(x)g(x)=\lim _{x\to c}{\frac {g(x)}{1/f(x)}}\!$
$\infty -\infty $ $\lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=\infty \!$ $\lim _{x\to c}(f(x)-g(x))=\lim _{x\to c}{\frac {1/g(x)-1/f(x)}{1/(f(x)g(x))}}\!$ $\lim _{x\to c}(f(x)-g(x))=\ln \lim _{x\to c}{\frac {e^{f(x)}}{e^{g(x)}}}\!$
$0^{0}$ $\lim _{x\to c}f(x)=0^{+},\lim _{x\to c}g(x)=0\!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!$
$1^{\infty }$ $\lim _{x\to c}f(x)=1,\ \lim _{x\to c}g(x)=\infty \!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!$
$\infty ^{0}$ $\lim _{x\to c}f(x)=\infty ,\ \lim _{x\to c}g(x)=0\!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {g(x)}{1/\ln f(x)}}\!$ $\lim _{x\to c}f(x)^{g(x)}=\exp \lim _{x\to c}{\frac {\ln f(x)}{1/g(x)}}\!$
See also
• Defined and undefined
• Division by zero
• Extended real number line
• Indeterminate equation
• Indeterminate system
• Indeterminate (variable)
• L'Hôpital's rule
References
1. Weisstein, Eric W. "Indeterminate". mathworld.wolfram.com. Retrieved 2019-12-02.
2. Louis M. Rotando; Henry Korn (January 1977). "The indeterminate form 00". Mathematics Magazine. 50 (1): 41–42. doi:10.2307/2689754. JSTOR 2689754.
3. "Undefined vs Indeterminate in Mathematics". www.cut-the-knot.org. Retrieved 2019-12-02.
4. "Table of equivalent infinitesimals" (PDF). Vaxa Software.
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| Wikipedia |
1 + 2 + 3 + 4 + ⋯
The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number
$\sum _{k=1}^{n}k={\frac {n(n+1)}{2}},$
which increases without bound as n goes to infinity. Because the sequence of partial sums fails to converge to a finite limit, the series does not have a sum.
Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of −+1/12, which is expressed by a famous formula:[2]
$1+2+3+4+\cdots =-{\frac {1}{12}},$
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis, quantum field theory, and string theory.[3]
In a monograph on moonshine theory, University of Alberta mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".[4]
Partial sums
Main article: Triangular number
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc. The nth partial sum is given by a simple formula:
$\sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.$
This equation was known to the Pythagoreans as early as the sixth century BCE.[5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series 1 + 2 + 3 + 4 + ⋯ also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the term test.
Summability
Among the classical divergent series, 1 + 2 + 3 + 4 + ⋯ is relatively difficult to manipulate into a finite value. Many summation methods are used to assign numerical values to divergent series, some more powerful than others. For example, Cesàro summation is a well-known method that sums Grandi's series, the mildly divergent series 1 − 1 + 1 − 1 + ⋯, to 1/2. Abel summation is a more powerful method that not only sums Grandi's series to 1/2, but also sums the trickier series 1 − 2 + 3 − 4 + ⋯ to 1/4.
Unlike the above series, 1 + 2 + 3 + 4 + ⋯ is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞.[6] Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value; see below. More advanced methods are required, such as zeta function regularization or Ramanujan summation. It is also possible to argue for the value of −+1/12 using some rough heuristics related to these methods.
Heuristics
Srinivasa Ramanujan presented two derivations of "1 + 2 + 3 + 4 + ⋯ = −+1/12" in chapter 8 of his first notebook.[7][8][9] The simpler, less rigorous derivation proceeds in two steps, as follows.
The first key insight is that the series of positive numbers 1 + 2 + 3 + 4 + ⋯ closely resembles the alternating series 1 − 2 + 3 − 4 + ⋯. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.[10]
In order to transform the series 1 + 2 + 3 + 4 + ⋯ into 1 − 2 + 3 − 4 + ⋯, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is 4 + 8 + 12 + 16 + ⋯, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it c = 1 + 2 + 3 + 4 + ⋯. Then multiply this equation by 4 and subtract the second equation from the first:
${\begin{alignedat}{7}c={}&&1+2&&{}+3+4&&{}+5+6+\cdots \\4c={}&&4&&{}+8&&{}+12+\cdots \\c-4c={}&&1-2&&{}+3-4&&{}+5-6+\cdots \end{alignedat}}$
The second key insight is that the alternating series 1 − 2 + 3 − 4 + ⋯ is the formal power series expansion of the function 1/(1 + x)2 but with x defined as 1. (This can be seen by equating 1/1 + x to the alternating sum of the nonnegative powers of x, and then differentiating and negating both sides of the equation.) Accordingly, Ramanujan writes
$-3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}.$
Dividing both sides by −3, one gets c = −+1/12.
Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, appending a single zero to the front of the series can lead to a different result.[1]
One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.[11] In the series 1 + 2 + 3 + 4 + ⋯, each term n is just a number. If the term n is promoted to a function n−s, where s is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable s can be set to −1 later. The implementation of this strategy is called zeta function regularization.
Zeta function regularization
In zeta function regularization, the series $ \sum _{n=1}^{\infty }n$ is replaced by the series $ \sum _{n=1}^{\infty }n^{-s}$. The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s). On the other hand, the Dirichlet series diverges when the real part of s is less than or equal to 1, so, in particular, the series 1 + 2 + 3 + 4 + ⋯ that results from setting s = –1 does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of s by analytic continuation. One can then define the zeta-regularized sum of 1 + 2 + 3 + 4 + ⋯ to be ζ(−1).
From this point, there are a few ways to prove that ζ(−1) = −+1/12. One method, along the lines of Euler's reasoning,[12] uses the relationship between the Riemann zeta function and the Dirichlet eta function η(s). The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities:
${\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s).\end{alignedat}}$
The identity $(1-2^{1-s})\zeta (s)=\eta (s)$ continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s = −1, one gets −3ζ(−1) = η(−1). Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series,[13] which is a one-sided limit:
$-3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.$
Dividing both sides by −3, one gets ζ(−1) = −+1/12.
Cutoff regularization
The series 1 + 2 + 3 + 4 + ⋯
After smoothing
The method of regularization using a cutoff function can "smooth" the series to arrive at −+1/12. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler–Maclaurin formula. Instead, the method operates directly on conservative transformations of the series, using methods from real analysis.
The idea is to replace the ill-behaved discrete series $\textstyle \sum _{n=0}^{N}n$ with a smoothed version
$\sum _{n=0}^{\infty }nf\left({\frac {n}{N}}\right),$
where f is a cutoff function with appropriate properties. The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that f is smooth, bounded, and compactly supported. One can then prove that this smoothed sum is asymptotic to −+1/12 + CN2, where C is a constant that depends on f. The constant term of the asymptotic expansion does not depend on f: it is necessarily the same value given by analytic continuation, −+1/12.[1]
Ramanujan summation
The Ramanujan sum of 1 + 2 + 3 + 4 + ⋯ is also −+1/12. Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. ... I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = −+1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. ..."[14]
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f, the classical Ramanujan sum of the series $\textstyle \sum _{k=1}^{\infty }f(k)$ is defined as
$c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),$
where f(2k−1) is the (2k − 1)-th derivative of f and B2k is the 2k-th Bernoulli number: B2 = 1/6, B4 = −+1/30, and so on. Setting f(x) = x, the first derivative of f is 1, and every other term vanishes, so[15]
$c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.$
To avoid inconsistencies, the modern theory of Ramanujan summation requires that f is "regular" in the sense that the higher-order derivatives of f decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property.[15] The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like 0 + 2 + 0 + 4 + ⋯, because no regular function takes those values. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.[16]
Failure of stable linear summation methods
See also: 1 + 1 + 1 + 1 + ⋯
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If
$1+2+3+\cdots =x,$
then adding 0 to both sides gives
$0+1+2+3+\cdots =0+x=x$
by stability. By linearity, one may subtract the second equation from the first (subtracting each component of the second line from the first line in columns) to give
$1+1+1+\cdots =x-x=0.$
Adding 0 to both sides again gives
$0+1+1+1+\cdots =0,$
and subtracting the last two series gives
$1+0+0+0+\cdots =0,$
contradicting stability.
Therefore, every method that gives a finite value to the sum 1 + 2 + 3 + ⋯ is not stable or not linear.[17]
Physics
In bosonic string theory, the attempt is to compute the possible energy levels of a string, in particular, the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of D − 2 independent quantum harmonic oscillators, one for each transverse direction, where D is the dimension of spacetime. If the fundamental oscillation frequency is ω, then the energy in an oscillator contributing to the n-th harmonic is nħω/2. So using the divergent series, the sum over all harmonics is −ħω(D − 2)/24. Ultimately it is this fact, combined with the Goddard–Thorn theorem, which leads to bosonic string theory failing to be consistent in dimensions other than 26.[18]
The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension.[19] An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term −1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.[20]
A similar calculation is involved in three dimensions, using the Epstein zeta-function in place of the Riemann zeta function.[21]
History
It is unclear whether Leonhard Euler summed the series to −+1/12. According to Morris Kline, Euler's early work on divergent series relied on function expansions, from which he concluded 1 + 2 + 3 + 4 + ⋯ = ∞.[22] According to Raymond Ayoub, the fact that the divergent zeta series is not Abel-summable prevented Euler from using the zeta function as freely as the eta function, and he "could not have attached a meaning" to the series.[23] Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to negative integers.[24][25][26] In the primary literature, the series 1 + 2 + 3 + 4 + ⋯ is mentioned in Euler's 1760 publication De seriebus divergentibus alongside the divergent geometric series 1 + 2 + 4 + 8 + ⋯. Euler hints that series of this type have finite, negative sums, and he explains what this means for geometric series, but he does not return to discuss 1 + 2 + 3 + 4 + ⋯. In the same publication, Euler writes that the sum of 1 + 1 + 1 + 1 + ⋯ is infinite.[27]
In popular media
David Leavitt's 2007 novel The Indian Clerk includes a scene where Hardy and Littlewood discuss the meaning of this series. They conclude that Ramanujan has rediscovered ζ(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.[28]
Simon McBurney's 2007 play A Disappearing Number focuses on the series in the opening scene. The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling", namely 1 + 2 + 3 + 4 + ⋯ = −+1/12. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. It's terrifying, but it's real."[29][30]
In January 2014, Numberphile produced a YouTube video on the series, which gathered over 1.5 million views in its first month.[31] The 8-minute video is narrated by Tony Padilla, a physicist at the University of Nottingham. Padilla begins with 1 − 1 + 1 − 1 + ⋯ and 1 − 2 + 3 − 4 + ⋯ and relates the latter to 1 + 2 + 3 + 4 + ⋯ using a term-by-term subtraction similar to Ramanujan's argument.[32] Numberphile also released a 21-minute version of the video featuring Nottingham physicist Ed Copeland, who describes in more detail how 1 − 2 + 3 − 4 + ⋯ = 1/4 as an Abel sum, and 1 + 2 + 3 + 4 + ⋯ = −+1/12 as ζ(−1).[33] After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.[34]
In The New York Times coverage of the Numberphile video, mathematician Edward Frenkel commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."[31]
Coverage of this topic in Smithsonian magazine describes the Numberphile video as misleading and notes that the interpretation of the sum as −+1/12 relies on a specialized meaning for the equals sign, from the techniques of analytic continuation, in which equals means is associated with.[35]
References
1. Tao, Terence (April 10, 2010), The Euler–Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014.
2. Lepowsky, J. (1999). "Vertex operator algebras and the zeta function". In Naihuan Jing and Kailash C. Misra (ed.). Recent Developments in Quantum Affine Algebras and Related Topics. Contemporary Mathematics. Vol. 248. pp. 327–340. arXiv:math/9909178. Bibcode:1999math......9178L..
3. Tong, David (February 23, 2012). "String Theory". pp. 28–48. arXiv:0908.0333 [hep-th].
4. Gannon, Terry (April 2010), Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge University Press, p. 140, ISBN 978-0521141888.
5. Pengelley, David J. (2002). "The bridge between the continuous and the discrete via original sources". In Otto Bekken; et al. (eds.). Study the Masters: The Abel-Fauvel Conference. National Center for Mathematics Education, University of Gothenburg, Sweden. p. 3. ISBN 978-9185143009..
6. Hardy 1949, p. 10.
7. Ramanujan's Notebooks, retrieved January 26, 2014
8. Abdi, Wazir Hasan (1992), Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician, National, p. 41
9. Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 135–136
10. Euler, Leonhard (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive. Retrieved 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Mémoires de l'Académie des Sciences de Berlin (in French). 17: 83–106.
11. Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 475–476. ISBN 0-486-66165-2.
12. Stopple, Jeffrey (2003), A Primer of Analytic Number Theory: From Pythagoras to Riemann, p. 202, ISBN 0-521-81309-3.
13. Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. pp. 490–492. ISBN 0-486-66165-2.
14. Aiyangar, Srinivasa Ramanujan (7 September 1995). Ramanujan: Letters and Commentary. p. 53. ISBN 9780821891254.
15. Berndt, Bruce C. (1985), Ramanujan's Notebooks: Part 1, Springer-Verlag, pp. 13, 134.
16. Hardy 1949, p. 346.
17. Natiello, Mario A.; Solari, Hernan Gustavo (July 2015), "On the removal of infinities from divergent series", Philosophy of Mathematics Education Journal, 29: 1–11, hdl:11336/46148.
18. Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", Physics Letters B, 190 (1–2): 137–139, Bibcode:1987PhLB..190..137B, doi:10.1016/0370-2693(87)90854-9.
19. See v:Quantum mechanics/Casimir effect in one dimension.
20. Zee 2003, pp. 65–67.
21. Zeidler, Eberhard (2007), "Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists", Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists, Springer: 305–306, Bibcode:2006qftb.book.....Z, ISBN 9783540347644.
22. Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–314, doi:10.2307/2690371, JSTOR 2690371.
23. Ayoub, Raymond (December 1974), "Euler and the Zeta Function" (PDF), The American Mathematical Monthly, 81 (10): 1067–1086, doi:10.2307/2319041, JSTOR 2319041, retrieved February 14, 2014.
24. Lefort, Jean, "Les séries divergentes chez Euler" (PDF), L'Ouvert (in French), IREM de Strasbourg (31): 15–25, archived from the original (PDF) on February 22, 2014, retrieved February 14, 2014.
25. Kaneko, Masanobu; Kurokawa, Nobushige; Wakayama, Masato (2003), "A variation of Euler's approach to values of the Riemann zeta function" (PDF), Kyushu Journal of Mathematics, 57 (1): 175–192, arXiv:math/0206171, doi:10.2206/kyushujm.57.175, S2CID 54514141, archived from the original (PDF) on 2014-02-02, retrieved January 31, 2014.
26. Sondow, Jonathan (February 1994), "Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series", Proceedings of the American Mathematical Society, 120 (4): 421–424, doi:10.1090/S0002-9939-1994-1172954-7, retrieved February 14, 2014.
27. Barbeau, E. J.; Leah, P. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6.
28. Leavitt, David (2007), The Indian Clerk, Bloomsbury, pp. 61–62.
29. Complicite (April 2012), A Disappearing Number, Oberon, ISBN 9781849432993.
30. Thomas, Rachel (December 1, 2008), "A disappearing number", Plus, retrieved February 5, 2014.
31. Overbye, Dennis (February 3, 2014), "In the End, It All Adds Up to –1/12", The New York Times, retrieved February 3, 2014.
32. ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = –1/12 on YouTube.
33. Sum of Natural Numbers (second proof and extra footage) on YouTube.
34. Padilla, Tony, What do we get if we sum all the natural numbers?, retrieved February 3, 2014.
35. Schultz, Colin (2014-01-31). "The Great Debate Over Whether 1 + 2 + 3 + 4... + ∞ = −1/12". Smithsonian. Retrieved 2016-05-16.
Bibliography
• Berndt, Bruce C.; Srinivasa Ramanujan Aiyangar; Rankin, Robert A. (1995). Ramanujan: letters and commentary. American Mathematical Society. ISBN 0-8218-0287-9.
• Hardy, G. H. (1949). Divergent Series. Clarendon Press.
• Zee, A. (2003). Quantum field theory in a nutshell. Princeton UP. ISBN 0-691-01019-6.
Further reading
• Zwiebach, Barton (2004). A First Course in String Theory. Cambridge UP. ISBN 0-521-83143-1. See p. 293.
• Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.
• Watson, G. N. (April 1929), "Theorems stated by Ramanujan (VIII): Theorems on Divergent Series", Journal of the London Mathematical Society, 1, 4 (2): 82–86, doi:10.1112/jlms/s1-4.14.82
External links
Wikiversity has learning resources about divergent series
• Lamb E. (2014), "Does 1+2+3... Really Equal –1/12?", Scientific American Blogs.
• This Week's Finds in Mathematical Physics (Week 124), (Week 126), (Week 147), (Week 213)
• Euler's Proof That 1 + 2 + 3 + ⋯ = −1/12 – by John Baez
• John Baez (September 19, 2008). "My Favorite Numbers: 24" (PDF).
• The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation by Terence Tao
• A recursive evaluation of zeta of negative integers by Luboš Motl
• Link to Numberphile video 1 + 2 + 3 + 4 + 5 + ... = –1/12
• Sum of Natural Numbers (second proof and extra footage) includes demonstration of Euler's method.
• What do we get if we sum all the natural numbers? response to comments about video by Tony Padilla
• Related article from New York Times
• Why –1/12 is a gold nugget follow-up Numberphile video with Edward Frenkel
• Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12 by Brydon Cais from University of Arizona
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
1 − 2 + 4 − 8 + ⋯
In mathematics, 1 − 2 + 4 − 8 + ⋯ is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2.
$\sum _{k=0}^{n}(-2)^{k}$
As a series of real numbers it diverges, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely 1/3, which is the limit of the series using the 2-adic metric.
Historical arguments
Gottfried Leibniz considered the divergent alternating series 1 − 2 + 4 − 8 + 16 − ⋯ as early as 1673. He argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite:
Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity
Leibniz did not quite assert that the series had a sum, but he did infer an association with 1/3 following Mercator's method.[1][2] The attitude that a series could equal some finite quantity without actually adding up to it as a sum would be commonplace in the 18th century, although no distinction is made in modern mathematics.[3]
After Christian Wolff read Leibniz's treatment of Grandi's series in mid-1712,[4] Wolff was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as 1 − 2 + 4 − 8 + 16 − ⋯. Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either 4m + 1/3 or −4n + 1/3. The mean of these values is 2m − 2n + 1/3, and assuming that m = n at infinity yields 1/3 as the value of the series. Leibniz's intuition prevented him from straining his solution this far, and he wrote back that Wolff's idea was interesting but invalid for several reasons. The arithmetic means of neighboring partial sums do not converge to any particular value, and for all finite cases one has n = 2m, not n = m. Generally, the terms of a summable series should decrease to zero; even 1 − 1 + 1 − 1 + ⋯ could be expressed as a limit of such series. Leibniz counsels Wolff to reconsider so that he "might produce something worthy of science and himself."[5]
Modern methods
Geometric series
Any summation method possessing the properties of regularity, linearity, and stability will sum a geometric series
$\sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}.$
In this case a = 1 and r = −2, so the sum is 1/3.
Euler summation
In his 1755 Institutiones, Leonhard Euler effectively took what is now called the Euler transform of 1 − 2 + 4 − 8 + ⋯, arriving at the convergent series 1/2 − 1/4 + 1/8 − 1/16 + ⋯. Since the latter sums to 1/3, Euler concluded that 1 − 2 + 4 − 8 + ... = 1/3.[6] His ideas on infinite series do not quite follow the modern approach; today one says that 1 − 2 + 4 − 8 + ... is Euler summable and that its Euler sum is 1/3.[7]
The Euler transform begins with the sequence of positive terms:
a0 = 1,
a1 = 2,
a2 = 4,
a3 = 8,...
The sequence of forward differences is then
Δa0 = a1 − a0 = 2 − 1 = 1,
Δa1 = a2 − a1 = 4 − 2 = 2,
Δa2 = a3 − a2 = 8 − 4 = 4,
Δa3 = a4 − a3 = 16 − 8 = 8,...
which is just the same sequence. Hence the iterated forward difference sequences all start with Δna0 = 1 for every n. The Euler transform is the series
${\frac {a_{0}}{2}}-{\frac {\Delta a_{0}}{4}}+{\frac {\Delta ^{2}a_{0}}{8}}-{\frac {\Delta ^{3}a_{0}}{16}}+\cdots ={\frac {1}{2}}-{\frac {1}{4}}+{\frac {1}{8}}-{\frac {1}{16}}+\cdots .$
This is a convergent geometric series whose sum is 1/3 by the usual formula.
Borel summation
The Borel sum of 1 − 2 + 4 − 8 + ⋯ is also 1/3; when Émile Borel introduced the limit formulation of Borel summation in 1896, this was one of his first examples after 1 − 1 + 1 − 1 + ⋯[8]
p-adic numbers
The sequence of partial sums associated with $1-2+4-8\ldots $ in the 2-adic metric is
$1,-1,3,-5,11,\ldots $
and when expressed in base 2 using two's complement,
${\overline {0}}1,{\overline {1}}1,{\overline {0}}11,{\overline {1}}011,{\overline {0}}1011,\ldots $
and the limit of this sequence is ${\overline {01}}1={\frac {1}{3}}$ in the 2-adic metric. Thus $1-2+4-8\ldots ={\frac {1}{3}}$.
See also
• 1 + 2 + 4 + 8 + ⋯
Notes
1. Leibniz pp. 205-207
2. Knobloch pp. 124–125. The quotation is from De progressionibus intervallorum tangentium a vertice, in the original Latin: "Nunc fere cum neutrum liceat, aut potius cum non possit determinari utrum liceat, natura medium eligit, et totum aequatur finito."
3. Ferraro and Panza p. 21
4. Wolff's first reference to the letter published in the Acta Eruditorum appears in a letter written from Halle, Saxony-Anhalt dated 12 June 1712; Gerhardt pp. 143–146.
5. The quotation is Moore's (pp. 2–3) interpretation; Leibniz's letter is in Gerhardt pp. 147–148, dated 13 July 1712 from Hanover.
6. Euler p.234
7. See Korevaar p. 325
8. Smail p. 7.
References
• Euler, Leonhard (1755). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum.
• Ferraro, Giovanni; Panza, Marco (February 2003). "Developing into series and returning from series: A note on the foundations of eighteenth-century analysis". Historia Mathematica. 30 (1): 17–46. doi:10.1016/S0315-0860(02)00017-4.
• Gerhardt, C. I. (1860). Briefwechsel zwischen Leibniz und Christian Wolf aus den handschriften der Koeniglichen Bibliothek zu Hannover. Halle: H. W. Schmidt.
• Knobloch, Eberhard (2006). "Beyond Cartesian limits: Leibniz's passage from algebraic to "transcendental" mathematics". Historia Mathematica. 33: 113–131. doi:10.1016/j.hm.2004.02.001.
• Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
• Leibniz, Gottfried (2003). Probst, S.; Knobloch, E.; Gädeke, N. (eds.). Sämtliche Schriften und Briefe, Reihe 7, Band 3: 1672–1676: Differenzen, Folgen, Reihen. Akademie Verlag. ISBN 3-05-004003-3. Archived from the original on 2013-10-17. Retrieved 2007-03-08.
• Moore, Charles (1938). Summable Series and Convergence Factors. AMS. LCC QA1 .A5225 V.22.
• Smail, Lloyd (1925). History and Synopsis of the Theory of Summable Infinite Processes. University of Oregon Press. LCC QA295 .S64.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
One-form (differential geometry)
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle.[1] Equivalently, a one-form on a manifold $M$ is a smooth mapping of the total space of the tangent bundle of $M$ to $\mathbb {R} $ whose restriction to each fibre is a linear functional on the tangent space.[2] Symbolically,
$\alpha :TM\rightarrow {\mathbb {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbb {R} },$
"One-form" redirects here. Not to be confused with One-form (linear algebra).
where $\alpha _{x}$ is linear.
Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:
$\alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n},$
where the $f_{i}$ are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.
Examples
The most basic non-trivial differential one-form is the "change in angle" form $d\theta .$ This is defined as the derivative of the angle "function" $\theta (x,y)$ (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function. Taking the derivative yields the following formula for the total derivative:
${\begin{aligned}d\theta &=\partial _{x}\left(\operatorname {atan2} (y,x)\right)dx+\partial _{y}\left(\operatorname {atan2} (y,x)\right)dy\\&=-{\frac {y}{x^{2}+y^{2}}}dx+{\frac {x}{x^{2}+y^{2}}}dy\end{aligned}}$
While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative $y$-axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number times $2\pi .$
In the language of differential geometry, this derivative is a one-form, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, that is, a function), and in fact it generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.
Differential of a function
Main article: Differential of a function
Let $U\subseteq \mathbb {R} $ be open (for example, an interval $(a,b)$), and consider a differentiable function $f:U\to \mathbb {R} ,$ with derivative $f'.$ The differential $df$ of $f,$ at a point $x_{0}\in U,$ is defined as a certain linear map of the variable $dx.$ Specifically, $df(x_{0},\cdot ):dx\mapsto f'(x_{0})dx.$ (The meaning of the symbol $dx$ is thus revealed: it is simply an argument, or independent variable, of the linear function $df(x_{0},\cdot ).$) Hence the map $x\mapsto df(x)$ sends each point $x$ to a linear functional $df(x,\cdot ).$ This is the simplest example of a differential (one-)form.
In terms of the de Rham cochain complex, one has an assignment from zero-forms (scalar functions) to one-forms; that is, $f\mapsto df.$
See also
• Differential form – Expression that may appear after an integral sign
• Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
• Reciprocal lattice – Fourier transform of a real-space lattice, important in solid-state physics
• Tensor – Algebraic object with geometric applications
References
1. "2 Introducing Differential Geometry‣ General Relativity by David Tong". www.damtp.cam.ac.uk. Retrieved 2022-10-04.
2. McInerney, Andrew (2013-07-09). First Steps in Differential Geometry: Riemannian, Contact, Symplectic. Springer Science & Business Media. pp. 136–155. ISBN 978-1-4614-7732-7.
Tensors
Glossary of tensor theory
Scope
Mathematics
• Coordinate system
• Differential geometry
• Dyadic algebra
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• Tensor calculus
• Physics
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Notation
• Abstract index notation
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Tensor
definitions
• Tensor (intrinsic definition)
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Operations
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• Hodge star operator
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Related
abstractions
• Affine connection
• Basis
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• Connection form
• Covariance and contravariance of vectors
• Differential form
• Dimension
• Exterior form
• Fiber bundle
• Geodesic
• Levi-Civita connection
• Linear map
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• Matrix
• Multivector
• Pseudotensor
• Spinor
• Vector
• Vector space
Notable tensors
Mathematics
• Kronecker delta
• Levi-Civita symbol
• Metric tensor
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• Ricci curvature
• Riemann curvature tensor
• Torsion tensor
• Weyl tensor
Physics
• Moment of inertia
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• Spin tensor
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• stress–energy tensor
• Einstein tensor
• EM tensor
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Mathematicians
• Élie Cartan
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• Elwin Bruno Christoffel
• Albert Einstein
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• Hermann Grassmann
• Tullio Levi-Civita
• Gregorio Ricci-Curbastro
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Manifolds (Glossary)
Basic concepts
• Topological manifold
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• Smooth atlas
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Main results (list)
• Atiyah–Singer index
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Maps
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• in Lie theory
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Types of
manifolds
• Closed
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Tensors
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• bundle
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Covectors
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• bundle
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• flow
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• Tensor field
• density
• Volume form
• Wedge product
Bundles
• Adjoint
• Affine
• Associated
• Cotangent
• Dual
• Fiber
• (Co) Fibration
• Jet
• Lie algebra
• (Stable) Normal
• Principal
• Spinor
• Subbundle
• Tangent
• Tensor
• Vector
Connections
• Affine
• Cartan
• Ehresmann
• Form
• Generalized
• Koszul
• Levi-Civita
• Principal
• Vector
• Parallel transport
Related
• Classification of manifolds
• Gauge theory
• History
• Morse theory
• Moving frame
• Singularity theory
Generalizations
• Banach manifold
• Diffeology
• Diffiety
• Fréchet manifold
• K-theory
• Orbifold
• Secondary calculus
• over commutative algebras
• Sheaf
• Stratifold
• Supermanifold
• Stratified space
| Wikipedia |
1-planar graph
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of the graph.
Coloring
1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors.[1] Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six.[2] The example of the complete graph K6, which is 1-planar, shows that 1-planar graphs may sometimes require six colors. However, the proof that six colors are always enough is more complicated.
Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and faces of a planar graph in such a way that no two adjacent vertices have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously be done using eight colors by applying the four color theorem to the given graph and its dual graph separately, using two disjoint sets of four colors. However, fewer colors may be obtained by forming an auxiliary graph that has a vertex for each vertex or face of the given planar graph, and in which two auxiliary graph vertices are adjacent whenever they correspond to adjacent features of the given planar graph. A vertex coloring of the auxiliary graph corresponds to a vertex-face coloring of the original planar graph. This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors.[2] The graph K6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a triangular prism, then its eleven vertices and faces require six colors, because no three of them may be given a single color.[3]
Edge density
Every 1-planar graph with n vertices has at most 4n − 8 edges.[4] More strongly, each 1-planar drawing has at most n − 2 crossings; removing one edge from each crossing pair of edges leaves a planar graph, which can have at most 3n − 6 edges, from which the 4n − 8 bound on the number of edges in the original 1-planar graph immediately follows.[5] However, unlike planar graphs (for which all maximal planar graphs on a given vertex set have the same number of edges as each other), there exist maximal 1-planar graphs (graphs to which no additional edges can be added while preserving 1-planarity) that have significantly fewer than 4n − 8 edges.[6] The bound of 4n − 8 on the maximum possible number of edges in a 1-planar graph can be used to show that the complete graph K7 on seven vertices is not 1-planar, because this graph has 21 edges and in this case 4n − 8 = 20 < 21.[7]
A 1-planar graph is said to be an optimal 1-planar graph if it has exactly 4n − 8 edges, the maximum possible. In a 1-planar embedding of an optimal 1-planar graph, the uncrossed edges necessarily form a quadrangulation (a polyhedral graph in which every face is a quadrilateral). Every quadrangulation gives rise to an optimal 1-planar graph in this way, by adding the two diagonals to each of its quadrilateral faces. It follows that every optimal 1-planar graph is Eulerian (all of its vertices have even degree), that the minimum degree in such a graph is six, and that every optimal 1-planar graph has at least eight vertices of degree exactly six. Additionally, every optimal 1-planar graph is 4-vertex-connected, and every 4-vertex cut in such a graph is a separating cycle in the underlying quadrangulation.[8]
The graphs that have straight 1-planar drawings (that is, drawings in which each edge is represented by a line segment, and in which each line segment is crossed by at most one other edge) have a slightly tighter bound of 4n − 9 on the maximum number of edges, achieved by infinitely many graphs.[9]
Complete multipartite graphs
A complete classification of the 1-planar complete graphs, complete bipartite graphs, and more generally complete multipartite graphs is known. Every complete bipartite graph of the form K2,n is 1-planar, as is every complete tripartite graph of the form K1,1,n. Other than these infinite sets of examples, the only complete multipartite 1-planar graphs are K6, K1,1,1,6, K1,1,2,3, K2,2,2,2, K1,1,1,2,2, and their subgraphs. The minimal non-1-planar complete multipartite graphs are K3,7, K4,5, K1,3,4, K2,3,3, and K1,1,1,1,3. For instance, the complete bipartite graph K3,6 is 1-planar because it is a subgraph of K1,1,1,6, but K3,7 is not 1-planar.[7]
Computational complexity
It is NP-complete to test whether a given graph is 1-planar,[10][11] and it remains NP-complete even for the graphs formed from planar graphs by adding a single edge[12] and for graphs of bounded bandwidth.[13] The problem is fixed-parameter tractable when parameterized by cyclomatic number or by tree-depth, so it may be solved in polynomial time when those parameters are bounded.[13]
In contrast to Fáry's theorem for planar graphs, not every 1-planar graph may be drawn 1-planarly with straight line segments for its edges.[14][15] However, testing whether a 1-planar drawing may be straightened in this way can be done in polynomial time.[16] Additionally, every 3-vertex-connected 1-planar graph has a 1-planar drawing in which at most one edge, on the outer face of the drawing, has a bend in it. This drawing can be constructed in linear time from a 1-planar embedding of the graph.[17] The 1-planar graphs have bounded book thickness,[18] but some 1-planar graphs including K2,2,2,2 have book thickness at least four.[19]
1-planar graphs have bounded local treewidth, meaning that there is a (linear) function f such that the 1-planar graphs of diameter d have treewidth at most f(d); the same property holds more generally for the graphs that can be embedded onto a surface of bounded genus with a bounded number of crossings per edge. They also have separators, small sets of vertices the removal of which decomposes the graph into connected components whose size is a constant fraction of the size of the whole graph. Based on these properties, numerous algorithms for planar graphs, such as Baker's technique for designing approximation algorithms, can be extended to 1-planar graphs. For instance, this method leads to a polynomial-time approximation scheme for the maximum independent set of a 1-planar graph.[20]
Generalizations and related concepts
The class of graphs analogous to outerplanar graphs for 1-planarity are called the outer-1-planar graphs. These are graphs that can be drawn in a disk, with the vertices on the boundary of the disk, and with at most one crossing per edge. These graphs can always be drawn (in an outer-1-planar way) with straight edges and right angle crossings.[21] By using dynamic programming on the SPQR tree of a given graph, it is possible to test whether it is outer-1-planar in linear time.[22][23] The triconnected components of the graph (nodes of the SPQR tree) can consist only of cycle graphs, bond graphs, and four-vertex complete graphs, from which it also follows that outer-1-planar graphs are planar and have treewidth at most three.
The 1-planar graphs include the 4-map graphs, graphs formed from the adjacencies of regions in the plane with at most four regions meeting in any point. Conversely, every optimal 1-planar graph is a 4-map graph. However, 1-planar graphs that are not optimal 1-planar may not be map graphs.[24]
1-planar graphs have been generalized to k-planar graphs, graphs for which each edge is crossed at most k times (0-planar graphs are exactly the planar graphs). Ringel defined the local crossing number of G to be the least non-negative integer k such that G has a k-planar drawing. Because the local crossing number is the maximum degree of the intersection graph of the edges of an optimal drawing, and the thickness (minimum number of planar graphs into which the edges can be partitioned) can be seen as the chromatic number of an intersection graph of an appropriate drawing, it follows from Brooks' theorem that the thickness is at most one plus the local crossing number.[25] The k-planar graphs with n vertices have at most O(k1/2n) edges,[26] and treewidth O((kn)1/2).[27] A shallow minor of a k-planar graph, with depth d, is itself a (2d + 1)k-planar graph, so the shallow minors of 1-planar graphs and of k-planar graphs are also sparse graphs, implying that the 1-planar and k-planar graphs have bounded expansion.[28]
Nonplanar graphs may also be parameterized by their crossing number, the minimum number of pairs of edges that cross in any drawing of the graph. A graph with crossing number k is necessarily k-planar, but not necessarily vice versa. For instance, the Heawood graph has crossing number 3, but it is not necessary for its three crossings to all occur on the same edge of the graph, so it is 1-planar, and can in fact be drawn in a way that simultaneously optimizes the total number of crossings and the crossings per edge.
Another related concept for nonplanar graphs is graph skewness, the minimal number of edges that must be removed to make a graph planar.
References
1. Ringel, Gerhard (1965), "Ein Sechsfarbenproblem auf der Kugel", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German), 29 (1–2): 107–117, doi:10.1007/BF02996313, MR 0187232, S2CID 123286264.
2. Borodin, O. V. (1984), "Solution of the Ringel problem on vertex-face coloring of planar graphs and coloring of 1-planar graphs", Metody Diskretnogo Analiza (41): 12–26, 108, MR 0832128.
3. Albertson, Michael O.; Mohar, Bojan (2006), "Coloring vertices and faces of locally planar graphs" (PDF), Graphs and Combinatorics, 22 (3): 289–295, doi:10.1007/s00373-006-0653-4, MR 2264852, S2CID 1028234.
4. Schumacher, H. (1986), "Zur Struktur 1-planarer Graphen", Mathematische Nachrichten (in German), 125: 291–300, doi:10.1002/mana.19861250122, MR 0847368.
5. Czap, Július; Hudák, Dávid (2013), "On drawings and decompositions of 1-planar graphs", Electronic Journal of Combinatorics, 20 (2), P54, doi:10.37236/2392.
6. Brandenburg, Franz Josef; Eppstein, David; Gleißner, Andreas; Goodrich, Michael T.; Hanauer, Kathrin; Reislhuber, Josef (2013), "On the density of maximal 1-planar graphs", in Didimo, Walter; Patrignani, Maurizio (eds.), Proc. 20th Int. Symp. Graph Drawing.
7. Czap, Július; Hudák, Dávid (2012), "1-planarity of complete multipartite graphs", Discrete Applied Mathematics, 160 (4–5): 505–512, doi:10.1016/j.dam.2011.11.014, MR 2876333.
8. Suzuki, Yusuke (2010), "Re-embeddings of maximum 1-planar graphs", SIAM Journal on Discrete Mathematics, 24 (4): 1527–1540, doi:10.1137/090746835, MR 2746706.
9. Didimo, Walter (2013), "Density of straight-line 1-planar graph drawings", Information Processing Letters, 113 (7): 236–240, doi:10.1016/j.ipl.2013.01.013, MR 3017985.
10. Grigoriev, Alexander; Bodlaender, Hans L. (2007), "Algorithms for graphs embeddable with few crossings per edge", Algorithmica, 49 (1): 1–11, doi:10.1007/s00453-007-0010-x, hdl:1874/17980, MR 2344391, S2CID 8174422.
11. Korzhik, Vladimir P.; Mohar, Bojan (2009), "Minimal obstructions for 1-immersions and hardness of 1-planarity testing", in Tollis, Ioannis G.; Patrignani, Maurizio (eds.), Graph Drawing: 16th International Symposium, GD 2008, Heraklion, Crete, Greece, September 21-24, 2008, Revised Papers, Lecture Notes in Computer Science, vol. 5417, Springer, pp. 302–312, arXiv:1110.4881, doi:10.1007/978-3-642-00219-9_29, S2CID 13436158.
12. Cabello, Sergio; Mohar, Bojan (2012), Adding one edge to planar graphs makes crossing number and 1-planarity hard, arXiv:1203.5944, Bibcode:2012arXiv1203.5944C. Expanded version of a paper from the 17th ACM Symposium on Computational Geometry, 2010.
13. Bannister, Michael J.; Cabello, Sergio; Eppstein, David (2013), "Parameterized complexity of 1-planarity", Algorithms and Data Structures Symposium (WADS 2013), arXiv:1304.5591, Bibcode:2013arXiv1304.5591B, doi:10.7155/jgaa.00457, S2CID 4417962.
14. Eggleton, Roger B. (1986), "Rectilinear drawings of graphs", Utilitas Mathematica, 29: 149–172, MR 0846198.
15. Thomassen, Carsten (1988), "Rectilinear drawings of graphs", Journal of Graph Theory, 12 (3): 335–341, doi:10.1002/jgt.3190120306, MR 0956195.
16. Hong, Seok-Hee; Eades, Peter; Liotta, Giuseppe; Poon, Sheung-Hung (2012), "Fáry's theorem for 1-planar graphs", in Gudmundsson, Joachim; Mestre, Julián; Viglas, Taso (eds.), Computing and Combinatorics: 18th Annual International Conference, COCOON 2012, Sydney, Australia, August 20-22, 2012, Proceedings, Lecture Notes in Computer Science, vol. 7434, Springer, pp. 335–346, doi:10.1007/978-3-642-32241-9_29.
17. Alam, Md. Jawaherul; Brandenburg, Franz J.; Kobourov, Stephen G. (2013), "Straight-line grid drawings of 3-connected 1-planar graphs", Graph Drawing: 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers (PDF), Lecture Notes in Computer Science, vol. 8242, pp. 83–94, doi:10.1007/978-3-319-03841-4_8.
18. Bekos, Michael A.; Bruckdorfer, Till; Kaufmann, Michael; Raftopoulou, Chrysanthi (2015), "1-Planar graphs have constant book thickness", Algorithms – ESA 2015, Lecture Notes in Computer Science, vol. 9294, Springer, pp. 130–141, doi:10.1007/978-3-662-48350-3_12.
19. Bekos, Michael; Kaufmann, Michael; Zielke, Christian (2015), "The book embedding problem from a SAT-solving perspective", Proc. 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015), pp. 113–125.
20. Grigoriev & Bodlaender (2007). Grigoriev and Bodlaender state their results only for graphs with a known 1-planar embedding, and use a tree decomposition of a planarization of the embedding with crossings replaced by degree-four vertices; however, their methods straightforwardly imply bounded local treewidth of the original 1-planar graph, allowing Baker's method to be applied directly to it without knowing the embedding.
21. Dehkordi, Hooman Reisi; Eades, Peter (2012), "Every outer-1-plane graph has a right angle crossing drawing", International Journal of Computational Geometry & Applications, 22 (6): 543–557, doi:10.1142/S021819591250015X, MR 3042921.
22. Hong, Seok-Hee; Eades, Peter; Katoh, Naoki; Liotta, Giuseppe; Schweitzer, Pascal; Suzuki, Yusuke (2013), "A linear-time algorithm for testing outer-1-planarity", in Wismath, Stephen; Wolff, Alexander (eds.), 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, Lecture Notes in Computer Science, vol. 8242, pp. 71–82, doi:10.1007/978-3-319-03841-4_7.
23. Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Gleißner, Andreas; Hanauer, Kathrin; Neuwirth, Daniel; Reislhuber, Josef (2013), "Recognizing outer 1-planar graphs in linear time", in Wismath, Stephen; Wolff, Alexander (eds.), 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, Lecture Notes in Computer Science, vol. 8242, pp. 107–118, doi:10.1007/978-3-319-03841-4_10.
24. Chen, Zhi-Zhong; Grigni, Michelangelo; Papadimitriou, Christos H. (2002), "Map graphs", Journal of the ACM, 49 (2): 127–138, arXiv:cs/9910013, doi:10.1145/506147.506148, MR 2147819, S2CID 2657838.
25. Kainen, Paul (1973), "Thickness and coarseness of graphs", Abh. Math. Sem. Univ. Hamburg, 39: 88–95, doi:10.1007/BF02992822, MR 0335322, S2CID 121667358.
26. Pach, János; Tóth, Géza (1997), "Graphs drawn with few crossings per edge", Combinatorica, 17 (3): 427–439, doi:10.1007/BF01215922, MR 1606052, S2CID 20480170.
27. Dujmović, Vida; Eppstein, David; Wood, David R. (2015), "Genus, treewidth, and local crossing number", Proc. 23rd International Symposium on Graph Drawing and Network Visualization (GD 2015), pp. 77–88, arXiv:1506.04380, Bibcode:2015arXiv150604380D.
28. Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28, Springer, Theorem 14.4, p. 321, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058.
Further reading
• Kobourov, Stephen; Liotta, Giuseppe; Montecchiani, Fabrizio (2017), "An annotated bibliography on 1-planarity", Computer Science Review, 25: 49–67, arXiv:1703.02261, Bibcode:2017arXiv170302261K, doi:10.1016/j.cosrev.2017.06.002, S2CID 7732463
| Wikipedia |
Square root of 2
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as ${\sqrt {2}}$ or $2^{1/2}$. It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property.
"Pythagoras's constant" redirects here. Not to be confused with Pythagoras number.
Square root of 2
The square root of 2 is equal to the length of the hypotenuse of an isosceles right triangle with legs of length 1.
Representations
Decimal1.4142135623730950488...
Continued fraction$1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}}}$
Binary1.01101010000010011110...
Hexadecimal1.6A09E667F3BCC908B2F...
Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length;[1] this follows from the Pythagorean theorem. It was probably the first number known to be irrational.[2] The fraction 99/70 (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator.
Sequence A002193 in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places:[3]
1.41421356237309504880168872420969807856967187537694807317667973799
History
The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of ${\sqrt {2}}$ in four sexagesimal figures, 1 24 51 10, which is accurate to about six decimal digits,[4] and is the closest possible three-place sexagesimal representation of ${\sqrt {2}}$:
$1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}={\frac {305470}{216000}}=1.41421{\overline {296}}.$
Another early approximation is given in ancient Indian mathematical texts, the Sulbasutras (c. 800–200 BC), as follows: Increase the length [of the side] by its third and this third by its own fourth less the thirty-fourth part of that fourth.[5] That is,
$1+{\frac {1}{3}}+{\frac {1}{3\times 4}}-{\frac {1}{3\times 4\times 34}}={\frac {577}{408}}=1.41421{\overline {56862745098039}}.$
This approximation is the seventh in a sequence of increasingly accurate approximations based on the sequence of Pell numbers, which can be derived from the continued fraction expansion of ${\sqrt {2}}$. Despite having a smaller denominator, it is only slightly less accurate than the Babylonian approximation.
Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned. For a while, the Pythagoreans treated as an official secret the discovery that the square root of two is irrational, and, according to legend, Hippasus was murdered for divulging it.[1][6][7][8] The square root of two is occasionally called Pythagoras's number or Pythagoras's constant, for example by Conway & Guy (1996).[9]
Ancient Roman architecture
In ancient Roman architecture, Vitruvius describes the use of the square root of 2 progression or ad quadratum technique. It consists basically in a geometric, rather than arithmetic, method to double a square, in which the diagonal of the original square is equal to the side of the resulting square. Vitruvius attributes the idea to Plato. The system was employed to build pavements by creating a square tangent to the corners of the original square at 45 degrees of it. The proportion was also used to design atria by giving them a length equal to a diagonal taken from a square, whose sides are equivalent to the intended atrium's width.[10]
Decimal value
Computation algorithms
Further information: Methods of computing square roots
There are many algorithms for approximating ${\sqrt {2}}$ as a ratio of integers or as a decimal. The most common algorithm for this, which is used as a basis in many computers and calculators, is the Babylonian method[11] for computing square roots. It goes as follows:
First, pick a guess, $a_{0}>0$; the value of the guess affects only how many iterations are required to reach an approximation of a certain accuracy. Then, using that guess, iterate through the following recursive computation:
$a_{n+1}={\frac {a_{n}+{\frac {2}{a_{n}}}}{2}}={\frac {a_{n}}{2}}+{\frac {1}{a_{n}}}.$
The more iterations through the algorithm (that is, the more computations performed and the greater "n"), the better the approximation. Each iteration roughly doubles the number of correct digits. Starting with $a_{0}=1$, the results of the algorithm are as follows:
• 1 (a0)
• 3/2 = 1.5 (a1)
• 17/12 = 1.416... (a2)
• 577/408 = 1.414215... (a3)
• 665857/470832 = 1.4142135623746... (a4)
Rational approximations
A simple rational approximation 99/70 (≈ 1.4142857) is sometimes used. Despite having a denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. +0.72×10−4).
The next two better rational approximations are 140/99 (≈ 1.4141414...) with a marginally smaller error (approx. −0.72×10−4), and 239/169 (≈ 1.4142012) with an error of approx −0.12×10−4.
The rational approximation of the square root of two derived from four iterations of the Babylonian method after starting with a0 = 1 (665,857/470,832) is too large by about 1.6×10−12; its square is ≈ 2.0000000000045.
Records in computation
In 1997, the value of ${\sqrt {2}}$ was calculated to 137,438,953,444 decimal places by Yasumasa Kanada's team. In February 2006, the record for the calculation of ${\sqrt {2}}$ was eclipsed with the use of a home computer. Shigeru Kondo calculated one trillion decimal places in 2010.[12] Among mathematical constants with computationally challenging decimal expansions, only π, e, and the golden ratio have been calculated more precisely as of March 2022.[13] Such computations aim to check empirically whether such numbers are normal.
This is a table of recent records in calculating the digits of ${\sqrt {2}}$.[13]
DateNameNumber of digits
January 5, 2022Tizian Hanselmann10000000001000
June 28, 2016Ron Watkins10000000000000
April 3, 2016Ron Watkins5000000000000
January 20, 2016Ron Watkins2000000000100
February 9, 2012Alexander Yee2000000000050
March 22, 2010Shigeru Kondo1000000000000
Proofs of irrationality
A short proof of the irrationality of ${\sqrt {2}}$ can be obtained from the rational root theorem, that is, if $p(x)$ is a monic polynomial with integer coefficients, then any rational root of $p(x)$ is necessarily an integer. Applying this to the polynomial $p(x)=x^{2}-2$, it follows that ${\sqrt {2}}$ is either an integer or irrational. Because ${\sqrt {2}}$ is not an integer (2 is not a perfect square), ${\sqrt {2}}$ must therefore be irrational. This proof can be generalized to show that any square root of any natural number that is not a perfect square is irrational.
For other proofs that the square root of any non-square natural number is irrational, see Quadratic irrational number or Infinite descent.
Proof by infinite descent
One proof of the number's irrationality is the following proof by infinite descent. It is also a proof of a negation by refutation: it proves the statement "${\sqrt {2}}$ is not rational" by assuming that it is rational and then deriving a falsehood.
1. Assume that ${\sqrt {2}}$ is a rational number, meaning that there exists a pair of integers whose ratio is exactly ${\sqrt {2}}$.
2. If the two integers have a common factor, it can be eliminated using the Euclidean algorithm.
3. Then ${\sqrt {2}}$ can be written as an irreducible fraction ${\frac {a}{b}}$ such that a and b are coprime integers (having no common factor) which additionally means that at least one of a or b must be odd.
4. It follows that ${\frac {a^{2}}{b^{2}}}=2$ and $a^{2}=2b^{2}$. ( (a/b)n = an/bn ) ( a2 and b2 are integers)
5. Therefore, a2 is even because it is equal to 2b2. (2b2 is necessarily even because it is 2 times another whole number.)
6. It follows that a must be even (as squares of odd integers are never even).
7. Because a is even, there exists an integer k that fulfills $a=2k$.
8. Substituting 2k from step 7 for a in the second equation of step 4: $2b^{2}=a^{2}=(2k)^{2}=4k^{2}$, which is equivalent to $b^{2}=2k^{2}$.
9. Because 2k2 is divisible by two and therefore even, and because $2k^{2}=b^{2}$, it follows that b2 is also even which means that b is even.
10. By steps 5 and 8, a and b are both even, which contradicts step 3 (that ${\frac {a}{b}}$ is irreducible).
Since we have derived a falsehood, the assumption (1) that ${\sqrt {2}}$ is a rational number must be false. This means that ${\sqrt {2}}$ is not a rational number; that is to say, ${\sqrt {2}}$ is irrational.
This proof was hinted at by Aristotle, in his Analytica Priora, §I.23.[14] It appeared first as a full proof in Euclid's Elements, as proposition 117 of Book X. However, since the early 19th century, historians have agreed that this proof is an interpolation and not attributable to Euclid.[15]
Proof by unique factorization
As with the proof by infinite descent, we obtain $a^{2}=2b^{2}$. Being the same quantity, each side has the same prime factorization by the fundamental theorem of arithmetic, and in particular, would have to have the factor 2 occur the same number of times. However, the factor 2 appears an odd number of times on the right, but an even number of times on the left—a contradiction.
Geometric proof
A simple proof is attributed to Stanley Tennenbaum when he was a student in the early 1950s.[16][17] Given two squares with integer sides respectively a and b, one of which has twice the area of the other, place two copies of the smaller square in the larger as shown in Figure 1. The square overlap region in the middle ($(2b-a)^{2}$) must equal the sum of the two uncovered squares ($2(a-b)^{2}$). However, these squares on the diagonal have positive integer sides that are smaller than the original squares. Repeating this process, there are arbitrarily small squares one twice the area of the other, yet both having positive integer sides, which is impossible since positive integers cannot be less than 1.
Tom M. Apostol made another geometric reductio ad absurdum argument showing that ${\sqrt {2}}$ is irrational.[18] It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the same algebraic proof as in the previous paragraph, viewed geometrically in another way.
Let △ ABC be a right isosceles triangle with hypotenuse length m and legs n as shown in Figure 2. By the Pythagorean theorem, ${\frac {m}{n}}={\sqrt {2}}$. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and ∠BAC and ∠DAE coincide. Therefore, the triangles ABC and ADE are congruent by SAS.
Because ∠EBF is a right angle and ∠BEF is half a right angle, △ BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n. By symmetry, DF = m − n, and △ FDC is also a right isosceles triangle. It also follows that FC = n − (m − n) = 2n − m.
Hence, there is an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore, m and n cannot be both integers; hence, ${\sqrt {2}}$ is irrational.
Constructive proof
While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let a and b be positive integers such that 1<a/b< 3/2 (as 1<2< 9/4 satisfies these bounds). Now 2b2 and a2 cannot be equal, since the first has an odd number of factors 2 whereas the second has an even number of factors 2. Thus |2b2 − a2| ≥ 1. Multiplying the absolute difference |√2 − a/b| by b2(√2 + a/b) in the numerator and denominator, we get[19]
$\left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac {|2b^{2}-a^{2}|}{b^{2}\!\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac {1}{b^{2}\!\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac {1}{3b^{2}}},$
the latter inequality being true because it is assumed that 1<a/b< 3/2, giving a/b + √2 ≤ 3 (otherwise the quantitative apartness can be trivially established). This gives a lower bound of 1/3b2 for the difference |√2 − a/b|, yielding a direct proof of irrationality in its constructively stronger form, not relying on the law of excluded middle; see Errett Bishop (1985, p. 18). This proof constructively exhibits an explicit discrepancy between ${\sqrt {2}}$ and any rational.
Proof by Pythagorean triples
This proof uses the following property of primitive Pythagorean triples:
If a, b, and c are coprime positive integers such that a2 + b2 = c2, then c is never even.[20]
This lemma can be used to show that two identical perfect squares can never be added to produce another perfect square.
Suppose the contrary that ${\sqrt {2}}$ is rational. Therefore,
${\sqrt {2}}={a \over b}$
where $a,b\in \mathbb {Z} $ and $\gcd(a,b)=1$
Squaring both sides,
$2={a^{2} \over b^{2}}$
$2b^{2}=a^{2}$
$b^{2}+b^{2}=a^{2}$
Here, (b, b, a) is a primitive Pythagorean triple, and from the lemma a is never even. However, this contradicts the equation 2b2 = a2 which implies that a must be even.
Multiplicative inverse
The multiplicative inverse (reciprocal) of the square root of two (i.e., the square root of 1/2) is a widely used constant.
${\frac {1}{\sqrt {2}}}={\frac {\sqrt {2}}{2}}=\sin 45^{\circ }=\cos 45^{\circ }=$ 0.70710678118654752440084436210484903928483593768847... (sequence A010503 in the OEIS)
One-half of ${\sqrt {2}}$, also the reciprocal of ${\sqrt {2}}$, is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates
$\left({\frac {\sqrt {2}}{2}},{\frac {\sqrt {2}}{2}}\right)\!.$
This number satisfies
${\tfrac {1}{2}}{\sqrt {2}}={\sqrt {\tfrac {1}{2}}}={\frac {1}{\sqrt {2}}}=\cos 45^{\circ }=\sin 45^{\circ }.$
Properties
One interesting property of ${\sqrt {2}}$ is
$\!\ {1 \over {{\sqrt {2}}-1}}={\sqrt {2}}+1$
since
$\left({\sqrt {2}}+1\right)\!\left({\sqrt {2}}-1\right)=2-1=1.$
This is related to the property of silver ratios.
${\sqrt {2}}$ can also be expressed in terms of copies of the imaginary unit i using only the square root and arithmetic operations, if the square root symbol is interpreted suitably for the complex numbers i and −i:
${\frac {{\sqrt {i}}+i{\sqrt {i}}}{i}}{\text{ and }}{\frac {{\sqrt {-i}}-i{\sqrt {-i}}}{-i}}$
${\sqrt {2}}$ is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1, x1 = c and xn+1 = cxn for n > 1, the limit of xn as n → ∞ will be called (if this limit exists) f(c). Then ${\sqrt {2}}$ is the only number c > 1 for which f(c) = c2. Or symbolically:
${\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{~\cdot ^{~\cdot ^{~\cdot }}}}}=2.$
${\sqrt {2}}$ appears in Viète's formula for π:
$2^{m}{\sqrt {2-{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}\to \pi {\text{ as }}m\to \infty $
for m square roots and only one minus sign.[21]
Similar in appearance but with a finite number of terms, ${\sqrt {2}}$ appears in various trigonometric constants:[22]
${\begin{aligned}\sin {\frac {\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {3\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}&\quad \sin {\frac {11\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}\\[6pt]\sin {\frac {\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}&\quad \sin {\frac {7\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {3\pi }{8}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\\[6pt]\sin {\frac {3\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}&\quad \sin {\frac {\pi }{4}}&={\tfrac {1}{2}}{\sqrt {2}}&\quad \sin {\frac {13\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}\\[6pt]\sin {\frac {\pi }{8}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2}}}}&\quad \sin {\frac {9\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}&\quad \sin {\frac {7\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\\[6pt]\sin {\frac {5\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}&\quad \sin {\frac {5\pi }{16}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}&\quad \sin {\frac {15\pi }{32}}&={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}\end{aligned}}$
It is not known whether ${\sqrt {2}}$ is a normal number, which is a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.[23]
Representations
Series and product
The identity cos π/4 = sin π/4 = 1/√2, along with the infinite product representations for the sine and cosine, leads to products such as
${\frac {1}{\sqrt {2}}}=\prod _{k=0}^{\infty }\left(1-{\frac {1}{(4k+2)^{2}}}\right)=\left(1-{\frac {1}{4}}\right)\!\left(1-{\frac {1}{36}}\right)\!\left(1-{\frac {1}{100}}\right)\cdots $
and
${\sqrt {2}}=\prod _{k=0}^{\infty }{\frac {(4k+2)^{2}}{(4k+1)(4k+3)}}=\left({\frac {2\cdot 2}{1\cdot 3}}\right)\!\left({\frac {6\cdot 6}{5\cdot 7}}\right)\!\left({\frac {10\cdot 10}{9\cdot 11}}\right)\!\left({\frac {14\cdot 14}{13\cdot 15}}\right)\cdots $
or equivalently,
${\sqrt {2}}=\prod _{k=0}^{\infty }\left(1+{\frac {1}{4k+1}}\right)\left(1-{\frac {1}{4k+3}}\right)=\left(1+{\frac {1}{1}}\right)\!\left(1-{\frac {1}{3}}\right)\!\left(1+{\frac {1}{5}}\right)\!\left(1-{\frac {1}{7}}\right)\cdots .$
The number can also be expressed by taking the Taylor series of a trigonometric function. For example, the series for cos π/4 gives
${\frac {1}{\sqrt {2}}}=\sum _{k=0}^{\infty }{\frac {(-1)^{k}\left({\frac {\pi }{4}}\right)^{2k}}{(2k)!}}.$
The Taylor series of √1 + x with x = 1 and using the double factorial n!! gives
${\sqrt {2}}=\sum _{k=0}^{\infty }(-1)^{k+1}{\frac {(2k-3)!!}{(2k)!!}}=1+{\frac {1}{2}}-{\frac {1}{2\cdot 4}}+{\frac {1\cdot 3}{2\cdot 4\cdot 6}}-{\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8}}+\cdots =1+{\frac {1}{2}}-{\frac {1}{8}}+{\frac {1}{16}}-{\frac {5}{128}}+{\frac {7}{256}}+\cdots .$
The convergence of this series can be accelerated with an Euler transform, producing
${\sqrt {2}}=\sum _{k=0}^{\infty }{\frac {(2k+1)!}{2^{3k+1}(k!)^{2}}}={\frac {1}{2}}+{\frac {3}{8}}+{\frac {15}{64}}+{\frac {35}{256}}+{\frac {315}{4096}}+{\frac {693}{16384}}+\cdots .$
It is not known whether ${\sqrt {2}}$ can be represented with a BBP-type formula. BBP-type formulas are known for π√2 and √2 ln(1+√2), however.[24]
The number can be represented by an infinite series of Egyptian fractions, with denominators defined by 2n th terms of a Fibonacci-like recurrence relation a(n) = 34a(n−1) − a(n−2), a(0) = 0, a(1) = 6.[25]
${\sqrt {2}}={\frac {3}{2}}-{\frac {1}{2}}\sum _{n=0}^{\infty }{\frac {1}{a(2^{n})}}={\frac {3}{2}}-{\frac {1}{2}}\left({\frac {1}{6}}+{\frac {1}{204}}+{\frac {1}{235416}}+\dots \right)$
Continued fraction
The square root of two has the following continued fraction representation:
${\sqrt {2}}=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}.$
The convergents p/q formed by truncating this representation form a sequence of fractions that approximate the square root of two to increasing accuracy, and that are described by the Pell numbers (i.e., p2 − 2q2 = ±1). The first convergents are: 1/1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408 and the convergent following p/q is p + 2q/p + q. The convergent p/q differs from ${\sqrt {2}}$ by almost exactly 1/2√2q2, which follows from:
$\left|{\sqrt {2}}-{\frac {p}{q}}\right|={\frac {|2q^{2}-p^{2}|}{q^{2}\!\left({\sqrt {2}}+{\frac {p}{q}}\right)}}={\frac {1}{q^{2}\!\left({\sqrt {2}}+{\frac {p}{q}}\right)}}\thickapprox {\frac {1}{2{\sqrt {2}}q^{2}}}$
Nested square
The following nested square expressions converge to $ {\sqrt {2}}$:
${\begin{aligned}{\sqrt {2}}&={\tfrac {3}{2}}-2\left({\tfrac {1}{4}}-\left({\tfrac {1}{4}}-{\bigl (}{\tfrac {1}{4}}-\cdots {\bigr )}^{2}\right)^{2}\right)^{2}\\[10mu]&={\tfrac {3}{2}}-4\left({\tfrac {1}{8}}+\left({\tfrac {1}{8}}+{\bigl (}{\tfrac {1}{8}}+\cdots {\bigr )}^{2}\right)^{2}\right)^{2}.\end{aligned}}$
Applications
Paper size
In 1786, German physics professor Georg Christoph Lichtenberg[26] found that any sheet of paper whose long edge is ${\sqrt {2}}$ times longer than its short edge could be folded in half and aligned with its shorter side to produce a sheet with exactly the same proportions as the original. This ratio of lengths of the longer over the shorter side guarantees that cutting a sheet in half along a line results in the smaller sheets having the same (approximate) ratio as the original sheet. When Germany standardised paper sizes at the beginning of the 20th century, they used Lichtenberg's ratio to create the "A" series of paper sizes.[26] Today, the (approximate) aspect ratio of paper sizes under ISO 216 (A4, A0, etc.) is 1:${\sqrt {2}}$.
Proof:
Let $S=$ shorter length and $L=$ longer length of the sides of a sheet of paper, with
$R={\frac {L}{S}}={\sqrt {2}}$ as required by ISO 216.
Let $R'={\frac {L'}{S'}}$ be the analogous ratio of the halved sheet, then
$R'={\frac {S}{L/2}}={\frac {2S}{L}}={\frac {2}{(L/S)}}={\frac {2}{\sqrt {2}}}={\sqrt {2}}=R$.
Physical sciences
There are some interesting properties involving the square root of 2 in the physical sciences:
• The square root of two is the frequency ratio of a tritone interval in twelve-tone equal temperament music.
• The square root of two forms the relationship of f-stops in photographic lenses, which in turn means that the ratio of areas between two successive apertures is 2.
• The celestial latitude (declination) of the Sun during a planet's astronomical cross-quarter day points equals the tilt of the planet's axis divided by ${\sqrt {2}}$.
See also
• List of mathematical constants
• Square root of 3, √3
• Square root of 5, √5
• Gelfond–Schneider constant, 2√2
• Silver ratio, 1 + √2
Notes
1. Weisstein, Eric W. "Pythagoras's Constant". mathworld.wolfram.com. Retrieved 2020-08-10.
2. Fowler, David H. (2001), "The story of the discovery of incommensurability, revisited", Neusis (10): 45–61, MR 1891736
3. "A002193 - OEIS". oeis.org. Retrieved 2020-08-10.
4. Fowler and Robson, p. 368.
Photograph, illustration, and description of the root(2) tablet from the Yale Babylonian Collection Archived 2012-08-13 at the Wayback Machine
High resolution photographs, descriptions, and analysis of the root(2) tablet (YBC 7289) from the Yale Babylonian Collection
5. Henderson.
6. Stephanie J. Morris, "The Pythagorean Theorem" Archived 2013-05-30 at the Wayback Machine, Dept. of Math. Ed., University of Georgia.
7. Brian Clegg, "The Dangerous Ratio ..." Archived 2013-06-27 at the Wayback Machine, Nrich.org, November 2004.
8. Kurt von Fritz, "The discovery of incommensurability by Hippasus of Metapontum", Annals of Mathematics, 1945.
9. Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, p. 25
10. Williams, Kim; Ostwald, Michael (2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser. p. 204. ISBN 9783319001371.
11. Although the term "Babylonian method" is common in modern usage, there is no direct evidence showing how the Babylonians computed the approximation of ${\sqrt {2}}$ seen on tablet YBC 7289. Fowler and Robson offer informed and detailed conjectures.
Fowler and Robson, p. 376. Flannery, p. 32, 158.
12. "Constants and Records of Computation". Numbers.computation.free.fr. 2010-08-12. Archived from the original on 2012-03-01. Retrieved 2012-09-07.
13. "Records set by y-cruncher". Archived from the original on 2022-04-07. Retrieved 2022-04-07.
14. All that Aristotle says, while writing about proofs by contradiction, is that "the diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate".
15. The edition of the Greek text of the Elements published by E. F. August in Berlin in 1826–1829 already relegates this proof to an Appendix. The same thing occurs with J. L. Heiberg's edition (1883–1888).
16. Proof 8‴ Archived 2016-04-22 at the Wayback Machine
17. Yanofsky, N. (2016). "Paradoxes, Contradictions, and the Limits of Science". Archived from the original on 2016-06-30.
18. Tom M. Apostol (Nov 2000), "Irrationality of The Square Root of Two -- A Geometric Proof", The American Mathematical Monthly, 107 (9): 841–842, doi:10.2307/2695741, JSTOR 2695741
19. See Katz, Karin Usadi; Katz, Mikhail G. (2011), "Meaning in Classical Mathematics: Is it at Odds with Intuitionism?", Intellectica, 56 (2): 223–302 (see esp. Section 2.3, footnote 15), arXiv:1110.5456, Bibcode:2011arXiv1110.5456U
20. Sierpiński, Wacław (2003), Pythagorean Triangles, Dover, pp. 4–6, ISBN 978-0-486-43278-6
21. Courant, Richard; Robbins, Herbert (1941), What is mathematics? An Elementary Approach to Ideas and Methods, London: Oxford University Press, p. 124
22. Julian D. A. Wiseman Sin and cos in surds Archived 2009-05-06 at the Wayback Machine
23. Good & Gover (1967).
24. Bailey, David H. (13 February 2011). "A Compendium of BBP-Type Formulas for Mathematical Constants" (PDF). Archived (PDF) from the original on 2011-06-10. Retrieved 2010-04-30.
25. Sloane, N. J. A. (ed.). "Sequence A082405 (a(n) = 34*a(n-1) - a(n-2); a(0)=0, a(1)=6)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-09-05.
26. Houston, Keith (2016). The Book: A Cover-to-Cover Exploration of the Most Powerful Object of Our Time. W. W. Norton & Company. p. 324. ISBN 978-0393244809.
References
• Apostol, Tom M. (2000), "Irrationality of the square root of two – A geometric proof", American Mathematical Monthly, 107 (9): 841–842, doi:10.2307/2695741, JSTOR 2695741.
• Aristotle (2007), Analytica priora, eBooks@Adelaide
• Bishop, Errett (1985), Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math. 39, Amer. Math. Soc., Providence, RI.
• Flannery, David (2005), The Square Root of Two, Springer-Verlag, ISBN 0-387-20220-X.
• Fowler, David; Robson, Eleanor (1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context", Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209.
• Good, I. J.; Gover, T. N. (1967), "The generalized serial test and the binary expansion of ${\sqrt {2}}$", Journal of the Royal Statistical Society, Series A, 130 (1): 102–107, doi:10.2307/2344040, JSTOR 2344040.
• Henderson, David W. (2000), "Square roots in the Śulba Sūtras", in Gorini, Catherine A. (ed.), Geometry At Work: Papers in Applied Geometry, Cambridge University Press, pp. 39–45, ISBN 978-0-88385-164-7.
External links
• Gourdon, X.; Sebah, P. (2001), "Pythagoras' Constant: ${\sqrt {2}}$", Numbers, Constants and Computation.
• The Square Root of Two to 5 million digits by Jerry Bonnell and Robert J. Nemiroff. May, 1994.
• Square root of 2 is irrational, a collection of proofs
• Grime, James; Bowley, Roger. "The Square Root ${\sqrt {2}}$ of Two". Numberphile. Brady Haran.
• √2 Search Engine 2 billion searchable digits of √2, π and e
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| Wikipedia |
Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities $a$ and $b$ with $a>b>0$,
${\frac {a+b}{a}}={\frac {a}{b}}=\varphi $
Golden ratio (φ)
Line segments in the golden ratio
Representations
Decimal1.618033988749894...[1]
Algebraic form${\frac {1+{\sqrt {5}}}{2}}$
Continued fraction$1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}}}$
Binary1.10011110001101110111...
Hexadecimal1.9E3779B97F4A7C15...
where the Greek letter phi ($\varphi $ or $\phi $) denotes the golden ratio.[lower-alpha 1] The constant $\varphi $ satisfies the quadratic equation $\varphi ^{2}=\varphi +1$ and is an irrational number with a value of[1]
$\varphi ={\frac {1+{\sqrt {5}}}{2}}=$1.618033988749....
The golden ratio was called the extreme and mean ratio by Euclid,[2] and the divine proportion by Luca Pacioli,[3] and also goes by several other names.[lower-alpha 2]
Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron.[7] A golden rectangle—that is, a rectangle with an aspect ratio of $\varphi $—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data.[8] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.
Calculation
Two quantities $a$ and $b$ are in the golden ratio $\varphi $ if[9]
${\frac {a+b}{a}}={\frac {a}{b}}=\varphi .$
One method for finding a closed form for $\varphi $ starts with the left fraction. Simplifying the fraction and substituting the reciprocal $b/a=1/\varphi $,
${\frac {a+b}{a}}={\frac {a}{a}}+{\frac {b}{a}}=1+{\frac {b}{a}}=1+{\frac {1}{\varphi }}.$
Therefore,
$1+{\frac {1}{\varphi }}=\varphi .$
Multiplying by $\varphi $ gives
$\varphi +1=\varphi ^{2}$
which can be rearranged to
${\varphi }^{2}-\varphi -1=0.$
The quadratic formula yields two solutions:
${\frac {1+{\sqrt {5}}}{2}}=1.618033\dots $ and ${\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .$
Because $\varphi $ is a ratio between positive quantities, $\varphi $ is necessarily the positive root.[10] The negative root is in fact the negative inverse $-{\frac {1}{\varphi }}$, which shares many properties with the golden ratio.
History
See also: Mathematics and art and Fibonacci number § History
According to Mario Livio,
Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. ... Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.[11]
— The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry;[12] the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.[13] According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans.[14] Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[15][lower-alpha 3] and contains its first known definition which proceeds as follows:[16]
A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.[17][lower-alpha 4]
The golden ratio was studied peripherally over the next millennium. Abu Kamil (c. 850–930) employed it in his geometric calculations of pentagons and decagons; his writings influenced that of Fibonacci (Leonardo of Pisa) (c. 1170–1250), who used the ratio in related geometry problems but did not observe that it was connected to the Fibonacci numbers.[19]
Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids.[20][21] Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section').[22] Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[23] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.[24]
German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[25] this was rediscovered by Johannes Kepler in 1608.[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about $0.6180340$" in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[27] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:
Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.[28]
Eighteenth-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula".[29] Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.[30] James Sully used the equivalent English term in 1875.[31]
By 1910, inventor Mark Barr began using the Greek letter phi ($\varphi $) as a symbol for the golden ratio.[32][lower-alpha 5] It has also been represented by tau ($\tau $), the first letter of the ancient Greek τομή ('cut' or 'section').[35]
The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[36] This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.[37]
Mathematics
Irrationality
The golden ratio is an irrational number. Below are two short proofs of irrationality:
Contradiction from an expression in lowest terms
Recall that:
the whole is the longer part plus the shorter part;
the whole is to the longer part as the longer part is to the shorter part.
If we call the whole $n$ and the longer part $m,$ then the second statement above becomes
$n$ is to $m$ as $m$ is to $n-m.$
To say that the golden ratio $\varphi $ is rational means that $\varphi $ is a fraction $n/m$ where $n$ and $m$ are integers. We may take $n/m$ to be in lowest terms and $n$ and $m$ to be positive. But if $n/m$ is in lowest terms, then the equally valued $m/(n-m)$ is in still lower terms. That is a contradiction that follows from the assumption that $\varphi $ is rational.
By irrationality of √5
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If $\varphi ={\tfrac {1}{2}}(1+{\sqrt {5}})$ is rational, then $2\varphi -1={\sqrt {5}}$ is also rational, which is a contradiction if it is already known that the square root of all non-square natural numbers are irrational.
Minimal polynomial
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial
$x^{2}-x-1.$
This quadratic polynomial has two roots, $\varphi $ and $-\varphi ^{-1}.$
The golden ratio is also closely related to the polynomial
$x^{2}+x-1,$
which has roots $-\varphi $ and $\varphi ^{-1}.$ As the root of a quadratic polynomial, the golden ratio is a constructible number.[38]
Golden ratio conjugate and powers
The conjugate root to the minimal polynomial $x^{2}-x-1$ is
$-{\frac {1}{\varphi }}=1-\varphi ={\frac {1-{\sqrt {5}}}{2}}=-0.618033\dots .$
The absolute value of this quantity ($0.618\ldots $) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, $b/a$).
This illustrates the unique property of the golden ratio among positive numbers, that
${\frac {1}{\varphi }}=\varphi -1,$
or its inverse:
${\frac {1}{1/\varphi }}={\frac {1}{\varphi }}+1.$
The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with $\varphi $:
${\begin{aligned}\varphi ^{2}&=\varphi +1=2.618033\dots ,\\[5mu]{\frac {1}{\varphi }}&=\varphi -1=0.618033\dots .\end{aligned}}$
The sequence of powers of $\varphi $ contains these values $0.618033\ldots ,$ $1.0,$ $1.618033\ldots ,$ $2.618033\ldots ;$ ;} more generally, any power of $\varphi $ is equal to the sum of the two immediately preceding powers:
$\varphi ^{n}=\varphi ^{n-1}+\varphi ^{n-2}=\varphi \cdot \operatorname {F} _{n}+\operatorname {F} _{n-1}.$
As a result, one can easily decompose any power of $\varphi $ into a multiple of $\varphi $ and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of $\varphi $:
If $\lfloor n/2-1\rfloor =m,$ then:
${\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-3}+\cdots +\varphi ^{n-1-2m}+\varphi ^{n-2-2m}\\[5mu]\varphi ^{n}-\varphi ^{n-1}&=\varphi ^{n-2}.\end{aligned}}$
Continued fraction and square root
See also: Lucas number § Continued fractions for powers of the golden ratio
The formula $\varphi =1+1/\varphi $ can be expanded recursively to obtain a continued fraction for the golden ratio:[39]
$\varphi =[1;1,1,1,\dots ]=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}$
It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
$\varphi ^{-1}=[0;1,1,1,\dots ]=0+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}$
The convergents of these continued fractions ($1/1,$ $2/1,$ $2/1,$ $3/2,$ $5/3,$ $8/5,$ $13/8,$ ... or $1/1,$ $1/2,$ $2/3,$ $3/5,$ $5/8,$ $8/13,$ ...) are ratios of successive Fibonacci numbers. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the Hurwitz inequality for Diophantine approximations, which states that for every irrational $\xi $, there are infinitely many distinct fractions $p/q$ such that,
$\left|\xi -{\frac {p}{q}}\right|<{\frac {1}{{\sqrt {5}}q^{2}}}.$
This means that the constant ${\sqrt {5}}$ cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.[40]
A continued square root form for $\varphi $ can be obtained from $\varphi ^{2}=1+\varphi $, yielding:[41]
$\varphi ={\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}.$
Relationship to Fibonacci and Lucas numbers
Further information: Fibonacci number § Relation to the golden ratio
See also: Lucas number § Relationship to Fibonacci numbers
A Fibonacci spiral (top) which approximates the golden spiral, using Fibonacci sequence square sizes up to 21. A different approximation to the golden spiral is generated (bottom) from stacking squares whose lengths of sides are numbers belonging to the sequence of Lucas numbers, here up to 76.
Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence $0,1$:
$0,$ $1,$ $1,$ $2,$ $3,$ $5,$ $8,$ $13,$ $21,$ $34,$ $55,$ $89,$ $\ldots $(OEIS: A000045).
The sequence of Lucas numbers (not to be confused with the generalized Lucas sequences, of which this is part) is like the Fibonacci sequence, in which each term is the sum of the previous two, however instead starts with $2,1$:
$2,$ $1,$ $3,$ $4,$ $7,$ $11,$ $18,$ $29,$ $47,$ $76,$ $123,$ $199,$ $\ldots $(OEIS: A000032).
Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers:[42]
$\lim _{n\to \infty }{\frac {F_{n+1}}{F_{n}}}=\lim _{n\to \infty }{\frac {L_{n+1}}{L_{n}}}=\varphi .$
In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates $\varphi $.
For example, ${\frac {F_{16}}{F_{15}}}={\frac {987}{610}}=1.6180327\ldots ,$ and ${\frac {L_{16}}{L_{15}}}={\frac {2207}{1364}}=1.6180351\ldots .$
These approximations are alternately lower and higher than $\varphi ,$ and converge to $\varphi $ as the Fibonacci and Lucas numbers increase.
Closed-form expressions for the Fibonacci and Lucas sequences that involve the golden ratio are:
$F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}={{\varphi ^{n}-(-\varphi )^{-n}} \over {\sqrt {5}}},$
$L\left(n\right)=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n}\,.$
Combining both formulas above, one obtains a formula for $\varphi ^{n}$ that involves both Fibonacci and Lucas numbers:
$\varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}\,.$
Between Fibonacci and Lucas numbers one can deduce $L_{2n}=5F_{n}^{2}+2(-1)^{n}=L_{n}^{2}-2(-1)^{n},$ which simplifies to express the limit of the quotient of Lucas numbers by Fibonacci numbers as equal to the square root of five:
$\lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}.$
Indeed, much stronger statements are true:
$\vert L_{n}-{\sqrt {5}}F_{n}\vert ={\frac {2}{\varphi ^{n}}}\to 0,$
$(L_{3n}/2)^{2}=5(F_{3n}/2)^{2}+(-1)^{n}.$
These values describe $\varphi $ as a fundamental unit of the algebraic number field $\mathbb {Q} ({\sqrt {5}})$.
Successive powers of the golden ratio obey the Fibonacci recurrence, i.e. $\varphi ^{n+1}=\varphi ^{n}+\varphi ^{n-1}.$
The reduction to a linear expression can be accomplished in one step by using:
$\varphi ^{n}=F_{n}\varphi +F_{n-1}.$
This identity allows any polynomial in $\varphi $ to be reduced to a linear expression, as in:
${\begin{aligned}3\varphi ^{3}-5\varphi ^{2}+4&=3(\varphi ^{2}+\varphi )-5\varphi ^{2}+4\\[5mu]&=3[(\varphi +1)+\varphi ]-5(\varphi +1)+4\\[5mu]&=\varphi +2\approx 3.618033.\end{aligned}}$
Consecutive Fibonacci numbers can also be used to obtain a similar formula for the golden ratio, here by infinite summation:
$\sum _{n=1}^{\infty }|F_{n}\varphi -F_{n+1}|=\varphi .$
In particular, the powers of $\varphi $ themselves round to Lucas numbers (in order, except for the first two powers, $\varphi ^{0}$ and $\varphi $, are in reverse order):
${\begin{aligned}\varphi ^{0}&=1,\\[5mu]\varphi ^{1}&=1.618033989\ldots \approx 2,\\[5mu]\varphi ^{2}&=2.618033989\ldots \approx 3,\\[5mu]\varphi ^{3}&=4.236067978\ldots \approx 4,\\[5mu]\varphi ^{4}&=6.854101967\ldots \approx 7,\end{aligned}}$
and so forth.[43] The Lucas numbers also directly generate powers of the golden ratio; for $n\geq 2$:
$\varphi ^{n}=L_{n}-(-\varphi )^{-n}.$
Rooted in their interconnecting relationship with the golden ratio is the notion that the sum of third consecutive Fibonacci numbers equals a Lucas number, that is $L_{n}=F_{n-1}+F_{n+1}$; and, importantly, that $L_{n}={\frac {F_{2n}}{F_{n}}}$.
Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.
Geometry
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.
Construction
Dividing a line segment by interior division (top) and exterior division (bottom) according to the golden ratio.
Dividing by interior division
1. Having a line segment $AB,$ construct a perpendicular $BC$ at point $B,$ with $BC$ half the length of $AB.$ Draw the hypotenuse $AC.$
2. Draw an arc with center $C$ and radius $BC.$ This arc intersects the hypotenuse $AC$ at point $D.$
3. Draw an arc with center $A$ and radius $AD.$ This arc intersects the original line segment $AB$ at point $S.$ Point $S$ divides the original line segment $AB$ into line segments $AS$ and $SB$ with lengths in the golden ratio.
Dividing by exterior division
1. Draw a line segment $AS$ and construct off the point $S$ a segment $SC$ perpendicular to $AS$ and with the same length as $AS.$
2. Do bisect the line segment $AS$ with $M.$
3. A circular arc around $M$ with radius $MC$ intersects in point $B$ the straight line through points $A$ and $S$ (also known as the extension of $AS$). The ratio of $AS$ to the constructed segment $SB$ is the golden ratio.
Application examples you can see in the articles Pentagon with a given side length, Decagon with given circumcircle and Decagon with a given side length.
Both of the above displayed different algorithms produce geometric constructions that determine two aligned line segments where the ratio of the longer one to the shorter one is the golden ratio.
Golden angle
Main article: Golden angle
When two angles that make a full circle have measures in the golden ratio, the smaller is called the golden angle, with measure $ g\colon $
${\begin{aligned}{\frac {2\pi -g}{g}}&={\frac {2\pi }{2\pi -g}}=\varphi ,\\[8mu]2\pi -g&={\frac {2\pi }{\varphi }}\approx 222.5^{\circ },\\[8mu]g&={\frac {2\pi }{\varphi ^{2}}}\approx 137.5^{\circ }.\end{aligned}}$
This angle occurs in patterns of plant growth as the optimal spacing of leaf shoots around plant stems so that successive leaves do not block sunlight from the leaves below them.[44]
Pentagon and pentagram
In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are $a,$ and short edges are $b,$ then Ptolemy's theorem gives $a^{2}=b^{2}+ab.$ Dividing both sides by $ab$ yields (see § Calculation above),
${\frac {a}{b}}={\frac {a+b}{a}}=\varphi .$
The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by $\varphi $. Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is $\varphi ,$ as the four-color illustration shows.
Pentagonal and pentagrammic geometry permits us to calculate the following values for $\varphi $:
${\begin{aligned}\varphi &=1+2\sin(\pi /10)=1+2\sin 18^{\circ },\\[5mu]\varphi &={\tfrac {1}{2}}\csc(\pi /10)={\tfrac {1}{2}}\csc 18^{\circ },\\[5mu]\varphi &=2\cos(\pi /5)=2\cos 36^{\circ },\\[5mu]\varphi &=2\sin(3\pi /10)=2\sin 54^{\circ }.\end{aligned}}$
Golden triangle and golden gnomon
Main article: Golden triangle (mathematics)
The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle 36° and base angles 72°.[45] Its two equal sides are in the golden ratio to its base.[46] The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle 108° and base angle 36°. Its base is in the golden ratio to its two equal sides.[46] The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles,[46] as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.[47]
Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.[46]
If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.[46]
Penrose tilings
Main article: Penrose tiling
The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.[48] Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
• Penrose's original version of this tiling used four shapes: regular pentagons and pentagrams, "boat" figures with three points of a pentagram, and "diamond" shaped rhombi.[49]
• The kite and dart Penrose tiling uses kites with three interior angles of 72° and one interior angle of 144°, and darts, concave quadrilaterals with two interior angles of 36°, one of 72°, and one non-convex angle of 216°. Special matching rules restrict how the tiles can meet at any edge, resulting in seven combinations of tiles at any vertex. Both the kites and darts have sides of two lengths, in the golden ratio to each other. The areas of these two tile shapes are also in the golden ratio to each other.[48]
• The kite and dart can each be cut on their symmetry axes into a pair of golden triangles and golden gnomons, respectively. With suitable matching rules, these triangles, called in this context Robinson triangles, can be used as the prototiles for a form of the Penrose tiling.[48][50]
• The rhombic Penrose tiling contains two types of rhombus, a thin rhombus with angles of 36° and 144°, and a thick rhombus with angles of 72° and 108°. All side lengths are equal, but the ratio of the length of sides to the short diagonal in the thin rhombus equals $1:\varphi $, as does the ratio of the sides of to the long diagonal of the thick rhombus. As with the kite and dart tiling, the areas of the two rhombi are in the golden ratio to each other. Again, these rhombi can be decomposed into pairs of Robinson triangles.[48]
Original four-tile Penrose tiling
Rhombic Penrose tiling
Odom's construction
George Odom found a construction for $\varphi $ involving an equilateral triangle: if the line segment joining the midpoints of two sides is extended to intersect the circumcircle, then the two midpoints and the point of intersection with the circle are in golden proportion.[51]
Kepler triangle
Main article: Kepler triangle
Geometric progression of areas of squares on the sides of a Kepler triangle
An isosceles triangle formed from two Kepler triangles maximizes the ratio of its inradius to side length
The Kepler triangle, named after Johannes Kepler, is the unique right triangle with sides in geometric progression:
$1\mathbin {:} {\sqrt {\varphi }}\mathbin {:} \varphi $.
These side lengths are the three Pythagorean means of the two numbers $\varphi \pm 1$. The three squares on its sides have areas in the golden geometric progression $1\mathbin {:} \varphi \mathbin {:} \varphi ^{2}$.
Among isosceles triangles, the ratio of inradius to side length is maximized for the triangle formed by two reflected copies of the Kepler triangle, sharing the longer of their two legs.[52] The same isosceles triangle maximizes the ratio of the radius of a semicircle on its base to its perimeter.[53]
For a Kepler triangle with smallest side length $s$, the area and acute internal angles are:
${\begin{aligned}A&={\tfrac {s^{2}}{2}}{\sqrt {\varphi }},\\[5mu]\theta &=\sin ^{-1}{\frac {1}{\varphi }}\approx 38.1727^{\circ },\\[5mu]\theta &=\cos ^{-1}{\frac {1}{\varphi }}\approx 51.8273^{\circ }.\end{aligned}}$
Golden rectangle
Main article: Golden rectangle
The golden ratio proportions the adjacent side lengths of a golden rectangle in $1:\varphi $ ratio.[54] Stacking golden rectangles produces golden rectangles anew, and removing or adding squares from golden rectangles leaves rectangles still proportioned in $\varphi $ ratio. They can be generated by golden spirals, through successive Fibonacci and Lucas number-sized squares and quarter circles. They feature prominently in the icosahedron as well as in the dodecahedron (see section below for more detail).[55]
Golden rhombus
Main article: Golden rhombus
A golden rhombus is a rhombus whose diagonals are in proportion to the golden ratio, most commonly $1:\varphi $.[56] For a rhombus of such proportions, its acute angle and obtuse angles are:
${\begin{aligned}\alpha &=2\arctan {1 \over \varphi }\approx 63.43495^{\circ },\\[5mu]\beta &=2\arctan \varphi =\pi -\arctan 2=\arctan 1+\arctan 3\approx 116.56505^{\circ }.\end{aligned}}$
The lengths of its short and long diagonals $d$ and $D$, in terms of side length $a$ are:
${\begin{aligned}d&={2a \over {\sqrt {2+\varphi }}}=2{\sqrt {{3-\varphi } \over 5}}a\approx 1.05146a,\\[5mu]D&=2{\sqrt {{2+\varphi } \over 5}}a\approx 1.70130a.\end{aligned}}$
Its area, in terms of $a$,and $d$:
${\begin{aligned}A&=(\sin(\arctan 2))~a^{2}={2 \over {\sqrt {5}}}~a^{2}\approx 0.89443a^{2},\\[5mu]A&={{\varphi } \over 2}d^{2}\approx 0.80902d^{2}.\end{aligned}}$
Its inradius, in terms of side $a$:
$r={\frac {a}{\sqrt {5}}}.$
Golden rhombi form the faces of the rhombic triacontahedron, the two golden rhombohedra, the Bilinski dodecahedron,[57] and the rhombic hexecontahedron.[56]
Golden spiral
Main article: Golden spiral
Logarithmic spirals are self-similar spirals where distances covered per turn are in geometric progression. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the golden spiral. These spirals can be approximated by quarter-circles that grow by the golden ratio,[59] or their approximations generated from Fibonacci numbers,[60] often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the polar equation with $(r,\theta )$:
$r=\varphi ^{2\theta /\pi }.$
Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each 108° that it turns, instead of the 90° turning angle of the golden spiral.[58] Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.[59]
In the dodecahedron and icosahedron
The regular dodecahedron and its dual polyhedron the icosahedron are Platonic solids whose dimensions are related to the golden ratio. An icosahedron is made of $12$ regular pentagonal faces, whereas the icosahedron is made of $20$ equilateral triangles; both with $30$ edges.[61]
For a dodecahedron of side $a$, the radius of a circumscribed and inscribed sphere, and midradius are ($r_{u},$ $r_{i},$ and $r_{m},$ respectively):
$r_{u}=a\,{\frac {{\sqrt {3}}\varphi }{2}},$ $r_{i}=a\,{\frac {\varphi ^{2}}{2{\sqrt {3-\varphi }}}},$ and $r_{m}=a\,{\frac {\varphi ^{2}}{2}}.$
While for an icosahedron of side $a$, the radius of a circumscribed and inscribed sphere, and midradius are:
$r_{u}=a{\frac {\sqrt {\varphi {\sqrt {5}}}}{2}},$ $r_{i}=a{\frac {\varphi ^{2}}{2{\sqrt {3}}}},$ and $r_{m}=a{\frac {\varphi }{2}}.$
The volume and surface area of the dodecahedron can be expressed in terms of $\varphi $:
$A_{d}={\frac {15\varphi }{\sqrt {3-\varphi }}}$ and $V_{d}={\frac {5\varphi ^{3}}{6-2\varphi }}$.
As well as for the icosahedron:
$A_{i}=20{\frac {\varphi ^{2}}{2}}$ and $V_{i}={\frac {5}{6}}(1+\varphi ).$
These geometric values can be calculated from their Cartesian coordinates, which also can be given using formulas involving $\varphi $. The coordinates of the dodecahedron are displayed on the figure above, while those of the icosahedron are the cyclic permutations of:
$(0,\pm 1,\pm \varphi )$, $(\pm 1,\pm \varphi ,0)$, $(\pm \varphi ,0,\pm 1).$
Sets of three golden rectangles intersect perpendicularly inside dodecahedra and icosahedra, forming Borromean rings.[62][55] In dodecahedra, pairs of opposing vertices in golden rectangles meet the centers of pentagonal faces, and in icosahedra, they meet at its vertices. In all, the three golden rectangles contain $12$ vertices of the icosahedron, or equivalently, intersect the centers of $12$ of the dodecahedron's faces.[61]
A cube can be inscribed in a regular dodecahedron, with some of the diagonals of the pentagonal faces of the dodecahedron serving as the cube's edges; therefore, the edge lengths are in the golden ratio. The cube's volume is ${\tfrac {2}{2+\varphi }}$ times that of the dodecahedron's.[63] In fact, golden rectangles inside a dodecahedron are in golden proportions to an inscribed cube, such that edges of a cube and the long edges of a golden rectangle are themselves in $\varphi :\varphi ^{2}$ :\varphi ^{2}} ratio. On the other hand, the octahedron, which is the dual polyhedron of the cube, can inscribe an icosahedron, such that an icosahedron's $12$ vertices touch the $12$ edges of an octahedron at points that divide its edges in golden ratio.[64]
Other polyhedra are related to the dodecahedron and icosahedron or their symmetries, and therefore have corresponding relations to the golden ratio. These include the compound of five cubes, compound of five octahedra, compound of five tetrahedra, the compound of ten tetrahedra, rhombic triacontahedron, icosidodecahedron, truncated icosahedron, truncated dodecahedron, and rhombicosidodecahedron, rhombic enneacontahedron, and Kepler-Poinsot polyhedra, and rhombic hexecontahedron. In four dimensions, the dodecahedron and icosahedron appear as faces of the 120-cell and 600-cell, which again have dimensions related to the golden ratio.
Other properties
The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation $x^{2}-x-1=0$ or on $x^{2}-5=0$ (to compute ${\sqrt {5}}$ first). The time needed to compute $n$ digits of the golden ratio using Newton's method is essentially $O(M(n))$, where $M(n)$ is the time complexity of multiplying two $n$-digit numbers.[65] This is considerably faster than known algorithms for $\pi $ and $e$. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers $F_{25001}$ and $F_{25000},$ each over $5000$ digits, yields over $10{,}000$ significant digits of the golden ratio. The decimal expansion of the golden ratio $\varphi $[1] has been calculated to an accuracy of ten trillion ($1\times 10^{13}=10{,}000{,}000{,}000{,}000$) digits.[66]
In the complex plane, the fifth roots of unity $z=e^{2\pi ki/5}$ (for an integer $ k$) satisfying $z^{5}=1$ are the vertices of a pentagon. They do not form a ring of quadratic integers, however the sum of any fifth root of unity and its complex conjugate, $z+{\bar {z}},$ is a quadratic integer, an element of $ \mathbb {Z} [\varphi ].$ Specifically,
${\begin{aligned}e^{0}+e^{-0}&=2,\\[5mu]e^{2\pi i/5}+e^{-2\pi i/5}&=\varphi ^{-1}=-1+\varphi ,\\[5mu]e^{4\pi i/5}+e^{-4\pi i/5}&=-\varphi .\end{aligned}}$
This also holds for the remaining tenth roots of unity satisfying $z^{10}=1,$
${\begin{aligned}e^{\pi i}+e^{-\pi i}&=-2,\\[5mu]e^{\pi i/5}+e^{-\pi i/5}&=\varphi ,\\[5mu]e^{3\pi i/5}+e^{-3\pi i/5}&=-\varphi ^{-1}=1-\varphi .\end{aligned}}$
For the gamma function $\Gamma $, the only solutions to the equation $\Gamma (z-1)=\Gamma (z+1)$ are $z=\varphi $ and $z=-\varphi ^{-1}$.
When the golden ratio is used as the base of a numeral system (see golden ratio base, sometimes dubbed phinary or $\varphi $-nary), quadratic integers in the ring $\mathbb {Z} [\varphi ]$ – that is, numbers of the form $a+b\varphi $ for $a,b\in \mathbb {Z} $ – have terminating representations, but rational fractions have non-terminating representations.
The golden ratio also appears in hyperbolic geometry, as the maximum distance from a point on one side of an ideal triangle to the closer of the other two sides: this distance, the side length of the equilateral triangle formed by the points of tangency of a circle inscribed within the ideal triangle, is $4\log(\varphi ).$[67]
The golden ratio appears in the theory of modular functions as well. For $\left|q\right|<1$, let
$R(q)={\cfrac {q^{1/5}}{1+{\cfrac {q}{1+{\cfrac {q^{2}}{1+{\cfrac {q^{3}}{1+\ddots }}}}}}}}.$
Then
$R(e^{-2\pi })={\sqrt {\varphi {\sqrt {5}}}}-\varphi ,\quad R(-e^{-\pi })=\varphi ^{-1}-{\sqrt {2-\varphi ^{-1}}}$
and
$R(e^{-2\pi i/\tau })={\frac {1-\varphi R(e^{2\pi i\tau })}{\varphi +R(e^{2\pi i\tau })}}$
where $\operatorname {Im} \tau >0$ and $(e^{z})^{1/5}$ in the continued fraction should be evaluated as $e^{z/5}$. The function $\tau \mapsto R(e^{2\pi i\tau })$ is invariant under $\Gamma (5)$, a congruence subgroup of the modular group. Also for positive real numbers $a,b\in \mathbb {R} ^{+}$ and $ab=\pi ^{2},$ then[68]
${\begin{aligned}{\Bigl (}\varphi +R{{\bigl (}e^{-2a}{\bigr )}}{\Bigr )}{\Bigl (}\varphi +R{{\bigl (}e^{-2b}{\bigr )}}{\Bigr )}&=\varphi {\sqrt {5}},\\[5mu]{\Bigl (}\varphi ^{-1}-R{{\bigl (}{-e^{-a}}{\bigr )}}{\Bigr )}{\Bigl (}\varphi ^{-1}-R{{\bigl (}{-e^{-b}}{\bigr )}}{\Bigr )}&=\varphi ^{-1}{\sqrt {5}}.\end{aligned}}$
$\varphi $ is a Pisot–Vijayaraghavan number.[69]
Applications and observations
Architecture
Further information: Mathematics and architecture
The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."[70][71]
Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.
In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.[72]
Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.[73]
Art
Further information: Mathematics and art and History of aesthetics
Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings.[74] Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.[75][76]
Salvador Dalí, influenced by the works of Matila Ghyka,[77] explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspended above and behind Jesus and dominates the composition.[74][78]
A statistical study on 565 works of art of different great painters, performed in 1999, found that these artists had not used the golden ratio in the size of their canvases. The study concluded that the average ratio of the two sides of the paintings studied is $1.34,$ with averages for individual artists ranging from $1.04$ (Goya) to $1.46$ (Bellini).[79] On the other hand, Pablo Tosto listed over 350 works by well-known artists, including more than 100 which have canvasses with golden rectangle and ${\sqrt {5}}$ proportions, and others with proportions like ${\sqrt {2}},$ $3,$ $4,$ and $6.$[80]
Books and design
According to Jan Tschichold,
There was a time when deviations from the truly beautiful page proportions $2\mathbin {:} 3,$ $1\mathbin {:} {\sqrt {3}},$ and the Golden Section were rare. Many books produced between 1550 and 1770 show these proportions exactly, to within half a millimeter.[82]
According to some sources, the golden ratio is used in everyday design, for example in the proportions of playing cards, postcards, posters, light switch plates, and widescreen televisions.[83]
Flags
The aspect ratio (width to height ratio) of the flag of Togo was intended to be the golden ratio, according to its designer.[84]
Music
Ernő Lendvai analyzes Béla Bartók's works as being based on two opposing systems, that of the golden ratio and the acoustic scale,[85] though other music scholars reject that analysis.[86] French composer Erik Satie used the golden ratio in several of his pieces, including Sonneries de la Rose+Croix. The golden ratio is also apparent in the organization of the sections in the music of Debussy's Reflets dans l'eau (Reflections in Water), from Images (1st series, 1905), in which "the sequence of keys is marked out by the intervals 34, 21, 13 and 8, and the main climax sits at the phi position".[87]
The musicologist Roy Howat has observed that the formal boundaries of Debussy's La Mer correspond exactly to the golden section.[88] Trezise finds the intrinsic evidence "remarkable", but cautions that no written or reported evidence suggests that Debussy consciously sought such proportions.[89]
Music theorists including Hans Zender and Heinz Bohlen have experimented with the 833 cents scale, a musical scale based on using the golden ratio as its fundamental musical interval. When measured in cents, a logarithmic scale for musical intervals, the golden ratio is approximately 833.09 cents.[90]
Nature
Main article: Patterns in nature
See also: Fibonacci number § Nature
Johannes Kepler wrote that "the image of man and woman stems from the divine proportion. In my opinion, the propagation of plants and the progenitive acts of animals are in the same ratio".[91]
The psychologist Adolf Zeising noted that the golden ratio appeared in phyllotaxis and argued from these patterns in nature that the golden ratio was a universal law.[92] Zeising wrote in 1854 of a universal orthogenetic law of "striving for beauty and completeness in the realms of both nature and art".[93]
However, some have argued that many apparent manifestations of the golden ratio in nature, especially in regard to animal dimensions, are fictitious.[94]
Physics
The quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) has 8 predicted excitation states (with E8 symmetry), that when probed with neutron scattering, showed its lowest two were in golden ratio. Specifically, these quantum phase transitions during spin excitation, which occur at near absolute zero temperature, showed pairs of kinks in its ordered-phase to spin-flips in its paramagnetic phase; revealing, just below its critical field, a spin dynamics with sharp modes at low energies approaching the golden mean.[95]
Optimization
There is no known general algorithm to arrange a given number of nodes evenly on a sphere, for any of several definitions of even distribution (see, for example, Thomson problem or Tammes problem). However, a useful approximation results from dividing the sphere into parallel bands of equal surface area and placing one node in each band at longitudes spaced by a golden section of the circle, i.e. $360^{\circ }/\varphi \approx 222.5^{\circ }.$ This method was used to arrange the 1500 mirrors of the student-participatory satellite Starshine-3.[96]
The golden ratio is a critical element to golden-section search as well.
Disputed observations
Examples of disputed observations of the golden ratio include the following:
• Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.[97][98]
• The shells of mollusks such as the nautilus are often claimed to be in the golden ratio.[99] The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio,[100] or sometimes claimed that each new chamber is golden-proportioned relative to the previous one.[101] However, measurements of nautilus shells do not support this claim.[102]
• Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is $1.45.$[103]
• Studies by psychologists, starting with Gustav Fechner c. 1876,[104] have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.[105][74]
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio.[106] The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.[107]
Egyptian pyramids
The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or on whole-number ratios. The consensus of modern scholars is that this pyramid's proportions are not based on the golden ratio, because such a basis would be inconsistent both with what is known about Egyptian mathematics from the time of construction of the pyramid, and with Egyptian theories of architecture and proportion used in their other works.[108]
The Parthenon
The Parthenon's façade (c. 432 BC) as well as elements of its façade and elsewhere are said by some to be circumscribed by golden rectangles.[110] Other scholars deny that the Greeks had any aesthetic association with golden ratio. For example, Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation."[111] Midhat J. Gazalé affirms that "It was not until Euclid ... that the golden ratio's mathematical properties were studied."[112]
From measurements of 15 temples, 18 monumental tombs, 8 sarcophagi, and 58 grave stelae from the fifth century BC to the second century AD, one researcher concluded that the golden ratio was totally absent from Greek architecture of the classical fifth century BC, and almost absent during the following six centuries.[113] Later sources like Vitruvius (first century BC) exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.
Modern art
The Section d'Or ('Golden Section') was a collective of painters, sculptors, poets and critics associated with Cubism and Orphism.[114] Active from 1911 to around 1914, they adopted the name both to highlight that Cubism represented the continuation of a grand tradition, rather than being an isolated movement, and in homage to the mathematical harmony associated with Georges Seurat.[115] (Several authors have claimed that Seurat employed the golden ratio in his paintings, but Seurat’s writings and paintings suggest that he employed simple whole-number ratios and any approximation of the golden ratio was coincidental.)[116] The Cubists observed in its harmonies, geometric structuring of motion and form, "the primacy of idea over nature", "an absolute scientific clarity of conception".[117] However, despite this general interest in mathematical harmony, whether the paintings featured in the celebrated 1912 Salon de la Section d'Or exhibition used the golden ratio in any compositions is more difficult to determine. Livio, for example, claims that they did not,[118] and Marcel Duchamp said as much in an interview.[119] On the other hand, an analysis suggests that Juan Gris made use of the golden ratio in composing works that were likely, but not definitively, shown at the exhibition.[119][120] Art historian Daniel Robbins has argued that in addition to referencing the mathematical term, the exhibition's name also refers to the earlier Bandeaux d'Or group, with which Albert Gleizes and other former members of the Abbaye de Créteil had been involved.[121]
Piet Mondrian has been said to have used the golden section extensively in his geometrical paintings,[122] though other experts (including critic Yve-Alain Bois) have discredited these claims.[74][123]
See also
• List of works designed with the golden ratio
• Metallic mean
• Plastic number
• Sacred geometry
• Supergolden ratio
References
Explanatory footnotes
1. If the constraint on $a$ and $b$ each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. $\varphi $ is defined as the positive solution. The negative solution is $-\varphi ^{-1}={\tfrac {1}{2}}{\bigl (}1-{\sqrt {5}}{\bigr )}.$ The sum of the two solutions is $1$, and the product of the two solutions is $-1$.
2. Other names include the golden mean, golden section,[4] golden cut,[5] golden proportion, golden number,[6] medial section, and divine section.
3. Euclid, Elements, Book II, Proposition 11; Book IV, Propositions 10–11; Book VI, Proposition 30; Book XIII, Propositions 1–6, 8–11, 16–18.
4. "῎Ακρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττὸν."[18]
5. After Classical Greek sculptor Phidias (c. 490–430 BC);[33] Barr later wrote that he thought it unlikely that Phidias actually used the golden ratio.[34]
Citations
1. Sloane, N. J. A. (ed.). "Sequence A001622 (Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Euclid. "Book 6, Definition 3". Elements.
3. Pacioli, Luca (1509). De divina proportione. Venice: Luca Paganinem de Paganinus de Brescia (Antonio Capella).
4. Livio 2002, pp. 3, 81.
5. Summerson, John (1963). Heavenly Mansions and Other Essays on Architecture. New York: W.W. Norton. p. 37. And the same applies in architecture, to the rectangles representing these and other ratios (e.g., the 'golden cut'). The sole value of these ratios is that they are intellectually fruitful and suggest the rhythms of modular design.
6. Herz-Fischler 1998.
7. Herz-Fischler 1998, pp. 20–25.
8. Strogatz, Steven (2012-09-24). "Me, Myself, and Math: Proportion Control". The New York Times.
9. Schielack, Vincent P. (1987). "The Fibonacci Sequence and the Golden Ratio". The Mathematics Teacher. 80 (5): 357–358. doi:10.5951/MT.80.5.0357. JSTOR 27965402. This source contains an elementary derivation of the golden ratio's value.
10. Peters, J. M. H. (1978). "An Approximate Relation between π and the Golden Ratio". The Mathematical Gazette. 62 (421): 197–198. doi:10.2307/3616690. JSTOR 3616690. S2CID 125919525.
11. Livio 2002, p. 6.
12. Livio 2002, p. 4: "... line division, which Euclid defined for ... purely geometrical purposes ..."
13. Livio 2002, pp. 7–8.
14. Livio 2002, pp. 4–5.
15. Livio 2002, p. 78.
16. Hemenway, Priya (2005). Divine Proportion: Phi In Art, Nature, and Science. New York: Sterling. pp. 20–21. ISBN 9781402735226.
17. Livio 2002, p. 3.
18. Euclid (2007). Euclid's Elements of Geometry. Translated by Fitzpatrick, Richard. p. 156. ISBN 978-0615179841.
19. Livio 2002, pp. 88–96.
20. Mackinnon, Nick (1993). "The Portrait of Fra Luca Pacioli". The Mathematical Gazette. 77 (479): 130–219. doi:10.2307/3619717. JSTOR 3619717. S2CID 195006163.
21. Livio 2002, pp. 131–132.
22. Baravalle, H. V. (1948). "The geometry of the pentagon and the golden section". Mathematics Teacher. 41: 22–31. doi:10.5951/MT.41.1.0022.
23. Livio 2002, pp. 134–135.
24. Livio 2002, p. 141.
25. Schreiber, Peter (1995). "A Supplement to J. Shallit's Paper 'Origins of the Analysis of the Euclidean Algorithm'". Historia Mathematica. 22 (4): 422–424. doi:10.1006/hmat.1995.1033.
26. Livio 2002, pp. 151–152.
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30. Herz-Fischler 1998, pp. 167–170.
31. Posamentier & Lehmann 2011, p. 8.
32. Posamentier & Lehmann 2011, p. 285.
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Stakhov, Alexey P.; Olsen, Scott (2009). "§1.4.1 A Golden Rectangle with a Side Ratio of τ". The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. World Scientific. pp. 20–21. A credit card has a form of the golden rectangle
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103. Man, John (2002). Gutenberg: How One Man Remade the World with Word. Wiley. pp. 166–167. ISBN 9780471218234. The half-folio page (30.7 × 44.5 cm) was made up of two rectangles—the whole page and its text area—based on the so called 'golden section', which specifies a crucial relationship between short and long sides, and produces an irrational number, as pi is, but is a ratio of about 5:8.
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106. Osler, Carol (2000). "Support for Resistance: Technical Analysis and Intraday Exchange Rates" (PDF). Federal Reserve Bank of New York Economic Policy Review. 6 (2): 53–68. Archived (PDF) from the original on 2007-05-12. 38.2 percent and 61.8 percent retracements of recent rises or declines are common,
107. Batchelor, Roy; Ramyar, Richard (2005). Magic numbers in the Dow (Report). Cass Business School. pp. 13, 31. Popular press summaries can be found in: Stevenson, Tom (2006-04-10). "Not since the 'big is beautiful' days have giants looked better". The Daily Telegraph. "Technical failure". The Economist. 2006-09-23.
108. Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. ISBN 0-88920-324-5. The entire book surveys many alternative theories for this pyramid's shape. See Chapter 11, "Kepler triangle theory", pp. 80–91, for material specific to the Kepler triangle, and p. 166 for the conclusion that the Kepler triangle theory can be eliminated by the principle that "A theory must correspond to a level of mathematics consistent with what was known to the ancient Egyptians." See note 3, p. 229, for the history of Kepler's work with this triangle.
Rossi, Corinna (2004). Architecture and Mathematics in Ancient Egypt. Cambridge University Press. pp. 67–68. there is no direct evidence in any ancient Egyptian written mathematical source of any arithmetic calculation or geometrical construction which could be classified as the Golden Section ... convergence to $\varphi $, and $\varphi $ itself as a number, do not fit with the extant Middle Kingdom mathematical sources; see also extensive discussion of multiple alternative theories for the shape of the pyramid and other Egyptian architecture, pp. 7–56
Rossi, Corinna; Tout, Christopher A. (2002). "Were the Fibonacci series and the Golden Section known in ancient Egypt?". Historia Mathematica. 29 (2): 101–113. doi:10.1006/hmat.2001.2334. hdl:11311/997099.
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109. Livio 2002, pp. 74–75.
110. Van Mersbergen, Audrey M. (1998). "Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic". Communication Quarterly. 46 (2): 194–213. doi:10.1080/01463379809370095.
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118. Livio 2002, p. 169.
119. Camfield, William A. (March 1965). "Juan Gris and the golden section". The Art Bulletin. 47 (1): 128–134. doi:10.1080/00043079.1965.10788819.
120. Green, Christopher (1992). Juan Gris. Yale. pp. 37–38. ISBN 9780300053746.
Cottington, David (2004). Cubism and Its Histories. Manchester University Press. p. 112, 142.
121. Allard, Roger (June 1911). "Sur quelques peintres". Les Marches du Sud-Ouest: 57–64. Reprinted in Antliff, Mark; Leighten, Patricia, eds. (2008). A Cubism Reader, Documents and Criticism, 1906–1914. The University of Chicago Press. pp. 178–191.
122. Bouleau, Charles (1963). The Painter's Secret Geometry: A Study of Composition in Art. Harcourt, Brace & World. pp. 247–248.
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Works cited
• Herz-Fischler, Roger (1998) [1987]. A Mathematical History of the Golden Number. Dover. ISBN 9780486400075. (Originally titled A Mathematical History of Division in Extreme and Mean Ratio.)
• Livio, Mario (2002). The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books. ISBN 9780767908153.
• Posamentier, Alfred S.; Lehmann, Ingmar (2011). The Glorious Golden Ratio. Prometheus Books. ISBN 9-781-61614-424-1.
Further reading
• Doczi, György (1981). The Power of Limits: Proportional Harmonies in Nature, Art, and Architecture. Boston: Shambhala.
• Hargittai, István, ed. (1992). Fivefold Symmetry. World Scientific. ISBN 9789810206000.
• Huntley, H. E. (1970). The Divine Proportion: A Study in Mathematical Beauty. New York: Dover. ISBN 978-0-486-22254-7.
• Schaaf, William L., ed. (1967). The Golden Measure (PDF). California School Mathematics Study Group Reprint Series. Stanford University. Archived (PDF) from the original on 2015-04-25.
• Scimone, Aldo (1997). La Sezione Aurea. Storia culturale di un leitmotiv della Matematica. Palermo: Sigma Edizioni. ISBN 978-88-7231-025-0.
• Walser, Hans (2001) [Der Goldene Schnitt 1993]. The Golden Section. Peter Hilton trans. Washington, DC: The Mathematical Association of America. ISBN 978-0-88385-534-8.
External links
Wikimedia Commons has media related to Golden ratio.
• Weisstein, Eric W. "Golden Ratio". MathWorld.
• Bogomolny, Alexander (2018). "Golden Ratio in Geometry". Cut-the-Knot.
• Knott, Ron. "The Golden section ratio: Phi". Information and activities by a mathematics professor.
• The Myth That Will Not Go Away, by Keith Devlin, addressing multiple allegations about the use of the golden ratio in culture.
• Spurious golden spirals collected by Randall Munroe
• YouTube lecture on Zeno's mice problem and logarithmic spirals
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| Wikipedia |
1/2 + 1/4 + 1/8 + 1/16 + ⋯
In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =\sum _{n=1}^{\infty }\left({\frac {1}{2}}\right)^{n}=1.$
The series is related to philosophical questions considered in antiquity, particularly to Zeno's paradoxes.
Proof
As with any infinite series, the sum
${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots $
is defined to mean the limit of the partial sum of the first n terms
$s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n-1}}}+{\frac {1}{2^{n}}}$
as n approaches infinity. By various arguments,[lower-alpha 1] one can show that this finite sum is equal to
$s_{n}=1-{\frac {1}{2^{n}}}.$
As n approaches infinity, the term ${\frac {1}{2^{n}}}$ approaches 0 and so sn tends to 1.
History
Zeno's paradox
This series was used as a representation of many of Zeno's paradoxes.[1] For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but the tortoise would already have moved another five meters by then. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on. Zeno argued that the tortoise would always remain ahead of Achilles.
The Dichotomy paradox also states that to move a certain distance, you have to move half of it, then half of the remaining distance, and so on, therefore having infinitely many time intervals.[1] This can be easily resolved by noting that each time interval is a term of the infinite geometric series, and will sum to a finite number.
The Eye of Horus
The parts of the Eye of Horus were once thought to represent the first six summands of the series.[2]
In a myriad ages it will not be exhausted
A version of the series appears in the ancient Taoist book Zhuangzi. The miscellaneous chapters "All Under Heaven" include the following sentence: "Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted."
See also
• 0.999...
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• Actual infinity
Notes
1. For example: multiplying sn by 2 yields $2s_{n}={\frac {2}{2}}+{\frac {2}{4}}+{\frac {2}{8}}+{\frac {2}{16}}+\cdots +{\frac {2}{2^{n}}}=1+\left[{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots +{\frac {1}{2^{n-1}}}\right]=1+\left[s_{n}-{\frac {1}{2^{n}}}\right].$ Subtracting sn from both sides, one concludes $s_{n}=1-{\frac {1}{2^{n}}}.$ Other arguments might proceed by mathematical induction, or by adding ${\frac {1}{2^{n}}}$ to both sides of $s_{n}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots +{\frac {1}{2^{n-1}}}+{\frac {1}{2^{n}}}$ and manipulating to show that the right side of the result is equal to 1.
References
1. Field, Paul and Weisstein, Eric W. "Zeno's Paradoxes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ZenosParadoxes.html
2. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN 978-1-84668-292-6.
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| Wikipedia |
1/3–2/3 conjecture
In order theory, a branch of mathematics, the 1/3–2/3 conjecture states that, if one is comparison sorting a set of items then, no matter what comparisons may have already been performed, it is always possible to choose the next comparison in such a way that it will reduce the number of possible sorted orders by a factor of 2/3 or better. Equivalently, in every finite partially ordered set that is not totally ordered, there exists a pair of elements x and y with the property that at least 1/3 and at most 2/3 of the linear extensions of the partial order place x earlier than y.
Example
The partial order formed by three elements a, b, and c with a single comparability relationship, a ≤ b, has three linear extensions, a ≤ b ≤ c, a ≤ c ≤ b, and c ≤ a ≤ b. In all three of these extensions, a is earlier than b. However, a is earlier than c in only two of them, and later than c in the third. Therefore, the pair of a and c have the desired property, showing that this partial order obeys the 1/3–2/3 conjecture.
This example shows that the constants 1/3 and 2/3 in the conjecture are tight; if q is any fraction strictly between 1/3 and 2/3, then there would not exist a pair x, y in which x is earlier than y in a number of partial orderings that is between q and 1 − q times the total number of partial orderings.[1]
More generally, let P be any series composition of three-element partial orders and of one-element partial orders, such as the one in the figure. Then P forms an extreme case for the 1/3–2/3 conjecture in the sense that, for each pair x, y of elements, one of the two elements occurs earlier than the other in at most 1/3 of the linear extensions of P. Partial orders with this structure are necessarily series-parallel semiorders; they are the only known extreme cases for the conjecture and can be proven to be the only extreme cases with width two.[2]
Definitions
A partially ordered set is a set X together with a binary relation ≤ that is reflexive, antisymmetric, and transitive. A total order is a partial order in which every pair of elements is comparable. A linear extension of a finite partial order is a sequential ordering of the elements of X, with the property that if x ≤ y in the partial order, then x must come before y in the linear extension. In other words, it is a total order compatible with the partial order. If a finite partially ordered set is totally ordered, then it has only one linear extension, but otherwise it will have more than one. The 1/3–2/3 conjecture states that one can choose two elements x and y such that, among this set of possible linear extensions, between 1/3 and 2/3 of them place x earlier than y, and symmetrically between 1/3 and 2/3 of them place y earlier than x.[3]
There is an alternative and equivalent statement of the 1/3–2/3 conjecture in the language of probability theory. One may define a uniform probability distribution on the linear extensions in which each possible linear extension is equally likely to be chosen. The 1/3–2/3 conjecture states that, under this probability distribution, there exists a pair of elements x and y such that the probability that x is earlier than y in a random linear extension is between 1/3 and 2/3.[3]
In 1984, Jeff Kahn and Michael Saks defined δ(P), for any partially ordered set P, to be the largest real number δ such that P has a pair x, y with x earlier than y in a number of linear extensions that is between δ and 1 − δ of the total number of linear extensions. In this notation, the 1/3–2/3 conjecture states that every finite partial order that is not total has δ(P) ≥ 1/3.[4]
History
The 1/3–2/3 conjecture was formulated by Sergey Kislitsyn in 1968,[5] and later made independently by Michael Fredman[6] and by Nati Linial in 1984.[3] It was listed as a featured unsolved problem at the founding of the journal Order, and remains unsolved; being called "one of the most intriguing problems in the combinatorial theory of posets."[3]
A survey of the conjecture was produced in 1999.[7]
Partial results
The 1/3–2/3 conjecture is known to be true for certain special classes of partial orders, including partial orders of width two,[8] partial orders of height two,[9] partial orders with at most 11 elements,[10] partial orders in which each element is incomparable to at most six others,[11] series-parallel partial orders,[12] semiorders.[13] and polytrees.[14] In the limit as n goes to infinity, the proportion of n-element partial orders that obey the 1/3–2/3 conjecture approaches 100%.[10]
In 1995, Graham Brightwell, Stefan Felsner, and William Trotter proved that, for any finite partial order P that is not total, δ(P) ≥ 1/2 − √5/10 ≈ 0.276. Their results improve previous weaker bounds of the same type.[15] They use the probabilistic interpretation of δ(P) to extend its definition to certain infinite partial orders; in that context, they show that their bounds are optimal, in that there exist infinite partial orders with δ(P) = 1/2 − √5/10.
Applications
In 1984 Jeff Kahn and Saks proposed the following application for the problem: suppose one wishes to comparison sort a totally ordered set X, for which some partial order information is already known in the form of a partial order on X. In the worst case, each additional comparison between a pair x and y of elements may yield as little information as possible, by resolving the comparison in a way that leaves as many linear extensions as possible compatible with the comparison result. The 1/3–2/3 conjecture states that, at each step, one may choose a comparison to perform that reduces the remaining number of linear extensions by a factor of 2/3; therefore, if there are E linear extensions of the partial order given by the initial information, the sorting problem can be completed in at most log3/2E additional comparisons.[16]
However, this analysis neglects the complexity of selecting the optimal pair x and y to compare. Additionally, it may be possible to sort a partial order using a number of comparisons that is better than this analysis would suggest, because it may not be possible for this worst-case behavior to occur at each step of a sorting algorithm. In this direction, it has been conjectured that logφE comparisons may suffice, where φ denotes the golden ratio.[10]
A closely related class of comparison sorting problems was considered by Fredman in 1976, among them the problem of comparison sorting a set X when the sorted order of X is known to lie in some set S of permutations of X. Here S is not necessarily generated as the set of linear extensions of a partial order. Despite this added generality, Fredman showed that X can be sorted using log2|S| + O(|X|) comparisons, expressed in big O notation. This same bound applies as well to the case of partial orders and shows that log2E + O(n) comparisons suffice.[17]
Generalizations and related results
In 1984, Kahn and Saks conjectured that, in the limit as w tends to infinity, the value of δ(P) for partially ordered sets of width w should tend to 1/2. In particular, they expect that only partially ordered sets of width two can achieve the worst case value δ(P) = 1/3,[18] and in 1985 Martin Aigner stated this explicitly as a conjecture.[2] The smallest known value of δ(P) for posets of width three is 14/39,[19] and computer searches have shown that no smaller value is possible for width-3 posets with nine or fewer elements.[9] Another related conjecture, again based on computer searches, states that there is a gap between 1/3 and the other possible values of δ(P): whenever a partial order does not have δ(P) exactly 1/3, it has δ(P) ≥ 0.348843.[20]
Marcin Peczarski[10][11] has formulated a "gold partition conjecture" stating that in each partial order that is not a total order one can find two consecutive comparisons such that, if ti denotes the number of linear extensions remaining after i of the comparisons have been made, then (in each of the four possible outcomes of the comparisons) t0 ≥ t1 + t2. If this conjecture is true, it would imply the 1/3–2/3 conjecture: the first of the two comparisons must be between a pair that splits the remaining comparisons by at worst a 1/3–2/3 ratio. The gold partition conjecture would also imply that a partial order with E linear extensions can be sorted in at most logφE comparisons; the name of the conjecture is derived from this connection with the golden ratio.
It is #P-complete, given a finite partial order P and a pair of elements x and y, to calculate the proportion of the linear extensions of P that place x earlier than y.[21]
Notes
1. Kahn & Saks (1984); Brightwell, Felsner & Trotter (1995).
2. Aigner (1985).
3. Brightwell, Felsner & Trotter (1995).
4. Kahn & Saks (1984)
5. Kislitsyn (1968)
6. However, despite the close connection of Fredman (1976) to the problem of sorting partially ordered data and hence to the 1/3–2/3 conjecture, it is not mentioned in that paper.
7. Brightwell (1999)
8. Linial (1984), Theorem 2; Sah (2021).
9. Trotter, Gehrlein & Fishburn (1992).
10. Peczarski (2006).
11. Peczarski (2008).
12. Zaguia (2012).
13. Brightwell (1989).
14. Zaguia (2019).
15. Brightwell, Felsner & Trotter (1995); Kahn & Saks (1984); Khachiyan (1989); Kahn & Linial (1991); Felsner & Trotter (1993).
16. Kahn & Saks (1984)
17. Fredman (1976)
18. Kahn & Saks (1984).
19. Saks (1985).
20. Peczarski (2019).
21. Brightwell & Winkler (1991).
References
• Aigner, Martin (1985), "A note on merging", Order, 2 (3): 257–264, doi:10.1007/BF00333131, S2CID 118877843.
• Brightwell, Graham R. (1989), "Semiorders and the 1/3–2/3 conjecture", Order, 5 (4): 369–380, doi:10.1007/BF00353656, S2CID 86860160.
• Brightwell, Graham R. (1999), "Balanced pairs in partial orders", Discrete Mathematics, 201 (1–3): 25–52, doi:10.1016/S0012-365X(98)00311-2.
• Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841, doi:10.1007/BF01110378, MR 1368815, S2CID 14793475.
• Brightwell, Graham R.; Winkler, Peter (1991), "Counting linear extensions", Order, 8 (3): 225–242, doi:10.1007/BF00383444, S2CID 119697949.
• Felsner, Stefan; Trotter, William T. (1993), "Balancing pairs in partially ordered sets", Combinatorics, Paul Erdős is eighty, Bolyai Society Mathematical Studies, vol. 1, Budapest: János Bolyai Mathematical Society, pp. 145–157, MR 1249709.
• Fredman, M. L. (1976), "How good is the information theory bound in sorting?", Theoretical Computer Science, 1 (4): 355–361, doi:10.1016/0304-3975(76)90078-5
• Kahn, Jeff; Linial, Nati (1991), "Balancing extensions via Brunn-Minkowski", Combinatorica, 11 (4): 363–368, doi:10.1007/BF01275670, S2CID 206793172.
• Kahn, Jeff; Saks, Michael (1984), "Balancing poset extensions", Order, 1 (2): 113–126, doi:10.1007/BF00565647, S2CID 123370506.
• Khachiyan, Leonid (1989), "Optimal algorithms in convex programming decomposition and sorting", in Jaravlev, J. (ed.), Computers and Decision Problems (in Russian), Moscow: Nauka, pp. 161–205. As cited by Brightwell, Felsner & Trotter (1995).
• Kislitsyn, S. S. (1968), "A finite partially ordered set and its corresponding set of permutations", Mathematical Notes, 4 (5): 798–801, doi:10.1007/BF01111312, S2CID 120228193.
• Linial, Nati (1984), "The information-theoretic bound is good for merging", SIAM Journal on Computing, 13 (4): 795–801, doi:10.1137/0213049, S2CID 5149351.
• Peczarski, Marcin (2006), "The gold partition conjecture", Order, 23 (1): 89–95, doi:10.1007/s11083-006-9033-1, S2CID 42415160.
• Peczarski, Marcin (2008), "The gold partition conjecture for 6-thin posets", Order, 25 (2): 91–103, doi:10.1007/s11083-008-9081-9, S2CID 42034095.
• Peczarski, Marcin (2019), "The worst balanced partially ordered sets—ladders with broken rungs", Experimental Mathematics, 28 (2): 181–184, doi:10.1080/10586458.2017.1368050, MR 3955809, S2CID 125593629.
• Sah, Ashwin (2021), "Improving the ${\tfrac {1}{3}}$–${\tfrac {2}{3}}$ conjecture for width two posets", Combinatorica, 41 (1): 99–126, arXiv:1811.01500, doi:10.1007/s00493-020-4091-3, MR 4235316, S2CID 119604901
• Saks, Michael (1985), "Balancing linear extensions of ordered sets", Unsolved problems, Order, 2: 327–330, doi:10.1007/BF00333138, S2CID 189901558
• Trotter, William T.; Gehrlein, William V.; Fishburn, Peter C. (1992), "Balance theorems for height-2 posets", Order, 9 (1): 43–53, doi:10.1007/BF00419038, S2CID 16157076.
• Zaguia, Imed (2012), "The 1/3-2/3 Conjecture for N-free ordered sets", Electronic Journal of Combinatorics, 19 (2): P29, arXiv:1107.5626, Bibcode:2011arXiv1107.5626Z, doi:10.37236/2345, S2CID 1979845.
• Zaguia, Imed (2019), "The 1/3–2/3 conjecture for ordered sets whose cover graph is a forest", Order, 36 (2): 335–347, arXiv:1610.00809, doi:10.1007/s11083-018-9469-0, MR 3983482, S2CID 119631612.
| Wikipedia |
10-orthoplex
In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells 4-faces, 13440 5-faces, 15360 6-faces, 11520 7-faces, 5120 8-faces, and 1024 9-faces.
10-orthoplex
Decacross
Orthogonal projection
inside Petrie polygon
TypeRegular 10-polytope
FamilyOrthoplex
Schläfli symbol{38,4}
{37,31,1}
Coxeter-Dynkin diagrams
9-faces1024 {38}
8-faces5120 {37}
7-faces11520 {36}
6-faces15360 {35}
5-faces13440 {34}
4-faces8064 {33}
Cells3360 {3,3}
Faces960 {3}
Edges180
Vertices20
Vertex figure9-orthoplex
Petrie polygonIcosagon
Coxeter groupsC10, [38,4]
D10, [37,1,1]
Dual10-cube
PropertiesConvex, Hanner polytope
It has two constructed forms, the first being regular with Schläfli symbol {38,4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {37,31,1} or Coxeter symbol 711.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.
Alternate names
• Decacross is derived from combining the family name cross polytope with deca for ten (dimensions) in Greek
• Chilliaicositetraronnon as a 1024-facetted 10-polytope (polyronnon).
Construction
There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or [4,38] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or [37,1,1] symmetry group.
Cartesian coordinates
Cartesian coordinates for the vertices of a 10-orthoplex, centred at the origin are
(±1,0,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0,0), (0,0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]
A9 A5
— —
[10] [6]
A7 A3
— —
[8] [4]
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o4o - ka".
External links
• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Association for Symbolic Logic
The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Alonzo Church. The current president of the ASL is Phokion Kolaitis.[1]
Association for Symbolic Logic
AbbreviationASL
Formation1936
TypeScholarly society
PurposeResearch, Inquiry
HeadquartersStorrs, Connecticut
President
Phokion Kolaitis
Vice President
Natasha Dobrinen
Co-Secretary-Treasurer
Russell G. Miller
Co-Secretary-Treasurer
Reed Solomon
Websiteaslonline.org
Publications
The ASL publishes books and academic journals. Its three official journals are:
• Journal of Symbolic Logic (website) – publishes research in all areas of mathematical logic. Founded in 1936, ISSN 0022-4812.
• Bulletin of Symbolic Logic (website) – publishes primarily expository articles and reviews. Founded in 1995, ISSN 1079-8986.
• Review of Symbolic Logic (website) – publishes research relating to logic, philosophy, science, and their interactions. Founded in 2008, ISSN 1755-0203.
In addition, the ASL has a sponsored journal:
• Journal of Logic and Analysis (website) – publishes research on the interactions between mathematical logic and pure and applied analysis. Founded in 2009 as an open-access successor to the Springer journal Logic and Analysis. ISSN 1759-9008.
The organization played a part in publishing the collected writings of Kurt Gödel.[2]
Meetings
The ASL holds two main meetings every year, one in North America and one in Europe (the latter known as the Logic Colloquium). In addition, the ASL regularly holds joint meetings with both the American Mathematical Society ("AMS") and the American Philosophical Association ("APA"), and sponsors meetings in many different countries every year.
List of presidents
Name Term of office
1st President Curt John Ducasse 1936–1937
2nd President Haskell Curry 1938–1940
3rd President Cooper Harold Langford 1941–1943
4th President Alfred Tarski 1944–1946
5th President Ernest Nagel 1947–1949
6th President J. Barkley Rosser 1950–1952
7th President Willard Van Orman Quine 1953–1955
8th President Stephen Cole Kleene 1956–1958
9th President Frederic Fitch 1959–1961
10th President Leon Henkin 1962–1964
11th President William Craig 1965–1967
12th President Abraham Robinson 1968–1970
13th President Dana Scott 1971–1973
14th President Joseph R. Shoenfield 1974–1976
15th President Hilary Putnam 1977–1979
16th President Solomon Feferman 1980–1982
17th President Ruth Barcan Marcus 1983–1985
18th President Michael Morley 1986–1988
19th President Charles Parsons 1989–1991
20th President Yiannis Moschovakis 1992–1994
21st President George Boolos 1995–1996
22nd President Menachem Magidor 1996–1997
23rd President Donald A. Martin 1998–2000
24th President Richard Shore 2001–2003
25th President Alexander Kechris 2004–2006
26th President Penelope Maddy 2007–2009
27th President Alex Wilkie 2010–2012
28th President Alasdair Urquhart 2013–2015
29th President Ulrich Kohlenbach 2016–2018
30th President Julia Knight 2019–2021
31st President Phokion Kolaitis 2022–2024
[3]
Awards
The association periodically presents a number of prizes and awards.[4]
Karp Prize
The Karp Prize is awarded by the association every five years for an outstanding paper or book in the field of symbolic logic. It consists of a cash award and was established in 1973 in memory of Professor Carol Karp.[5]
YearRecipient(s)
1978Robert Vaught, University of California, Berkeley
1983Saharon Shelah, Hebrew University
1988Donald A. Martin, UCLA; John R. Steel, UCLA; W. Hugh Woodin, University of California, Berkeley
1993Ehud Hrushovski, MIT and Alex Wilkie, Oxford
1998Ehud Hrushovski, Hebrew University
2003Gregory Hjorth, UCLA and Alexander Kechris, Caltech
2008Zlil Sela, Hebrew University
2013Moti Gitik, Tel Aviv University; Ya'acov Peterzil, University of Haifa; Jonathan Pila, University of Oxford; Sergei Starchenko, University of Notre Dame; Alex Wilkie, University of Manchester
2018Matthias Aschenbrenner, UCLA; Lou van den Dries, University of Illinois at Urbana–Champaign; Joris van der Hoeven, École Polytechnique
Sacks Prize
The Sacks Prize is awarded for the most outstanding doctoral dissertation in mathematical logic. It consists of a cash award and was established in 1999 to honor Professor Gerald Sacks of MIT and Harvard.
Recipients include:[6]
YearRecipient(s)
1994Gregory Hjorth
1995Slawomir Solecki
1996Byunghan Kim
1997Ilijas Farah and Thomas Scanlon
1998no prize awarded
1999Denis Hirschfeldt and Rene Schipperus
2000Eric Jaligot
2001Matthias Aschenbrenner
2002no prize awarded
2003Itay Ben Yaacov
2004Joseph Mileti and Nathan Segerlind
2005Antonio Montalbán
2006Matteo Viale
2007Adrien Deloro and Wojciech Moczydlowski
2008Inessa Epstein and Dilip Raghavan
2009Isaac Goldbring and Grigor Sargsyan
2010Uri Andrews
2011Mingzhong Cai and Adam Day
2012Pierre Simon
2013Artem Chernikov and Nathanaël Mariaule
2014no prize awarded
2015Omer Ben-Neria and Martino Lupini
2016William Johnson and Ludovic Patey
2017Matthew Harrison-Trainor and Sebastien Vasey
2018Danny Nguyen
2019Gabriel Goldberg
2020James Walsh
2021Marcos Mazari Armida
2022Francesco Gallinaro and Patrick Lutz
Shoenfield Prize
Inaugurated in 2007, the Shoenfield Prize is awarded every three years in two categories, book and article, recognizing outstanding expository writing in the field of logic and honoring the name of Joseph R. Shoenfield.[7]
Recipients include:
YearRecipient(s)
2007John P. Burgess (book); Bohuslav Balcar and Thomas Jech (article)
2010John T. Baldwin (book); Rod Downey, Denis Hirschfeldt, André Nies, and Sebastiaan Terwijn (article)
2013Stevo Todorcevic (book); Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov (article)
2016Rod Downey and Denis Hirschfeldt (book); Lou van den Dries (article)
2019Pierre Simon (book); John Steel (article)
2022Paolo Mancosu, Sergio Galvan, and Richard Zach (book); Vasco Brattka (article)
References
1. "Council Members, Committees, and Representatives – Association for Symbolic Logic". Association for Symbolic Logic. Retrieved 4 May 2023.
2. Gödel Lecturers – Association for Symbolic Logic
3. Former Officers – Association for Symbolic Logic
4. "Prizes and Awards – Association for Symbolic Logic". Association of Symbolic Logic. Retrieved 24 January 2019.
5. Karp Prize Recipients – Association for Symbolic Logic
6. Sacks Prize Recipients – Association for Symbolic Logic
7. Shoenfield Prize Recipients – Association for Symbolic Logic
External links
• ASL website
Authority control
International
• ISNI
• 2
• VIAF
National
• France
• BnF data
• Germany
• Israel
• United States
• Czech Republic
• Australia
Other
• IdRef
| Wikipedia |
Myriagon
In geometry, a myriagon or 10000-gon is a polygon with 10000 sides. Several philosophers have used the regular myriagon to illustrate issues regarding thought.[1][2][3][4][5]
Regular myriagon
A regular myriagon
TypeRegular polygon
Edges and vertices10000
Schläfli symbol{10000}, t{5000}, tt{2500}, ttt{1250}, tttt{625}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D10000), order 2×10000
Internal angle (degrees)179.964°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular myriagon
A regular myriagon is represented by Schläfli symbol {10,000} and can be constructed as a truncated 5000-gon, t{5000}, or a twice-truncated 2500-gon, tt{2500}, or a thrice-truncated 1250-gon, ttt{1250), or a four-fold-truncated 625-gon, tttt{625}.
The measure of each internal angle in a regular myriagon is 179.964°. The area of a regular myriagon with sides of length a is given by
$A=2500a^{2}\cot {\frac {\pi }{10000}}$
The result differs from the area of its circumscribed circle by up to 40 parts per billion.
Because 10,000 = 24 × 54, the number of sides is neither a product of distinct Fermat primes nor a power of two. Thus the regular myriagon is not a constructible polygon. Indeed, it is not even constructible with the use of an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.
Symmetry
The regular myriagon has Dih10000 dihedral symmetry, order 20000, represented by 10000 lines of reflection. Dih10000 has 24 dihedral subgroups: (Dih5000, Dih2500, Dih1250, Dih625), (Dih2000, Dih1000, Dih500, Dih250, Dih125), (Dih400, Dih200, Dih100, Dih50, Dih25), (Dih80, Dih40, Dih20, Dih10, Dih5), and (Dih16, Dih8, Dih4, Dih2, Dih1). It also has 25 more cyclic symmetries as subgroups: (Z10000, Z5000, Z2500, Z1250, Z625), (Z2000, Z1000, Z500, Z250, Z125), (Z400, Z200, Z100, Z50, Z25), (Z80, Z40, Z20, Z10), and (Z16, Z8, Z4, Z2, Z1), with Zn representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.[6] r20000 represents full symmetry, and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.
These lower symmetries allows degrees of freedom in defining irregular myriagons. Only the g10000 subgroup has no degrees of freedom but can seen as directed edges.
Myriagram
A myriagram is a 10,000-sided star polygon. There are 1999 regular forms[lower-alpha 1] given by Schläfli symbols of the form {10000/n}, where n is an integer between 2 and 5,000 that is coprime to 10,000. There are also 3000 regular star figures in the remaining cases.
In popular culture
In the novella Flatland, the Chief Circle is assumed to have ten thousand sides, making him a myriagon.
See also
• Chiliagon
• Megagon
Notes
1. 5000 cases - 1 (convex) - 1,000 (multiples of 5) - 2,500 (multiples of 2)+ 500 (multiples of 2 and 5)
References
1. Meditation VI by Descartes (English translation).
2. Hippolyte Taine, On Intelligence: pp. 9–10
3. Jacques Maritain, An Introduction to Philosophy: p. 108
4. Alan Nelson (ed.), A Companion to Rationalism: p. 285
5. Paolo Fabiani, The philosophy of the imagination in Vico and Malebranche: p. 222
6. The Symmetries of Things, Chapter 20
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
666 (number)
666 (six hundred [and] sixty-six) is the natural number following 665 and preceding 667.
← 665 666 667 →
• List of numbers
• Integers
← 0 100 200 300 400 500 600 700 800 900 →
Cardinalsix hundred sixty-six
Ordinal666th
(six hundred sixty-sixth)
Factorization2 × 32 × 37
Greek numeralΧΞϚ´
Roman numeralDCLXVI
Greek prefixἑξακόσιοι ἑξήκοντα ἕξ
hexakósioi hexēkonta héx
Latin prefixsescenti sexaginta sex
Binary10100110102
Ternary2202003
Senary30306
Octal12328
Duodecimal47612
Hexadecimal29A16
Chinese numeral六百六十六
Devanagari numeral६६६
In Christianity, 666 is referred to in (most manuscripts of) chapter 13 of the Book of Revelation of the New Testament as the "number of the beast."[1][2][3]
In mathematics
666 is the sum of the first thirty-six natural numbers, which makes it a triangular number:[4]
$\sum _{i=1}^{36}i=1+2+3+\cdots +34+35+36=666$.
Since 36 is also triangular, 666 is a doubly triangular number.[5] Also, 36 = 15 + 21 where 15 and 21 are triangular as well, whose squares (152 = 225 and 212 = 441) add to 666 and have a difference of 216 = 6 × 6 × 6.
The number of integers which are relatively prime to 666 is also 216, $\phi (666)=216$;[6] and for an angle measured in degrees, $\sin {(666^{\circ })}=\cos {(216^{\circ })}=-{\tfrac {\varphi }{2}}$ (where here $\varphi $ is the golden ratio).[7][8][lower-alpha 1]
666 is also the sum of the squares of the first seven primes (22 + 32 + 52 + 72 + 112 + 132 + 172),[7][10] while the number of twin primes less than 66 + 666 is 666.[11]
A prime reciprocal magic square based on ${\tfrac {1}{149}}$ in decimal has a magic constant of 666. The twelfth pair of twin primes is (149, 151),[12] with 151 the thirty-sixth prime number.[lower-alpha 2]
666 is a Smith number and Harshad number in base-ten.[13][14] The 27th indexed unique prime in decimal features a "666" in the middle of its sequence of digits.[15][lower-alpha 3]
The Roman numeral for 666, DCLXVI, has exactly one occurrence of all symbols whose value is less than 1000 in decreasing order (D = 500, C = 100, L = 50, X = 10, V = 5, I = 1).[7]
In religion
Number of the beast
In the Textus Receptus manuscripts of the New Testament, the Book of Revelation (13:17–18) cryptically asserts 666 to be "the man's number" or "the number of a man" (depending on how the text is translated) associated with the Beast, an antagonistic creature that appears briefly about two-thirds into the apocalyptic vision. Some manuscripts of the original Koine Greek use the symbols χξϛ chi xi stigma (or χξϝ with a digamma), while other manuscripts spell out the number in words.[16]
In modern popular culture, 666 has become one of the most widely recognized symbols for the Antichrist or, alternatively, the devil. Earnest references to the number occur both among apocalypticist Christian groups and in explicitly anti-Christian subcultures. References in contemporary Western art or literature are, more likely than not, intentional references to the Beast symbolism. Such popular references are therefore too numerous to list.
It is common to see the symbolic role of the integer 666 transferred to the numerical digit sequence 6-6-6. Some people take the Satanic associations of 666 so seriously that they actively avoid things related to 666 or the digits 6-6-6. This is known as hexakosioihexekontahexaphobia.
The Number of the Beast is cited as 616 in some early biblical manuscripts, the earliest known instance being in Papyrus 115.[17][18]
Other occurrences
• In the Bible, 666 is the number of talents of gold Solomon collected each year (see 1 Kings 10:14 and 2 Chronicles 9:13).
• In the Bible, 666 is the number of Adonikam's descendants who return to Jerusalem and Judah from the Babylonian exile (see Ezra 2:13).
• Using gematria, Neron Caesar transliterated from Greek into Hebrew short-form spelling, נרון קסר, produces the number 666. The Latin spelling of "Nero Caesar" transliterated into Ktiv haser Hebrew, נרו קסר, produces the number 616. Thus, in the Bible, 666 may have been a coded reference to Nero, who was the Roman emperor from 55 to 68 AD.[19] Though historic protestants such as Andreas Helwig in 1612 proposed the application of the Isopsephy principle to the papal name Vicarius Filii Dei.[20]
In other fields
• Is the magic sum, or sum of the magic constants of a six by six magic square, any row or column of which adds up to 111.
• Is the sum of all the numbers on a roulette wheel (0 through 36).[19]
• Was a winning lottery number in the 1980 Pennsylvania Lottery scandal, in which equipment was tampered to favor a 4 or 6 as each of the three individual random digits.[21]
• Was the original name of the Macintosh SevenDust computer virus that was discovered in 1998. It is also the name of an extension that SevenDust can add to an uninfected Macintosh.[22]
• The number is a frequent visual element of Aryan Brotherhood tattoos.[23]
• Aleister Crowley adopted the title "the Beast 666".[24] As such, 666 is also associated with him, his work, and his religious philosophy of Thelema.
• Molar mass of the high-temperature superconductor YBa2Cu3O7.
• In Chinese numerology, the number is considered to be lucky and is often displayed in shop windows and neon signs.[25][26] In China, 666 can mean "everything goes smoothly" (the number six has the same pronunciation as the character 溜, which means "smooth".[27]
• Is commonly used by ISPs to blackhole traffic using BGP communities.[28]
• 666 Fifth Avenue in New York City, which was bought for $1.8 billion in 2007, was the most expensive real estate deal in New York's history.[29]
• The Number of the Beast, the 1982 album by English heavy metal band Iron Maiden, references 666 in its title and the album's title song.
See also
• Numerology
Notes
1. Other relevant trigonometric equalities include:[9]
• sin(6°) − sin(66°) = sin(666°),
• cos(6°) − cos(66°) = cos(666°), and
• tan(666°) × tan(216) = −1.
Where, sin(666°) = cos(216)° = −φ/2, also equivalently −cos(36°).
2. 149 and 151 also generate a twin-prime sum equal to 300, which is the 24th triangular number.
3. It is the number 11110...109877666779011...011111 (241 digits long), with an equal amount of digits (119) that precede and follow a central 666 string (that also contains all sixes in the number). It is the twenty-seventh indexed member within the sequence of unique primes (where 33 = 27).
References
1. Revelation 13
2. Beale, Gregory K. (1999). The Book of Revelation: A Commentary on the Greek Text. Grand Rapids, Michigan: Wm. B. Eerdmans Publishing. p. 718. ISBN 080282174X. Retrieved 9 July 2012.
3. "The Numerology of the Beast". people.math.harvard.edu. Retrieved 2021-03-30.
4. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
5. Sloane, N. J. A. (ed.). "Sequence A002817 (Doubly triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
6. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
7. Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books. pp. 145–146. ISBN 0-14-008029-5. OCLC 39262447. S2CID 118329153.
8. Wang, Steve C. (1994). "The Sign of the Devil...and the Sine of the Devil" (PDF). Journal of Recreational Mathematics. Baywood Publishing. 26 (3): 201–205. ISSN 0022-412X. OCLC 938842643. Archived (PDF) from the original on 2023-07-12.
9. Bogomolny, Alexander. "Beauty and the Beast - in Trigonometry". Cut The Knot (Interactive Mathematics Miscellany and Puzzles). Archived from the original on 2023-07-12. Retrieved 2023-07-11.
10. Sloane, N. J. A. (ed.). "Sequence A024450 (Sum of squares of the first n primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-12.
11. Caldwell, Chris K.; Honaker, Jr., G. L. "666". PrimeCurios!. PrimePages. Retrieved 2023-07-12.
66 + 666 = 47322, and the 666th pair of twin primes is (47147, 47149).
12. Sloane, N. J. A. (ed.). "Sequence A077800 (List of twin primes {p, p+2}.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
13. Sloane, N. J. A. (ed.). "Sequence A006753 (Smith (or joke) numbers: composite numbers n such that sum of digits of n is equal to the sum of digits of prime factors of n (counted with multiplicity).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
14. Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-12.
15. Sloane, N. J. A. (ed.). "Sequence A040017 (Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-11.
16. "Revelation 13:18". Stephanus New Testament. Bible Gateway. Retrieved 2006-06-22.
17. Novum Testamentum Graece, Nestle and Aland, 1991, footnote to verse 13:18 of Revelation, page 659: "-σιοι δέκα ἕξ" as found in C [C=Codex Ephraemi Rescriptus]; for English see Metzger's Textual Commentary on the Greek New Testament, note on verse 13:18 of Revelation, page 750: "the numeral 616 was also read ..."
18. "The Other Number of the Beast". Archived from the original on 2000-03-01.
19. "666 – professors explain Roulette and Nero in detail; numberphile.com". Archived from the original on 2013-03-31. Retrieved 2013-04-06.
20. Helwig, Andreas (1512). Antichristus Romanus. VVttenbergae, Typis Laurentij Seuberlichs.
21. Baer, John (1980-09-20). "Six Won $1.2 Million in Rigged Lottery, Pa. Says". Washington Post. ISSN 0190-8286. Retrieved 2021-03-18.
22. "Detailed Analysis - Mac/Sevendust-A - Viruses and Spyware - Advanced Network Threat Protection | ATP from Targeted Malware Attacks and Persistent Threats | sophos.com - Threat Center". www.sophos.com. Retrieved 2022-01-24.
23. Brook, John Lee (June 2011). Blood In, Blood Out: The Violent Empire of the Aryan Brotherhood. SCB Distributors. ISBN 978-1-900486-80-4. OCLC 793002272. Retrieved 10 January 2018.
24. "Aleister Crowley | Biography, Teachings, Reputation, & Facts | Britannica". www.britannica.com. Retrieved 2022-01-24.
25. Mah, Adeline Yen (2009). China: Land of Dragons and Emperors. ISBN 978-0375890994. Retrieved 2013-12-07.
26. "Know the Meaning of Numbers in Chinese Culture". au.ibtimes.com. Archived from the original on 2014-11-08. Retrieved 2014-10-30.
27. "666 – Good day, bad day or just another day?". www.newsgd.com. Retrieved 2014-10-30.
28. RFC 7999
29. "What you need to know about 666 Fifth Avenue". www.cbsnews.com. 19 June 2017. Retrieved 2021-07-29.
External links
• Media related to 666 (number) at Wikimedia Commons
• Weisstein, Eric W. "Beast Number". MathWorld.
• Watts, Peter; Grime, James. "666 – The Number of the Beast". Numberphile. Brady Haran. Archived from the original on 2013-03-31. Retrieved 2013-04-06.
• disastercountdown
• CNN(U.S.A):Pastor with 666 tattoo
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| Wikipedia |
Power of two
A power of two is a number of the form 2n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent.
In a context where only integers are considered, n is restricted to non-negative values,[1] so there are 1, 2, and 2 multiplied by itself a certain number of times.[2]
The first ten powers of 2 for non-negative values of n are:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ... (sequence A000079 in the OEIS)
Base of the binary numeral system
Because two is the base of the binary numeral system, powers of two are common in computer science. Written in binary, a power of two always has the form 100...000 or 0.00...001, just like a power of 10 in the decimal system.
Computer science
Two to the exponent of n, written as 2n, is the number of ways the bits in a binary word of length n can be arranged. A word, interpreted as an unsigned integer, can represent values from 0 (000...0002) to 2n − 1 (111...1112) inclusively. Corresponding signed integer values can be positive, negative and zero; see signed number representations. Either way, one less than a power of two is often the upper bound of an integer in binary computers. As a consequence, numbers of this form show up frequently in computer software. As an example, a video game running on an 8-bit system might limit the score or the number of items the player can hold to 255—the result of using a byte, which is 8 bits long, to store the number, giving a maximum value of 28 − 1 = 255. For example, in the original Legend of Zelda the main character was limited to carrying 255 rupees (the currency of the game) at any given time, and the video game Pac-Man famously has a kill screen at level 256.
Powers of two are often used to measure computer memory. A byte is now considered eight bits (an octet), resulting in the possibility of 256 values (28). (The term byte once meant (and in some cases, still means) a collection of bits, typically of 5 to 32 bits, rather than only an 8-bit unit.) The prefix kilo, in conjunction with byte, may be, and has traditionally been, used, to mean 1,024 (210). However, in general, the term kilo has been used in the International System of Units to mean 1,000 (103). Binary prefixes have been standardized, such as kibi (Ki) meaning 1,024. Nearly all processor registers have sizes that are powers of two, 32 or 64 being very common.
Powers of two occur in a range of other places as well. For many disk drives, at least one of the sector size, number of sectors per track, and number of tracks per surface is a power of two. The logical block size is almost always a power of two.
Numbers that are not powers of two occur in a number of situations, such as video resolutions, but they are often the sum or product of only two or three powers of two, or powers of two minus one. For example, 640 = 32 × 20, and 480 = 32 × 15. Put another way, they have fairly regular bit patterns.
Mersenne and Fermat primes
A prime number that is one less than a power of two is called a Mersenne prime. For example, the prime number 31 is a Mersenne prime because it is 1 less than 32 (25). Similarly, a prime number (like 257) that is one more than a positive power of two is called a Fermat prime—the exponent itself is a power of two. A fraction that has a power of two as its denominator is called a dyadic rational. The numbers that can be represented as sums of consecutive positive integers are called polite numbers; they are exactly the numbers that are not powers of two.
Euclid's Elements, Book IX
The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the nth term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.
Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted from the second and last term in the sequence, then as the excess of the second is to the first—so is the excess of the last to all those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31, 62, 124, 248. Therefore, the numbers 1, 2, 4, 8, 16, 31, 62, 124 and 248 add up to 496 and further these are all the numbers that divide 496. For suppose that p divides 496 and it is not amongst these numbers. Assume p q is equal to 16 × 31, or 31 is to q as p is to 16. Now p cannot divide 16 or it would be amongst the numbers 1, 2, 4, 8 or 16. Therefore, 31 cannot divide q. And since 31 does not divide q and q measures 496, the fundamental theorem of arithmetic implies that q must divide 16 and be amongst the numbers 1, 2, 4, 8 or 16. Let q be 4, then p must be 124, which is impossible since by hypothesis p is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.
Table of values
(sequence A000079 in the OEIS)
n 2n n 2n n 2n n 2n
01 1665,536 324,294,967,296 48281,474,976,710,656
12 17131,072 338,589,934,592 49562,949,953,421,312
24 18262,144 3417,179,869,184 501,125,899,906,842,624
38 19524,288 3534,359,738,368 512,251,799,813,685,248
416 201,048,576 3668,719,476,736 524,503,599,627,370,496
532 212,097,152 37137,438,953,472 539,007,199,254,740,992
664 224,194,304 38274,877,906,944 5418,014,398,509,481,984
7128 238,388,608 39549,755,813,888 5536,028,797,018,963,968
8256 2416,777,216 401,099,511,627,776 5672,057,594,037,927,936
9512 2533,554,432 412,199,023,255,552 57144,115,188,075,855,872
101,024 2667,108,864 424,398,046,511,104 58288,230,376,151,711,744
112,048 27134,217,728 438,796,093,022,208 59576,460,752,303,423,488
124,096 28268,435,456 4417,592,186,044,416 601,152,921,504,606,846,976
138,192 29536,870,912 4535,184,372,088,832 612,305,843,009,213,693,952
1416,384 301,073,741,824 4670,368,744,177,664 624,611,686,018,427,387,904
1532,768 312,147,483,648 47140,737,488,355,328 639,223,372,036,854,775,808
Last digits
Starting with 2 the last digit is periodic with period 4, with the cycle 2–4–8–6–, and starting with 4 the last two digits are periodic with period 20. These patterns are generally true of any power, with respect to any base. The pattern continues where each pattern has starting point 2k, and the period is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicative group of integers modulo n).
Powers of 1024
(sequence A140300 in the OEIS)
The first few powers of 210 are slightly larger than those same powers of 1000 (103). The powers of 210 values that have less than 25% deviation are listed below:
20=1= 10000(0% deviation)
210=1 024≈ 10001(2.4% deviation)
220=1 048 576≈ 10002(4.9% deviation)
230=1 073 741 824≈ 10003(7.4% deviation)
240=1 099 511 627 776≈ 10004(10.0% deviation)
250=1 125 899 906 842 624≈ 10005(12.6% deviation)
260=1 152 921 504 606 846 976≈ 10006(15.3% deviation)
270=1 180 591 620 717 411 303 424≈ 10007(18.1% deviation)
280=1 208 925 819 614 629 174 706 176≈ 10008(20.9% deviation)
290=1 237 940 039 285 380 274 899 124 224≈ 10009(23.8% deviation)
It takes approximately 17 powers of 1024 to reach 50% deviation and approximately 29 powers of 1024 to reach 100% deviation of the same powers of 1000.
Powers of two whose exponents are powers of two
Because data (specifically integers) and the addresses of data are stored using the same hardware, and the data is stored in one or more octets (23), double exponentials of two are common. The first 20 of them are:
n2n22n (sequence A001146 in the OEIS)digits
0121
1241
24162
382563
41665,5365
5324,294,967,29610
66418,446,744,073,709,551,61620
7128340,282,366,920,938,463,463,374,607,431,768,211,45639
8256115,792,089,237,316,195,423,570,9...4,039,457,584,007,913,129,639,93678
951213,407,807,929,942,597,099,574,02...1,946,569,946,433,649,006,084,096155
101,024179,769,313,486,231,590,772,930,5...6,304,835,356,329,624,224,137,216309
112,04832,317,006,071,311,007,300,714,87...8,193,555,853,611,059,596,230,656617
124,0961,044,388,881,413,152,506,691,752,...0,243,804,708,340,403,154,190,3361,234
138,1921,090,748,135,619,415,929,462,984,...1,997,186,505,665,475,715,792,8962,467
1416,3841,189,731,495,357,231,765,085,759,...2,460,447,027,290,669,964,066,8164,933
1532,7681,415,461,031,044,954,789,001,553,...7,541,122,668,104,633,712,377,8569,865
16 65,536 2,003,529,930,406,846,464,979,072,...2,339,445,587,895,905,719,156,73619,729
17 131,072 4,014,132,182,036,063,039,166,060,...1,850,665,812,318,570,934,173,69639,457
18 262,144 16,113,257,174,857,604,736,195,72...0,753,862,605,349,934,298,300,41678,914
19 524,288 259,637,056,783,100,077,612,659,6...1,369,814,364,528,226,185,773,056157,827
Also see tetration and lower hyperoperations.
Last digits for powers of two whose exponents are powers of two
All of these numbers end in 6. Starting with 16 the last two digits are periodic with period 4, with the cycle 16–56–36–96–, and starting with 16 the last three digits are periodic with period 20. These patterns are generally true of any power, with respect to any base. The pattern continues where each pattern has starting point 2k, and the period is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicative group of integers modulo n).
Facts about powers of two whose exponents are powers of two
In a connection with nimbers, these numbers are often called Fermat 2-powers.
The numbers $2^{2^{n}}$ form an irrationality sequence: for every sequence $x_{i}$ of positive integers, the series
$\sum _{i=0}^{\infty }{\frac {1}{2^{2^{i}}x_{i}}}={\frac {1}{2x_{0}}}+{\frac {1}{4x_{1}}}+{\frac {1}{16x_{2}}}+\cdots $
converges to an irrational number. Despite the rapid growth of this sequence, it is the slowest-growing irrationality sequence known.[3]
Powers of two whose exponents are powers of two in computer science
Several of these numbers represent the number of values representable using common computer data types. For example, a 32-bit word consisting of 4 bytes can represent 232 distinct values, which can either be regarded as mere bit-patterns, or are more commonly interpreted as the unsigned numbers from 0 to 232 − 1, or as the range of signed numbers between −231 and 231 − 1. For more about representing signed numbers see two's complement.
Selected powers of two
22 = 4
The number that is the square of two. Also the first power of two tetration of two.
28 = 256
The number of values represented by the 8 bits in a byte, more specifically termed as an octet. (The term byte is often defined as a collection of bits rather than the strict definition of an 8-bit quantity, as demonstrated by the term kilobyte.)
210 = 1,024
The binary approximation of the kilo-, or 1,000 multiplier, which causes a change of prefix. For example: 1,024 bytes = 1 kilobyte (or kibibyte).
212 = 4,096
The hardware page size of an Intel x86-compatible processor.
215 = 32,768
The number of non-negative values for a signed 16-bit integer.
216 = 65,536
The number of distinct values representable in a single word on a 16-bit processor, such as the original x86 processors.[4]
The maximum range of a short integer variable in the C#, Java, and SQL programming languages. The maximum range of a Word or Smallint variable in the Pascal programming language.
The number of binary relations on a 4-element set.
220 = 1,048,576
The binary approximation of the mega-, or 1,000,000 multiplier, which causes a change of prefix. For example: 1,048,576 bytes = 1 megabyte (or mebibyte).
224 = 16,777,216
The number of unique colors that can be displayed in truecolor, which is used by common computer monitors.
This number is the result of using the three-channel RGB system, with 8 bits for each channel, or 24 bits in total.
The size of the largest unsigned integer or address in computers with 24-bit registers or data buses.
229 = 536,870,912
The largest power of two with distinct digits in base ten.[5]
230 = 1,073,741,824
The binary approximation of the giga-, or 1,000,000,000 multiplier, which causes a change of prefix. For example, 1,073,741,824 bytes = 1 gigabyte (or gibibyte).
231 = 2,147,483,648
The number of non-negative values for a signed 32-bit integer. Since Unix time is measured in seconds since January 1, 1970, it will run out at 2,147,483,647 seconds or 03:14:07 UTC on Tuesday, 19 January 2038 on 32-bit computers running Unix, a problem known as the year 2038 problem.
232 = 4,294,967,296
The number of distinct values representable in a single word on a 32-bit processor.[6] Or, the number of values representable in a doubleword on a 16-bit processor, such as the original x86 processors.[4]
The range of an int variable in the Java, C#, and SQL programming languages.
The range of a Cardinal or Integer variable in the Pascal programming language.
The minimum range of a long integer variable in the C and C++ programming languages.
The total number of IP addresses under IPv4. Although this is a seemingly large number, the number of available 32-bit IPv4 addresses has been exhausted (but not for IPv6 addresses).
The number of binary operations with domain equal to any 4-element set, such as GF(4).
240 = 1,099,511,627,776
The binary approximation of the tera-, or 1,000,000,000,000 multiplier, which causes a change of prefix. For example, 1,099,511,627,776 bytes = 1 terabyte or tebibyte.
250 = 1,125,899,906,842,624
The binary approximation of the peta-, or 1,000,000,000,000,000 multiplier. 1,125,899,906,842,624 bytes = 1 petabyte or pebibyte.
253 = 9,007,199,254,740,992
The number until which all integer values can exactly be represented in IEEE double precision floating-point format. Also the first power of 2 to start with the digit 9 in decimal.
256 = 72,057,594,037,927,936
The number of different possible keys in the obsolete 56 bit DES symmetric cipher.
260 = 1,152,921,504,606,846,976
The binary approximation of the exa-, or 1,000,000,000,000,000,000 multiplier. 1,152,921,504,606,846,976 bytes = 1 exabyte or exbibyte.
263 = 9,223,372,036,854,775,808
The number of non-negative values for a signed 64-bit integer.
263 − 1, a common maximum value (equivalently the number of positive values) for a signed 64-bit integer in programming languages.
264 = 18,446,744,073,709,551,616
The number of distinct values representable in a single word on a 64-bit processor. Or, the number of values representable in a doubleword on a 32-bit processor. Or, the number of values representable in a quadword on a 16-bit processor, such as the original x86 processors.[4]
The range of a long variable in the Java and C# programming languages.
The range of a Int64 or QWord variable in the Pascal programming language.
The total number of IPv6 addresses generally given to a single LAN or subnet.
264 − 1, the number of grains of rice on a chessboard, according to the old story, where the first square contains one grain of rice and each succeeding square twice as many as the previous square. For this reason the number is sometimes known as the "chess number".
264 − 1 is also the number of moves required to complete the legendary 64-disk version of the Tower of Hanoi.
268 = 295,147,905,179,352,825,856
The first power of 2 to contain all decimal digits. (sequence A137214 in the OEIS)
270 = 1,180,591,620,717,411,303,424
The binary approximation of the zetta-, or 1,000,000,000,000,000,000,000 multiplier. 1,180,591,620,717,411,303,424 bytes = 1 zettabyte (or zebibyte).
280 = 1,208,925,819,614,629,174,706,176
The binary approximation of the yotta-, or 1,000,000,000,000,000,000,000,000 multiplier. 1,208,925,819,614,629,174,706,176 bytes = 1 yottabyte (or yobibyte).
286 = 77,371,252,455,336,267,181,195,264
286 is conjectured to be the largest power of two not containing a zero in decimal.[7]
296 = 79,228,162,514,264,337,593,543,950,336
The total number of IPv6 addresses generally given to a local Internet registry. In CIDR notation, ISPs are given a /32, which means that 128-32=96 bits are available for addresses (as opposed to network designation). Thus, 296 addresses.
2108 = 324,518,553,658,426,726,783,156,020,576,256
The largest known power of 2 not containing a 9 in decimal. (sequence A035064 in the OEIS)
2126 = 85,070,591,730,234,615,865,843,651,857,942,052,864
The largest known power of 2 not containing a pair of consecutive equal digits. (sequence A050723 in the OEIS)
2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456
The total number of IP addresses available under IPv6. Also the number of distinct universally unique identifiers (UUIDs).
2168 = 374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856
The largest known power of 2 not containing all decimal digits (the digit 2 is missing in this case). (sequence A137214 in the OEIS)
2192 = 6,277,101,735,386,680,763,835,789,423,207,666,416,102,355,444,464,034,512,896
The total number of different possible keys in the AES 192-bit key space (symmetric cipher).
2229 = 862,718,293,348,820,473,429,344,482,784,628,181,556,388,621,521,298,319,395,315,527,974,912
2229 is the largest known power of two containing the least number of zeros relative to its power. It is conjectured by Metin Sariyar that every digit 0 to 9 is inclined to appear an equal number of times in the decimal expansion of power of two as the power increases. (sequence A330024 in the OEIS)
2256 = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936
The total number of different possible keys in the AES 256-bit key space (symmetric cipher).
21,024 = 179,769,313,486,231,590,772,930,...,304,835,356,329,624,224,137,216
The maximum number that can fit in a 64-bit IEEE double-precision floating-point format (approximately 1.797×10308), and hence the maximum number that can be represented by many programs, for example Microsoft Excel.
216,384 = 1,189,731,495,357,231,765,085,75...,460,447,027,290,669,964,066,816
The maximum number that can fit in a 128-bit IEEE quadruple-precision floating-point format (approximately 1.189×104932).
2262,144 = 16,113,257,174,857,604,736,195,7...,753,862,605,349,934,298,300,416
The maximum number that can fit in a 256-bit IEEE octuple-precision floating-point format (approximately 1.611×1078913).
282,589,933 = 1,488,944,457,420,413,255,478,06...,074,037,951,210,325,217,902,592
One more than the largest known prime number as of June 2023. It has 24,862,048 digits.[8]
Powers of two in music theory
In musical notation, all unmodified note values have a duration equal to a whole note divided by a power of two; for example a half note (1/2), a quarter note (1/4), an eighth note (1/8) and a sixteenth note (1/16). Dotted or otherwise modified notes have other durations. In time signatures the lower numeral, the beat unit, which can be seen as the denominator of a fraction, is almost always a power of two.
If the ratio of frequencies of two pitches is a power of two, then the interval between those pitches is full octaves. In this case, the corresponding notes have the same name.
Other properties
The sum of all n-choose binomial coefficients is equal to 2n. Consider the set of all n-digit binary integers. Its cardinality is 2n. It is also the sums of the cardinalities of certain subsets: the subset of integers with no 1s (consisting of a single number, written as n 0s), the subset with a single 1, the subset with two 1s, and so on up to the subset with n 1s (consisting of the number written as n 1s). Each of these is in turn equal to the binomial coefficient indexed by n and the number of 1s being considered (for example, there are 10-choose-3 binary numbers with ten digits that include exactly three 1s).
Currently, powers of two are the only known almost perfect numbers.
The number of vertices of an n-dimensional hypercube is 2n. Similarly, the number of (n − 1)-faces of an n-dimensional cross-polytope is also 2n and the formula for the number of x-faces an n-dimensional cross-polytope has is $2^{x}{\tbinom {n}{x}}.$
The sum of the reciprocals of the powers of two is 1. The sum of the reciprocals of the squared powers of two (powers of four) is 1/3.
The smallest natural power of two whose decimal representation begins with 7 is[9]
$2^{46}=70\ 368\ 744\ 177\ 664.$
Every power of 2 (excluding 1) can be written as the sum of four square numbers in 24 ways. The powers of 2 are the natural numbers greater than 1 that can be written as the sum of four square numbers in the fewest ways.
As a real polynomial, an + bn is irreducible, if and only if n is a power of two. (If n is odd, then an + bn is divisible by a+n, and if n is even but not a power of 2, then n can be written as n=mp, where m is odd, and thus $a^{n}+b^{n}=(a^{p})^{m}+(b^{p})^{m}$, which is divisible by ap + bp.) But in the domain of complex numbers, the polynomial $a^{2n}+b^{2n}$ (where n>=1) can always be factorized as $a^{2n}+b^{2n}=(a^{n}+b^{n}i)\cdot (a^{n}-b^{n}i)$, even if n is a power of two.
See also
• 2048 (video game)
• Binary number
• Fermi–Dirac prime
• Geometric progression
• Gould's sequence
• Inchworm (song)
• Integer binary logarithm
• Octave (electronics)
• Power of 10
• Power of three
• Sum-free sequence
References
1. Lipschutz, Seymour (1982). Schaum's Outline of Theory and Problems of Essential Computer Mathematics. New York: McGraw-Hill. p. 3. ISBN 0-07-037990-4.
2. Sewell, Michael J. (1997). Mathematics Masterclasses. Oxford: Oxford University Press. p. 78. ISBN 0-19-851494-8.
3. Guy, Richard K. (2004), "E24 Irrationality sequences", Unsolved problems in number theory (3rd ed.), Springer-Verlag, p. 346, ISBN 0-387-20860-7, Zbl 1058.11001, archived from the original on 2016-04-28
4. Though they vary in word size, all x86 processors use the term "word" to mean 16 bits; thus, a 32-bit x86 processor refers to its native wordsize as a dword
5. Prime Curios!: 536870912 "Prime Curios! 536870912". Archived from the original on 2017-09-05. Retrieved 2017-09-05.
6. "Powers of 2 Table - - - - - - Vaughn's Summaries". www.vaughns-1-pagers.com. Archived from the original on August 12, 2015.
7. Weisstein, Eric W. "Zero." From MathWorld--A Wolfram Web Resource. "Zero". Archived from the original on 2013-06-01. Retrieved 2013-05-29.
8. "Mersenne Prime Discovery - 2^82589933-1 is Prime!". www.mersenne.org.
9. Paweł Strzelecki (1994). "O potęgach dwójki (About powers of two)" (in Polish). Delta. Archived from the original on 2016-05-09.
Sequences and series
Integer sequences
Basic
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Advanced (list)
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Properties of sequences
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Properties of series
Series
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Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
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Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
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• Category
Classes of natural numbers
Powers and related numbers
• Achilles
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Of the form a × 2b ± 1
• Cullen
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Other polynomial numbers
• Hilbert
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Recursively defined numbers
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Possessing a specific set of other numbers
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Expressible via specific sums
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Figurate numbers
2-dimensional
centered
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pyramidal
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Combinatorial numbers
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Primes
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Pseudoprimes
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Arithmetic functions and dynamics
Divisor functions
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Aliquot sequences
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Large numbers
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Expression
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| Wikipedia |
Large numbers
Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical mechanics. They are typically large positive integers, or more generally, large positive real numbers, but may also be other numbers in other contexts. Googology is the study of nomenclature and properties of large numbers.[1][2]
In the everyday world
See also: Scientific notation, Logarithmic scale, Orders of magnitude, and Names of large numbers
Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, or a 1 followed by nine zeros: 1 000 000 000. The reciprocal, 1.0 × 10−9, means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. In addition to scientific (powers of 10) notation, the following examples include (short scale) systematic nomenclature of large numbers.
Examples of large numbers describing everyday real-world objects include:
• The number of cells in the human body (estimated at 3.72 × 1013), or 37.2 trillion[3]
• The number of bits on a computer hard disk (as of 2023, typically about 1013, 1–2 TB), or 10 trillion
• The number of neuronal connections in the human brain (estimated at 1014), or 100 trillion
• The Avogadro constant is the number of “elementary entities” (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 – approximately 6.022×1023, or 602.2 sextillion.
• The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3 ± 3.6) × 1037, or 53±36 undecillion[4][5]
• The mass of Earth consists of about 4 × 1051, or 4 sexdecillion, nucleons
• The estimated number of atoms in the observable universe (1080), or 100 quinvigintillion
• The lower bound on the game-tree complexity of chess, also known as the “Shannon number” (estimated at around 10120), or 1 novemtrigintillion[6]
• Note that this value of the Shannon number is for Standard Chess. It has even larger values for larger-board chess variants such as Grant Acedrex, Tai Shogi, and Taikyoku Shogi.
Astronomical
Other large numbers, as regards length and time, are found in astronomy and cosmology. For example, the current Big Bang model suggests that the universe is 13.8 billion years (4.355 × 1017 seconds) old, and that the observable universe is 93 billion light years across (8.8 × 1026 metres), and contains about 5 × 1022 stars, organized into around 125 billion (1.25 × 1011) galaxies, according to Hubble Space Telescope observations. There are about 1080 atoms in the observable universe, by rough estimation.[7]
According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
$10^{10^{10^{10^{10^{1.1}}}}}{\mbox{ years}}$
which corresponds to the scale of an estimated Poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain inflationary model with an inflaton whose mass is 10−6 Planck masses.[8][9] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations on a set of fixed objects, grows very rapidly with the number of objects. Stirling's formula gives a precise asymptotic expression for this rate of growth.
Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms.
Gödel numbers, and similar numbers used to represent bit-strings in algorithmic information theory, are very large, even for mathematical statements of reasonable length. However, some pathological numbers are even larger than the Gödel numbers of typical mathematical propositions.
Logician Harvey Friedman has done work related to very large numbers, such as with Kruskal's tree theorem and the Robertson–Seymour theorem.
"Billions and billions"
To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b". Sagan never did, however, say "billions and billions". The public's association of the phrase and Sagan came from a Tonight Show skit. Parodying Sagan's effect, Johnny Carson quipped "billions and billions".[10] The phrase has, however, now become a humorous fictitious number—the Sagan. Cf., Sagan Unit.
Examples
• googol = $10^{100}$
• centillion = $10^{303}$ or $10^{600}$, depending on number naming system
• millinillion = $10^{3003}$ or $10^{6000}$, depending on number naming system
• The largest known Smith number = (101031−1) × (104594 + 3×102297 + 1)1476 ×103913210
• The largest known Mersenne prime = $2^{82,589,933}-1$[11]
• googolplex = $10^{\text{googol}}=10^{10^{100}}$
• Skewes's numbers: the first is approximately $10^{10^{10^{34}}}$, the second $10^{10^{10^{964}}}$
• Tritri {3, 3, 3} on the lower end of BEAF (Bowers Exploding Array Function). It can be written as 3{3}3, 3^^^3 or 3^^(3^^3), the latter 2 showing how Knuth's up-arrow notation begins to build grahams number.
• Tritet {4, 4, 4} on the lower end of BEAF (Bowers Exploding Array Function).
• Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation.
• Supertet {4, 4, 4, 4}, example of the numbers that can be generated through BEAF (Bowers Exploding Array Function). It can be written as 4{{{{4}}}}4, a more clear representation of the denotetration used to generate the number.
• Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number.
• Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007.
Standardized system of writing
A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.
To compare numbers in scientific notation, say 5×104 and 2×105, compare the exponents first, in this case 5 > 4, so 2×105 > 5×104. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×104 > 2×104 because 5 > 2.
Tetration with base 10 gives the sequence $10\uparrow \uparrow n=10\to n\to 2=(10\uparrow )^{n}1$, the power towers of numbers 10, where $(10\uparrow )^{n}$ denotes a functional power of the function $f(n)=10^{n}$ (the function also expressed by the suffix "-plex" as in googolplex, see the googol family).
These are very round numbers, each representing an order of magnitude in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.
More precisely, numbers in between can be expressed in the form $(10\uparrow )^{n}a$, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g. $10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829$, a number between $10\uparrow \uparrow 5$ and $10\uparrow \uparrow 6$ (note that $10\uparrow \uparrow n<(10\uparrow )^{n}a<10\uparrow \uparrow (n+1)$ if $1<a<10$). (See also extension of tetration to real heights.)
Thus googolplex is $10^{10^{100}}=(10\uparrow )^{2}100=(10\uparrow )^{3}2$
Another example:
$2\uparrow \uparrow \uparrow 4={\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\qquad \quad \ \ \ 65,536{\mbox{ copies of }}2\end{matrix}}\approx (10\uparrow )^{65,531}(6\times 10^{19,728})\approx (10\uparrow )^{65,533}4.3$ (between $10\uparrow \uparrow 65,533$ and $10\uparrow \uparrow 65,534$)
Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (n) one has to take the $log_{10}$ to get a number between 1 and 10. Thus, the number is between $10\uparrow \uparrow n$ and $10\uparrow \uparrow (n+1)$. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 1010, or the next, between 0 and 1.
Note that
$10^{(10\uparrow )^{n}x}=(10\uparrow )^{n}10^{x}$
I.e., if a number x is too large for a representation $(10\uparrow )^{n}x$ the power tower can be made one higher, replacing x by log10x, or find x from the lower-tower representation of the log10 of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).
If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g. $10\uparrow \uparrow (7.21\times 10^{8})$) can be used. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.
Examples:
$10\uparrow \uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}$ (between $10\uparrow \uparrow \uparrow 2$ and $10\uparrow \uparrow \uparrow 3$)
$10\uparrow \uparrow 10\uparrow \uparrow (10\uparrow )^{497}(9.73\times 10^{32})=(10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})$ (between $10\uparrow \uparrow \uparrow 4$ and $10\uparrow \uparrow \uparrow 5$)
Similarly to the above, if the exponent of $(10\uparrow )$ is not exactly given then giving a value at the right does not make sense, and instead of using the power notation of $(10\uparrow )$, it is possible to add $1$ to the exponent of $(10\uparrow \uparrow )$, to obtain e.g. $(10\uparrow \uparrow )^{3}(2.8\times 10^{12})$.
If the exponent of $(10\uparrow \uparrow )$ is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and instead of using the power notation of $(10\uparrow \uparrow )$ it is possible use the triple arrow operator, e.g. $10\uparrow \uparrow \uparrow (7.3\times 10^{6})$.
If the right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g. $10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})$ (between $10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 4$ and $10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 5$). This can be done recursively, so it is possible to have a power of the triple arrow operator.
Then it is possible to proceed with operators with higher numbers of arrows, written $\uparrow ^{n}$.
Compare this notation with the hyper operator and the Conway chained arrow notation:
$a\uparrow ^{n}b$ = ( a → b → n ) = hyper(a, n + 2, b)
An advantage of the first is that when considered as function of b, there is a natural notation for powers of this function (just like when writing out the n arrows): $(a\uparrow ^{n})^{k}b$. For example:
$(10\uparrow ^{2})^{3}b$ = ( 10 → ( 10 → ( 10 → b → 2 ) → 2 ) → 2 )
and only in special cases the long nested chain notation is reduced; for $''b''=1$ obtains:
$10\uparrow ^{3}3=(10\uparrow ^{2})^{3}1$ = ( 10 → 3 → 3 )
Since the b can also be very large, in general it can be written instead a number with a sequence of powers $(10\uparrow ^{n})^{k_{n}}$ with decreasing values of n (with exactly given integer exponents ${k_{n}}$) with at the end a number in ordinary scientific notation. Whenever a ${k_{n}}$ is too large to be given exactly, the value of ${k_{n+1}}$ is increased by 1 and everything to the right of $({n+1})^{k_{n+1}}$ is rewritten.
For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, $10\uparrow (10\uparrow \uparrow )^{5}a=(10\uparrow \uparrow )^{6}a$, and $10\uparrow (10\uparrow \uparrow \uparrow 3)=10\uparrow \uparrow (10\uparrow \uparrow 10+1)\approx 10\uparrow \uparrow \uparrow 3$. Thus is obtained the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10x are "almost equal" (for arithmetic of large numbers see also below).
If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it act, instead it is possible to simply use a standard value at the right, say 10, and the expression reduces to $10\uparrow ^{n}10=(10\to 10\to n)$ with an approximate n. For such numbers the advantage of using the upward arrow notation no longer applies, so the chain notation can be used instead.
The above can be applied recursively for this n, so the notation $\uparrow ^{n}$ is obtained in the superscript of the first arrow, etc., or a nested chain notation, e.g.:
(10 → 10 → (10 → 10 → $3\times 10^{5}$) ) = $10\uparrow ^{10\uparrow ^{3\times 10^{5}}10}10$
If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function $f(n)=10\uparrow ^{n}10$ = (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form $f^{m}(n)$ where m is given exactly and n is an integer which may or may not be given exactly (for example: $f^{2}(3\times 10^{5})$). If n is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form fm(1) = (10→10→m→2). For example, $(10\to 10\to 3\to 2)=10\uparrow ^{10\uparrow ^{10^{10}}10}10$
Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus $G<3\rightarrow 3\rightarrow 65\rightarrow 2<(10\to 10\to 65\to 2)=f^{65}(1)$, but also $G<f^{64}(4)<f^{65}(1)$.
If m in $f^{m}(n)$ is too large to give exactly, it is possible to use a fixed n, e.g. n = 1, and apply the above recursively to m, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of f this gives multiple levels of f. Introducing a function $g(n)=f^{n}(1)$ these levels become functional powers of g, allowing us to write a number in the form $g^{m}(n)$ where m is given exactly and n is an integer which may or may not be given exactly. For example, if (10→10→m→3) = gm(1). If n is large any of the above can be used for expressing it. Similarly a function h, etc. can be introduced. If many such functions are required, they can be numbered instead of using a new letter every time, e.g. as a subscript, such that there are numbers of the form $f_{k}^{m}(n)$ where k and m are given exactly and n is an integer which may or may not be given exactly. Using k=1 for the f above, k=2 for g, etc., obtains (10→10→n→k) = $f_{k}(n)=f_{k-1}^{n}(1)$. If n is large any of the above can be used to express it. Thus is obtained a nesting of forms ${f_{k}}^{m_{k}}$ where going inward the k decreases, and with as inner argument a sequence of powers $(10\uparrow ^{n})^{p_{n}}$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
When k is too large to be given exactly, the number concerned can be expressed as ${f_{n}}(10)$=(10→10→10→n) with an approximate n. Note that the process of going from the sequence $10^{n}$=(10→n) to the sequence $10\uparrow ^{n}10$=(10→10→n) is very similar to going from the latter to the sequence ${f_{n}}(10)$=(10→10→10→n): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions ${f_{qk}}^{m_{qk}}$, nested in lexicographical order with q the most significant number, but with decreasing order for q and for k; as inner argument yields a sequence of powers $(10\uparrow ^{n})^{p_{n}}$ with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
For a number too large to write down in the Conway chained arrow notation it size can be described by the length of that chain, for example only using elements 10 in the chain; in other words, one could specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number same techniques can be applied again.
Examples
Numbers expressible in decimal notation:
• 22 = 4
• 222 = 2 ↑↑ 3 = 16
• 33 = 27
• 44 = 256
• 55 = 3,125
• 66 = 46,656
• $2^{2^{2^{2}}}$ = 2 ↑↑ 4 = 2↑↑↑3 = 65,536
• 77 = 823,543
• 106 = 1,000,000 = 1 million
• 88 = 16,777,216
• 99 = 387,420,489
• 109 = 1,000,000,000 = 1 billion
• 1010 = 10,000,000,000
• 1012 = 1,000,000,000,000 = 1 trillion
• 333 = 3 ↑↑ 3 = 7,625,597,484,987 ≈ 7.63 × 1012
• 1015 = 1,000,000,000,000,000 = 1 million billion = 1 quadrillion
• 1018 = 1,000,000,000,000,000,000 = 1 billion billion = 1 quintilion
Numbers expressible in scientific notation:
• Approximate number of atoms in the observable universe = 1080 = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
• googol = 10100 = 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
• 444 = 4 ↑↑ 3 = 2512 ≈ 1.34 × 10154 ≈ (10 ↑)2 2.2
• Approximate number of Planck volumes composing the volume of the observable universe = 8.5 × 10184
• 555 = 5 ↑↑ 3 = 53125 ≈ 1.91 × 102184 ≈ (10 ↑)2 3.3
• $2^{2^{2^{2^{2}}}}=2\uparrow \uparrow 5=2^{65,536}\approx 2.0\times 10^{19,728}\approx (10\uparrow )^{2}4.3$
• 666 = 6 ↑↑ 3 ≈ 2.66 × 1036,305 ≈ (10 ↑)2 4.6
• 777 = 7 ↑↑ 3 ≈ 3.76 × 10695,974 ≈ (10 ↑)2 5.8
• 888 = 8 ↑↑ 3 ≈ 6.01 × 1015,151,335 ≈ (10 ↑)2 7.2
• $M_{82,589,933}\approx 1.49\times 10^{24,862,047}\approx 10^{10^{7.3955}}=(10\uparrow )^{2}\ 7.3955$, the 51st and as of January 2021 the largest known Mersenne prime.
• 999 = 9 ↑↑ 3 ≈ 4.28 × 10369,693,099 ≈ (10 ↑)2 8.6
• 101010 =10 ↑↑ 3 = 1010,000,000,000 = (10 ↑)3 1
• $3^{3^{3^{3}}}=3\uparrow \uparrow 4\approx 1.26\times 10^{3,638,334,640,024}\approx (10\uparrow )^{3}1.10$
Numbers expressible in (10 ↑)n k notation:
• googolplex = $10^{10^{100}}=(10\uparrow )^{3}2$
• $2^{2^{2^{2^{2^{2}}}}}=2\uparrow \uparrow 6=2^{2^{65,536}}\approx 2^{(10\uparrow )^{2}4.3}\approx 10^{(10\uparrow )^{2}4.3}=(10\uparrow )^{3}4.3$
• $10^{10^{10^{10}}}=10\uparrow \uparrow 4=(10\uparrow )^{4}1$
• $3^{3^{3^{3^{3}}}}=3\uparrow \uparrow 5\approx 3^{10^{3.6\times 10^{12}}}\approx (10\uparrow )^{4}1.10$
• $2^{2^{2^{2^{2^{2^{2}}}}}}=2\uparrow \uparrow 7\approx (10\uparrow )^{4}4.3$
• 10 ↑↑ 5 = (10 ↑)5 1
• 3 ↑↑ 6 ≈ (10 ↑)5 1.10
• 2 ↑↑ 8 ≈ (10 ↑)5 4.3
• 10 ↑↑ 6 = (10 ↑)6 1
• 10 ↑↑↑ 2 = 10 ↑↑ 10 = (10 ↑)10 1
• 2 ↑↑↑↑ 3 = 2 ↑↑↑ 4 = 2 ↑↑ 65,536 ≈ (10 ↑)65,533 4.3 is between 10 ↑↑ 65,533 and 10 ↑↑ 65,534
Bigger numbers:
• 3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑ 3) ≈ 3 ↑↑ 7.6 × 1012 ≈ 10 ↑↑ 7.6 × 1012 is between (10 ↑↑)2 2 and (10 ↑↑)2 3
• $10\uparrow \uparrow \uparrow 3=(10\uparrow \uparrow )^{3}1$ = ( 10 → 3 → 3 )
• $(10\uparrow \uparrow )^{2}11$
• $(10\uparrow \uparrow )^{2}10^{\,\!10^{10^{3.81\times 10^{17}}}}$
• $10\uparrow \uparrow \uparrow 4=(10\uparrow \uparrow )^{4}1$ = ( 10 → 4 → 3 )
• $(10\uparrow \uparrow )^{2}(10\uparrow )^{497}(9.73\times 10^{32})$
• $10\uparrow \uparrow \uparrow 5=(10\uparrow \uparrow )^{5}1$ = ( 10 → 5 → 3 )
• $10\uparrow \uparrow \uparrow 6=(10\uparrow \uparrow )^{6}1$ = ( 10 → 6 → 3 )
• $10\uparrow \uparrow \uparrow 7=(10\uparrow \uparrow )^{7}1$ = ( 10 → 7 → 3 )
• $10\uparrow \uparrow \uparrow 8=(10\uparrow \uparrow )^{8}1$ = ( 10 → 8 → 3 )
• $10\uparrow \uparrow \uparrow 9=(10\uparrow \uparrow )^{9}1$ = ( 10 → 9 → 3 )
• $10\uparrow \uparrow \uparrow \uparrow 2=10\uparrow \uparrow \uparrow 10=(10\uparrow \uparrow )^{10}1$ = ( 10 → 2 → 4 ) = ( 10 → 10 → 3 )
• The first term in the definition of Graham's number, g1 = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) ≈ 3 ↑↑↑ (10 ↑↑ 7.6 × 1012) ≈ 10 ↑↑↑ (10 ↑↑ 7.6 × 1012) is between (10 ↑↑↑)2 2 and (10 ↑↑↑)2 3 (See Graham's number#Magnitude)
• $10\uparrow \uparrow \uparrow \uparrow 3=(10\uparrow \uparrow \uparrow )^{3}1$ = (10 → 3 → 4)
• $4\uparrow \uparrow \uparrow \uparrow 4$ = ( 4 → 4 → 4 ) $\approx (10\uparrow \uparrow \uparrow )^{2}(10\uparrow \uparrow )^{3}154$
• $10\uparrow \uparrow \uparrow \uparrow 4=(10\uparrow \uparrow \uparrow )^{4}1$ = ( 10 → 4 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow 5=(10\uparrow \uparrow \uparrow )^{5}1$ = ( 10 → 5 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow 6=(10\uparrow \uparrow \uparrow )^{6}1$ = ( 10 → 6 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow 7=(10\uparrow \uparrow \uparrow )^{7}1=$ = ( 10 → 7 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow 8=(10\uparrow \uparrow \uparrow )^{8}1=$ = ( 10 → 8 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow 9=(10\uparrow \uparrow \uparrow )^{9}1=$ = ( 10 → 9 → 4 )
• $10\uparrow \uparrow \uparrow \uparrow \uparrow 2=10\uparrow \uparrow \uparrow \uparrow 10=(10\uparrow \uparrow \uparrow )^{10}1$ = ( 10 → 2 → 5 ) = ( 10 → 10 → 4 )
• ( 2 → 3 → 2 → 2 ) = ( 2 → 3 → 8 )
• ( 3 → 2 → 2 → 2 ) = ( 3 → 2 → 9 ) = ( 3 → 3 → 8 )
• ( 10 → 10 → 10 ) = ( 10 → 2 → 11 )
• ( 10 → 2 → 2 → 2 ) = ( 10 → 2 → 100 )
• ( 10 → 10 → 2 → 2 ) = ( 10 → 2 → $10^{10}$ ) = $10\uparrow ^{10^{10}}10$
• The second term in the definition of Graham's number, g2 = 3 ↑g1 3 > 10 ↑g1 – 1 10.
• ( 10 → 10 → 3 → 2 ) = (10 → 10 → (10 → 10 → $10^{10}$) ) = $10\uparrow ^{10\uparrow ^{10^{10}}10}10$
• g3 = (3 → 3 → g2) > (10 → 10 → g2 – 1) > (10 → 10 → 3 → 2)
• g4 = (3 → 3 → g3) > (10 → 10 → g3 – 1) > (10 → 10 → 4 → 2)
• ...
• g9 = (3 → 3 → g8) is between (10 → 10 → 9 → 2) and (10 → 10 → 10 → 2)
• ( 10 → 10 → 10 → 2 )
• g10 = (3 → 3 → g9) is between (10 → 10 → 10 → 2) and (10 → 10 → 11 → 2)
• ...
• g63 = (3 → 3 → g62) is between (10 → 10 → 63 → 2) and (10 → 10 → 64 → 2)
• ( 10 → 10 → 64 → 2 )
• Graham's number, g64[12]
• ( 10 → 10 → 65 → 2 )
• ( 10 → 10 → 10 → 3 )
• ( 10 → 10 → 10 → 4 )
• ( 10 → 10 → 10 → 10 )
• ( 10 → 10 → 10 → 10 → 10 )
• ( 10 → 10 → 10 → 10 → 10 → 10 )
• ( 10 → 10 → 10 → 10 → 10 → 10 → 10 → ... → 10 → 10 → 10 → 10 → 10 → 10 → 10 → 10 ) where there are ( 10 → 10 → 10 ) "10"s
Other notations
Some notations for extremely large numbers:
• Knuth's up-arrow notation/hyperoperators/Ackermann function, including tetration
• Conway chained arrow notation
• Steinhaus-Moser notation; apart from the method of construction of large numbers, this also involves a graphical notation with polygons. Alternative notations, like a more conventional function notation, can also be used with the same functions.
• Fast-growing hierarchy
These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever-faster-increasing functions can easily be constructed recursively by applying these functions with large integers as argument.
A function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.
Comparison of base values
The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.
$100^{12}=10^{24}$, with base 10 the exponent is doubled.
$100^{100^{12}}=10^{2*10^{24}}$, ditto.
$100^{100^{100^{12}}}\approx 10^{10^{2*10^{24}+0.30103}}$, the highest exponent is very little more than doubled (increased by log102).
• $100\uparrow \uparrow 2=10^{200}$
• $100\uparrow \uparrow 3=10^{2\times 10^{200}}$
• $100\uparrow \uparrow 4=(10\uparrow )^{2}(2\times 10^{200}+0.3)=(10\uparrow )^{2}(2\times 10^{200})=(10\uparrow )^{3}200.3=(10\uparrow )^{4}2.3$
• $100\uparrow \uparrow n=(10\uparrow )^{n-2}(2\times 10^{200})=(10\uparrow )^{n-1}200.3=(10\uparrow )^{n}2.3<10\uparrow \uparrow (n+1)$ (thus if n is large it seems fair to say that $100\uparrow \uparrow n$ is "approximately equal to" $10\uparrow \uparrow n$)
• $100\uparrow \uparrow \uparrow 2=(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow )^{100}2.3$
• $100\uparrow \uparrow \uparrow 3=10\uparrow \uparrow (10\uparrow )^{98}(2\times 10^{200})=10\uparrow \uparrow (10\uparrow )^{100}2.3$
• $100\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{98}(2\times 10^{200})=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{100}2.3<10\uparrow \uparrow \uparrow (n+1)$ (compare $10\uparrow \uparrow \uparrow n=(10\uparrow \uparrow )^{n-2}(10\uparrow )^{10}1<10\uparrow \uparrow \uparrow (n+1)$; thus if n is large it seems fair to say that $100\uparrow \uparrow \uparrow n$ is "approximately equal to" $10\uparrow \uparrow \uparrow n$)
• $100\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3$ (compare $10\uparrow \uparrow \uparrow \uparrow 2=(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1$)
• $100\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3$ (compare $10\uparrow \uparrow \uparrow \uparrow 3=10\uparrow \uparrow \uparrow (10\uparrow \uparrow )^{8}(10\uparrow )^{10}1$)
• $100\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{98}(10\uparrow )^{100}2.3$ (compare $10\uparrow \uparrow \uparrow \uparrow n=(10\uparrow \uparrow \uparrow )^{n-2}(10\uparrow \uparrow )^{8}(10\uparrow )^{10}1$; if n is large this is "approximately" equal)
Accuracy
For a number $10^{n}$, one unit change in n changes the result by a factor 10. In a number like $10^{\,\!6.2\times 10^{3}}$, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor $10^{50}$ too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).
For very large numbers
In the case of an approximation of an extremely large number, the relative error may be large, yet there may still be a sense in which one wants to consider the numbers as "close in magnitude". For example, consider
$10^{10}$ and $10^{9}$
The relative error is
$1-{\frac {10^{9}}{10^{10}}}=1-{\frac {1}{10}}=90\%$
a large relative error. However, one can also consider the relative error in the logarithms; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%.
The point is that exponential functions magnify relative errors greatly – if a and b have a small relative error,
$10^{a}$ and $10^{b}$
the relative error is larger, and
$10^{10^{a}}$ and $10^{10^{b}}$
will have an even larger relative error. The question then becomes: on which level of iterated logarithms do to compare two numbers? There is a sense in which one may want to consider
$10^{10^{10}}$ and $10^{10^{9}}$
to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:
$\log _{10}(\log _{10}(10^{10^{10}}))=10$ and $\log _{10}(\log _{10}(10^{10^{9}}))=9$
Such comparisons of iterated logarithms are common, e.g., in analytic number theory.
Classes
One solution to the problem of comparing large numbers is to define classes of numbers, such as the system devised by Robert Munafo,[13] which is based on different "levels" of perception of an average person. Class 0 – numbers between zero and six – is defined to contain numbers that are easily subitized, that is, numbers that show up very frequently in daily life and are almost instantly comparable. Class 1 – numbers between six and 1,000,000=106 – is defined to contain numbers whose decimal expressions are easily subitized, that is, numbers who are easily comparable not by cardinality, but "at a glance" given the decimal expansion.
Each class after these are defined in terms of iterating this base-10 exponentiation, to simulate the effect of another "iteration" of human indistinguishibility. For example, class 5 is defined to include numbers between 101010106 and 10101010106, which are numbers where X becomes humanly indistinguishable from X2 [14] (taking iterated logarithms of such X yields indistinguishibility firstly between log(X) and 2log(X), secondly between log(log(X)) and 1+log(log(X)), and finally an extremely long decimal expansion whose length can't be subitized).
Approximate arithmetic
There are some general rules relating to the usual arithmetic operations performed on very large numbers:
• The sum and the product of two very large numbers are both "approximately" equal to the larger one.
• $(10^{a})^{\,\!10^{b}}=10^{a10^{b}}=10^{10^{b+\log _{10}a}}$
Hence:
• A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large $n$ there is $n^{n}\approx 10^{n}$ (see e.g. the computation of mega) and also $2^{n}\approx 10^{n}$. Thus $2\uparrow \uparrow 65536\approx 10\uparrow \uparrow 65533$, see table.
Systematically creating ever-faster-increasing sequences
Main article: fast-growing hierarchy
Given a strictly increasing integer sequence/function $f_{0}(n)$ (n≥1), it is possible to produce a faster-growing sequence $f_{1}(n)=f_{0}^{n}(n)$ (where the superscript n denotes the nth functional power). This can be repeated any number of times by letting $f_{k}(n)=f_{k-1}^{n}(n)$, each sequence growing much faster than the one before it. Thus it is possible to define $f_{\omega }(n)=f_{n}(n)$, which grows much faster than any $f_{k}$ for finite k (here ω is the first infinite ordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.
For example, starting with f0(n) = n + 1:
• f1(n) = f0n(n) = n + n = 2n
• f2(n) = f1n(n) = 2nn > (2 ↑) n for n ≥ 2 (using Knuth up-arrow notation)
• f3(n) = f2n(n) > (2 ↑)n n ≥ 2 ↑2 n for n ≥ 2
• fk+1(n) > 2 ↑k n for n ≥ 2, k < ω
• fω(n) = fn(n) > 2 ↑n – 1 n > 2 ↑n − 2 (n + 3) − 3 = A(n, n) for n ≥ 2, where A is the Ackermann function (of which fω is a unary version)
• fω+1(64) > fω64(6) > Graham's number (= g64 in the sequence defined by g0 = 4, gk+1 = 3 ↑gk 3)
• This follows by noting fω(n) > 2 ↑n – 1 n > 3 ↑n – 2 3 + 2, and hence fω(gk + 2) > gk+1 + 2
• fω(n) > 2 ↑n – 1 n = (2 → n → n-1) = (2 → n → n-1 → 1) (using Conway chained arrow notation)
• fω+1(n) = fωn(n) > (2 → n → n-1 → 2) (because if gk(n) = X → n → k then X → n → k+1 = gkn(1))
• fω+k(n) > (2 → n → n-1 → k+1) > (n → n → k)
• fω2(n) = fω+n(n) > (n → n → n) = (n → n → n→ 1)
• fω2+k(n) > (n → n → n → k)
• fω3(n) > (n → n → n → n)
• fωk(n) > (n → n → ... → n → n) (Chain of k+1 n's)
• fω2(n) = fωn(n) > (n → n → ... → n → n) (Chain of n+1 n's)
In some noncomputable sequences
The busy beaver function Σ is an example of a function which grows faster than any computable function. Its value for even relatively small input is huge. The values of Σ(n) for n = 1, 2, 3, 4 are 1, 4, 6, 13 (sequence A028444 in the OEIS). Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 10↑↑15.
Infinite numbers
Main article: cardinal number
See also: large cardinal, Mahlo cardinal, and totally indescribable cardinal
Although all the numbers discussed above are very large, they are all still decidedly finite. Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite set of natural numbers, and aleph-one is the next greatest cardinal number. ${\mathfrak {c}}$ is the cardinality of the reals. The proposition that ${\mathfrak {c}}=\aleph _{1}$ is known as the continuum hypothesis.
See also
• Arbitrary-precision arithmetic
• List of arbitrary-precision arithmetic software
• Dirac large numbers hypothesis
• Exponential growth
• History of large numbers
• Human scale
• Indefinite and fictitious numbers
• Largest number
• Infinity
• Law of large numbers
• Myriads (10,000) in East Asia
• Names of large numbers
• Power of two
• Power of 10
• Tetration
References
1. One Million Things: A Visual Encyclopedia
2. «The study of large numbers is called googology»
3. Bianconi, Eva; Piovesan, Allison; Facchin, Federica; Beraudi, Alina; Casadei, Raffaella; Frabetti, Flavia; Vitale, Lorenza; Pelleri, Maria Chiara; Tassani, Simone (Nov–Dec 2013). "An estimation of the number of cells in the human body". Annals of Human Biology. 40 (6): 463–471. doi:10.3109/03014460.2013.807878. hdl:11585/152451. ISSN 1464-5033. PMID 23829164. S2CID 16247166.
4. Landenmark HK, Forgan DH, Cockell CS (June 2015). "An Estimate of the Total DNA in the Biosphere". PLOS Biology. 13 (6): e1002168. doi:10.1371/journal.pbio.1002168. PMC 4466264. PMID 26066900.
5. Nuwer R (18 July 2015). "Counting All the DNA on Earth". The New York Times. New York. ISSN 0362-4331. Retrieved 2015-07-18.
6. Shannon, Claude (March 1950). "XXII. Programming a Computer for Playing Chess" (PDF). Philosophical Magazine. Series 7. 41 (314). Archived from the original (PDF) on 2010-07-06. Retrieved 2019-01-25.
7. Atoms in the Universe. Universe Today. 30-07-2009. Retrieved 02-03-13.
8. Information Loss in Black Holes and/or Conscious Beings?, Don N. Page, Heat Kernel Techniques and Quantum Gravity (1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics. arXiv:hep-th/9411193. ISBN 0-9630728-3-8.
9. How to Get A Googolplex
10. Carl Sagan takes questions more from his 'Wonder and Skepticism' CSICOP 1994 keynote, Skeptical Inquirer Archived December 21, 2016, at the Wayback Machine
11. "GIMPS Discovers Largest Known Prime Number". Great Internet Mersenne Prime Search. 2018-12-21.
12. Regarding the comparison with the previous value: $10\uparrow ^{n}10<3\uparrow ^{n+1}3$, so starting the 64 steps with 1 instead of 4 more than compensates for replacing the numbers 3 by 10
13. "Large Numbers at MROB". www.mrob.com. Retrieved 2021-05-13.
14. "Large Numbers (page 2) at MROB". www.mrob.com. Retrieved 2021-05-13.
Large numbers
Examples
in
numerical
order
• Thousand
• Ten thousand
• Hundred thousand
• Million
• Ten million
• Hundred million
• Billion
• Trillion
• Quadrillion
• Quintillion
• Sextillion
• Septillion
• Octillion
• Nonillion
• Decillion
• Eddington number
• Googol
• Shannon number
• Googolplex
• Skewes's number
• Moser's number
• Graham's number
• TREE(3)
• SSCG(3)
• BH(3)
• Rayo's number
• Transfinite numbers
Expression
methods
Notations
• Scientific notation
• Knuth's up-arrow notation
• Conway chained arrow notation
• Steinhaus–Moser notation
Operators
• Hyperoperation
• Tetration
• Pentation
• Ackermann function
• Grzegorczyk hierarchy
• Fast-growing hierarchy
Related
articles
(alphabetical
order)
• Busy beaver
• Extended real number line
• Indefinite and fictitious numbers
• Infinitesimal
• Largest known prime number
• List of numbers
• Long and short scales
• Number systems
• Number names
• Orders of magnitude
• Power of two
• Power of three
• Power of 10
• Sagan Unit
• Names
• History
Hyperoperations
Primary
• Successor (0)
• Addition (1)
• Multiplication (2)
• Exponentiation (3)
• Tetration (4)
• Pentation (5)
Inverse for left argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Root extraction (3)
• Super-root (4)
Inverse for right argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Logarithm (3)
• Super-logarithm (4)
Related articles
• Ackermann function
• Conway chained arrow notation
• Grzegorczyk hierarchy
• Knuth's up-arrow notation
• Steinhaus–Moser notation
| Wikipedia |
Numerical digit
A numerical digit (often shortened to just digit) is a single symbol used alone (such as "1") or in combinations (such as "15"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin digiti meaning fingers)[1] of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal (ancient Latin adjective decem meaning ten)[2] digits.
For a given numeral system with an integer base, the number of different digits required is given by the absolute value of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the binary system (base 2) requires two digits (0 and 1).
Overview
In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.
Digital values
Each digit in a number system represents an integer. For example, in decimal the digit "1" represents the integer one, and in the hexadecimal system, the letter "A" represents the number ten. A positional number system has one unique digit for each integer from zero up to, but not including, the radix of the number system.
Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 can be expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 can be expressed by three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.
Computation of place values
The decimal numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages,[3] to denote the "ones place" or "units place",[4][5][6] which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10.34 (written in base 10),
the 0 is immediately to the left of the separator, so it is in the ones or units place, and is called the units digit or ones digit;[7][8][9]
the 1 to the left of the ones place is in the tens place, and is called the tens digit;[10]
the 3 is to the right of the ones place, so it is in the tenths place, and is called the tenths digit;[11]
the 4 to the right of the tenths place is in the hundredths place, and is called the hundredths digit.[11]
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. The zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place.
The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a complement to the logic behind numeral systems. The calculation involves the multiplication of the given digit by the base raised by the exponent n − 1, where n represents the position of the digit from the separator; the value of n is positive (+), but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative (−) n. For example, in the number 10.34 (written in base 10),
the 1 is second to the left of the separator, so based on calculation, its value is,
$n-1=2-1=1$
$1\times 10^{1}=10$
the 4 is second to the right of the separator, so based on calculation its value is,
$n=-2$
$4\times 10^{-2}={\frac {4}{100}}$
History
Main article: History of the Hindu–Arabic numeral system
Glyphs used to represent digits of the Hindu–Arabic numeral system.
European (descended from the Western Arabic) 0123456789
Arabic-Indic ٠١٢٣٤٥٦٧٨٩
Eastern Arabic-Indic (Persian and Urdu) ۰۱۲۳۴۵۶۷۸۹
Devanagari (Hindi) ०१२३४५६७८९
Tamil ௧௨௩௪௫௬௭௮௯
The first true written positional numeral system is considered to be the Hindu–Arabic numeral system. This system was established by the 7th century in India,[12] but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876.[13] The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.[12]
By the 13th century, Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). They began to enter common use in the 15th century.[14] By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.
Other historical numeral systems using digits
The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was vigesimal (base 20), so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.
The Thai numeral system is identical to the Hindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common in Thailand than it once was, but they are still used alongside Arabic numerals.
The rod numerals, the written forms of counting rods once used by Chinese and Japanese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. The Suzhou numerals are variants of rod numerals.
Rod numerals (vertical)
0 1 2 3 4 5 6 7 8 9
–0 –1 –2 –3 –4 –5 –6 –7 –8 –9
Modern digital systems
In computer science
The binary (base 2), octal (base 8), and hexadecimal (base 16) systems, extensively used in computer science, all follow the conventions of the Hindu–Arabic numeral system.[15] The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7". The hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively.[16] When the binary system is used, the term "bit(s)" is typically used as an alternative for "digit(s)", being a portmanteau of the term "binary digit". Similar terms exist for other number systems, such as "trit(s)" for a ternary system and "dit(s) for the decimal system, although less frequently used.
Unusual systems
The ternary and balanced ternary systems have sometimes been used. They are both base 3 systems.[17]
Balanced ternary is unusual in having the digit values 1, 0 and –1. Balanced ternary turns out to have some useful properties and the system has been used in the experimental Russian Setun computers.[18]
Several authors in the last 300 years have noted a facility of positional notation that amounts to a modified decimal representation. Some advantages are cited for use of numerical digits that represent negative values. In 1840 Augustin-Louis Cauchy advocated use of signed-digit representation of numbers, and in 1928 Florian Cajori presented his collection of references for negative numerals. The concept of signed-digit representation has also been taken up in computer design.
Digits in mathematics
Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics.[19] Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.
Digital roots
Main article: Digital root
The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained.[20]
Casting out nines
Main article: Casting out nines
Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let $f(x)$ represent the digital root of $x$, as described above. Casting out nines makes use of the fact that if $A+B=C$, then $f(f(A)+f(B))=f(C)$. In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal, the original addition must have been faulty.[21]
Repunits and repdigits
Main article: Repunit
Repunits are integers that are represented with only the digit 1. For example, 1111 (one thousand, one hundred and eleven) is a repunit. Repdigits are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The primality of repunits is of interest to mathematicians.[22]
Palindromic numbers and Lychrel numbers
Main article: Palindromic number
Palindromic numbers are numbers that read the same when their digits are reversed.[23] A Lychrel number is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed.[24] The question of whether there are any Lychrel numbers in base 10 is an open problem in recreational mathematics; the smallest candidate is 196.[25]
History of ancient numbers
Main article: History of writing ancient numbers
Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.[26]
To preserve numerical information, tallies carved in wood, bone, and stone have been used since prehistoric times.[27] Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods.
A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC.[28] This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC, written numbers were dissociated from the things being counted and became abstract numerals.
Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions.[29] This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.[30]
Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees).[31]
History of modern numbers
In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply.[32] This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in digital signal processing.[33]
The oldest Greek system was that of the Attic numerals,[34] but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals).[35] Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.[36]
The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until positional notation came into common use in the 16th century.[37]
The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.[38] They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.[39]
The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers.[40] Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.
Some authorities believe that positional arithmetic began with the wide use of counting rods in China.[41] The earliest written positional records seem to be rod calculus results in China around 400. Zero was first used in India in the 7th century CE by Brahmagupta.[42]
The modern positional Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773.[43]
From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century.[44] The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201.[45] In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.[46]
The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz.[47] Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I Ching from China.[48] Binary numbers came into common use in the 20th century because of computer applications.[47]
Numerals in most popular systems
West Arabic 0 1 2 3 4 5 6 7 8 9
Asomiya (Assamese); Bengali ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ৯
Devanagari ० १ २ ३ ४ ५ ६ ७ ८ ९
East Arabic ٠ ١ ٢ ٣ ٤ ٥ ٦ ٧ ٨ ٩
Persian ٠ ١ ٢ ٣ ۴ ۵ ۶ ٧ ٨ ٩
Gurmukhi ੦ ੧ ੨ ੩ ੪ ੫ ੬ ੭ ੮ ੯
Urdu
Chinese (everyday) 〇 一 二 三 四 五 六 七 八 九
Chinese (Traditional) 零 壹 貳 叄 肆 伍 陸 柒 捌 玖
Chinese (Simplified) 零 壹 贰 叁 肆 伍 陆 柒 捌 玖
Chinese (Suzhou) 〇 〡 〢 〣 〤 〥 〦 〧 〨 〩
Ge'ez (Ethiopic) ፩ ፪ ፫ ፬ ፭ ፮ ፯ ፰ ፱
Gujarati ૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯
Hieroglyphic Egyptian 𓏺 𓏻 𓏼 𓏽 𓏾 𓏿 𓐀 𓐁 𓐂
Japanese 零/〇 一 二 三 四 五 六 七 八 九
Kannada ೦ ೧ ೨ ೩ ೪ ೫ ೬ ೭ ೮ ೯
Khmer (Cambodia) ០ ១ ២ ៣ ៤ ៥ ៦ ៧ ៨ ៩
Lao ໐ ໑ ໒ ໓ ໔ ໕ ໖ ໗ ໘ ໙
Limbu ᥆ ᥇ ᥈ ᥉ ᥊ ᥋ ᥌ ᥍ ᥎ ᥏
Malayalam ൦ ൧ ൨ ൩ ൪ ൫ ൬ ൭ ൮ ൯
Mongolian ᠐ ᠑ ᠒ ᠓ ᠔ ᠕ ᠖ ᠗ ᠘ ᠙
Burmese ၀ ၁ ၂ ၃ ၄ ၅ ၆ ၇ ၈ ၉
Oriya ୦ ୧ ୨ ୩ ୪ ୫ ୬ ୭ ୮ ୯
Roman I II III IV V VI VII VIII IX
Shan ႐ ႑ ႒ ႓ ႔ ႕ ႖ ႗ ႘ ႙
Sinhala 𑇡 𑇢 𑇣 𑇤 𑇥 𑇦 𑇧 𑇨 𑇩
Tamil ௦ ௧ ௨ ௩ ௪ ௫ ௬ ௭ ௮ ௯
Telugu ౦ ౧ ౨ ౩ ౪ ౫ ౬ ౭ ౮ ౯
Thai ๐ ๑ ๒ ๓ ๔ ๕ ๖ ๗ ๘ ๙
Tibetan ༠ ༡ ༢ ༣ ༤ ༥ ༦ ༧ ༨ ༩
New Tai Lue ᧐ ᧑ ᧒ ᧓ ᧔ ᧕ ᧖ ᧗ ᧘ ᧙
Javanese ꧐ ꧑ ꧒ ꧓ ꧔ ꧕ ꧖ ꧗ ꧘ ꧙
Additional numerals
1 5 10 20 30 40 50 60 70 80 90 100 500 1000 10000 108
Chinese
(simple)
一 五 十 二十 三十 四十 五十 六十 七十 八十 九十 百 五百 千 万 亿
Chinese
(complex)
壹 伍 拾 贰拾 叁拾 肆拾 伍拾 陆拾 柒拾 捌拾 玖拾 佰 伍佰 仟 萬 億
Ge'ez
(Ethiopic)
፩ ፭ ፲ ፳ ፴ ፵ ፶ ፷ ፸ ፹ ፺ ፻ ፭፻ ፲፻ ፼ ፼፼
Roman I V X XX XXX XL L LX LXX LXXX XC C D M X
See also
• Hexadecimal
• Binary digit (bit), Quantum binary digit (qubit)
• Ternary digit (trit), Quantum ternary digit (qutrit)
• Decimal digit (dit)
• Hexadecimal digit (Hexit)
• Natural digit (nat, nit)
• Naperian digit (nepit)
• Significant digit
• Large numbers
• Text figures
• Abacus
• History of large numbers
• List of numeral system topics
Numeral notation in various scripts
• Arabic numerals
• Armenian numerals
• Babylonian numerals
• Balinese numerals
• Bengali numerals
• Burmese numerals
• Chinese numerals
• Cistercian numerals
• Dzongkha numerals
• Eastern Arabic numerals
• Georgian numerals
• Greek numerals
• Gujarati numerals
• Gurmukhi numerals
• Hebrew numerals
• Hokkien numerals
• Indian numerals
• Japanese numerals
• Javanese numerals
• Khmer numerals
• Korean numerals
• Lao numerals
• Mayan numerals
• Mongolian numerals
• Quipu
• Rod numerals
• Roman numerals
• Sinhala numerals
• Suzhou numerals
• Tamil numerals
• Thai numerals
• Vietnamese numerals
References
1. ""Digit" Origin". dictionary.com. Retrieved 23 May 2015.
2. ""Decimal" Origin". dictionary.com. Retrieved 23 May 2015.
3. Weisstein, Eric W. "Decimal Point". mathworld.wolfram.com. Retrieved 2020-07-22.
4. Snyder, Barbara Bode (1991). Practical math for the technician : the basics. Englewood Cliffs, N.J.: Prentice Hall. p. 225. ISBN 0-13-251513-X. OCLC 22345295. units or ones place
5. Andrew Jackson Rickoff (1888). Numbers Applied. D. Appleton & Company. pp. 5–. units' or ones' place
6. John William McClymonds; D. R. Jones (1905). Elementary Arithmetic. R.L. Telfer. pp. 17–18. units' or ones' place
7. Richard E. Johnson; Lona Lee Lendsey; William E. Slesnick (1967). Introductory Algebra for College Students. Addison-Wesley Publishing Company. p. 30. units' or ones', digit
8. R. C. Pierce; W. J. Tebeaux (1983). Operational Mathematics for Business. Wadsworth Publishing Company. p. 29. ISBN 978-0-534-01235-9. ones or units digit
9. Max A. Sobel (1985). Harper & Row algebra one. Harper & Row. p. 282. ISBN 978-0-06-544000-3. ones, or units, digit
10. Max A. Sobel (1985). Harper & Row algebra one. Harper & Row. p. 277. ISBN 978-0-06-544000-3. every two-digit number can be expressed as 10t+u when t is the tens digit
11. Taggart, Robert (2000). Mathematics. Decimals and percents. Portland, Me.: J. Weston Walch. pp. 51–54. ISBN 0-8251-4178-8. OCLC 47352965.
12. O'Connor, J. J. and Robertson, E. F. Arabic Numerals. January 2001. Retrieved on 2007-02-20.
13. Bill Casselman (February 2007). "All for Nought". Feature Column. AMS.
14. Bradley, Jeremy. "How Arabic Numbers Were Invented". www.theclassroom.com. Retrieved 2020-07-22.
15. Ravichandran, D. (2001-07-01). Introduction To Computers And Communication. Tata McGraw-Hill Education. pp. 24–47. ISBN 978-0-07-043565-0.
16. "Hexadecimals". www.mathsisfun.com. Retrieved 2020-07-22.
17. (PDF). 2019-10-30 https://web.archive.org/web/20191030114823/http://bit-player.org/wp-content/extras/bph-publications/AmSci-2001-11-Hayes-ternary.pdf. Archived from the original (PDF) on 2019-10-30. Retrieved 2020-07-22. {{cite web}}: Missing or empty |title= (help)
18. "Development of ternary computers at Moscow State University. Russian Virtual Computer Museum". www.computer-museum.ru. Retrieved 2020-07-22.
19. Kirillov, A.A. "What are numbers?" (PDF). math.upenn. p. 2. True, if you open a modern mathematical journal and try to read any article, it is very probable that you will see no numbers at all.
20. Weisstein, Eric W. "Digital Root". mathworld.wolfram.com. Retrieved 2020-07-22.
21. Weisstein, Eric W. "Casting Out Nines". mathworld.wolfram.com. Retrieved 2020-07-22.
22. Weisstein, Eric W. "Repunit". MathWorld.
23. Weisstein, Eric W. "Palindromic Number". mathworld.wolfram.com. Retrieved 2020-07-22.
24. Weisstein, Eric W. "Lychrel Number". mathworld.wolfram.com. Retrieved 2020-07-22.
25. Garcia, Stephan Ramon; Miller, Steven J. (2019-06-13). 100 Years of Math Milestones: The Pi Mu Epsilon Centennial Collection. American Mathematical Soc. pp. 104–105. ISBN 978-1-4704-3652-0.
26. Saxe, Geoffrey B. (2012). Cultural development of mathematical ideas : Papua New Guinea studies. Esmonde, Indigo. Cambridge: Cambridge University Press. pp. 44–45. ISBN 978-1-139-55157-1. OCLC 811060760. The Okspamin body system includes 27 body parts...
27. Tuniz, C. (Claudio) (24 May 2016). Humans : an unauthorized biography. Tiberi Vipraio, Patrizia, Haydock, Juliet. Switzerland. p. 101. ISBN 978-3-319-31021-3. OCLC 951076018. ...even notches cut into sticks made out of wood, bone or other materials dating back 30,000 years (often referred to as "notched tallies").{{cite book}}: CS1 maint: location missing publisher (link)
28. Ifrah, Georges (1985). From one to zero : a universal history of numbers. New York: Viking. p. 154. ISBN 0-670-37395-8. OCLC 11237558. And so , by the beginning of the third millennium B . C . , the Sumerians and Elamites had adopted the practice of recording numerical information on small , usually rectangular clay tablets
29. London Encyclopædia, Or, Universal Dictionary of Science, Art, Literature, and Practical Mechanics: Comprising a Popular View of the Present State of Knowledge; Illustrated by Numerous Engravings and Appropriate Diagrams. T. Tegg. 1845. p. 226.
30. Neugebauer, O. (2013-11-11). Astronomy and History Selected Essays. Springer Science & Business Media. ISBN 978-1-4612-5559-8.
31. "Sexagesimal System". Springer Reference. 2011. doi:10.1007/springerreference_78190. {{cite book}}: |work= ignored (help)
32. Knuth, Donald Ervin (1998). The art of computer programming. Reading, Mass.: Addison-Wesley Pub. Co. ISBN 0-201-03809-9. OCLC 823849. The advantages of a modular representation are that addition, subtraction, and multiplication are very simple
33. Echtle, Klaus; Hammer, Dieter; Powell, David (1994-09-21). Dependable Computing - EDCC-1: First European Dependable Computing Conference, Berlin, Germany, October 4-6, 1994. Proceedings. Springer Science & Business Media. p. 439. ISBN 978-3-540-58426-1.
34. Woodhead, A. G. (Arthur Geoffrey) (1981). The study of Greek inscriptions (2nd ed.). Cambridge: Cambridge University Press. pp. 109–110. ISBN 0-521-23188-4. OCLC 7736343.
35. Ushakov, Igor (22 June 2012). In the Beginning Was the Number (2). Lulu.com. ISBN 978-1-105-88317-0.
36. Chrisomalis, Stephen (2010). Numerical notation : a comparative history. Cambridge: Cambridge University Press. p. 157. ISBN 978-0-511-67683-3. OCLC 630115876. The first safely dated instance in which the use of Hebrew alphabetic numerals is certain is on coins from the reign of Hasmonean king Alexander Janneus(103 to 76 BC)...
37. Silvercloud, Terry David (2007). The Shape of God: Secrets, Tales, and Legends of the Dawn Warriors. Terry David Silvercloud. p. 152. ISBN 978-1-4251-0836-6.
38. Wheeler, Ruric E.; Wheeler, Ed R. (2001), Modern Mathematics, Kendall Hunt, p. 130, ISBN 9780787290627.
39. Swami, Devamrita (2002). Searching for Vedic India. The Bhaktivedanta Book Trust. ISBN 978-0-89213-350-5. Maya astronomy finely calculated both the duration of the solar year and the synodical revolution of Venus
40. "Quipu | Incan counting tool". Encyclopedia Britannica. Retrieved 2020-07-23.
41. Chen, Sheng-Hong (2018-06-21). Computational Geomechanics and Hydraulic Structures. Springer. p. 8. ISBN 978-981-10-8135-4. … definitely before 400 BC they possessed a similar positional notation based on the ancient counting rods.
42. "Foundations of mathematics - The reexamination of infinity". Encyclopedia Britannica. Retrieved 2020-07-23.
43. The Encyclopedia Britannica. 1899. p. 626.
44. Struik, Dirk J. (Dirk Jan) (1967). A concise history of mathematics (3d rev. ed.). New York: Dover Publications. ISBN 0-486-60255-9. OCLC 635553.
45. Sigler, Laurence (2003-11-11). Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation. Springer Science & Business Media. ISBN 978-0-387-40737-1.
46. Deming, David (2010). Science and technology in world history. Volume 1, The ancient world and classical civilization. Jefferson, N.C.: McFarland & Co. p. 86. ISBN 978-0-7864-5657-4. OCLC 650873991.
47. Yanushkevich, Svetlana N. (2008). Introduction to logic design. Shmerko, Vlad P. Boca Raton: CRC Press. p. 56. ISBN 978-1-4200-6094-2. OCLC 144226528.
48. Sloane, Sarah (2005). The I Ching for writers : finding the page inside you. Novato, Calif.: New World Library. p. 9. ISBN 1-57731-496-4. OCLC 56672043.
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| Wikipedia |
10
10 (ten) is the even natural number following 9 and preceding 11. It is the first double-digit number. Ten is the base of the decimal numeral system, the most common system of denoting numbers in both spoken and written language.
← 9 10 11 →
← 10 11 12 13 14 15 16 17 18 19 →
• List of numbers
• Integers
← 0 10 20 30 40 50 60 70 80 90 →
Cardinalten
Ordinal10th
(tenth)
Numeral systemdecimal
Factorization2 × 5
Divisors1, 2, 5, 10
Greek numeralΙ´
Roman numeralX
Roman numeral (unicode)X, x
Greek prefixdeca-/deka-
Latin prefixdeci-
Binary10102
Ternary1013
Senary146
Octal128
DuodecimalA12
HexadecimalA16
Chinese numeral十，拾
Hebrewי (Yod)
Khmer១០
Tamil௰
Thai๑๐
Devanāgarī१०
Bengali১০
Arabic & Kurdish & Iranian١٠
Malayalam൰
Anthropology
Usage and terms
• A collection of ten items (most often ten years) is called a decade.
• The ordinal adjective is decimal; the distributive adjective is denary.
• Increasing a quantity by one order of magnitude is most widely understood to mean multiplying the quantity by ten.
• To reduce something by one tenth is to decimate. (In ancient Rome, the killing of one in ten soldiers in a cohort was the punishment for cowardice or mutiny; or, one-tenth of the able-bodied men in a village as a form of retribution, thus causing a labor shortage and threat of starvation in agrarian societies.)
Other
• The number of kingdoms in Five Dynasties and Ten Kingdoms period.
• The house number of 10 Downing Street.
• The number of Provinces in Canada.
• Number of dots in a tetractys.
• The number of the French department Aube.
In mathematics
Ten is the fifth composite number. It is also the smallest noncototient, a number that cannot be expressed as the difference between any integer and the total number of coprimes below it.[1] It is the second discrete semiprime ($2\times 5$), as well as the second member of the $2\times q$ discrete semiprime family. Ten is the only number whose sum and difference of its prime divisors yield prime numbers ($2+5=7$ and $5-2=3$). In general, powers of 10 contain $n^{2}$ divisors, where $n$ is the number of digits: 10 has 22 = 4 divisors, 100 has 32 = 9 divisors, 1,000 has 42 = 16 divisors, 10,000 has 52 = 25 divisors, and so forth. Ten is the smallest number whose status as a possible friendly number is unknown.[2]
As important sums,
• $10=1+2+3+4$ the sum of the first four positive integers.
• $10=2+3+5=2\times 5$, the sum of the first three prime numbers, and the smallest semiprime that is the sum of all the distinct prime numbers from its lower factor through its higher factor.[3]
• $10=3+7=5+5$, the smallest number that can be written as the sum of two prime numbers in two different ways.
• $10=1^{2}+3^{2}$, the sum of the squares of the first two odd numbers.
The factorial of ten is equal to the product of the factorials of the first three odd primes, $10!=3!\cdot 5!\cdot 7!$.[4] Ten is also the first number whose fourth power can be written as a sum of two squares in two different ways ($80^{2}+60^{2}$ and $96^{2}+28^{2}$).
Ten has an aliquot sum σ(10) of 8 and is accordingly the first discrete semiprime to be in deficit, with all subsequent discrete semiprimes in deficit.[5] The aliquot sequence for 10 comprises five members (10, 8, 7, 1, 0) with this number being the second composite member of the 7-aliquot tree.[6] 10 is also the eighth Perrin number, preceded in the sequence by (5, 5, 7).[7] In the sequence of triangular numbers, indexed powers of 10 in this sequence generate the following sequence of triangular numbers in decimal: 55 (10th), 5,050 (100th), 500,500 (1,000th), ...[8][lower-alpha 1]
While 55 is the tenth triangular number, it is also the tenth Fibonacci number, and the largest such number to also be a triangular number.[9] Ten is also the first non-trivial decagonal number,[10] the third centered triangular number[11] and tetrahedral number,[12] and the fifth semi-meandric number.[13]
10 is the fourth telephone number, and the number of Young tableaux with four cells.[14] It is the number of $n$-queens problem solutions for $n=5$.[15]
A $10\times 10$ magic square has a magic constant of 505.[16]
There are ten small Pisot numbers that do not exceed the golden ratio.[17]
According to conjecture, ten is the average sum of the proper divisors of the natural numbers $\mathbb {N} $ if the size of the numbers approaches infinity.[18]
In geometry
A polygon with ten sides is called a decagon. As a constructible polygon with a compass and straight-edge, it has an internal angle of $12^{2}=144$ degrees and a central angle of $6^{2}=36$ degrees. All regular $n$-sided polygons with up to ten sides are able to tile a plane-vertex alongside other regular polygons alone; the first regular polygon unable to do so is the eleven-sided hendecagon.[19] A decagon can fill a plane-vertex alongside two regular pentagons, and alongside a fifteen-sided pentadecagon and triangle. Ten of the eleven regular and semiregular (or Archimedean) tilings of the plane are Wythoffian, the elongated triangular tiling is the only exception.[20]
The regular decagon is the Petrie polygon of the regular dodecahedron and icosahedron, and it is the largest face that an Archimedean solid can contain, as with the truncated dodecahedron and the truncated icosidodecahedron. The decagon is the hemi-face of the icosidodecahedron, such that a plane dissection yields two mirrored pentagonal rotundae. A regular ten-pointed {10/3} decagram is the hemi-face of the great icosidodecahedron, as well as the Petrie polygon of two regular Kepler–Poinsot polyhedra. Ten non-prismatic uniform polyhedra contain regular decagons as faces (U26, U28, U33, U37, U39, ...), and ten contain regular decagrams as faces (U42, U45, U58, U59, U63, ...). The decagonal prism is also the largest prism that is a facet inside four-dimensional uniform polychora.
There are ten regular star polychora in the fourth dimension.[21] All of these polychora have orthographic projections in the $\mathrm {H} _{3}$ Coxeter plane that contain various decagrammic symmetries, which include the regular {10/3} form as well as its three alternate compound forms.
$\mathrm {M} _{10}$ is a multiply transitive permutation group on 10 points. It is an almost simple group, of order 720 = 24·32·5 = 2·3·4·5·6 = 8·9·10. It functions as a point stabilizer of degree 11 inside the smallest sporadic group $\mathrm {M} _{11}$, a Mathieu group which has an irreducible faithful complex representation in 10 dimensions.
$\mathrm {E} _{10}$ is an infinite-dimensional Kac–Moody algebra which has the even Lorentzian unimodular lattice II9,1 of dimension 10 as its root lattice. It is the first $\mathrm {E} _{n}$ Lie algebra with a negative Cartan matrix determinant, of −1.
There are precisely ten affine Coxeter groups that admit a formal description of reflections across $n$ dimensions in Euclidean space. These contain infinite facets whose quotient group of their normal abelian subgroups is finite. They include the one-dimensional Coxeter group ${\tilde {I}}_{1}$ [∞], which represents the apeirogonal tiling, as well as the five affine Coxeter groups ${\tilde {G}}_{2}$, ${\tilde {F}}_{4}$, ${\tilde {E}}_{6}$, ${\tilde {E}}_{7}$, and ${\tilde {E}}_{8}$ that are associated with the five exceptional Lie algebras. They also include the four general affine Coxeter groups ${\tilde {A}}_{n}$, ${\tilde {B}}_{n}$, ${\tilde {C}}_{n}$, and ${\tilde {D}}_{n}$ that are associated with simplex, cubic and demihypercubic honeycombs, or tessellations. Regarding Coxeter groups in hyperbolic space, there are infinitely many such groups; however, ten is the highest rank for paracompact hyperbolic solutions, with a representation in nine dimensions. There also exist hyperbolic Lorentzian cocompact groups where removing any permutation of two nodes in its Coxeter–Dynkin diagram leaves a finite or Euclidean graph. The tenth dimension is the highest dimensional representation for such solutions, which share a root symmetry in eleven dimensions. These are of particular interest in M-theory of string theory.
List of basic calculations
Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 50 100 1000
10 × x 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 500 1000 10000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
10 ÷ x 10 5 3.3 2.5 2 1.6 1.428571 1.25 1.1 1 0.90 0.83 0.769230 0.714285 0.6
x ÷ 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 1.2 1.3 1.4 1.5
Exponentiation 1 2 3 4 5 6 7 8 9 10
10x 10 100 1000 10000 100000 1000000 10000000 100000000 1000000000 10000000000
x10 1 1024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401
In science
The SI prefix for 10 is "deca-".
The meaning "10" is part of the following terms:
• decapoda, an order of crustaceans with ten feet.
• decane, a hydrocarbon with 10 carbon atoms.
Also, the number 10 plays a role in the following:
• The atomic number of neon.
• The number of hydrogen atoms in butane, a hydrocarbon.
• The number of spacetime dimensions in some superstring theories.
The metric system is based on the number 10, so converting units is done by adding or removing zeros (e.g. 1 centimeter = 10 millimeters, 1 decimeter = 10 centimeters, 1 meter = 100 centimeters, 1 dekameter = 10 meters, 1 kilometer = 1,000 meters).
Astronomy
• The New General Catalogue object NGC 10, a magnitude 12.5 spiral galaxy in the constellation Sculptor.
• Messier object M10, a magnitude 6.4 globular cluster in the constellation Ophiuchus.
In religion and philosophy
• References in the Bible, Judaism and Christianity:
• The Ten Commandments of Exodus[22] and Deuteronomy[23] are considered a cornerstone of Judaism and Christianity.
• People traditionally tithed one-tenth of their produce. The practice of tithing is still common in Christian churches today, though it is disputed in some circles as to whether or not it is required of Christians.
• In Deuteronomy 26:12, the Torah commands Jews to give one-tenth of their produce to the poor (Maaser Ani). From this verse and from an earlier verse (Deut. 14:22) there derives a practice for Jews to give one-tenth of all earnings to the poor.[24]
• Ten Plagues were inflicted on Egypt in Exodus 7–12.
• Jews observe the annual Ten Days of Repentance beginning on Rosh Hashanah and ending on Yom Kippur.
• In Jewish liturgy, Ten Martyrs are singled out as a group.
• There are said to be Ten Lost Tribes of Israel (those other than Judah and Benjamin).
• There are Ten Sephirot in the Kabbalistic Tree of Life.
• In Judaism, ten men are the required quorum, called a minyan, for prayer services.
• In Genesis 18:23-32, Abraham pleads on behalf of Sodom and Gomorrah, asking to save the cities if there are enough righteous people there. He starts at 10 per city, and ends with 10 total in all cities.
• Interpretations of Genesis in Talmudic and Midrashic teachings suggest that on the first day, God drew forth ten primal elements from the abyss in order to construct all of Creation: Heaven (or Fire), Earth, Chaos, Void, Light, Darkness, Wind (or Spirit), Water, Day, and Night. See also Bereshit (parsha).
• Jesus tells the Parable of the Ten Virgins in Matthew 25:1–13.
• In Pythagoreanism, the number 10 played an important role and was symbolized by the tetractys.
• In Hinduism, Lord Vishnu appeared on the earth in 10 incarnations, popularly known as Dashaavathar.
• In Sikhism, there are ten human Gurus.
In money
Most countries issue coins and bills with a denomination of 10 (See e.g. 10 dollar note). Of these, the U.S. dime, with the value of ten cents, or one tenth of a dollar, derives its name from the meaning "one-tenth" − see Dime (United States coin)#Denomination history and etymology.
In music
• The interval of a major tenth is an octave plus a major third.
• The interval of a minor tenth is an octave plus a minor third.
• "Ten lords a-leaping" is the gift on the tenth day of Christmas in the carol "The Twelve Days of Christmas".
In sports and games
• Decathlon is a combined event in athletics consisting of ten track and field events.
• In association football, the number 10 is traditionally worn by the team's advanced playmaker. This use has led to "Number 10" becoming a synonym for the player in that particular role, even if they do not wear that number.[25]
• In gridiron football, a team has a limited number of downs to advance the ball ten yards or more from where it was on its last first down; doing this is referred to as gaining another first down.
• In auto racing, driving a car at ten-tenths is driving as fast as possible, on the limit.
• In a regular basketball game, two teams playing against each other have 5 members each, for a total of 10 players on court. Under FIBA, WNBA, and NCAA women's rules, each quarter runs for 10 minutes.
• In blackjack, the Ten, Jack, Queen and King are all worth 10 points.
• In boxing, if the referee counts to 10 whether the boxer is unconscious or not, it will declare a winner by knockout.
• In men's field lacrosse, each team has 10 players on the field at any given time, except in penalty situations.
• Ten-ball is a pool game played with a cue ball and ten numbered balls.
• In most rugby league competitions, the number 10 is worn by one of the two starting props. One exception to this rule is the Super League, which uses static squad numbering.
• In rugby union, the starting fly-half wears the 10 shirt.
• In ten-pin bowling, 10 pins are arranged in a triangular pattern and there are 10 frames per game.
In technology
• Ten-codes are commonly used on emergency service radio systems.
• Ten refers to the "meter band" on the radio spectrum between 28 and 29.7 MHz, used by amateur radio.
• ASCII and Unicode code point for line feed.
• In MIDI, Channel 10 is reserved for unpitched percussion instruments.
• In the Rich Text Format specification, all language codes for regional variants of the Spanish language are congruent to 10 mod 256.
• In macOS, the F10 function key tiles all the windows of the current application and grays the windows of other applications.
• The IP addresses in the range 10.0.0.0/8 (meaning the interval between 10.0.0.0 and 10.255.255.255) are reserved for use by private networks by RFC 1918.
Age 10
• This is generally the age when a child enters the preteen stage and also a denarian (someone within the age range of 10–19).
• The ESRB recommends video games with an E10+ rating to children aged 10 and up.
In other fields
• Blake Edwards' 1979 movie 10.
• Series on HBO entitled 1st & Ten which aired between December 1984 and January 1991.
• Series on ESPN and ESPN2 entitled 1st and 10 which launched on ESPN in October 2003 to 2008 and moved to ESPN2 since 2008.
• In astrology, Capricorn is the 10th astrological sign of the Zodiac.
• In Chinese astrology, the 10 Heavenly Stems, refer to a cyclic number system that is used also for time reckoning.
• A 1977 short documentary film Powers of Ten depicts the relative scale of the Universe in factors of ten (orders of magnitude).
• CBS (parent company Paramount Global also owns another entity on this list, Network 10) has a game show called Power of 10, where the player's prize goes up and down by either the previous or next power of ten.
• "Ten Chances" is one of the pricing games on The Price is Right.
• There are ten official inkblots in the Rorschach inkblot test.
• The traditional Snellen chart uses 10 different letters.
• Network 10 is an Australian television network. The Sydney member of the network has the three-letter call-sign TEN and used to broadcast in analogue on VHF Channel 10. Paramount Global owns this network since December 4, 2019.
• Number Ten (also called Ella) is a character in the book series Lorien Legacies. The sixth book, The Fate of Ten, is named after her.
• A Cartoon Network franchise Ben 10, which has a number on its title.
See also
• Mathematics portal
• List of highways numbered 10
References
1. 19 is another number that is the first member of a sequence displaying a similar uniform property. The 19th triangular number is 190, the 199th triangular number is 19900, ... (sequence A186076 in the OEIS)
1. "Sloane's A005278 : Noncototients". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
2. Sloane, N. J. A. (ed.). "Sequence A074902 (Known friendly numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
3. Sloane, N. J. A. (ed.). "Sequence A055233 (Composite numbers equal to the sum of the primes from their smallest prime factor to their largest prime factor.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
4. "10". PrimeCurios!. PrimePages. Retrieved 2023-01-14.
5. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
6. Sloane, N. J. A. (1975). "Aliquot sequences". Mathematics of Computation. OEIS Foundation. 29 (129): 101–107. Retrieved 2022-12-08.
7. Sloane, N. J. A. (ed.). "Sequence A001608 (Perrin sequence (or Ondrej Such sequence))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
8. Sloane, N. J. A. (ed.). "Sequence A037156". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
For n = 0; a(0) = 1 = 1 * 1 = 1
For n = 1; a(1) = 1 + 2 + ...... + 10 = 11 * 5 = 55
For n = 2; a(2) = 1 + 2 + .... + 100 = 101 * 50 = 5050
For n = 3; a(3) = 1 + 2 + .. + 1000 = 1001 * 500 = 500500
...
9. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
10. "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
11. "Sloane's A005448 : Centered triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
12. "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
13. Sloane, N. J. A. (ed.). "Sequence A000682 (Semi-meanders: number of ways a semi-infinite directed curve can cross a straight line n times.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
14. Sloane, N. J. A. (ed.). "Sequence A000085 (Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with four cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-17.
15. Sloane, N. J. A. (ed.). "Sequence A000170 (Number of ways of placing n nonattacking queens on an n X n board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
16. Andrews, W.S. (1917). Magic Squares and Cubes (2nd ed.). Open Court Publishing. p. 30.
17. M.J. Bertin; A. Decomps-Guilloux; M. Grandet-Hugot; M. Pathiaux-Delefosse; J.P. Schreiber (1992). Pisot and Salem Numbers. Birkhäuser. ISBN 3-7643-2648-4.
18. Sloane, N. J. A. (ed.). "Sequence A297575 (Numbers whose sum of divisors is divisible by 10.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-08.
19. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 230, 231. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
20. Grünbaum, Branko; Shephard, G. C. (1987). "Section 2.1: Regular and uniform tilings". Tilings and Patterns. New York: W. H. Freeman and Company. p. 64. doi:10.2307/2323457. ISBN 0-7167-1193-1. JSTOR 2323457. OCLC 13092426. S2CID 119730123.
21. Coxeter, H. S. M (1948). "Chapter 14: Star-polytopes". Regular Polytopes. London: Methuen & Co. LTD. p. 263.
22. Exodus 20:2–13
23. Deuteronomy 5:6–17
24. Archived February 23, 2006, at the Wayback Machine
25. Khalil Garriot (21 June 2014). "Mystery solved: Why do the best soccer players wear No. 10?". Yahoo. Retrieved 19 May 2015.
External links
Wikimedia Commons has media related to 10 (number).
Look up ten in Wiktionary, the free dictionary.
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| Wikipedia |
String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force. Thus, string theory is a theory of quantum gravity.
String theory
Fundamental objects
• String
• Cosmic string
• Brane
• D-brane
Perturbative theory
• Bosonic
• Superstring (Type I, Type II, Heterotic)
Non-perturbative results
• S-duality
• T-duality
• U-duality
• M-theory
• F-theory
• AdS/CFT correspondence
Phenomenology
• Phenomenology
• Cosmology
• Landscape
Mathematics
• Geometric Langlands correspondence
• Mirror symmetry
• Monstrous moonshine
• Vertex algebra
Related concepts
• Theory of everything
• Conformal field theory
• Quantum gravity
• Supersymmetry
• Supergravity
• Twistor string theory
• N = 4 supersymmetric Yang–Mills theory
• Kaluza–Klein theory
• Multiverse
• Holographic principle
Theorists
• Aganagić
• Arkani-Hamed
• Atiyah
• Banks
• Berenstein
• Bousso
• Cleaver
• Curtright
• Dijkgraaf
• Distler
• Douglas
• Duff
• Dvali
• Ferrara
• Fischler
• Friedan
• Gates
• Gliozzi
• Gopakumar
• Green
• Greene
• Gross
• Gubser
• Gukov
• Guth
• Hanson
• Harvey
• Hořava
• Horowitz
• Gibbons
• Kachru
• Kaku
• Kallosh
• Kaluza
• Kapustin
• Klebanov
• Knizhnik
• Kontsevich
• Klein
• Linde
• Maldacena
• Mandelstam
• Marolf
• Martinec
• Minwalla
• Moore
• Motl
• Mukhi
• Myers
• Nanopoulos
• Năstase
• Nekrasov
• Neveu
• Nielsen
• van Nieuwenhuizen
• Novikov
• Olive
• Ooguri
• Ovrut
• Polchinski
• Polyakov
• Rajaraman
• Ramond
• Randall
• Randjbar-Daemi
• Roček
• Rohm
• Sagnotti
• Scherk
• Schwarz
• Seiberg
• Sen
• Shenker
• Siegel
• Silverstein
• Sơn
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• Steinhardt
• Strominger
• Sundrum
• Susskind
• 't Hooft
• Townsend
• Trivedi
• Turok
• Vafa
• Veneziano
• Verlinde
• Verlinde
• Wess
• Witten
• Yau
• Yoneya
• Zamolodchikov
• Zamolodchikov
• Zaslow
• Zumino
• Zwiebach
• History
• Glossary
String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has contributed a number of advances to mathematical physics, which have been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, and condensed matter physics, and it has stimulated a number of major developments in pure mathematics. Because string theory potentially provides a unified description of gravity and particle physics, it is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter. Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details.
String theory was first studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in favor of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. The earliest version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in 11 dimensions known as M-theory. In late 1997, theorists discovered an important relationship called the anti-de Sitter/conformal field theory correspondence (AdS/CFT correspondence), which relates string theory to another type of physical theory called a quantum field theory.
One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, which has complicated efforts to develop theories of particle physics based on string theory. These issues have led some in the community to criticize these approaches to physics, and to question the value of continued research on string theory unification.
Fundamentals
Overview
In the 20th century, two theoretical frameworks emerged for formulating the laws of physics. The first is Albert Einstein's general theory of relativity, a theory that explains the force of gravity and the structure of spacetime at the macro-level. The other is quantum mechanics, a completely different formulation, which uses known probability principles to describe physical phenomena at the micro-level. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the universe, from elementary particles to atoms to the evolution of stars and the universe as a whole.[1]
In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of quantum gravity.[1] The general theory of relativity is formulated within the framework of classical physics, whereas the other fundamental forces are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity.[2] In addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, and the early universe.[lower-alpha 1]
String theory is a theoretical framework that attempts to address these questions and many others. The starting point for string theory is the idea that the point-like particles of particle physics can also be modeled as one-dimensional objects called strings. String theory describes how strings propagate through space and interact with each other. In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, and it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass, charge, and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. In string theory, one of the vibrational states of the string gives rise to the graviton, a quantum mechanical particle that carries gravitational force. Thus string theory is a theory of quantum gravity.[3]
One of the main developments of the past several decades in string theory was the discovery of certain 'dualities', mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, and this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as M-theory.[4]
Studies of string theory have also yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black holes, and work on string theory has attempted to clarify these issues. In late 1997 this line of work culminated in the discovery of the anti-de Sitter/conformal field theory correspondence or AdS/CFT.[5] This is a theoretical result that relates string theory to other physical theories which are better understood theoretically. The AdS/CFT correspondence has implications for the study of black holes and quantum gravity, and it has been applied to other subjects, including nuclear[6] and condensed matter physics.[7][8]
Since string theory incorporates all of the fundamental interactions, including gravity, many physicists hope that it will eventually be developed to the point where it fully describes our universe, making it a theory of everything. One of the goals of current research in string theory is to find a solution of the theory that reproduces the observed spectrum of elementary particles, with a small cosmological constant, containing dark matter and a plausible mechanism for cosmic inflation. While there has been progress toward these goals, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of details.[9]
One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. The scattering of strings is most straightforwardly defined using the techniques of perturbation theory, but it is not known in general how to define string theory nonperturbatively.[10] It is also not clear whether there is any principle by which string theory selects its vacuum state, the physical state that determines the properties of our universe.[11] These problems have led some in the community to criticize these approaches to the unification of physics and question the value of continued research on these problems.[12]
Strings
The application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields.[13]
In quantum field theory, one typically computes the probabilities of various physical events using the techniques of perturbation theory. Developed by Richard Feynman and others in the first half of the twentieth century, perturbative quantum field theory uses special diagrams called Feynman diagrams to organize computations. One imagines that these diagrams depict the paths of point-like particles and their interactions.[13]
The starting point for string theory is the idea that the point-like particles of quantum field theory can also be modeled as one-dimensional objects called strings.[14] The interaction of strings is most straightforwardly defined by generalizing the perturbation theory used in ordinary quantum field theory. At the level of Feynman diagrams, this means replacing the one-dimensional diagram representing the path of a point particle by a two-dimensional (2D) surface representing the motion of a string.[15] Unlike in quantum field theory, string theory does not have a full non-perturbative definition, so many of the theoretical questions that physicists would like to answer remain out of reach.[16]
In theories of particle physics based on string theory, the characteristic length scale of strings is assumed to be on the order of the Planck length, or 10−35 meters, the scale at which the effects of quantum gravity are believed to become significant.[15] On much larger length scales, such as the scales visible in physics laboratories, such objects would be indistinguishable from zero-dimensional point particles, and the vibrational state of the string would determine the type of particle. One of the vibrational states of a string corresponds to the graviton, a quantum mechanical particle that carries the gravitational force.[3]
The original version of string theory was bosonic string theory, but this version described only bosons, a class of particles that transmit forces between the matter particles, or fermions. Bosonic string theory was eventually superseded by theories called superstring theories. These theories describe both bosons and fermions, and they incorporate a theoretical idea called supersymmetry. In theories with supersymmetry, each boson has a counterpart which is a fermion, and vice versa.[17]
There are several versions of superstring theory: type I, type IIA, type IIB, and two flavors of heterotic string theory (SO(32) and E8×E8). The different theories allow different types of strings, and the particles that arise at low energies exhibit different symmetries. For example, the type I theory includes both open strings (which are segments with endpoints) and closed strings (which form closed loops), while types IIA, IIB and heterotic include only closed strings.[18]
Extra dimensions
In everyday life, there are three familiar dimensions (3D) of space: height, width and length. Einstein's general theory of relativity treats time as a dimension on par with the three spatial dimensions; in general relativity, space and time are not modeled as separate entities but are instead unified to a four-dimensional (4D) spacetime. In this framework, the phenomenon of gravity is viewed as a consequence of the geometry of spacetime.[19]
In spite of the fact that the Universe is well described by 4D spacetime, there are several reasons why physicists consider theories in other dimensions. In some cases, by modeling spacetime in a different number of dimensions, a theory becomes more mathematically tractable, and one can perform calculations and gain general insights more easily.[lower-alpha 2] There are also situations where theories in two or three spacetime dimensions are useful for describing phenomena in condensed matter physics.[13] Finally, there exist scenarios in which there could actually be more than 4D of spacetime which have nonetheless managed to escape detection.[20]
String theories require extra dimensions of spacetime for their mathematical consistency. In bosonic string theory, spacetime is 26-dimensional, while in superstring theory it is 10-dimensional, and in M-theory it is 11-dimensional. In order to describe real physical phenomena using string theory, one must therefore imagine scenarios in which these extra dimensions would not be observed in experiments.[21]
Compactification is one way of modifying the number of dimensions in a physical theory. In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles.[22] In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions. A standard analogy for this is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length. However, as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling on the surface of the hose would move in two dimensions.
Compactification can be used to construct models in which spacetime is effectively four-dimensional. However, not every way of compactifying the extra dimensions produces a model with the right properties to describe nature. In a viable model of particle physics, the compact extra dimensions must be shaped like a Calabi–Yau manifold.[22] A Calabi–Yau manifold is a special space which is typically taken to be six-dimensional in applications to string theory. It is named after mathematicians Eugenio Calabi and Shing-Tung Yau.[23]
Another approach to reducing the number of dimensions is the so-called brane-world scenario. In this approach, physicists assume that the observable universe is a four-dimensional subspace of a higher dimensional space. In such models, the force-carrying bosons of particle physics arise from open strings with endpoints attached to the four-dimensional subspace, while gravity arises from closed strings propagating through the larger ambient space. This idea plays an important role in attempts to develop models of real-world physics based on string theory, and it provides a natural explanation for the weakness of gravity compared to the other fundamental forces.[24]
Dualities
A notable fact about string theory is that the different versions of the theory all turn out to be related in highly nontrivial ways. One of the relationships that can exist between different string theories is called S-duality. This is a relationship that says that a collection of strongly interacting particles in one theory can, in some cases, be viewed as a collection of weakly interacting particles in a completely different theory. Roughly speaking, a collection of particles is said to be strongly interacting if they combine and decay often and weakly interacting if they do so infrequently. Type I string theory turns out to be equivalent by S-duality to the SO(32) heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality.[25]
Another relationship between different string theories is T-duality. Here one considers strings propagating around a circular extra dimension. T-duality states that a string propagating around a circle of radius R is equivalent to a string propagating around a circle of radius 1/R in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, a string has momentum as it propagates around a circle, and it can also wind around the circle one or more times. The number of times the string winds around a circle is called the winding number. If a string has momentum p and winding number n in one description, it will have momentum n and winding number p in the dual description. For example, type IIA string theory is equivalent to type IIB string theory via T-duality, and the two versions of heterotic string theory are also related by T-duality.[25]
In general, the term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way. Two theories related by a duality need not be string theories. For example, Montonen–Olive duality is an example of an S-duality relationship between quantum field theories. The AdS/CFT correspondence is an example of a duality that relates string theory to a quantum field theory. If two theories are related by a duality, it means that one theory can be transformed in some way so that it ends up looking just like the other theory. The two theories are then said to be dual to one another under the transformation. Put differently, the two theories are mathematically different descriptions of the same phenomena.[26]
Branes
Main article: Brane
In string theory and other related theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. For instance, a point particle can be viewed as a brane of dimension zero, while a string can be viewed as a brane of dimension one. It is also possible to consider higher-dimensional branes. In dimension p, these are called p-branes. The word brane comes from the word "membrane" which refers to a two-dimensional brane.[27]
Branes are dynamical objects which can propagate through spacetime according to the rules of quantum mechanics. They have mass and can have other attributes such as charge. A p-brane sweeps out a (p+1)-dimensional volume in spacetime called its worldvolume. Physicists often study fields analogous to the electromagnetic field which live on the worldvolume of a brane.[27]
In string theory, D-branes are an important class of branes that arise when one considers open strings. As an open string propagates through spacetime, its endpoints are required to lie on a D-brane. The letter "D" in D-brane refers to a certain mathematical condition on the system known as the Dirichlet boundary condition. The study of D-branes in string theory has led to important results such as the AdS/CFT correspondence, which has shed light on many problems in quantum field theory.[27]
Branes are frequently studied from a purely mathematical point of view, and they are described as objects of certain categories, such as the derived category of coherent sheaves on a complex algebraic variety, or the Fukaya category of a symplectic manifold.[28] The connection between the physical notion of a brane and the mathematical notion of a category has led to important mathematical insights in the fields of algebraic and symplectic geometry[29] and representation theory.[30]
M-theory
Prior to 1995, theorists believed that there were five consistent versions of superstring theory (type I, type IIA, type IIB, and two versions of heterotic string theory). This understanding changed in 1995 when Edward Witten suggested that the five theories were just special limiting cases of an eleven-dimensional theory called M-theory. Witten's conjecture was based on the work of a number of other physicists, including Ashoke Sen, Chris Hull, Paul Townsend, and Michael Duff. His announcement led to a flurry of research activity now known as the second superstring revolution.[31]
Unification of superstring theories
In the 1970s, many physicists became interested in supergravity theories, which combine general relativity with supersymmetry. Whereas general relativity makes sense in any number of dimensions, supergravity places an upper limit on the number of dimensions.[32] In 1978, work by Werner Nahm showed that the maximum spacetime dimension in which one can formulate a consistent supersymmetric theory is eleven.[33] In the same year, Eugene Cremmer, Bernard Julia, and Joël Scherk of the École Normale Supérieure showed that supergravity not only permits up to eleven dimensions but is in fact most elegant in this maximal number of dimensions.[34][35]
Initially, many physicists hoped that by compactifying eleven-dimensional supergravity, it might be possible to construct realistic models of our four-dimensional world. The hope was that such models would provide a unified description of the four fundamental forces of nature: electromagnetism, the strong and weak nuclear forces, and gravity. Interest in eleven-dimensional supergravity soon waned as various flaws in this scheme were discovered. One of the problems was that the laws of physics appear to distinguish between clockwise and counterclockwise, a phenomenon known as chirality. Edward Witten and others observed this chirality property cannot be readily derived by compactifying from eleven dimensions.[35]
In the first superstring revolution in 1984, many physicists turned to string theory as a unified theory of particle physics and quantum gravity. Unlike supergravity theory, string theory was able to accommodate the chirality of the standard model, and it provided a theory of gravity consistent with quantum effects.[35] Another feature of string theory that many physicists were drawn to in the 1980s and 1990s was its high degree of uniqueness. In ordinary particle theories, one can consider any collection of elementary particles whose classical behavior is described by an arbitrary Lagrangian. In string theory, the possibilities are much more constrained: by the 1990s, physicists had argued that there were only five consistent supersymmetric versions of the theory.[35]
Although there were only a handful of consistent superstring theories, it remained a mystery why there was not just one consistent formulation.[35] However, as physicists began to examine string theory more closely, they realized that these theories are related in intricate and nontrivial ways. They found that a system of strongly interacting strings can, in some cases, be viewed as a system of weakly interacting strings. This phenomenon is known as S-duality. It was studied by Ashoke Sen in the context of heterotic strings in four dimensions[36][37] and by Chris Hull and Paul Townsend in the context of the type IIB theory.[38] Theorists also found that different string theories may be related by T-duality. This duality implies that strings propagating on completely different spacetime geometries may be physically equivalent.[39]
At around the same time, as many physicists were studying the properties of strings, a small group of physicists were examining the possible applications of higher dimensional objects. In 1987, Eric Bergshoeff, Ergin Sezgin, and Paul Townsend showed that eleven-dimensional supergravity includes two-dimensional branes.[40] Intuitively, these objects look like sheets or membranes propagating through the eleven-dimensional spacetime. Shortly after this discovery, Michael Duff, Paul Howe, Takeo Inami, and Kellogg Stelle considered a particular compactification of eleven-dimensional supergravity with one of the dimensions curled up into a circle.[41] In this setting, one can imagine the membrane wrapping around the circular dimension. If the radius of the circle is sufficiently small, then this membrane looks just like a string in ten-dimensional spacetime. Duff and his collaborators showed that this construction reproduces exactly the strings appearing in type IIA superstring theory.[42]
Speaking at a string theory conference in 1995, Edward Witten made the surprising suggestion that all five superstring theories were in fact just different limiting cases of a single theory in eleven spacetime dimensions. Witten's announcement drew together all of the previous results on S- and T-duality and the appearance of higher-dimensional branes in string theory.[43] In the months following Witten's announcement, hundreds of new papers appeared on the Internet confirming different parts of his proposal.[44] Today this flurry of work is known as the second superstring revolution.[45]
Initially, some physicists suggested that the new theory was a fundamental theory of membranes, but Witten was skeptical of the role of membranes in the theory. In a paper from 1996, Hořava and Witten wrote "As it has been proposed that the eleven-dimensional theory is a supermembrane theory but there are some reasons to doubt that interpretation, we will non-committally call it the M-theory, leaving to the future the relation of M to membranes."[46] In the absence of an understanding of the true meaning and structure of M-theory, Witten has suggested that the M should stand for "magic", "mystery", or "membrane" according to taste, and the true meaning of the title should be decided when a more fundamental formulation of the theory is known.[47]
Matrix theory
In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mechanics.[48]
One important example of a matrix model is the BFSS matrix model proposed by Tom Banks, Willy Fischler, Stephen Shenker, and Leonard Susskind in 1997. This theory describes the behavior of a set of nine large matrices. In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototype for a correct formulation of M-theory and a tool for investigating the properties of M-theory in a relatively simple setting.[48]
The development of the matrix model formulation of M-theory has led physicists to consider various connections between string theory and a branch of mathematics called noncommutative geometry. This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra.[49] In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum field theory, a special kind of physical theory in which spacetime is described mathematically using noncommutative geometry.[50] This established a link between matrix models and M-theory on the one hand, and noncommutative geometry on the other hand. It quickly led to the discovery of other important links between noncommutative geometry and various physical theories.[51][52]
Black holes
In general relativity, a black hole is defined as a region of spacetime in which the gravitational field is so strong that no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to arise when massive stars undergo gravitational collapse, and many galaxies are thought to contain supermassive black holes at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for theorists attempting to understand the quantum aspects of gravity. String theory has proved to be an important tool for investigating the theoretical properties of black holes because it provides a framework in which theorists can study their thermodynamics.[53]
Bekenstein–Hawking formula
In the branch of physics called statistical mechanics, entropy is a measure of the randomness or disorder of a physical system. This concept was studied in the 1870s by the Austrian physicist Ludwig Boltzmann, who showed that the thermodynamic properties of a gas could be derived from the combined properties of its many constituent molecules. Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise definition of entropy as the natural logarithm of the number of different states of the molecules (also called microstates) that give rise to the same macroscopic features.[54]
In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the 1970s, the physicist Jacob Bekenstein suggested that the entropy of a black hole is instead proportional to the surface area of its event horizon, the boundary beyond which matter and radiation are lost to its gravitational attraction.[55] When combined with ideas of the physicist Stephen Hawking,[56] Bekenstein's work yielded a precise formula for the entropy of a black hole. The Bekenstein–Hawking formula expresses the entropy S as
$S={\frac {c^{3}kA}{4\hbar G}}$
where c is the speed of light, k is the Boltzmann constant, ħ is the reduced Planck constant, G is Newton's constant, and A is the surface area of the event horizon.[57]
Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features. The Bekenstein–Hawking entropy formula gives the expected value of the entropy of a black hole, but by the 1990s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory.[58]
Derivation within string theory
In a paper from 1996, Andrew Strominger and Cumrun Vafa showed how to derive the Beckenstein–Hawking formula for certain black holes in string theory.[59] Their calculation was based on the observation that D-branes—which look like fluctuating membranes when they are weakly interacting—become dense, massive objects with event horizons when the interactions are strong. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and charge for the resulting black hole. Their calculation reproduced the Bekenstein–Hawking formula exactly, including the factor of 1/4.[60] Subsequent work by Strominger, Vafa, and others refined the original calculations and gave the precise values of the "quantum corrections" needed to describe very small black holes.[61][62]
The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference was that Strominger and Vafa considered only extremal black holes in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.[63] Strominger and Vafa also restricted attention to black holes in five-dimensional spacetime with unphysical supersymmetry.[64]
Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity. Indeed, in 1998, Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry.[65] In collaboration with several other authors in 2010, he showed that some results on black hole entropy could be extended to non-extremal astrophysical black holes.[66][67]
AdS/CFT correspondence
One approach to formulating string theory and studying its properties is provided by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. This is a theoretical result which implies that string theory is in some cases equivalent to a quantum field theory. In addition to providing insights into the mathematical structure of string theory, the AdS/CFT correspondence has shed light on many aspects of quantum field theory in regimes where traditional calculational techniques are ineffective.[6] The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997.[68] Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov,[69] and by Edward Witten.[70] By 2010, Maldacena's article had over 7000 citations, becoming the most highly cited article in the field of high energy physics.[lower-alpha 3]
Overview of the correspondence
In the AdS/CFT correspondence, the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called anti-de Sitter space.[6] In very elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points (the metric) is different from the notion of distance in ordinary Euclidean geometry. It is closely related to hyperbolic space, which can be viewed as a disk as illustrated on the left.[71] This image shows a tessellation of a disk by triangles and squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.[72]
One can imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensional anti-de Sitter space.[71] It looks like a solid cylinder in which any cross section is a copy of the hyperbolic disk. Time runs along the vertical direction in this picture. The surface of this cylinder plays an important role in the AdS/CFT correspondence. As with the hyperbolic plane, anti-de Sitter space is curved in such a way that any point in the interior is actually infinitely far from this boundary surface.[72]
This construction describes a hypothetical universe with only two space dimensions and one time dimension, but it can be generalized to any number of dimensions. Indeed, hyperbolic space can have more than two dimensions and one can "stack up" copies of hyperbolic space to get higher-dimensional models of anti-de Sitter space.[71]
An important feature of anti-de Sitter space is its boundary (which looks like a cylinder in the case of three-dimensional anti-de Sitter space). One property of this boundary is that, within a small region on the surface around any given point, it looks just like Minkowski space, the model of spacetime used in nongravitational physics.[73] One can therefore consider an auxiliary theory in which "spacetime" is given by the boundary of anti-de Sitter space. This observation is the starting point for AdS/CFT correspondence, which states that the boundary of anti-de Sitter space can be regarded as the "spacetime" for a quantum field theory. The claim is that this quantum field theory is equivalent to a gravitational theory, such as string theory, in the bulk anti-de Sitter space in the sense that there is a "dictionary" for translating entities and calculations in one theory into their counterparts in the other theory. For example, a single particle in the gravitational theory might correspond to some collection of particles in the boundary theory. In addition, the predictions in the two theories are quantitatively identical so that if two particles have a 40 percent chance of colliding in the gravitational theory, then the corresponding collections in the boundary theory would also have a 40 percent chance of colliding.[74]
Applications to quantum gravity
The discovery of the AdS/CFT correspondence was a major advance in physicists' understanding of string theory and quantum gravity. One reason for this is that the correspondence provides a formulation of string theory in terms of quantum field theory, which is well understood by comparison. Another reason is that it provides a general framework in which physicists can study and attempt to resolve the paradoxes of black holes.[53]
In 1975, Stephen Hawking published a calculation which suggested that black holes are not completely black but emit a dim radiation due to quantum effects near the event horizon.[56] At first, Hawking's result posed a problem for theorists because it suggested that black holes destroy information. More precisely, Hawking's calculation seemed to conflict with one of the basic postulates of quantum mechanics, which states that physical systems evolve in time according to the Schrödinger equation. This property is usually referred to as unitarity of time evolution. The apparent contradiction between Hawking's calculation and the unitarity postulate of quantum mechanics came to be known as the black hole information paradox.[75]
The AdS/CFT correspondence resolves the black hole information paradox, at least to some extent, because it shows how a black hole can evolve in a manner consistent with quantum mechanics in some contexts. Indeed, one can consider black holes in the context of the AdS/CFT correspondence, and any such black hole corresponds to a configuration of particles on the boundary of anti-de Sitter space.[76] These particles obey the usual rules of quantum mechanics and in particular evolve in a unitary fashion, so the black hole must also evolve in a unitary fashion, respecting the principles of quantum mechanics.[77] In 2005, Hawking announced that the paradox had been settled in favor of information conservation by the AdS/CFT correspondence, and he suggested a concrete mechanism by which black holes might preserve information.[78]
Applications to nuclear physics
In addition to its applications to theoretical problems in quantum gravity, the AdS/CFT correspondence has been applied to a variety of problems in quantum field theory. One physical system that has been studied using the AdS/CFT correspondence is the quark–gluon plasma, an exotic state of matter produced in particle accelerators. This state of matter arises for brief instants when heavy ions such as gold or lead nuclei are collided at high energies. Such collisions cause the quarks that make up atomic nuclei to deconfine at temperatures of approximately two trillion kelvin, conditions similar to those present at around 10−11 seconds after the Big Bang.[79]
The physics of the quark–gluon plasma is governed by a theory called quantum chromodynamics, but this theory is mathematically intractable in problems involving the quark–gluon plasma.[lower-alpha 4] In an article appearing in 2005, Đàm Thanh Sơn and his collaborators showed that the AdS/CFT correspondence could be used to understand some aspects of the quark-gluon plasma by describing it in the language of string theory.[80] By applying the AdS/CFT correspondence, Sơn and his collaborators were able to describe the quark-gluon plasma in terms of black holes in five-dimensional spacetime. The calculation showed that the ratio of two quantities associated with the quark-gluon plasma, the shear viscosity and volume density of entropy, should be approximately equal to a certain universal constant. In 2008, the predicted value of this ratio for the quark-gluon plasma was confirmed at the Relativistic Heavy Ion Collider at Brookhaven National Laboratory.[7][81]
Applications to condensed matter physics
The AdS/CFT correspondence has also been used to study aspects of condensed matter physics. Over the decades, experimental condensed matter physicists have discovered a number of exotic states of matter, including superconductors and superfluids. These states are described using the formalism of quantum field theory, but some phenomena are difficult to explain using standard field theoretic techniques. Some condensed matter theorists including Subir Sachdev hope that the AdS/CFT correspondence will make it possible to describe these systems in the language of string theory and learn more about their behavior.[7]
So far some success has been achieved in using string theory methods to describe the transition of a superfluid to an insulator. A superfluid is a system of electrically neutral atoms that flows without any friction. Such systems are often produced in the laboratory using liquid helium, but recently experimentalists have developed new ways of producing artificial superfluids by pouring trillions of cold atoms into a lattice of criss-crossing lasers. These atoms initially behave as a superfluid, but as experimentalists increase the intensity of the lasers, they become less mobile and then suddenly transition to an insulating state. During the transition, the atoms behave in an unusual way. For example, the atoms slow to a halt at a rate that depends on the temperature and on Planck's constant, the fundamental parameter of quantum mechanics, which does not enter into the description of the other phases. This behavior has recently been understood by considering a dual description where properties of the fluid are described in terms of a higher dimensional black hole.[8]
Phenomenology
In addition to being an idea of considerable theoretical interest, string theory provides a framework for constructing models of real-world physics that combine general relativity and particle physics. Phenomenology is the branch of theoretical physics in which physicists construct realistic models of nature from more abstract theoretical ideas. String phenomenology is the part of string theory that attempts to construct realistic or semi-realistic models based on string theory.
Partly because of theoretical and mathematical difficulties and partly because of the extremely high energies needed to test these theories experimentally, there is so far no experimental evidence that would unambiguously point to any of these models being a correct fundamental description of nature. This has led some in the community to criticize these approaches to unification and question the value of continued research on these problems.[12]
Particle physics
The currently accepted theory describing elementary particles and their interactions is known as the standard model of particle physics. This theory provides a unified description of three of the fundamental forces of nature: electromagnetism and the strong and weak nuclear forces. Despite its remarkable success in explaining a wide range of physical phenomena, the standard model cannot be a complete description of reality. This is because the standard model fails to incorporate the force of gravity and because of problems such as the hierarchy problem and the inability to explain the structure of fermion masses or dark matter.
String theory has been used to construct a variety of models of particle physics going beyond the standard model. Typically, such models are based on the idea of compactification. Starting with the ten- or eleven-dimensional spacetime of string or M-theory, physicists postulate a shape for the extra dimensions. By choosing this shape appropriately, they can construct models roughly similar to the standard model of particle physics, together with additional undiscovered particles.[82] One popular way of deriving realistic physics from string theory is to start with the heterotic theory in ten dimensions and assume that the six extra dimensions of spacetime are shaped like a six-dimensional Calabi–Yau manifold. Such compactifications offer many ways of extracting realistic physics from string theory. Other similar methods can be used to construct realistic or semi-realistic models of our four-dimensional world based on M-theory.[83]
Cosmology
The Big Bang theory is the prevailing cosmological model for the universe from the earliest known periods through its subsequent large-scale evolution. Despite its success in explaining many observed features of the universe including galactic redshifts, the relative abundance of light elements such as hydrogen and helium, and the existence of a cosmic microwave background, there are several questions that remain unanswered. For example, the standard Big Bang model does not explain why the universe appears to be the same in all directions, why it appears flat on very large distance scales, or why certain hypothesized particles such as magnetic monopoles are not observed in experiments.[84]
Currently, the leading candidate for a theory going beyond the Big Bang is the theory of cosmic inflation. Developed by Alan Guth and others in the 1980s, inflation postulates a period of extremely rapid accelerated expansion of the universe prior to the expansion described by the standard Big Bang theory. The theory of cosmic inflation preserves the successes of the Big Bang while providing a natural explanation for some of the mysterious features of the universe.[85] The theory has also received striking support from observations of the cosmic microwave background, the radiation that has filled the sky since around 380,000 years after the Big Bang.[86]
In the theory of inflation, the rapid initial expansion of the universe is caused by a hypothetical particle called the inflaton. The exact properties of this particle are not fixed by the theory but should ultimately be derived from a more fundamental theory such as string theory.[87] Indeed, there have been a number of attempts to identify an inflaton within the spectrum of particles described by string theory and to study inflation using string theory. While these approaches might eventually find support in observational data such as measurements of the cosmic microwave background, the application of string theory to cosmology is still in its early stages.[88]
Connections to mathematics
In addition to influencing research in theoretical physics, string theory has stimulated a number of major developments in pure mathematics. Like many developing ideas in theoretical physics, string theory does not at present have a mathematically rigorous formulation in which all of its concepts can be defined precisely. As a result, physicists who study string theory are often guided by physical intuition to conjecture relationships between the seemingly different mathematical structures that are used to formalize different parts of the theory. These conjectures are later proved by mathematicians, and in this way, string theory serves as a source of new ideas in pure mathematics.[89]
Mirror symmetry
Main article: Mirror symmetry (string theory)
After Calabi–Yau manifolds had entered physics as a way to compactify extra dimensions in string theory, many physicists began studying these manifolds. In the late 1980s, several physicists noticed that given such a compactification of string theory, it is not possible to reconstruct uniquely a corresponding Calabi–Yau manifold.[90] Instead, two different versions of string theory, type IIA and type IIB, can be compactified on completely different Calabi–Yau manifolds giving rise to the same physics. In this situation, the manifolds are called mirror manifolds, and the relationship between the two physical theories is called mirror symmetry.[28]
Regardless of whether Calabi–Yau compactifications of string theory provide a correct description of nature, the existence of the mirror duality between different string theories has significant mathematical consequences. The Calabi–Yau manifolds used in string theory are of interest in pure mathematics, and mirror symmetry allows mathematicians to solve problems in enumerative geometry, a branch of mathematics concerned with counting the numbers of solutions to geometric questions.[28][91]
Enumerative geometry studies a class of geometric objects called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety defined using a certain polynomial of degree three in four variables. A celebrated result of nineteenth-century mathematicians Arthur Cayley and George Salmon states that there are exactly 27 straight lines that lie entirely on such a surface.[92]
Generalizing this problem, one can ask how many lines can be drawn on a quintic Calabi–Yau manifold, such as the one illustrated above, which is defined by a polynomial of degree five. This problem was solved by the nineteenth-century German mathematician Hermann Schubert, who found that there are exactly 2,875 such lines. In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250.[93]
By the year 1991, most of the classical problems of enumerative geometry had been solved and interest in enumerative geometry had begun to diminish.[94] The field was reinvigorated in May 1991 when physicists Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parks showed that mirror symmetry could be used to translate difficult mathematical questions about one Calabi–Yau manifold into easier questions about its mirror.[95] In particular, they used mirror symmetry to show that a six-dimensional Calabi–Yau manifold can contain exactly 317,206,375 curves of degree three.[94] In addition to counting degree-three curves, Candelas and his collaborators obtained a number of more general results for counting rational curves which went far beyond the results obtained by mathematicians.[96]
Originally, these results of Candelas were justified on physical grounds. However, mathematicians generally prefer rigorous proofs that do not require an appeal to physical intuition. Inspired by physicists' work on mirror symmetry, mathematicians have therefore constructed their own arguments proving the enumerative predictions of mirror symmetry.[lower-alpha 5] Today mirror symmetry is an active area of research in mathematics, and mathematicians are working to develop a more complete mathematical understanding of mirror symmetry based on physicists' intuition.[102] Major approaches to mirror symmetry include the homological mirror symmetry program of Maxim Kontsevich[29] and the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow.[103]
Monstrous moonshine
Main article: Monstrous moonshine
Group theory is the branch of mathematics that studies the concept of symmetry. For example, one can consider a geometric shape such as an equilateral triangle. There are various operations that one can perform on this triangle without changing its shape. One can rotate it through 120°, 240°, or 360°, or one can reflect in any of the lines labeled S0, S1, or S2 in the picture. Each of these operations is called a symmetry, and the collection of these symmetries satisfies certain technical properties making it into what mathematicians call a group. In this particular example, the group is known as the dihedral group of order 6 because it has six elements. A general group may describe finitely many or infinitely many symmetries; if there are only finitely many symmetries, it is called a finite group.[104]
Mathematicians often strive for a classification (or list) of all mathematical objects of a given type. It is generally believed that finite groups are too diverse to admit a useful classification. A more modest but still challenging problem is to classify all finite simple groups. These are finite groups that may be used as building blocks for constructing arbitrary finite groups in the same way that prime numbers can be used to construct arbitrary whole numbers by taking products.[lower-alpha 6] One of the major achievements of contemporary group theory is the classification of finite simple groups, a mathematical theorem that provides a list of all possible finite simple groups.[104]
This classification theorem identifies several infinite families of groups as well as 26 additional groups which do not fit into any family. The latter groups are called the "sporadic" groups, and each one owes its existence to a remarkable combination of circumstances. The largest sporadic group, the so-called monster group, has over 1053 elements, more than a thousand times the number of atoms in the Earth.[105]
A seemingly unrelated construction is the j-function of number theory. This object belongs to a special class of functions called modular functions, whose graphs form a certain kind of repeating pattern.[106] Although this function appears in a branch of mathematics that seems very different from the theory of finite groups, the two subjects turn out to be intimately related. In the late 1970s, mathematicians John McKay and John Thompson noticed that certain numbers arising in the analysis of the monster group (namely, the dimensions of its irreducible representations) are related to numbers that appear in a formula for the j-function (namely, the coefficients of its Fourier series).[107] This relationship was further developed by John Horton Conway and Simon Norton[108] who called it monstrous moonshine because it seemed so far fetched.[109]
In 1992, Richard Borcherds constructed a bridge between the theory of modular functions and finite groups and, in the process, explained the observations of McKay and Thompson.[110][111] Borcherds' work used ideas from string theory in an essential way, extending earlier results of Igor Frenkel, James Lepowsky, and Arne Meurman, who had realized the monster group as the symmetries of a particular version of string theory.[112] In 1998, Borcherds was awarded the Fields medal for his work.[113]
Since the 1990s, the connection between string theory and moonshine has led to further results in mathematics and physics.[105] In 2010, physicists Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa discovered connections between a different sporadic group, the Mathieu group M24, and a certain version of string theory.[114] Miranda Cheng, John Duncan, and Jeffrey A. Harvey proposed a generalization of this moonshine phenomenon called umbral moonshine,[115] and their conjecture was proved mathematically by Duncan, Michael Griffin, and Ken Ono.[116] Witten has also speculated that the version of string theory appearing in monstrous moonshine might be related to a certain simplified model of gravity in three spacetime dimensions.[117]
History
Early results
Some of the structures reintroduced by string theory arose for the first time much earlier as part of the program of classical unification started by Albert Einstein. The first person to add a fifth dimension to a theory of gravity was Gunnar Nordström in 1914, who noted that gravity in five dimensions describes both gravity and electromagnetism in four. Nordström attempted to unify electromagnetism with his theory of gravitation, which was however superseded by Einstein's general relativity in 1919. Thereafter, German mathematician Theodor Kaluza combined the fifth dimension with general relativity, and only Kaluza is usually credited with the idea. In 1926, the Swedish physicist Oskar Klein gave a physical interpretation of the unobservable extra dimension—it is wrapped into a small circle. Einstein introduced a non-symmetric metric tensor, while much later Brans and Dicke added a scalar component to gravity. These ideas would be revived within string theory, where they are demanded by consistency conditions.
String theory was originally developed during the late 1960s and early 1970s as a never completely successful theory of hadrons, the subatomic particles like the proton and neutron that feel the strong interaction. In the 1960s, Geoffrey Chew and Steven Frautschi discovered that the mesons make families called Regge trajectories with masses related to spins in a way that was later understood by Yoichiro Nambu, Holger Bech Nielsen and Leonard Susskind to be the relationship expected from rotating strings. Chew advocated making a theory for the interactions of these trajectories that did not presume that they were composed of any fundamental particles, but would construct their interactions from self-consistency conditions on the S-matrix. The S-matrix approach was started by Werner Heisenberg in the 1940s as a way of constructing a theory that did not rely on the local notions of space and time, which Heisenberg believed break down at the nuclear scale. While the scale was off by many orders of magnitude, the approach he advocated was ideally suited for a theory of quantum gravity.
Working with experimental data, R. Dolen, D. Horn and C. Schmid developed some sum rules for hadron exchange. When a particle and antiparticle scatter, virtual particles can be exchanged in two qualitatively different ways. In the s-channel, the two particles annihilate to make temporary intermediate states that fall apart into the final state particles. In the t-channel, the particles exchange intermediate states by emission and absorption. In field theory, the two contributions add together, one giving a continuous background contribution, the other giving peaks at certain energies. In the data, it was clear that the peaks were stealing from the background—the authors interpreted this as saying that the t-channel contribution was dual to the s-channel one, meaning both described the whole amplitude and included the other.
The result was widely advertised by Murray Gell-Mann, leading Gabriele Veneziano to construct a scattering amplitude that had the property of Dolen–Horn–Schmid duality, later renamed world-sheet duality. The amplitude needed poles where the particles appear, on straight-line trajectories, and there is a special mathematical function whose poles are evenly spaced on half the real line—the gamma function— which was widely used in Regge theory. By manipulating combinations of gamma functions, Veneziano was able to find a consistent scattering amplitude with poles on straight lines, with mostly positive residues, which obeyed duality and had the appropriate Regge scaling at high energy. The amplitude could fit near-beam scattering data as well as other Regge type fits and had a suggestive integral representation that could be used for generalization.
Over the next years, hundreds of physicists worked to complete the bootstrap program for this model, with many surprises. Veneziano himself discovered that for the scattering amplitude to describe the scattering of a particle that appears in the theory, an obvious self-consistency condition, the lightest particle must be a tachyon. Miguel Virasoro and Joel Shapiro found a different amplitude now understood to be that of closed strings, while Ziro Koba and Holger Nielsen generalized Veneziano's integral representation to multiparticle scattering. Veneziano and Sergio Fubini introduced an operator formalism for computing the scattering amplitudes that was a forerunner of world-sheet conformal theory, while Virasoro understood how to remove the poles with wrong-sign residues using a constraint on the states. Claud Lovelace calculated a loop amplitude, and noted that there is an inconsistency unless the dimension of the theory is 26. Charles Thorn, Peter Goddard and Richard Brower went on to prove that there are no wrong-sign propagating states in dimensions less than or equal to 26.
In 1969–70, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind recognized that the theory could be given a description in space and time in terms of strings. The scattering amplitudes were derived systematically from the action principle by Peter Goddard, Jeffrey Goldstone, Claudio Rebbi, and Charles Thorn, giving a space-time picture to the vertex operators introduced by Veneziano and Fubini and a geometrical interpretation to the Virasoro conditions.
In 1971, Pierre Ramond added fermions to the model, which led him to formulate a two-dimensional supersymmetry to cancel the wrong-sign states. John Schwarz and André Neveu added another sector to the fermi theory a short time later. In the fermion theories, the critical dimension was 10. Stanley Mandelstam formulated a world sheet conformal theory for both the bose and fermi case, giving a two-dimensional field theoretic path-integral to generate the operator formalism. Michio Kaku and Keiji Kikkawa gave a different formulation of the bosonic string, as a string field theory, with infinitely many particle types and with fields taking values not on points, but on loops and curves.
In 1974, Tamiaki Yoneya discovered that all the known string theories included a massless spin-two particle that obeyed the correct Ward identities to be a graviton. John Schwarz and Joël Scherk came to the same conclusion and made the bold leap to suggest that string theory was a theory of gravity, not a theory of hadrons. They reintroduced Kaluza–Klein theory as a way of making sense of the extra dimensions. At the same time, quantum chromodynamics was recognized as the correct theory of hadrons, shifting the attention of physicists and apparently leaving the bootstrap program in the dustbin of history.
String theory eventually made it out of the dustbin, but for the following decade, all work on the theory was completely ignored. Still, the theory continued to develop at a steady pace thanks to the work of a handful of devotees. Ferdinando Gliozzi, Joël Scherk, and David Olive realized in 1977 that the original Ramond and Neveu Schwarz-strings were separately inconsistent and needed to be combined. The resulting theory did not have a tachyon and was proven to have space-time supersymmetry by John Schwarz and Michael Green in 1984. The same year, Alexander Polyakov gave the theory a modern path integral formulation, and went on to develop conformal field theory extensively. In 1979, Daniel Friedan showed that the equations of motions of string theory, which are generalizations of the Einstein equations of general relativity, emerge from the renormalization group equations for the two-dimensional field theory. Schwarz and Green discovered T-duality, and constructed two superstring theories—IIA and IIB related by T-duality, and type I theories with open strings. The consistency conditions had been so strong, that the entire theory was nearly uniquely determined, with only a few discrete choices.
First superstring revolution
In the early 1980s, Edward Witten discovered that most theories of quantum gravity could not accommodate chiral fermions like the neutrino. This led him, in collaboration with Luis Álvarez-Gaumé, to study violations of the conservation laws in gravity theories with anomalies, concluding that type I string theories were inconsistent. Green and Schwarz discovered a contribution to the anomaly that Witten and Alvarez-Gaumé had missed, which restricted the gauge group of the type I string theory to be SO(32). In coming to understand this calculation, Edward Witten became convinced that string theory was truly a consistent theory of gravity, and he became a high-profile advocate. Following Witten's lead, between 1984 and 1986, hundreds of physicists started to work in this field, and this is sometimes called the first superstring revolution.
During this period, David Gross, Jeffrey Harvey, Emil Martinec, and Ryan Rohm discovered heterotic strings. The gauge group of these closed strings was two copies of E8, and either copy could easily and naturally include the standard model. Philip Candelas, Gary Horowitz, Andrew Strominger and Edward Witten found that the Calabi–Yau manifolds are the compactifications that preserve a realistic amount of supersymmetry, while Lance Dixon and others worked out the physical properties of orbifolds, distinctive geometrical singularities allowed in string theory. Cumrun Vafa generalized T-duality from circles to arbitrary manifolds, creating the mathematical field of mirror symmetry. Daniel Friedan, Emil Martinec and Stephen Shenker further developed the covariant quantization of the superstring using conformal field theory techniques. David Gross and Vipul Periwal discovered that string perturbation theory was divergent. Stephen Shenker showed it diverged much faster than in field theory suggesting that new non-perturbative objects were missing.
In the 1990s, Joseph Polchinski discovered that the theory requires higher-dimensional objects, called D-branes and identified these with the black-hole solutions of supergravity. These were understood to be the new objects suggested by the perturbative divergences, and they opened up a new field with rich mathematical structure. It quickly became clear that D-branes and other p-branes, not just strings, formed the matter content of the string theories, and the physical interpretation of the strings and branes was revealed—they are a type of black hole. Leonard Susskind had incorporated the holographic principle of Gerardus 't Hooft into string theory, identifying the long highly excited string states with ordinary thermal black hole states. As suggested by 't Hooft, the fluctuations of the black hole horizon, the world-sheet or world-volume theory, describes not only the degrees of freedom of the black hole, but all nearby objects too.
Second superstring revolution
In 1995, at the annual conference of string theorists at the University of Southern California (USC), Edward Witten gave a speech on string theory that in essence united the five string theories that existed at the time, and giving birth to a new 11-dimensional theory called M-theory. M-theory was also foreshadowed in the work of Paul Townsend at approximately the same time. The flurry of activity that began at this time is sometimes called the second superstring revolution.[31]
During this period, Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated matrix theory, a full holographic description of M-theory using IIA D0 branes.[48] This was the first definition of string theory that was fully non-perturbative and a concrete mathematical realization of the holographic principle. It is an example of a gauge-gravity duality and is now understood to be a special case of the AdS/CFT correspondence. Andrew Strominger and Cumrun Vafa calculated the entropy of certain configurations of D-branes and found agreement with the semi-classical answer for extreme charged black holes.[59] Petr Hořava and Witten found the eleven-dimensional formulation of the heterotic string theories, showing that orbifolds solve the chirality problem. Witten noted that the effective description of the physics of D-branes at low energies is by a supersymmetric gauge theory, and found geometrical interpretations of mathematical structures in gauge theory that he and Nathan Seiberg had earlier discovered in terms of the location of the branes.
In 1997, Juan Maldacena noted that the low energy excitations of a theory near a black hole consist of objects close to the horizon, which for extreme charged black holes looks like an anti-de Sitter space.[68] He noted that in this limit the gauge theory describes the string excitations near the branes. So he hypothesized that string theory on a near-horizon extreme-charged black-hole geometry, an anti-de Sitter space times a sphere with flux, is equally well described by the low-energy limiting gauge theory, the N = 4 supersymmetric Yang–Mills theory. This hypothesis, which is called the AdS/CFT correspondence, was further developed by Steven Gubser, Igor Klebanov and Alexander Polyakov,[69] and by Edward Witten,[70] and it is now well-accepted. It is a concrete realization of the holographic principle, which has far-reaching implications for black holes, locality and information in physics, as well as the nature of the gravitational interaction.[53] Through this relationship, string theory has been shown to be related to gauge theories like quantum chromodynamics and this has led to a more quantitative understanding of the behavior of hadrons, bringing string theory back to its roots.
Criticism
Number of solutions
To construct models of particle physics based on string theory, physicists typically begin by specifying a shape for the extra dimensions of spacetime. Each of these different shapes corresponds to a different possible universe, or "vacuum state", with a different collection of particles and forces. String theory as it is currently understood has an enormous number of vacuum states, typically estimated to be around 10500, and these might be sufficiently diverse to accommodate almost any phenomenon that might be observed at low energies.[118]
Many critics of string theory have expressed concerns about the large number of possible universes described by string theory. In his book Not Even Wrong, Peter Woit, a lecturer in the mathematics department at Columbia University, has argued that the large number of different physical scenarios renders string theory vacuous as a framework for constructing models of particle physics. According to Woit,
The possible existence of, say, 10500 consistent different vacuum states for superstring theory probably destroys the hope of using the theory to predict anything. If one picks among this large set just those states whose properties agree with present experimental observations, it is likely there still will be such a large number of these that one can get just about whatever value one wants for the results of any new observation.[119]
Some physicists believe this large number of solutions is actually a virtue because it may allow a natural anthropic explanation of the observed values of physical constants, in particular the small value of the cosmological constant.[119] The anthropic principle is the idea that some of the numbers appearing in the laws of physics are not fixed by any fundamental principle but must be compatible with the evolution of intelligent life. In 1987, Steven Weinberg published an article in which he argued that the cosmological constant could not have been too large, or else galaxies and intelligent life would not have been able to develop.[120] Weinberg suggested that there might be a huge number of possible consistent universes, each with a different value of the cosmological constant, and observations indicate a small value of the cosmological constant only because humans happen to live in a universe that has allowed intelligent life, and hence observers, to exist.[121]
String theorist Leonard Susskind has argued that string theory provides a natural anthropic explanation of the small value of the cosmological constant.[122] According to Susskind, the different vacuum states of string theory might be realized as different universes within a larger multiverse. The fact that the observed universe has a small cosmological constant is just a tautological consequence of the fact that a small value is required for life to exist.[123] Many prominent theorists and critics have disagreed with Susskind's conclusions.[124] According to Woit, "in this case [anthropic reasoning] is nothing more than an excuse for failure. Speculative scientific ideas fail not just when they make incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything."[125]
Compatibility with dark energy
It remains unknown whether string theory is compatible with a metastable, positive cosmological constant. Some putative examples of such solutions do exist, such as the model described by Kachru et al. in 2003.[126] In 2018, a group of four physicists advanced a controversial conjecture which would imply that no such universe exists. This is contrary to some popular models of dark energy such as Λ-CDM, which requires a positive vacuum energy. However, string theory is likely compatible with certain types of quintessence, where dark energy is caused by a new field with exotic properties.[127]
Background independence
One of the fundamental properties of Einstein's general theory of relativity is that it is background independent, meaning that the formulation of the theory does not in any way privilege a particular spacetime geometry.[128]
One of the main criticisms of string theory from early on is that it is not manifestly background-independent. In string theory, one must typically specify a fixed reference geometry for spacetime, and all other possible geometries are described as perturbations of this fixed one. In his book The Trouble With Physics, physicist Lee Smolin of the Perimeter Institute for Theoretical Physics claims that this is the principal weakness of string theory as a theory of quantum gravity, saying that string theory has failed to incorporate this important insight from general relativity.[129]
Others have disagreed with Smolin's characterization of string theory. In a review of Smolin's book, string theorist Joseph Polchinski writes
[Smolin] is mistaking an aspect of the mathematical language being used for one of the physics being described. New physical theories are often discovered using a mathematical language that is not the most suitable for them... In string theory, it has always been clear that the physics is background-independent even if the language being used is not, and the search for a more suitable language continues. Indeed, as Smolin belatedly notes, [AdS/CFT] provides a solution to this problem, one that is unexpected and powerful.[130]
Polchinski notes that an important open problem in quantum gravity is to develop holographic descriptions of gravity which do not require the gravitational field to be asymptotically anti-de Sitter.[130] Smolin has responded by saying that the AdS/CFT correspondence, as it is currently understood, may not be strong enough to resolve all concerns about background independence.[131]
Sociology of science
Since the superstring revolutions of the 1980s and 1990s, string theory has been one of dominant paradigms of high energy theoretical physics.[132] Some string theorists have expressed the view that there does not exist an equally successful alternative theory addressing the deep questions of fundamental physics. In an interview from 1987, Nobel laureate David Gross made the following controversial comments about the reasons for the popularity of string theory:
The most important [reason] is that there are no other good ideas around. That's what gets most people into it. When people started to get interested in string theory they didn't know anything about it. In fact, the first reaction of most people is that the theory is extremely ugly and unpleasant, at least that was the case a few years ago when the understanding of string theory was much less developed. It was difficult for people to learn about it and to be turned on. So I think the real reason why people have got attracted by it is because there is no other game in town. All other approaches of constructing grand unified theories, which were more conservative to begin with, and only gradually became more and more radical, have failed, and this game hasn't failed yet.[133]
Several other high-profile theorists and commentators have expressed similar views, suggesting that there are no viable alternatives to string theory.[134]
Many critics of string theory have commented on this state of affairs. In his book criticizing string theory, Peter Woit views the status of string theory research as unhealthy and detrimental to the future of fundamental physics. He argues that the extreme popularity of string theory among theoretical physicists is partly a consequence of the financial structure of academia and the fierce competition for scarce resources.[135] In his book The Road to Reality, mathematical physicist Roger Penrose expresses similar views, stating "The often frantic competitiveness that this ease of communication engenders leads to bandwagon effects, where researchers fear to be left behind if they do not join in."[136] Penrose also claims that the technical difficulty of modern physics forces young scientists to rely on the preferences of established researchers, rather than forging new paths of their own.[137] Lee Smolin expresses a slightly different position in his critique, claiming that string theory grew out of a tradition of particle physics which discourages speculation about the foundations of physics, while his preferred approach, loop quantum gravity, encourages more radical thinking. According to Smolin,
String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted to it. If it has so far failed, the principal reason is that its intrinsic flaws are closely tied to its strengths—and, of course, the story is unfinished, since string theory may well turn out to be part of the truth. The real question is not why we have expended so much energy on string theory but why we haven't expended nearly enough on alternative approaches.[138]
Smolin goes on to offer a number of prescriptions for how scientists might encourage a greater diversity of approaches to quantum gravity research.[139]
Notes
1. For example, physicists are still working to understand the phenomenon of quark confinement, the paradoxes of black holes, and the origin of dark energy.
2. For example, in the context of the AdS/CFT correspondence, theorists often formulate and study theories of gravity in unphysical numbers of spacetime dimensions.
3. "Top Cited Articles during 2010 in hep-th". Retrieved 25 July 2013.
4. More precisely, one cannot apply the methods of perturbative quantum field theory.
5. Two independent mathematical proofs of mirror symmetry were given by Givental[97][98] and Lian et al.[99][100][101]
6. More precisely, a nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself. The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups.
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121. Woit, p. 243
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Bibliography
• Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String theory and M-theory: A modern introduction. Cambridge University Press. ISBN 978-0-521-86069-7.
• Duff, Michael (1998). "The theory formerly known as strings". Scientific American. 278 (2): 64–9. Bibcode:1998SciAm.278b..64D. doi:10.1038/scientificamerican0298-64.
• Gannon, Terry. Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms, and Physics. Cambridge University Press.
• Hori, Kentaro; Katz, Sheldon; Klemm, Albrecht; Pandharipande, Rahul; Thomas, Richard; Vafa, Cumrun; Vakil, Ravi; Zaslow, Eric, eds. (2003). Mirror Symmetry (PDF). Clay Mathematics Monographs. Vol. 1. American Mathematical Society. ISBN 978-0-8218-2955-4. Archived from the original (PDF) on 2006-09-19.
• Maldacena, Juan (2005). "The Illusion of Gravity" (PDF). Scientific American. 293 (5): 56–63. Bibcode:2005SciAm.293e..56M. doi:10.1038/scientificamerican1105-56. PMID 16318027. Archived from the original (PDF) on November 1, 2014. Retrieved 29 December 2016.
• Penrose, Roger (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf. ISBN 978-0-679-45443-4.
• Smolin, Lee (2006). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. New York: Houghton Mifflin Co. ISBN 978-0-618-55105-7.
• Wald, Robert (1984). General Relativity. University of Chicago Press. ISBN 978-0-226-87033-5.
• Woit, Peter (2006). Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. Basic Books. p. 105. ISBN 978-0-465-09275-8.
• Yau, Shing-Tung; Nadis, Steve (2010). The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. Basic Books. ISBN 978-0-465-02023-2.
• Zwiebach, Barton (2009). A First Course in String Theory. Cambridge University Press. ISBN 978-0-521-88032-9.
Further reading
Popular science
• Greene, Brian (2003). The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. New York: W.W. Norton & Company. ISBN 978-0-393-05858-1.
• Greene, Brian (2004). The Fabric of the Cosmos: Space, Time, and the Texture of Reality. New York: Alfred A. Knopf. Bibcode:2004fcst.book.....G. ISBN 978-0-375-41288-2.
• Penrose, Roger (2005). The Road to Reality: A Complete Guide to the Laws of the Universe. Knopf. ISBN 978-0-679-45443-4.
• Smolin, Lee (2006). The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next. New York: Houghton Mifflin Co. ISBN 978-0-618-55105-7.
• Woit, Peter (2006). Not Even Wrong: The Failure of String Theory And the Search for Unity in Physical Law. London: Jonathan Cape &: New York: Basic Books. ISBN 978-0-465-09275-8.
Textbooks
• Becker, K.; Becker, M.; Schwarz, J.H. (2006). String Theory and M-Theory: A Modern Introduction. Cambridge University Press. ISBN 978-0521860697.
• Blumenhagen, R.; Lüst, D.; Theisen, S. (2012). Basic Concepts of String Theory. Springer. ISBN 978-3642294969.
• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 1: Introduction. Cambridge University Press. ISBN 978-1107029118.
• Green, Michael; Schwarz, John; Witten, Edward (2012). Superstring theory. Vol. 2: Loop amplitudes, anomalies and phenomenology. Cambridge University Press. ISBN 978-1107029132.
• Ibáñez, L.E.; Uranga, A.M. (2012). String Theory and Particle Physics: An Introduction to String Phenomenology. Cambridge University Press. ISBN 978-0521517522.
• Kiritsis, E. (2019). String Theory in a Nutshell. Princeton University Press. ISBN 978-0691155791.
• Ortín, T. (2015). Gravity and Strings. Cambridge University Press. ISBN 978-0521768139.
• Polchinski, Joseph (1998). String Theory Vol. 1: An Introduction to the Bosonic String. Cambridge University Press. ISBN 978-0-521-63303-1.
• Polchinski, Joseph (1998). String Theory Vol. 2: Superstring Theory and Beyond. Cambridge University Press. ISBN 978-0-521-63304-8.
• West, P. (2012). Introduction to Strings and Branes. Cambridge University Press. ISBN 978-0521817479.
• Zwiebach, Barton (2009). A First Course in String Theory. Cambridge University Press. ISBN 978-0-521-88032-9.
External links
Look up string theory in Wiktionary, the free dictionary.
Wikimedia Commons has media related to String theory.
Wikiquote has quotations related to String theory.
Websites
• Not Even Wrong—A blog critical of string theory
• Why String Theory—An introduction to string theory.
Video
• bbc-horizon: parallel-uni — 2002 feature documentary by BBC Horizon, episode Parallel Universes focus on history and emergence of M-theory, and scientists involved.
• pbs.org-nova: elegant-uni — 2003 Emmy Award-winning, three-hour miniseries by Nova with Brian Greene, adapted from his The Elegant Universe (original PBS broadcast dates: October 28, 8–10 p.m. and November 4, 8–9 p.m., 2003).
String theory
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| Wikipedia |
110-vertex Iofinova–Ivanov graph
The 110-vertex Iofinova–Ivanov graph is, in graph theory, a semi-symmetric cubic graph with 110 vertices and 165 edges.
110-vertex Iofinova–Ivanov graph
Vertices110
Edges165
Radius7
Diameter7
Girth10
Automorphisms1320 (PGL2(11))
Chromatic number2
Chromatic index3
Propertiessemi-symmetric
bipartite
cubic
Hamiltonian
Table of graphs and parameters
Properties
Iofinova and Ivanov proved in 1985 the existence of five and only five semi-symmetric cubic bipartite graphs whose automorphism groups act primitively on each partition.[1] The smallest has 110 vertices. The others have 126, 182, 506 and 990.[2] The 126-vertex Iofinova–Ivanov graph is also known as the Tutte 12-cage.
The diameter of the 110-vertex Iofinova–Ivanov graph, the greatest distance between any pair of vertices, is 7. Its radius is likewise 7. Its girth is 10.
It is 3-connected and 3-edge-connected: to make it disconnected at least three edges, or at least three vertices, must be removed.
Coloring
The chromatic number of the 110-vertex Iofina-Ivanov graph is 2: its vertices can be 2-colored so that no two vertices of the same color are joined by an edge. Its chromatic index is 3: its edges can be 3-colored so that no two edges of the same color met at a vertex.
Algebraic properties
The characteristic polynomial of the 110-vertex Iofina-Ivanov graph is $(x-3)x^{20}(x+3)(x^{4}-8x^{2}+11)^{12}(x^{4}-6x^{2}+6)^{10}$. The symmetry group of the 110-vertex Iofina-Ivanov is the projective linear group PGL2(11), with 1320 elements.[3]
Semi-symmetry
Few graphs show semi-symmetry: most edge-transitive graphs are also vertex-transitive. The smallest semi-symmetric graph is the Folkman graph, with 20 vertices, which is 4-regular. The three smallest cubic semi-symmetric graphs are the Gray graph, with 54 vertices, this the smallest of the Iofina-Ivanov graphs with 110, and the Ljubljana graph with 112.[4][5] It is only for the five Iofina-Ivanov graphs that the symmetry group acts primitively on each partition of the vertices.
References
1. Han and Lu. "Affine primitive groups and Semisymmetric graphs". combinatorics.org. Retrieved 12 August 2015.
2. Weisstein, Eric. "Iofinova-Ivanov Graphs". Wolfram MathWorld. Wolfram. Retrieved 11 August 2015.
3. Iofinova and Ivanov (2013). Investigations in Algebraic Theory of Combinatorial Objects. Springer. p. 470. ISBN 9789401719728. Retrieved 12 August 2015.
4. Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), "The Ljubljana Graph" (PDF), IMFM Preprints, Ljubljana: Institute of Mathematics, Physics and Mechanics, vol. 40, no. 845
5. Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), "A census of semisymmetric cubic graphs on up to 768 vertices", Journal of Algebraic Combinatorics, 23 (3): 255–294, doi:10.1007/s10801-006-7397-3.
Bibliography
• Iofinova, M. E. and Ivanov, A. A. Bi-Primitive Cubic Graphs. In Investigations in the Algebraic Theory of Combinatorial Objects. pp. 123–134, 2002. (Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, pp. 137–152, 1985.)
• Ivanov, A. A. Computation of Lengths of Orbits of a Subgroup in a Transitive Permutation Group. In Methods for Complex System Studies. Moscow: VNIISI, pp. 3–7, 1983.
• Ivanov, A. V. On Edge But Not Vertex Transitive Regular Graphs. In Combinatorial Design Theory (Ed. C. J. Colbourn and R. Mathon). Amsterdam, Netherlands: North-Holland, pp. 273–285, 1987.
| Wikipedia |
Hendecagon
In geometry, a hendecagon (also undecagon[1][2] or endecagon[3]) or 11-gon is an eleven-sided polygon. (The name hendecagon, from Greek hendeka "eleven" and –gon "corner", is often preferred to the hybrid undecagon, whose first part is formed from Latin undecim "eleven".[4])
Regular hendecagon
A regular hendecagon
TypeRegular polygon
Edges and vertices11
Schläfli symbol{11}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D11), order 2×11
Internal angle (degrees)≈147.273°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular hendecagon
A regular hendecagon is represented by Schläfli symbol {11}.
A regular hendecagon has internal angles of 147.27 degrees (=147 ${\tfrac {3}{11}}$ degrees).[5] The area of a regular hendecagon with side length a is given by[2]
$A={\frac {11}{4}}a^{2}\cot {\frac {\pi }{11}}\simeq 9.36564\,a^{2}.$
As 11 is not a Fermat prime, the regular hendecagon is not constructible with compass and straightedge.[6] Because 11 is not a Pierpont prime, construction of a regular hendecagon is still impossible even with the usage of an angle trisector.
Close approximations to the regular hendecagon can be constructed. For instance, the ancient Greek mathematicians approximated the side length of a hendecagon inscribed in a unit circle as being 14/25 units long.[7]
The hendecagon can be constructed exactly via neusis construction[8] and also via two-fold origami.[9]
Approximate construction
Hendecagon inscribed in a circle, a continuation of the basic construction according to T. Drummond as animation.
Corresponds to the copper engraving by Anton Ernst Burkhard of Birckenstein.
Hendecagon, copper engraving by 1698 by Anton Ernst Burkhard of Birckenstein
The following construction description is given by T. Drummond from 1800:[10]
"Draw the radius A B, bisect it in C—with an opening of the compasses equal to half the radius, upon A and C as centres describe the arcs C D I and A D—with the distance I D upon I describe the arc D O and draw the line C O, which will be the extent of one side of a hendecagon sufficiently exact for practice."
On a unit circle:
• Constructed hendecagon side length $b=0.563692\ldots $
• Theoretical hendecagon side length $a=2\sin({\frac {\pi }{11}})=0.563465\ldots $
• Absolute error $\delta =b-a=2.27\ldots \cdot 10^{-4}$ – if AB is 10 m then this error is approximately 2.3 mm.
Symmetry
The regular hendecagon has Dih11 symmetry, order 22. Since 11 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z11, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the hendecagon. John Conway labels these by a letter and group order.[11] Full symmetry of the regular form is r22 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g11 subgroup has no degrees of freedom but can seen as directed edges.
Use in coinage
The Canadian dollar coin, the loonie, is similar to, but not exactly, a regular hendecagonal prism,[12] as are the Indian 2-rupee coin[13] and several other lesser-used coins of other nations.[14] The cross-section of a loonie is actually a Reuleaux hendecagon. The United States Susan B. Anthony dollar has a hendecagonal outline along the inside of its edges.[15]
Related figures
The hendecagon shares the same set of 11 vertices with four regular hendecagrams:
{11/2}
{11/3}
{11/4}
{11/5}
See also
• 10-simplex - can be seen as a complete graph in a regular hendecagonal orthogonal projection
References
1. Haldeman, Cyrus B. (1922), "Construction of the regular undecagon by a sextic curve", Discussions, American Mathematical Monthly, 29 (10), doi:10.2307/2299029, JSTOR 2299029.
2. Loomis, Elias (1859), Elements of Plane and Spherical Trigonometry: With Their Applications to Mensuration, Surveying, and Navigation, Harper, p. 65.
3. Brewer, Ebenezer Cobham (1877), Errors of speech and of spelling, London: W. Tegg and co., p. iv.
4. Hendecagon – from Wolfram MathWorld
5. McClain, Kay (1998), Glencoe mathematics: applications and connections, Glencoe/McGraw-Hill, p. 357, ISBN 9780028330549.
6. As Gauss proved, a polygon with a prime number p of sides can be constructed if and only if p − 1 is a power of two, which is not true for 11. See Kline, Morris (1990), Mathematical Thought From Ancient to Modern Times, vol. 2, Oxford University Press, pp. 753–754, ISBN 9780199840427.
7. Heath, Sir Thomas Little (1921), A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus, The Clarendon Press, p. 329.
8. Benjamin, Elliot; Snyder, C. Mathematical Proceedings of the Cambridge Philosophical Society156.3 (May 2014): 409-424.; https://dx.doi.org/10.1017/S0305004113000753
9. Lucero, J. C. (2018). "Construction of a regular hendecagon by two-fold origami". Crux Mathematicorum. 44: 207–213.
10. T. Drummond, (1800) The Young Ladies and Gentlemen's AUXILIARY, in Taking Heights and Distances ..., Construction description pp. 15–16 Fig. 40: scroll from page 69 ... to page 76 Part I. Second Edition, retrieved on 26 March 2016
11. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
12. Mossinghoff, Michael J. (2006), "A $1 problem" (PDF), American Mathematical Monthly, 113 (5): 385–402, doi:10.2307/27641947, JSTOR 27641947
13. Cuhaj, George S.; Michael, Thomas (2012), 2013 Standard Catalog of World Coins 2001 to Date, Krause Publications, p. 402, ISBN 9781440229657.
14. Cuhaj, George S.; Michael, Thomas (2011), Unusual World Coins (6th ed.), Krause Publications, pp. 23, 222, 233, 526, ISBN 9781440217128.
15. U.S. House of Representatives, 1978, p. 7.
Works cited
• United States House of Representatives (1978). Proposed Smaller One-Dollar Coin. Washington, D.C.: Government Printing Office.
External links
• Properties of an Undecagon (hendecagon) With interactive animation
• Weisstein, Eric W. "Hendecagon". MathWorld.
• Regular hendecagons
• Regular hendecagon, an approximate construction
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
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• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
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• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
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• Infinite skew
• Isogonal
• Isotoxal
• Magic
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| Wikipedia |
Demihypercube
In geometry, demihypercubes (also called n-demicubes, n-hemicubes, and half measure polytopes) are a class of n-polytopes constructed from alternation of an n-hypercube, labeled as hγn for being half of the hypercube family, γn. Half of the vertices are deleted and new facets are formed. The 2n facets become 2n (n−1)-demicubes, and 2n (n−1)-simplex facets are formed in place of the deleted vertices.[1]
Not to be confused with Hemicube (geometry).
They have been named with a demi- prefix to each hypercube name: demicube, demitesseract, etc. The demicube is identical to the regular tetrahedron, and the demitesseract is identical to the regular 16-cell. The demipenteract is considered semiregular for having only regular facets. Higher forms don't have all regular facets but are all uniform polytopes.
The vertices and edges of a demihypercube form two copies of the halved cube graph.
An n-demicube has inversion symmetry if n is even.
Discovery
Thorold Gosset described the demipenteract in his 1900 publication listing all of the regular and semiregular figures in n-dimensions above three. He called it a 5-ic semi-regular. It also exists within the semiregular k21 polytope family.
The demihypercubes can be represented by extended Schläfli symbols of the form h{4,3,...,3} as half the vertices of {4,3,...,3}. The vertex figures of demihypercubes are rectified n-simplexes.
Constructions
They are represented by Coxeter-Dynkin diagrams of three constructive forms:
1. ... (As an alternated orthotope) s{21,1,...,1}
2. ... (As an alternated hypercube) h{4,3n−1}
3. .... (As a demihypercube) {31,n−3,1}
H.S.M. Coxeter also labeled the third bifurcating diagrams as 1k1 representing the lengths of the three branches and led by the ringed branch.
An n-demicube, n greater than 2, has n(n−1)/2 edges meeting at each vertex. The graphs below show less edges at each vertex due to overlapping edges in the symmetry projection.
n 1k1 Petrie
polygon
Schläfli symbol Coxeter diagrams
A1n
Bn
Dn
Elements Facets:
Demihypercubes &
Simplexes
Vertex figure
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 8-faces 9-faces
2 1−1,1 demisquare
(digon)
s{2}
h{4}
{31,−1,1}
2 2
2 edges
--
3 101 demicube
(tetrahedron)
s{21,1}
h{4,3}
{31,0,1}
4 6 4 (6 digons)
4 triangles
Triangle
(Rectified triangle)
4 111 demitesseract
(16-cell)
s{21,1,1}
h{4,3,3}
{31,1,1}
8 24 32 16 8 demicubes
(tetrahedra)
8 tetrahedra
Octahedron
(Rectified tetrahedron)
5 121 demipenteract
s{21,1,1,1}
h{4,33}{31,2,1}
16 80 160 120 26 10 16-cells
16 5-cells
Rectified 5-cell
6 131 demihexeract
s{21,1,1,1,1}
h{4,34}{31,3,1}
32 240 640 640 252 44 12 demipenteracts
32 5-simplices
Rectified hexateron
7 141 demihepteract
s{21,1,1,1,1,1}
h{4,35}{31,4,1}
64 672 2240 2800 1624 532 78 14 demihexeracts
64 6-simplices
Rectified 6-simplex
8 151 demiocteract
s{21,1,1,1,1,1,1}
h{4,36}{31,5,1}
128 1792 7168 10752 8288 4032 1136 144 16 demihepteracts
128 7-simplices
Rectified 7-simplex
9 161 demienneract
s{21,1,1,1,1,1,1,1}
h{4,37}{31,6,1}
256 4608 21504 37632 36288 23520 9888 2448 274 18 demiocteracts
256 8-simplices
Rectified 8-simplex
10 171 demidekeract
s{21,1,1,1,1,1,1,1,1}
h{4,38}{31,7,1}
512 11520 61440 122880 142464 115584 64800 24000 5300 532 20 demienneracts
512 9-simplices
Rectified 9-simplex
...
n 1n−3,1 n-demicube s{21,1,...,1}
h{4,3n−2}{31,n−3,1}
...
...
...
2n−1 2n (n−1)-demicubes
2n−1 (n−1)-simplices
Rectified (n−1)-simplex
In general, a demicube's elements can be determined from the original n-cube: (with Cn,m = mth-face count in n-cube = 2n−m n!/(m!(n−m)!))
• Vertices: Dn,0 = 1/2 Cn,0 = 2n−1 (Half the n-cube vertices remain)
• Edges: Dn,1 = Cn,2 = 1/2 n(n−1) 2n−2 (All original edges lost, each square faces create a new edge)
• Faces: Dn,2 = 4 * Cn,3 = 2/3 n(n−1)(n−2) 2n−3 (All original faces lost, each cube creates 4 new triangular faces)
• Cells: Dn,3 = Cn,3 + 23 Cn,4 (tetrahedra from original cells plus new ones)
• Hypercells: Dn,4 = Cn,4 + 24 Cn,5 (16-cells and 5-cells respectively)
• ...
• [For m = 3,...,n−1]: Dn,m = Cn,m + 2m Cn,m+1 (m-demicubes and m-simplexes respectively)
• ...
• Facets: Dn,n−1 = 2n + 2n−1 ((n−1)-demicubes and (n−1)-simplices respectively)
Symmetry group
The stabilizer of the demihypercube in the hyperoctahedral group (the Coxeter group $BC_{n}$ [4,3n−1]) has index 2. It is the Coxeter group $D_{n},$ [3n−3,1,1] of order $2^{n-1}n!$, and is generated by permutations of the coordinate axes and reflections along pairs of coordinate axes.[2]
Orthotopic constructions
Constructions as alternated orthotopes have the same topology, but can be stretched with different lengths in n-axes of symmetry.
The rhombic disphenoid is the three-dimensional example as alternated cuboid. It has three sets of edge lengths, and scalene triangle faces.
See also
• Hypercube honeycomb
• Semiregular E-polytope
References
1. Regular and semi-regular polytopes III, p. 315-316
2. "week187". math.ucr.edu. Retrieved 20 April 2018.
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
External links
• Olshevsky, George. "Half measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
120-cell honeycomb
In the geometry of hyperbolic 4-space, the 120-cell honeycomb is one of five compact regular space-filling tessellations (or honeycombs). With Schläfli symbol {5,3,3,3}, it has three 120-cells around each face. Its dual is the order-5 5-cell honeycomb, {3,3,3,5}.
120-cell honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{5,3,3,3}
Coxeter diagram
4-faces {5,3,3}
Cells {5,3}
Faces {5}
Face figure {3}
Edge figure {3,3}
Vertex figure {3,3,3}
DualOrder-5 5-cell honeycomb
Coxeter groupH4, [5,3,3,3]
PropertiesRegular
Related honeycombs
It is related to the order-4 120-cell honeycomb, {5,3,3,4}, and order-5 120-cell honeycomb, {5,3,3,5}.
It is topologically similar to the finite 5-cube, {4,3,3,3}, and 5-simplex, {3,3,3,3}.
It is analogous to the 120-cell, {5,3,3}, and dodecahedron, {5,3}.
See also
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
| Wikipedia |
13th root
Extracting the 13th root of a number is a famous category for the mental calculation world records.[1] The challenge consists of being given a large number (possibly over 100 digits) and asked to return the number that, when taken to the 13th power, equals the given number. For example, the 13th root of 8,192 is 2 and the 13th root of 96,889,010,407 is 7.
Properties of the challenge
Extracting the 13th root has certain properties. One is that the 13th root of a number is much smaller: a 13th root will have approximately 1/13th the number of digits. Thus, the 13th root of a 100-digit number only has 8 digits and the 13th root of a 200-digit number will have 16 digits. Furthermore, the last digit of the 13th root is always the same as the last digit of the power.
For the 13th root of a 100-digit number there are 7,992,563 possibilities, in the range 41,246,264 – 49,238,826. This is considered a relatively easy calculation. There are 393,544,396,177,593 possibilities, in the range 2,030,917,620,904,736 – 2,424,462,017,082,328, for the 13th root of a 200-digit number. This is considered a difficult calculation.
Records
The Guinness Book of World Records has published records for extracting the 13th root of a 100-digit number. All world records for mentally extracting a 13th root have been for numbers with an integer root:
• The first record was 23 minutes by De Grote (Mexico).
• The most published time was at one time 88.8 seconds by Klein (Netherlands).
• Mittring calculated it in 39 seconds.
• Alexis Lemaire has broken this record with 13.55 seconds.[2] This is the last official world record for extracting the 13th root of a 100-digit number.
• Mittring attempted to break this record with 11.8 seconds, but it was rejected by all organizations (Saxonia Record club, Guinness, 13th root group).
• Lemaire broke this record unofficially 6 times, twice within 4 seconds: the best was 3.625 seconds.
• Lemaire has also set the first world record for the 13th root of a 200-digit number: 513.55 seconds and 742 attempts on April 6, 2005, and broken it with 267.77 seconds and 577 attempts on June 3, 2005.
• The same day, Lemaire has also set in front of official witnesses an unofficial record of 113 seconds and 40 attempts.
• On February 27, 2007, he set a world record of 1 minute and 47 seconds
• He broke this record on July 24, 2007, with a time of 1 minute and 17 seconds (77.99 seconds) at the Museum of History of Science, University of Oxford, UK
• Lemaire broke his record on November 15, 2007, with a time of 72.4 seconds
• Lemaire broke his record on December 10, 2007, with a time of 70.2 seconds
References
1. "13th Root : Definition & Problems With Answers". StudyBay. 2020-06-02. Retrieved 2020-10-10.
2. "13th Root : Definition & Problems With Answers". StudyBay. 2020-06-02. Retrieved 2020-10-10.
| Wikipedia |
15 puzzle
The 15 puzzle (also called Gem Puzzle, Boss Puzzle, Game of Fifteen, Mystic Square and many others) is a sliding puzzle which has 15 square tiles numbered 1 to 15 in a frame that is 4 tile positions high and 4 positions wide, with one unoccupied position. Tiles in the same row or column of the open position can be moved by sliding them horizontally or vertically, respectively. The goal of the puzzle is to place the tiles in numerical order (from left to right, top to bottom).
"Magic 15" redirects here. For the numbered grid where each row and column sums to 15, see Magic square.
Named after the number of tiles in the frame, the 15 puzzle may also be called a 16 puzzle, alluding to its total tile capacity. Similar names are used for different sized variants of the 15 puzzle, such as the 8 puzzle, which has 8 tiles in a 3×3 frame.
The n puzzle is a classical problem for modelling algorithms involving heuristics. Commonly used heuristics for this problem include counting the number of misplaced tiles and finding the sum of the taxicab distances between each block and its position in the goal configuration.[1] Note that both are admissible. That is, they never overestimate the number of moves left, which ensures optimality for certain search algorithms such as A*.[1]
Mathematics
Solvability
Johnson & Story (1879) used a parity argument to show that half of the starting positions for the n puzzle are impossible to resolve, no matter how many moves are made. This is done by considering a function of the tile configuration that is invariant under any valid move, and then using this to partition the space of all possible labelled states into two equivalence classes of reachable and unreachable states.
The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance (number of rows plus number of columns) of the empty square from the lower right corner. This is an invariant because each move changes both the parity of the permutation and the parity of the taxicab distance. In particular, if the empty square is in the lower right corner, then the puzzle is solvable if and only if the permutation of the remaining pieces is even.
Johnson & Story (1879) also showed that on boards of size m × n, where m and n are both larger or equal to 2, all even permutations are solvable. It can be proven by induction on m and n, starting with m = n = 2. Archer (1999) gave another proof, based on defining equivalence classes via a Hamiltonian path.
Wilson (1974) studied the generalisation of the 15 puzzle to arbitrary finite graphs, the original problem being the case of a 4×4 grid graph. The problem has some degenerate cases where the answer is either trivial or a simple combination of the answers to the same problem on some subgraphs. Namely, for paths and polygons, the puzzle has no freedom; if the graph is disconnected, only the connected component of the vertex with the "empty space" is relevant; and if there is an articulation vertex, the problem reduces to the same puzzle on each of the biconnected components of that vertex. Excluding these cases, Wilson showed that other than one exceptional graph on 7 vertices, it is possible to obtain all permutations unless the graph is bipartite, in which case exactly the even permutations can be obtained. The exceptional graph is a regular hexagon with one diagonal and a vertex at the center added; only 1/6 of its permutations can be attained.
For larger versions of the n puzzle, finding a solution is easy, but the problem of finding the shortest solution is NP-hard. It is also NP-hard to approximate the fewest slides within an additive constant, but there is a polynomial-time constant-factor approximation.[2][3] For the 15 puzzle, lengths of optimal solutions range from 0 to 80 single-tile moves (there are 17 configurations requiring 80 moves)[4][5] or 43 multi-tile moves;[6] the 8 puzzle always can be solved in no more than 31 single-tile moves or 24 multi-tile moves (integer sequence A087725). The multi-tile metric counts subsequent moves of the empty tile in the same direction as one.[6]
The number of possible positions of the 24 puzzle is 25!/2 ≈ 7.76×1024 which is too many to calculate God's number. In 2011, lower bounds of 152 single-tile moves or 41 multi-tile moves had been established, as well as upper bounds of 208 single-tile moves or 109 multi-tile moves.[7][8][9][10] In 2016, the upper bound was improved to 205 single-tile moves.[11]
The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed);[12][13][14] this groupoid acts on configurations.
Group theory
Because the combinations of the 15 puzzle can be generated by 3-cycles, it can be proved that the 15 puzzle can be represented by the alternating group $A_{15}$.[15] In fact, any $2k-1$ sliding puzzle with square tiles of equal size can be represented by $A_{2k-1}$.
Alternate proof
In an alternate view of the problem, the invariant can be considered as the sum of two components. The first component is the parity of the number of inversions in the current order of the 15 numbered pieces, and the second component is the parity of the difference in the row number of the empty square from the row number of the last row (referred to as row distance from the last row). This invariant remains constant throughout the puzzle-solving process.
The validity of this invariant is based on the following observations: each column move, involving the movement of a piece within the same column, changes both the parity of the number of inversions (by ±1 or ±3) and the parity of the row distance from the last row (by ±1). Conversely, each row move, which entails moving a piece within the same row, does not affect either of the two parities. Analysing the solved state of the puzzle reveals that the sum of these parities is always even.
By employing an inductive reasoning, it can be proven that any state of the puzzle in which the above sum is odd cannot be solved. In particular, when the empty square is located in the lower right corner or anywhere in the last row, the puzzle is solvable if and only if the number of inversions of the numbered pieces is even.
History
The puzzle was "invented" by Noyes Palmer Chapman,[16] a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34 (see magic square). Copies of the improved Fifteen Puzzle made their way to Syracuse, New York, by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, Rhode Island, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle and, by December 1879, selling them both locally and in Boston, Massachusetts. Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late January 1880, Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle.[16]
The game became a craze in the U.S. in 1880.[17]
Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey.[16]
Sam Loyd
Sam Loyd claimed from 1891 until his death in 1911 that he had invented the puzzle. However, Loyd had nothing to do with the invention or initial popularity of the puzzle, and in any case, the craze was in 1880, not the early 1870s. Loyd's first article about the puzzle was published in 1886, and it was not until 1891 that he first claimed to be the inventor.[16][18]
Some later interest was fueled by Loyd's offer of a $1,000 prize to anyone who could provide a solution for achieving a particular combination specified by Loyd, namely reversing the 14 and 15, which Loyd called the 14-15 puzzle.[1] This is impossible, as had been shown over a decade earlier by Johnson & Story (1879), because it requires a transformation from an even to an odd permutation.
Miscellaneous
The Minus Cube, manufactured in the USSR, is a 3D puzzle with similar operations to the 15 puzzle.
Chess world champion Bobby Fischer was an expert at solving the 15-Puzzle.[19] He had been timed to be able to solve it within 25 seconds; Fischer demonstrated this on November 8, 1972, on The Tonight Show Starring Johnny Carson.[20][21]
See also
• Combination puzzles
• Jeu de taquin, an operation on skew Young tableaux similar to the moves of the 15 puzzle
• Klotski
• Mechanical puzzles
• Pebble motion problems
• Rubik's Cube
• Three cups problem
Notes
1. Korf, R. E. (2000), "Recent Progress in the Design and Analysis of Admissible Heuristic Functions" (PDF), in Choueiry, B. Y.; Walsh, T. (eds.), Abstraction, Reformulation, and Approximation (PDF), SARA 2000. Lecture Notes in Computer Science, vol. 1864, Springer, Berlin, Heidelberg, pp. 45–55, doi:10.1007/3-540-44914-0_3, ISBN 978-3-540-67839-7, archived from the original (PDF) on 2010-08-16, retrieved 2010-04-26
2. Ratner, Daniel; Warmuth, Manfred (1986). "Finding a Shortest Solution for the N × N Extension of the 15-PUZZLE Is Intractable" (PDF). National Conference on Artificial Intelligence. Archived (PDF) from the original on 2012-03-09.
3. Ratner, Daniel; Warmuth, Manfred (1990). "The (n2−1)-puzzle and related relocation problems". Journal of Symbolic Computation. 10 (2): 111–137. doi:10.1016/S0747-7171(08)80001-6.
4. Richard E. Korf, Linear-time disk-based implicit graph search, Journal of the ACM Volume 55 Issue 6 (December 2008), Article 26, pp. 29-30. "For the 4 × 4 Fifteen Puzzle, there are 17 different states at a depth of 80 moves from an initial state with the blank in the corner, while for the 2 × 8 Fifteen Puzzle there is a unique state at the maximum state of 140 moves from the initial state."
5. A. Brüngger, A. Marzetta, K. Fukuda and J. Nievergelt, The parallel search bench ZRAM and its applications, Annals of Operations Research 90 (1999), pp. 45–63.
:"Gasser found 9 positions, requiring 80 moves...We have now proved that the hardest 15-puzzle positions require, in fact, 80 moves. We have also discovered two previously unknown positions, requiring exactly 80 moves to be solved."
6. "The Fifteen Puzzle can be solved in 43 "moves"". Domain of the Cube Forum
7. "24 puzzle new lower bound: 152". Domain of the Cube Forum
8. "m × n puzzle (current state of the art)". Sliding Tile Puzzle Corner.
9. "208s for 5x5". Domain of the Cube Forum.
10. "5x5 can be solved in 109 MTM". Domain of the Cube Forum.
11. "5x5 sliding puzzle can be solved in 205 moves". Domain of the Cube Forum.
12. Jim Belk (2008) Puzzles, Groups, and Groupoids, The Everything Seminar
13. The 15-puzzle groupoid (1), Never Ending Books
14. The 15-puzzle groupoid (2), Never Ending Books
15. Beeler, Robert. "The Fifteen Puzzle: A Motivating Example for the Alternating Group" (PDF). faculty.etsu.edu/. East Tennessee State University. Archived from the original (PDF) on 7 January 2021. Retrieved 26 December 2020.
16. The 15 Puzzle, by Jerry Slocum & Dic Sonneveld, 2006. ISBN 1-890980-15-3
17. Slocum & Singmaster (2009, p. 15)
18. Barry R. Clarke, Puzzles for Pleasure, pp.10-12, Cambridge University Press, 1994 ISBN 0-521-46634-2.
19. Clifford A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, p. 262, Sterling Publishing, 2009 ISBN 1402757964.
20. "Bobby Fischer solves a 15 puzzle in 17 seconds on Carson Tonight Show - 11/08/1972", The Tonight Show, 8 November 1972, Johnny Carson Productions, retrieved 1 August 2021.
21. Adam Spencer, Adam Spencer's Big Book of Numbers: Everything you wanted to know about the numbers 1 to 100, p. 58, Brio Books, 2014 ISBN 192113433X.
References
• Archer, Aaron F. (1999), "A modern treatment of the 15 puzzle", The American Mathematical Monthly, 106 (9): 793–799, CiteSeerX 10.1.1.19.1491, doi:10.2307/2589612, ISSN 0002-9890, JSTOR 2589612, MR 1732661
• Johnson, Wm. Woolsey; Story, William E. (1879), "Notes on the "15" Puzzle", American Journal of Mathematics, 2 (4): 397–404, doi:10.2307/2369492, ISSN 0002-9327, JSTOR 2369492
• Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 177–80, Simon & Schuster.
• Slocum, Jerry; Singmaster, David (2009). The Cube: The Ultimate Guide to the World's Best-Selling Puzzle—Secrets, Stories, Solutions. Black Dog & Leventhal. ISBN 978-1579128050.
• Wilson, Richard M. (1974), "Graph puzzles, homotopy, and the alternating group", Journal of Combinatorial Theory, Series B, 16: 86–96, doi:10.1016/0095-8956(74)90098-7, ISSN 0095-8956, MR 0332555
External links
Wikimedia Commons has media related to 15 puzzle.
• The history of the 15 puzzle
• Fifteen Puzzle Solution
• Maximal number of moves required for the m X n generalization of the 15 puzzle
• 15-Puzzle Optimal Solver with download (from Herbert Kociemba)
| Wikipedia |
Hexadecagon
In mathematics, a hexadecagon (sometimes called a hexakaidecagon or 16-gon) is a sixteen-sided polygon.[1]
Regular hexadecagon
A regular hexadecagon
TypeRegular polygon
Edges and vertices16
Schläfli symbol{16}, t{8}, tt{4}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D16), order 2×16
Internal angle (degrees)157.5°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular hexadecagon
A regular hexadecagon is a hexadecagon in which all angles are equal and all sides are congruent. Its Schläfli symbol is {16} and can be constructed as a truncated octagon, t{8}, and a twice-truncated square tt{4}. A truncated hexadecagon, t{16}, is a triacontadigon, {32}.
Construction
As 16 = 24 (a power of two), a regular hexadecagon is constructible using compass and straightedge: this was already known to ancient Greek mathematicians.[2]
Construction of a regular hexadecagon
at a given circumcircle
Construction of a regular hexadecagon
at a given side length, animation. (The construction is very similar to that of octagon at a given side length.)
Measurements
Each angle of a regular hexadecagon is 157.5 degrees, and the total angle measure of any hexadecagon is 2520 degrees.
The area of a regular hexadecagon with edge length t is
${\begin{aligned}A=4t^{2}\cot {\frac {\pi }{16}}=&4t^{2}\left(1+{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}\right)\\=&4t^{2}({\sqrt {2}}+1)({\sqrt {4-2{\sqrt {2}}}}+1).\end{aligned}}$
Because the hexadecagon has a number of sides that is a power of two, its area can be computed in terms of the circumradius R by truncating Viète's formula:
$A=R^{2}\cdot {\frac {2}{1}}\cdot {\frac {2}{\sqrt {2}}}\cdot {\frac {2}{\sqrt {2+{\sqrt {2}}}}}=4R^{2}{\sqrt {2-{\sqrt {2}}}}.$
Since the area of the circumcircle is $\pi R^{2},$ the regular hexadecagon fills approximately 97.45% of its circumcircle.
Symmetry
Symmetry
The 14 symmetries of a regular hexadecagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.
The regular hexadecagon has Dih16 symmetry, order 32. There are 4 dihedral subgroups: Dih8, Dih4, Dih2, and Dih1, and 5 cyclic subgroups: Z16, Z8, Z4, Z2, and Z1, the last implying no symmetry.
On the regular hexadecagon, there are 14 distinct symmetries. John Conway labels full symmetry as r32 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders.[3]
The most common high symmetry hexadecagons are d16, an isogonal hexadecagon constructed by eight mirrors can alternate long and short edges, and p16, an isotoxal hexadecagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexadecagon.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g16 subgroup has no degrees of freedom but can seen as directed edges.
Dissection
16-cube projection 112 rhomb dissection
Regular
Isotoxal
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. [4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular hexadecagon, m=8, and it can be divided into 28: 4 squares and 3 sets of 8 rhombs. This decomposition is based on a Petrie polygon projection of an 8-cube, with 28 of 1792 faces. The list OEIS: A006245 enumerates the number of solutions as 1232944, including up to 16-fold rotations and chiral forms in reflection.
Dissection into 28 rhombs
8-cube
Skew hexadecagon
3 regular skew zig-zag hexadecagon
{8}#{ } {8⁄3}#{ } {8⁄5}#{ }
A regular skew hexadecagon is seen as zig-zagging edges of an octagonal antiprism, an octagrammic antiprism, and an octagrammic crossed-antiprism.
A skew hexadecagon is a skew polygon with 24 vertices and edges but not existing on the same plane. The interior of such an hexadecagon is not generally defined. A skew zig-zag hexadecagon has vertices alternating between two parallel planes.
A regular skew hexadecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew hexadecagon and can be seen in the vertices and side edges of an octagonal antiprism with the same D8d, [2+,16] symmetry, order 32. The octagrammic antiprism, s{2,16/3} and octagrammic crossed-antiprism, s{2,16/5} also have regular skew octagons.
Petrie polygons
The regular hexadecagon is the Petrie polygon for many higher-dimensional polytopes, shown in these skew orthogonal projections, including:
A15 B8 D9 2B2 (4D)
15-simplex
8-orthoplex
8-cube
611
161
8-8 duopyramid
8-8 duoprism
Related figures
A hexadecagram is a 16-sided star polygon, represented by symbol {16/n}. There are three regular star polygons, {16/3}, {16/5}, {16/7}, using the same vertices, but connecting every third, fifth or seventh points. There are also three compounds: {16/2} is reduced to 2{8} as two octagons, {16/4} is reduced to 4{4} as four squares and {16/6} reduces to 2{8/3} as two octagrams, and finally {16/8} is reduced to 8{2} as eight digons.
Compound and star hexadecagons
Form Convex polygon Compound Star polygon Compound
Image
{16/1} or {16}
{16/2} or 2{8}
{16/3}
{16/4} or 4{4}
Interior angle 157.5°135°112.5°90°
Form Star polygon Compound Star polygon Compound
Image
{16/5}
{16/6} or 2{8/3}
{16/7}
{16/8} or 8{2}
Interior angle 67.5°45°22.5°0°
Deeper truncations of the regular octagon and octagram can produce isogonal (vertex-transitive) intermediate hexadecagram forms with equally spaced vertices and two edge lengths.[5]
A truncated octagon is a hexadecagon, t{8}={16}. A quasitruncated octagon, inverted as {8/7}, is a hexadecagram: t{8/7}={16/7}. A truncated octagram {8/3} is a hexadecagram: t{8/3}={16/3} and a quasitruncated octagram, inverted as {8/5}, is a hexadecagram: t{8/5}={16/5}.
Isogonal truncations of octagon and octagram
Quasiregular Isogonal Quasiregular
t{8}={16}
t{8/7}={16/7}
t{8/3}={16/3}
t{8/5}={16/5}
In art
In the early 16th century, Raphael was the first to construct a perspective image of a regular hexadecagon: the tower in his painting The Marriage of the Virgin has 16 sides, elaborating on an eight-sided tower in a previous painting by Pietro Perugino.[6]
Hexadecagrams (16-sided star polygons) are included in the Girih patterns in the Alhambra.[7]
Irregular hexadecagons
An octagonal star can be seen as a concave hexadecagon:
See also
• Octagram
• Rhumbline network
References
1. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics, Second Edition. CRC Press. p. 1365. ISBN 9781420035223.
2. Koshy, Thomas (2007), Elementary Number Theory with Applications (2nd ed.), Academic Press, p. 142, ISBN 9780080547091.
3. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
6. Speiser, David (2011), "Architecture, mathematics and theology in Raphael's paintings", in Williams, Kim (ed.), Crossroads: History of Science, History of Art. Essays by David Speiser, vol. II, Springer, pp. 29–39, doi:10.1007/978-3-0348-0139-3_3. Originally published in Nexus III: Architecture and Mathematics, Kim Williams, ed. (Ospedaletto, Pisa: Pacini Editore, 2000), pp. 147–156.
7. Hankin, E. Hanbury (May 1925), "Examples of methods of drawing geometrical arabesque patterns", The Mathematical Gazette, 12 (176): 370–373, doi:10.2307/3604213.
External links
• Weisstein, Eric W. "Hexadecagon". MathWorld.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
17-animal inheritance puzzle
The 17-animal inheritance puzzle is a mathematical puzzle involving unequal but fair allocation of indivisible goods, usually stated in terms of inheritance of a number of large animals (17 camels, 17 horses, 17 elephants, etc.) which must be divided in some stated proportion among a number of beneficiaries.
17 indivisible camels
Despite often being framed as a puzzle, it is more an anecdote about a curious calculation than a problem with a clear mathematical solution.[1] Beyond recreational mathematics and mathematics education, the story has been repeated as a parable with varied metaphorical meanings.
Although an ancient origin for the puzzle has often been claimed, it has not been documented. Instead, a version of the puzzle can be traced back to the works of Mulla Muhammad Mahdi Naraqi, an 18th-century Iranian philosopher. It entered the western recreational mathematics literature in the late 19th century. Several mathematicians have formulated different generalizations of the puzzle to numbers other than 17.
Statement
According to the statement of the puzzle, a man dies leaving 17 camels (or other animals) to his three sons, to be divided in the following proportions: the eldest son should inherit 1⁄2 of the man's property, the middle son should inherit 1⁄3, and the youngest son should inherit 1⁄9. How should they divide the camels, noting that only a whole live camel has value?[2]
Solution
As usually stated, to solve the puzzle, the three sons ask for the help of another man, often a priest, judge, or other local official. This man solves the puzzle in the following way: he lends the three sons his own camel, so that there are now 18 camels to be divided. That leaves nine camels for the eldest son, six camels for the middle son, and two camels for the youngest son, in the proportions demanded for the inheritance. These 17 camels leave one camel left over, which the judge takes back as his own.[2]
Some sources point out an additional feature of this solution: each son is satisfied, because he receives more camels than his originally-stated inheritance. The eldest son was originally promised only 8+1⁄2 camels, but receives nine; the middle son was promised 5+2⁄3, but receives six; and the youngest was promised 1+8⁄9, but receives two.[3]
History
Similar problems of unequal division go back to ancient times, but without the twist of the loan and return of the extra camel. For instance, the Rhind Mathematical Papyrus features a problem in which many loaves of bread are to be divided in four different specified proportions.[2][4] The 17 animals puzzle can be seen as an example of a "completion to unity" problem, of a type found in other examples on this papyrus, in which a set of fractions adding to less than one should be completed, by adding more fractions, to make their total come out to exactly one.[5] Another similar case, involving fractional inheritance in the Roman empire, appears in the writings of Publius Juventius Celsus, attributed to a case decided by Salvius Julianus.[6][7] The problems of fairly subdividing indivisible elements into specified proportions, seen in these inheritance problems, also arise when allocating seats in electoral systems based on proportional representation.[8]
Many similar problems of division into fractions are known from mathematics in the medieval Islamic world,[1][4][9] but "it does not seem that the story of the 17 camels is part of classical Arab-Islamic mathematics".[9] Supposed origins of the problem in the works of al-Khwarizmi, Fibonacci or Tartaglia can also not be verified.[10] It has also been attributed to 16th-century Mughal Empire minister Birbal, but only as a "legendary tale".[11] The earliest documented appearance of the puzzle found by Pierre Ageron, using 17 camels, appears in the work of 18th-century Shiite Iranian philosopher Mulla Muhammad Mahdi Naraqi.[9] By 1850 it had already entered circulation in America, through a travelogue of Mesopotamia published by James Phillips Fletcher.[12][13] It appeared in The Mathematical Monthly in 1859,[10][14] and a version with 17 elephants and a claimed Chinese origin was included in Hanky Panky: A Book of Conjuring Tricks (London, 1872), edited by William Henry Cremer but often attributed to Wiljalba Frikell or Henry Llewellyn Williams.[2][10] The same puzzle subsequently appeared in the late 19th and early 20th centuries in the works of Henry Dudeney, Sam Loyd,[2] Édouard Lucas,[9] Professor Hoffmann,[15] and Émile Fourrey,[16] among others.[17][18][19][20] A version with 17 horses circulated as folklore in mid-20th-century America.[21]
A variant of the story has been told with 11 camels, to be divided into 1⁄2, 1⁄4, and 1⁄6.[22][23] Another variant of the puzzle appears in the book The Man Who Counted, a mathematical puzzle book originally published in Portuguese by Júlio César de Mello e Souza in 1938. This version starts with 35 camels, to be divided in the same proportions as in the 17-camel version. After the hero of the story lends a camel, and the 36 camels are divided among the three brothers, two are left over: one to be returned to the hero, and another given to him as a reward for his cleverness. The endnotes to the English translation of the book cite the 17-camel version of the problem to the works of Fourrey and Gaston Boucheny (1939).[10]
Beyond recreational mathematics, the story has been used as the basis for school mathematics lessons,[3][24] as a parable with varied morals in religion, law, economics, and politics,[19][25][26][27][28] and even as a lay-explanation for catalysis in chemistry.[29]
Generalizations
Paul Stockmeyer, a computer scientist, defines a class of similar puzzles for any number $n$ of animals, with the property that $n$ can be written as a sum of distinct divisors $d_{1},d_{2},\dots $ of $n+1$. In this case, one obtains a puzzle in which the fractions into which the $n$ animals should be divided are
${\frac {d_{1}}{n+1}},{\frac {d_{2}}{n+1}},\dots .$
Because the numbers $d_{i}$ have been chosen to divide $n+1$, all of these fractions simplify to unit fractions. When combined with the judge's share of the animals, $1/(n+1)$, they produce an Egyptian fraction representation of the number one.[2]
The numbers of camels that can be used as the basis for such a puzzle (that is, numbers $n$ that can be represented as sums of distinct divisors of $n+1$) form the integer sequence
1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, ...[30]
S. Naranan, an Indian physicist, seeks a more restricted class of generalized puzzles, with only three terms, and with $n+1$ equal to the least common multiple of the denominators of the three unit fractions, finding only seven possible triples of fractions that meet these conditions.[11]
Brazilian researchers Márcio Luís Ferreira Nascimento and Luiz Barco generalize the problem further, as in the variation with 35 camels, to instances in which more than one camel may be lent and the number returned may be larger than the number lent.[10]
See also
• The monkey and the coconuts, a more complicated fair-division puzzle
• Mathematics of apportionment, general methods for rounding fractional subdivisions into integer numbers of items
References
1. Sesiano, Jacques (2014), "Le partage des chameaux", Récréations Mathématiques au Moyen Âge (in French), Lausanne: Presses Polytechniques et Universitaires Romandes, pp. 198–200, archived from the original on 2023-03-25, retrieved 2023-03-25
2. Stockmeyer, Paul K. (September 2013), "Of camels, inheritance, and unit fractions", Math Horizons, 21 (1): 8–11, doi:10.4169/mathhorizons.21.1.8, JSTOR 10.4169/mathhorizons.21.1.8, MR 3313765, S2CID 125145732
3. Ben-Chaim, David; Shalitin, Yechiel; Stupel, Moshe (February 2019), "Historical mathematical problems suitable for classroom activities", The Mathematical Gazette, 103 (556): 12–19, doi:10.1017/mag.2019.2, S2CID 86506133
4. Finkel, Joshua (1955), "A mathematical conundrum in the Ugaritic Keret poem", Hebrew Union College Annual, 26: 109–149, JSTOR 23506151
5. Anne, Premchand (1998), "Egyptian fractions and the inheritance problem", The College Mathematics Journal, 29 (4): 296–300, doi:10.1080/07468342.1998.11973958, JSTOR 2687685, MR 1648474
6. Cajori, Florian (1894), A History of Mathematics, MacMillan and Co., pp. 79–80
7. Smith, David Eugene (1917), "On the origin of certain typical problems", The American Mathematical Monthly, 24 (2): 64–71, doi:10.2307/2972701, JSTOR 2972701, MR 1518704
8. Çarkoğlu, Ali; Erdoğan, Emre (1998), "Fairness in the apportionment of seats in the Turkish legislature: is there room for improvement?", New Perspectives on Turkey, 19: 97–124, doi:10.1017/s0896634600003046, S2CID 148547260
9. Ageron, Pierre (2013), "Le partage des dix-sept chameaux et autres arithmétiques attributes à l'immam 'Alî: Mouvance et circulation de récits de la tradition musulmane chiite" (PDF), Revue d'histoire des mathématiques (in French), 19 (1): 1–41, archived (PDF) from the original on 2023-03-24, retrieved 2023-03-24; see in particular pp. 13–14.
10. Nascimento, Márcio Luís Ferreira; Barco, Luiz (September 2016), "The man who loved to count and the incredible story of the 35 camels", Journal of Mathematics and the Arts, 10 (1–4): 35–43, doi:10.1080/17513472.2016.1221211, S2CID 54030575, archived from the original on 2023-03-25, retrieved 2023-03-25
11. Naranan, S. (1973), "An "elephantine" equation", Mathematics Magazine, 46 (5): 276–278, doi:10.2307/2688266, JSTOR 2688266, MR 1572070
12. Fletcher, James Phillips (1850), Notes from Nineveh: And Travels in Mesopotamia, Assyria and Syria, Lea & Blanchard, p. 206
13. Maxham, Ephraim; Wing, Daniel Ripley (24 October 1850), "A Wise Judge", The Eastern Mail, Waterville, Maine, vol. 4, no. 14, p. 3, archived from the original on 2023-03-24, retrieved 2023-03-24
14. "Problem", Notes and queries, The Mathematical Monthly, 1 (11): 362, August 1859, archived from the original on 2023-03-25, retrieved 2023-03-25
15. Professor Hoffmann (1893), "No. XI—An Unmanageable Legacy", Puzzles Old and New, London: Frederick Warne and Co., p. 147; solution, pp. 191–192
16. Fourrey, Émile (1899), "Curieux partages", Récréations arithmétiques (in French), Paris: Librairie Nony, p. 159
17. Morrell, E. W. (February 1897), "Problems for solution: arithmetic, no. 76", The American Mathematical Monthly, 4 (2): 61, doi:10.2307/2970050, JSTOR 2970050
18. White, William F. (1908), "Puzzle of the camels", A Scrap-Book of Elementary Mathematics: Notes, Recreations, Essays, The Open Court Publishing Company, p. 193
19. Wolff, Sir Henry Drummond (1908), "A Parsee inspiration", Rambling Recollections, vol. II, London: MacMillan and Co., p. 56
20. Wentworth, George; Smith, David Eugene (1909), Complete Arithmetic, Wentworth–Smith Mathematical Series, Ginn and Company, p. 467
21. Browne, Ray B. (Fall 1961), "Riddles from Tippecanoe County, Indiana", Midwest Folklore, 11 (3): 155–160, JSTOR 4317919
22. Van Vleck, J. H. (January 1929), "The new quantum mechanics", Chemical Reviews, 5 (4): 467–507, doi:10.1021/cr60020a006
23. Seibert, Thomas M. (December 1987), "The arguments of a judge", Argumentation: Analysis and Practices, De Gruyter, pp. 119–122, doi:10.1515/9783110869170, ISBN 978-3-11-013027-0
24. Coyle, Stephen (November 2000), "Fractions give me the hump", Mathematics in School, 29 (5): 40, JSTOR 30215451
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| Wikipedia |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example,
• a 0-dimensional simplex is a point,
• a 1-dimensional simplex is a line segment,
• a 2-dimensional simplex is a triangle,
• a 3-dimensional simplex is a tetrahedron, and
• a 4-dimensional simplex is a 5-cell.
Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points $u_{0},\dots ,u_{k}$ are affinely independent, which means that the k vectors $u_{1}-u_{0},\dots ,u_{k}-u_{0}$ are linearly independent. Then, the simplex determined by them is the set of points
$C=\left\{\theta _{0}u_{0}+\dots +\theta _{k}u_{k}~{\Bigg |}~\sum _{i=0}^{k}\theta _{i}=1{\mbox{ and }}\theta _{i}\geq 0{\mbox{ for }}i=0,\dots ,k\right\}.$
A regular simplex[1] is a simplex that is also a regular polytope. A regular k-simplex may be constructed from a regular (k − 1)-simplex by connecting a new vertex to all original vertices by the common edge length.
The standard simplex or probability simplex[2] is the k − 1 dimensional simplex whose vertices are the k standard unit vectors in $\mathbb {R} ^{k}$, or in other words
$\left\{x\in \mathbb {R} ^{k}:x_{0}+\dots +x_{k-1}=1,x_{i}\geq 0{\text{ for }}i=0,\dots ,k-1\right\}.$
In topology and combinatorics, it is common to "glue together" simplices to form a simplicial complex. The associated combinatorial structure is called an abstract simplicial complex, in which context the word "simplex" simply means any finite set of vertices.
History
The concept of a simplex was known to William Kingdon Clifford, who wrote about these shapes in 1886 but called them "prime confines". Henri Poincaré, writing about algebraic topology in 1900, called them "generalized tetrahedra". In 1902 Pieter Hendrik Schoute described the concept first with the Latin superlative simplicissimum ("simplest") and then with the same Latin adjective in the normal form simplex ("simple").[3]
The regular simplex family is the first of three regular polytope families, labeled by Donald Coxeter as αn, the other two being the cross-polytope family, labeled as βn, and the hypercubes, labeled as γn. A fourth family, the tessellation of n-dimensional space by infinitely many hypercubes, he labeled as δn.[4]
Elements
The convex hull of any nonempty subset of the n + 1 points that define an n-simplex is called a face of the simplex. Faces are simplices themselves. In particular, the convex hull of a subset of size m + 1 (of the n + 1 defining points) is an m-simplex, called an m-face of the n-simplex. The 0-faces (i.e., the defining points themselves as sets of size 1) are called the vertices (singular: vertex), the 1-faces are called the edges, the (n − 1)-faces are called the facets, and the sole n-face is the whole n-simplex itself. In general, the number of m-faces is equal to the binomial coefficient ${\tbinom {n+1}{m+1}}$.[5] Consequently, the number of m-faces of an n-simplex may be found in column (m + 1) of row (n + 1) of Pascal's triangle. A simplex A is a coface of a simplex B if B is a face of A. Face and facet can have different meanings when describing types of simplices in a simplicial complex.
The extended f-vector for an n-simplex can be computed by (1,1)n+1, like the coefficients of polynomial products. For example, a 7-simplex is (1,1)8 = (1,2,1)4 = (1,4,6,4,1)2 = (1,8,28,56,70,56,28,8,1).
The number of 1-faces (edges) of the n-simplex is the n-th triangle number, the number of 2-faces of the n-simplex is the (n − 1)th tetrahedron number, the number of 3-faces of the n-simplex is the (n − 2)th 5-cell number, and so on.
n-Simplex elements[6]
Δn Name Schläfli
Coxeter
0-
faces
(vertices)
1-
faces
(edges)
2-
faces
(faces)
3-
faces
(cells)
4-
faces
5-
faces
6-
faces
7-
faces
8-
faces
9-
faces
10-
faces
Sum
= 2n+1 − 1
Δ0 0-simplex
(point)
( )
1 1
Δ1 1-simplex
(line segment)
{ } = ( ) ∨ ( ) = 2⋅( )
2 1 3
Δ2 2-simplex
(triangle)
{3} = 3⋅( )
3 3 1 7
Δ3 3-simplex
(tetrahedron)
{3,3} = 4⋅( )
4 6 4 1 15
Δ4 4-simplex
(5-cell)
{33} = 5⋅( )
5 10 10 5 1 31
Δ5 5-simplex {34} = 6⋅( )
6 15 20 15 6 1 63
Δ6 6-simplex {35} = 7⋅( )
7 21 35 35 21 7 1 127
Δ7 7-simplex {36} = 8⋅( )
8 28 56 70 56 28 8 1 255
Δ8 8-simplex {37} = 9⋅( )
9 36 84 126 126 84 36 9 1 511
Δ9 9-simplex {38} = 10⋅( )
10 45 120 210 252 210 120 45 10 1 1023
Δ10 10-simplex {39} = 11⋅( )
11 55 165 330 462 462 330 165 55 11 1 2047
An n-simplex is the polytope with the fewest vertices that requires n dimensions. Consider a line segment AB as a shape in a 1-dimensional space (the 1-dimensional space is the line in which the segment lies). One can place a new point C somewhere off the line. The new shape, triangle ABC, requires two dimensions; it cannot fit in the original 1-dimensional space. The triangle is the 2-simplex, a simple shape that requires two dimensions. Consider a triangle ABC, a shape in a 2-dimensional space (the plane in which the triangle resides). One can place a new point D somewhere off the plane. The new shape, tetrahedron ABCD, requires three dimensions; it cannot fit in the original 2-dimensional space. The tetrahedron is the 3-simplex, a simple shape that requires three dimensions. Consider tetrahedron ABCD, a shape in a 3-dimensional space (the 3-space in which the tetrahedron lies). One can place a new point E somewhere outside the 3-space. The new shape ABCDE, called a 5-cell, requires four dimensions and is called the 4-simplex; it cannot fit in the original 3-dimensional space. (It also cannot be visualized easily.) This idea can be generalized, that is, adding a single new point outside the currently occupied space, which requires going to the next higher dimension to hold the new shape. This idea can also be worked backward: the line segment we started with is a simple shape that requires a 1-dimensional space to hold it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space.
More formally, an (n + 1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ). An (m + n + 1)-simplex can be constructed as a join of an m-simplex and an n-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An isosceles triangle is the join of a 1-simplex and a point: { } ∨ ( ). An equilateral triangle is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4 ⋅ ( ) or {3,3} and so on.
In some conventions,[7] the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes.
Symmetric graphs of regular simplices
These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
The standard simplex
The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by
$\Delta ^{n}=\left\{(t_{0},\dots ,t_{n})\in \mathbb {R} ^{n+1}~{\Bigg |}~\sum _{i=0}^{n}t_{i}=1{\text{ and }}t_{i}\geq 0{\text{ for }}i=0,\ldots ,n\right\}$
The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition.
The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where
e0 = (1, 0, 0, ..., 0),
e1 = (0, 1, 0, ..., 0),
⋮
en = (0, 0, 0, ..., 1).
A standard simplex is an example of a 0/1-polytope, with all coordinates as 0 or 1. It can also be seen one facet of a regular (n+1)-orthoplex.
There is a canonical map from the standard n-simplex to an arbitrary n-simplex with vertices (v0, ..., vn) given by
$(t_{0},\ldots ,t_{n})\mapsto \sum _{i=0}^{n}t_{i}v_{i}$
The coefficients ti are called the barycentric coordinates of a point in the n-simplex. Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation. It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.
More generally, there is a canonical map from the standard $(n-1)$-simplex (with n vertices) onto any polytope with n vertices, given by the same equation (modifying indexing):
$(t_{1},\ldots ,t_{n})\mapsto \sum _{i=1}^{n}t_{i}v_{i}$
These are known as generalized barycentric coordinates, and express every polytope as the image of a simplex: $\Delta ^{n-1}\twoheadrightarrow P.$
A commonly used function from Rn to the interior of the standard $(n-1)$-simplex is the softmax function, or normalized exponential function; this generalizes the standard logistic function.
Examples
• Δ0 is the point 1 in R1.
• Δ1 is the line segment joining (1, 0) and (0, 1) in R2.
• Δ2 is the equilateral triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) in R3.
• Δ3 is the regular tetrahedron with vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) and (0, 0, 0, 1) in R4.
• Δ4 is the regular 5-cell with vertices (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0) and (0, 0, 0, 0, 1) in R5.
Increasing coordinates
An alternative coordinate system is given by taking the indefinite sum:
${\begin{aligned}s_{0}&=0\\s_{1}&=s_{0}+t_{0}=t_{0}\\s_{2}&=s_{1}+t_{1}=t_{0}+t_{1}\\s_{3}&=s_{2}+t_{2}=t_{0}+t_{1}+t_{2}\\&\;\;\vdots \\s_{n}&=s_{n-1}+t_{n-1}=t_{0}+t_{1}+\cdots +t_{n-1}\\s_{n+1}&=s_{n}+t_{n}=t_{0}+t_{1}+\cdots +t_{n}=1\end{aligned}}$
This yields the alternative presentation by order, namely as nondecreasing n-tuples between 0 and 1:
$\Delta _{*}^{n}=\left\{(s_{1},\ldots ,s_{n})\in \mathbb {R} ^{n}\mid 0=s_{0}\leq s_{1}\leq s_{2}\leq \dots \leq s_{n}\leq s_{n+1}=1\right\}.$
Geometrically, this is an n-dimensional subset of $\mathbb {R} ^{n}$ (maximal dimension, codimension 0) rather than of $\mathbb {R} ^{n+1}$ (codimension 1). The facets, which on the standard simplex correspond to one coordinate vanishing, $t_{i}=0,$ here correspond to successive coordinates being equal, $s_{i}=s_{i+1},$ while the interior corresponds to the inequalities becoming strict (increasing sequences).
A key distinction between these presentations is the behavior under permuting coordinates – the standard simplex is stabilized by permuting coordinates, while permuting elements of the "ordered simplex" do not leave it invariant, as permuting an ordered sequence generally makes it unordered. Indeed, the ordered simplex is a (closed) fundamental domain for the action of the symmetric group on the n-cube, meaning that the orbit of the ordered simplex under the n! elements of the symmetric group divides the n-cube into $n!$ mostly disjoint simplices (disjoint except for boundaries), showing that this simplex has volume $1/n!$ Alternatively, the volume can be computed by an iterated integral, whose successive integrands are $1,x,x^{2}/2,x^{3}/3!,\dots ,x^{n}/n!$
A further property of this presentation is that it uses the order but not addition, and thus can be defined in any dimension over any ordered set, and for example can be used to define an infinite-dimensional simplex without issues of convergence of sums.
Projection onto the standard simplex
Especially in numerical applications of probability theory a projection onto the standard simplex is of interest. Given $(p_{i})_{i}$ with possibly negative entries, the closest point $\left(t_{i}\right)_{i}$ on the simplex has coordinates
$t_{i}=\max\{p_{i}+\Delta \,,0\},$
where $\Delta $ is chosen such that $ \sum _{i}\max\{p_{i}+\Delta \,,0\}=1.$
$\Delta $ can be easily calculated from sorting $p_{i}$.[8] The sorting approach takes $O(n\log n)$ complexity, which can be improved to $O(n)$ complexity via median-finding algorithms.[9] Projecting onto the simplex is computationally similar to projecting onto the $\ell _{1}$ ball.
Corner of cube
Finally, a simple variant is to replace "summing to 1" with "summing to at most 1"; this raises the dimension by 1, so to simplify notation, the indexing changes:
$\Delta _{c}^{n}=\left\{(t_{1},\ldots ,t_{n})\in \mathbb {R} ^{n}~{\Bigg |}~\sum _{i=1}^{n}t_{i}\leq 1{\text{ and }}t_{i}\geq 0{\text{ for all }}i\right\}.$
This yields an n-simplex as a corner of the n-cube, and is a standard orthogonal simplex. This is the simplex used in the simplex method, which is based at the origin, and locally models a vertex on a polytope with n facets.
Cartesian coordinates for a regular n-dimensional simplex in Rn
One way to write down a regular n-simplex in Rn is to choose two points to be the first two vertices, choose a third point to make an equilateral triangle, choose a fourth point to make a regular tetrahedron, and so on. Each step requires satisfying equations that ensure that each newly chosen vertex, together with the previously chosen vertices, forms a regular simplex. There are several sets of equations that can be written down and used for this purpose. These include the equality of all the distances between vertices; the equality of all the distances from vertices to the center of the simplex; the fact that the angle subtended through the new vertex by any two previously chosen vertices is $\pi /3$; and the fact that the angle subtended through the center of the simplex by any two vertices is $\arccos(-1/n)$.
It is also possible to directly write down a particular regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of Rn by e1 through en. Begin with the standard (n − 1)-simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular n-simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form (α/n, ..., α/n) for some real number α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular n-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for α. Solving this equation shows that there are two choices for the additional vertex:
${\frac {1}{n}}\left(1\pm {\sqrt {n+1}}\right)\cdot (1,\dots ,1).$
Either of these, together with the standard basis vectors, yields a regular n-simplex.
The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are:
${\frac {1}{\sqrt {2}}}\mathbf {e} _{i}-{\frac {1}{n{\sqrt {2}}}}{\bigg (}1\pm {\frac {1}{\sqrt {n+1}}}{\bigg )}\cdot (1,\dots ,1),$
for $1\leq i\leq n$, and
$\pm {\frac {1}{\sqrt {2(n+1)}}}\cdot (1,\dots ,1).$
Note that there are two sets of vertices described here. One set uses $+$ in each calculation. The other set uses $-$ in each calculation.
This simplex is inscribed in a hypersphere of radius ${\sqrt {n/(2(n+1))}}$.
A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are
${\sqrt {1+n^{-1}}}\cdot \mathbf {e} _{i}-n^{-3/2}({\sqrt {n+1}}\pm 1)\cdot (1,\dots ,1),$
where $1\leq i\leq n$, and
$\pm n^{-1/2}\cdot (1,\dots ,1).$
The side length of this simplex is $ {\sqrt {2(n+1)/n}}$.
A highly symmetric way to construct a regular n-simplex is to use a representation of the cyclic group Zn+1 by orthogonal matrices. This is an n × n orthogonal matrix Q such that Qn+1 = I is the identity matrix, but no lower power of Q is. Applying powers of this matrix to an appropriate vector v will produce the vertices of a regular n-simplex. To carry this out, first observe that for any orthogonal matrix Q, there is a choice of basis in which Q is a block diagonal matrix
$Q=\operatorname {diag} (Q_{1},Q_{2},\dots ,Q_{k}),$
where each Qi is orthogonal and either 2 × 2 or 1 × 1. In order for Q to have order n + 1, all of these matrices must have order dividing n + 1. Therefore each Qi is either a 1 × 1 matrix whose only entry is 1 or, if n is odd, −1; or it is a 2 × 2 matrix of the form
${\begin{pmatrix}\cos {\frac {2\pi \omega _{i}}{n+1}}&-\sin {\frac {2\pi \omega _{i}}{n+1}}\\\sin {\frac {2\pi \omega _{i}}{n+1}}&\cos {\frac {2\pi \omega _{i}}{n+1}}\end{pmatrix}},$
where each ωi is an integer between zero and n inclusive. A sufficient condition for the orbit of a point to be a regular simplex is that the matrices Qi form a basis for the non-trivial irreducible real representations of Zn+1, and the vector being rotated is not stabilized by any of them.
In practical terms, for n even this means that every matrix Qi is 2 × 2, there is an equality of sets
$\{\omega _{1},n+1-\omega _{1},\dots ,\omega _{n/2},n+1-\omega _{n/2}\}=\{1,\dots ,n\},$
and, for every Qi, the entries of v upon which Qi acts are not both zero. For example, when n = 4, one possible matrix is
${\begin{pmatrix}\cos(2\pi /5)&-\sin(2\pi /5)&0&0\\\sin(2\pi /5)&\cos(2\pi /5)&0&0\\0&0&\cos(4\pi /5)&-\sin(4\pi /5)\\0&0&\sin(4\pi /5)&\cos(4\pi /5)\end{pmatrix}}.$
Applying this to the vector (1, 0, 1, 0) results in the simplex whose vertices are
${\begin{pmatrix}1\\0\\1\\0\end{pmatrix}},{\begin{pmatrix}\cos(2\pi /5)\\\sin(2\pi /5)\\\cos(4\pi /5)\\\sin(4\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(4\pi /5)\\\sin(4\pi /5)\\\cos(8\pi /5)\\\sin(8\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(6\pi /5)\\\sin(6\pi /5)\\\cos(2\pi /5)\\\sin(2\pi /5)\end{pmatrix}},{\begin{pmatrix}\cos(8\pi /5)\\\sin(8\pi /5)\\\cos(6\pi /5)\\\sin(6\pi /5)\end{pmatrix}},$
each of which has distance √5 from the others. When n is odd, the condition means that exactly one of the diagonal blocks is 1 × 1, equal to −1, and acts upon a non-zero entry of v; while the remaining diagonal blocks, say Q1, ..., Q(n − 1) / 2, are 2 × 2, there is an equality of sets
$\left\{\omega _{1},-\omega _{1},\dots ,\omega _{(n-1)/2},-\omega _{n-1)/2}\right\}=\left\{1,\dots ,(n-1)/2,(n+3)/2,\dots ,n\right\},$
and each diagonal block acts upon a pair of entries of v which are not both zero. So, for example, when n = 3, the matrix can be
${\begin{pmatrix}0&-1&0\\1&0&0\\0&0&-1\\\end{pmatrix}}.$
For the vector (1, 0, 1/√2), the resulting simplex has vertices
${\begin{pmatrix}1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\1\\-1/\surd 2\end{pmatrix}},{\begin{pmatrix}-1\\0\\1/\surd 2\end{pmatrix}},{\begin{pmatrix}0\\-1\\-1/\surd 2\end{pmatrix}},$
each of which has distance 2 from the others.
Geometric properties
Volume
The volume of an n-simplex in n-dimensional space with vertices (v0, ..., vn) is
$\mathrm {Volume} ={\frac {1}{n!}}\left|\det {\begin{pmatrix}v_{1}-v_{0}&&v_{2}-v_{0}&&\cdots &&v_{n}-v_{0}\end{pmatrix}}\right|$
where each column of the n × n determinant is a vector that points from vertex v0 to another vertex vk.[10] This formula is particularly useful when $v_{0}$ is the origin.
The expression
$\mathrm {Volume} ={\frac {1}{n!}}\det \left[{\begin{pmatrix}v_{1}^{T}-v_{0}^{T}\\v_{2}^{T}-v_{0}^{T}\\\vdots \\v_{n}^{T}-v_{0}^{T}\end{pmatrix}}{\begin{pmatrix}v_{1}-v_{0}&v_{2}-v_{0}&\cdots &v_{n}-v_{0}\end{pmatrix}}\right]^{1/2}$
employs a Gram determinant and works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions, e.g., a triangle in $\mathbb {R} ^{3}$.
A more symmetric way to compute the volume of an n-simplex in $\mathbb {R} ^{n}$ is
$\mathrm {Volume} ={1 \over n!}\left|\det {\begin{pmatrix}v_{0}&v_{1}&\cdots &v_{n}\\1&1&\cdots &1\end{pmatrix}}\right|.$
Another common way of computing the volume of the simplex is via the Cayley–Menger determinant, which works even when the n-simplex's vertices are in a Euclidean space with more than n dimensions.[11]
Without the 1/n! it is the formula for the volume of an n-parallelotope. This can be understood as follows: Assume that P is an n-parallelotope constructed on a basis $(v_{0},e_{1},\ldots ,e_{n})$ of $\mathbb {R} ^{n}$. Given a permutation $\sigma $ of $\{1,2,\ldots ,n\}$, call a list of vertices $v_{0},\ v_{1},\ldots ,v_{n}$ a n-path if
$v_{1}=v_{0}+e_{\sigma (1)},\ v_{2}=v_{1}+e_{\sigma (2)},\ldots ,v_{n}=v_{n-1}+e_{\sigma (n)}$
(so there are n! n-paths and $v_{n}$ does not depend on the permutation). The following assertions hold:
If P is the unit n-hypercube, then the union of the n-simplexes formed by the convex hull of each n-path is P, and these simplexes are congruent and pairwise non-overlapping.[12] In particular, the volume of such a simplex is
${\frac {\operatorname {Vol} (P)}{n!}}={\frac {1}{n!}}.$
If P is a general parallelotope, the same assertions hold except that it is no longer true, in dimension > 2, that the simplexes need to be pairwise congruent; yet their volumes remain equal, because the n-parallelotope is the image of the unit n-hypercube by the linear isomorphism that sends the canonical basis of $\mathbb {R} ^{n}$ to $e_{1},\ldots ,e_{n}$. As previously, this implies that the volume of a simplex coming from a n-path is:
${\frac {\operatorname {Vol} (P)}{n!}}={\frac {\det(e_{1},\ldots ,e_{n})}{n!}}.$
Conversely, given an n-simplex $(v_{0},\ v_{1},\ v_{2},\ldots v_{n})$ of $\mathbf {R} ^{n}$, it can be supposed that the vectors $e_{1}=v_{1}-v_{0},\ e_{2}=v_{2}-v_{1},\ldots e_{n}=v_{n}-v_{n-1}$ form a basis of $\mathbf {R} ^{n}$. Considering the parallelotope constructed from $v_{0}$ and $e_{1},\ldots ,e_{n}$, one sees that the previous formula is valid for every simplex.
Finally, the formula at the beginning of this section is obtained by observing that
$\det(v_{1}-v_{0},v_{2}-v_{0},\ldots ,v_{n}-v_{0})=\det(v_{1}-v_{0},v_{2}-v_{1},\ldots ,v_{n}-v_{n-1}).$
From this formula, it follows immediately that the volume under a standard n-simplex (i.e. between the origin and the simplex in Rn+1) is
${1 \over (n+1)!}$
The volume of a regular n-simplex with unit side length is
${\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}$
as can be seen by multiplying the previous formula by xn+1, to get the volume under the n-simplex as a function of its vertex distance x from the origin, differentiating with respect to x, at $x=1/{\sqrt {2}}$ (where the n-simplex side length is 1), and normalizing by the length $dx/{\sqrt {n+1}}$ of the increment, $(dx/(n+1),\ldots ,dx/(n+1))$, along the normal vector.
Dihedral angles of the regular n-simplex
Any two (n − 1)-dimensional faces of a regular n-dimensional simplex are themselves regular (n − 1)-dimensional simplices, and they have the same dihedral angle of cos−1(1/n).[13][14]
This can be seen by noting that the center of the standard simplex is $ \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)$, and the centers of its faces are coordinate permutations of $ \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)$. Then, by symmetry, the vector pointing from $ \left({\frac {1}{n+1}},\dots ,{\frac {1}{n+1}}\right)$ to $ \left(0,{\frac {1}{n}},\dots ,{\frac {1}{n}}\right)$ is perpendicular to the faces. So the vectors normal to the faces are permutations of $(-n,1,\dots ,1)$, from which the dihedral angles are calculated.
Simplices with an "orthogonal corner"
An "orthogonal corner" means here that there is a vertex at which all adjacent edges are pairwise orthogonal. It immediately follows that all adjacent faces are pairwise orthogonal. Such simplices are generalizations of right triangles and for them there exists an n-dimensional version of the Pythagorean theorem:
The sum of the squared (n − 1)-dimensional volumes of the facets adjacent to the orthogonal corner equals the squared (n − 1)-dimensional volume of the facet opposite of the orthogonal corner.
$\sum _{k=1}^{n}|A_{k}|^{2}=|A_{0}|^{2}$
where $A_{1}\ldots A_{n}$ are facets being pairwise orthogonal to each other but not orthogonal to $A_{0}$, which is the facet opposite the orthogonal corner.[15]
For a 2-simplex the theorem is the Pythagorean theorem for triangles with a right angle and for a 3-simplex it is de Gua's theorem for a tetrahedron with an orthogonal corner.
Relation to the (n + 1)-hypercube
The Hasse diagram of the face lattice of an n-simplex is isomorphic to the graph of the (n + 1)-hypercube's edges, with the hypercube's vertices mapping to each of the n-simplex's elements, including the entire simplex and the null polytope as the extreme points of the lattice (mapped to two opposite vertices on the hypercube). This fact may be used to efficiently enumerate the simplex's face lattice, since more general face lattice enumeration algorithms are more computationally expensive.
The n-simplex is also the vertex figure of the (n + 1)-hypercube. It is also the facet of the (n + 1)-orthoplex.
Topology
Topologically, an n-simplex is equivalent to an n-ball. Every n-simplex is an n-dimensional manifold with corners.
Probability
Main article: Categorical distribution
In probability theory, the points of the standard n-simplex in (n + 1)-space form the space of possible probability distributions on a finite set consisting of n + 1 possible outcomes. The correspondence is as follows: For each distribution described as an ordered (n + 1)-tuple of probabilities whose sum is (necessarily) 1, we associate the point of the simplex whose barycentric coordinates are precisely those probabilities. That is, the kth vertex of the simplex is assigned to have the kth probability of the (n + 1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism.
Compounds
Since all simplices are self-dual, they can form a series of compounds;
• Two triangles form a hexagram {6/2}.
• Two tetrahedra form a compound of two tetrahedra or stella octangula.
• Two 5-cells form a compound of two 5-cells in four dimensions.
Algebraic topology
In algebraic topology, simplices are used as building blocks to construct an interesting class of topological spaces called simplicial complexes. These spaces are built from simplices glued together in a combinatorial fashion. Simplicial complexes are used to define a certain kind of homology called simplicial homology.
A finite set of k-simplexes embedded in an open subset of Rn is called an affine k-chain. The simplexes in a chain need not be unique; they may occur with multiplicity. Rather than using standard set notation to denote an affine chain, it is instead the standard practice to use plus signs to separate each member in the set. If some of the simplexes have the opposite orientation, these are prefixed by a minus sign. If some of the simplexes occur in the set more than once, these are prefixed with an integer count. Thus, an affine chain takes the symbolic form of a sum with integer coefficients.
Note that each facet of an n-simplex is an affine (n − 1)-simplex, and thus the boundary of an n-simplex is an affine (n − 1)-chain. Thus, if we denote one positively oriented affine simplex as
$\sigma =[v_{0},v_{1},v_{2},\ldots ,v_{n}]$
with the $v_{j}$ denoting the vertices, then the boundary $\partial \sigma $ of σ is the chain
$\partial \sigma =\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}].$
It follows from this expression, and the linearity of the boundary operator, that the boundary of the boundary of a simplex is zero:
$\partial ^{2}\sigma =\partial \left(\sum _{j=0}^{n}(-1)^{j}[v_{0},\ldots ,v_{j-1},v_{j+1},\ldots ,v_{n}]\right)=0.$
Likewise, the boundary of the boundary of a chain is zero: $\partial ^{2}\rho =0$.
More generally, a simplex (and a chain) can be embedded into a manifold by means of smooth, differentiable map $f\colon \mathbb {R} ^{n}\to M$. In this case, both the summation convention for denoting the set, and the boundary operation commute with the embedding. That is,
$f\left(\sum \nolimits _{i}a_{i}\sigma _{i}\right)=\sum \nolimits _{i}a_{i}f(\sigma _{i})$
where the $a_{i}$ are the integers denoting orientation and multiplicity. For the boundary operator $\partial $, one has:
$\partial f(\rho )=f(\partial \rho )$
where ρ is a chain. The boundary operation commutes with the mapping because, in the end, the chain is defined as a set and little more, and the set operation always commutes with the map operation (by definition of a map).
A continuous map $f:\sigma \to X$ to a topological space X is frequently referred to as a singular n-simplex. (A map is generally called "singular" if it fails to have some desirable property such as continuity and, in this case, the term is meant to reflect to the fact that the continuous map need not be an embedding.)[16]
Algebraic geometry
Since classical algebraic geometry allows one to talk about polynomial equations but not inequalities, the algebraic standard n-simplex is commonly defined as the subset of affine (n + 1)-dimensional space, where all coordinates sum up to 1 (thus leaving out the inequality part). The algebraic description of this set is
$\Delta ^{n}:=\left\{x\in \mathbb {A} ^{n+1}~{\Bigg |}~\sum _{i=1}^{n+1}x_{i}=1\right\},$
which equals the scheme-theoretic description $\Delta _{n}(R)=\operatorname {Spec} (R[\Delta ^{n}])$ with
$R[\Delta ^{n}]:=R[x_{1},\ldots ,x_{n+1}]\left/\left(1-\sum x_{i}\right)\right.$
the ring of regular functions on the algebraic n-simplex (for any ring $R$).
By using the same definitions as for the classical n-simplex, the n-simplices for different dimensions n assemble into one simplicial object, while the rings $R[\Delta ^{n}]$ assemble into one cosimplicial object $R[\Delta ^{\bullet }]$ (in the category of schemes resp. rings, since the face and degeneracy maps are all polynomial).
The algebraic n-simplices are used in higher K-theory and in the definition of higher Chow groups.
Applications
• In statistics, simplices are sample spaces of compositional data and are also used in plotting quantities that sum to 1, such as proportions of subpopulations, as in a ternary plot.
• In industrial statistics, simplices arise in problem formulation and in algorithmic solution. In the design of bread, the producer must combine yeast, flour, water, sugar, etc. In such mixtures, only the relative proportions of ingredients matters: For an optimal bread mixture, if the flour is doubled then the yeast should be doubled. Such mixture problem are often formulated with normalized constraints, so that the nonnegative components sum to one, in which case the feasible region forms a simplex. The quality of the bread mixtures can be estimated using response surface methodology, and then a local maximum can be computed using a nonlinear programming method, such as sequential quadratic programming.[17]
• In operations research, linear programming problems can be solved by the simplex algorithm of George Dantzig.
• In geometric design and computer graphics, many methods first perform simplicial triangulations of the domain and then fit interpolating polynomials to each simplex.[18]
• In chemistry, the hydrides of most elements in the p-block can resemble a simplex if one is to connect each atom. Neon does not react with hydrogen and as such is a point, fluorine bonds with one hydrogen atom and forms a line segment, oxygen bonds with two hydrogen atoms in a bent fashion resembling a triangle, nitrogen reacts to form a tetrahedron, and carbon forms a structure resembling a Schlegel diagram of the 5-cell. This trend continues for the heavier analogues of each element, as well as if the hydrogen atom is replaced by a halogen atom.
• In some approaches to quantum gravity, such as Regge calculus and causal dynamical triangulations, simplices are used as building blocks of discretizations of spacetime; that is, to build simplicial manifolds.
See also
• 3-sphere
• Aitchison geometry
• Causal dynamical triangulation
• Complete graph
• Delaunay triangulation
• Distance geometry
• Geometric primitive
• Hill tetrahedron
• Hypersimplex
• List of regular polytopes
• Metcalfe's law
• Other regular n-polytopes
• Cross-polytope
• Hypercube
• Tesseract
• Polytope
• Schläfli orthoscheme
• Simplex algorithm—a method for solving optimization problems with inequalities.
• Simplicial complex
• Simplicial homology
• Simplicial set
• Spectrahedron
• Ternary plot
Notes
1. Elte, E.L. (2006) [1912]. "IV. five dimensional semiregular polytope". The Semiregular Polytopes of the Hyperspaces. Simon & Schuster. ISBN 978-1-4181-7968-7.
2. Boyd & Vandenberghe 2004
3. Miller, Jeff, "Simplex", Earliest Known Uses of Some of the Words of Mathematics, retrieved 2018-01-08
4. Coxeter 1973, pp. 120–124, §7.2.
5. Coxeter 1973, p. 120.
6. Sloane, N. J. A. (ed.). "Sequence A135278 (Pascal's triangle with its left-hand edge removed)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. Kozlov, Dimitry, Combinatorial Algebraic Topology, 2008, Springer-Verlag (Series: Algorithms and Computation in Mathematics)
8. Yunmei Chen; Xiaojing Ye (2011). "Projection Onto A Simplex". arXiv:1101.6081 [math.OC].
9. MacUlan, N.; De Paula, G. G. (1989). "A linear-time median-finding algorithm for projecting a vector on the simplex of n". Operations Research Letters. 8 (4): 219. doi:10.1016/0167-6377(89)90064-3.
10. A derivation of a very similar formula can be found in Stein, P. (1966). "A Note on the Volume of a Simplex". American Mathematical Monthly. 73 (3): 299–301. doi:10.2307/2315353. JSTOR 2315353.
11. Colins, Karen D. "Cayley-Menger Determinant". MathWorld.
12. Every n-path corresponding to a permutation $\scriptstyle \sigma $ is the image of the n-path $\scriptstyle v_{0},\ v_{0}+e_{1},\ v_{0}+e_{1}+e_{2},\ldots v_{0}+e_{1}+\cdots +e_{n}$ by the affine isometry that sends $\scriptstyle v_{0}$ to $\scriptstyle v_{0}$, and whose linear part matches $\scriptstyle e_{i}$ to $\scriptstyle e_{\sigma (i)}$ for all i. hence every two n-paths are isometric, and so is their convex hulls; this explains the congruence of the simplexes. To show the other assertions, it suffices to remark that the interior of the simplex determined by the n-path $\scriptstyle v_{0},\ v_{0}+e_{\sigma (1)},\ v_{0}+e_{\sigma (1)}+e_{\sigma (2)}\ldots v_{0}+e_{\sigma (1)}+\cdots +e_{\sigma (n)}$ is the set of points $\scriptstyle v_{0}+(x_{1}+\cdots +x_{n})e_{\sigma (1)}+\cdots +(x_{n-1}+x_{n})e_{\sigma (n-1)}+x_{n}e_{\sigma (n)}$, with $\scriptstyle 0<x_{i}<1$ and $\scriptstyle x_{1}+\cdots +x_{n}<1.$ Hence the components of these points with respect to each corresponding permuted basis are strictly ordered in the decreasing order. That explains why the simplexes are non-overlapping. The fact that the union of the simplexes is the whole unit n-hypercube follows as well, replacing the strict inequalities above by "$\scriptstyle \leq $". The same arguments are also valid for a general parallelotope, except the isometry between the simplexes.
13. Parks, Harold R.; Wills, Dean C. (October 2002). "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex". American Mathematical Monthly. 109 (8): 756–8. doi:10.2307/3072403. JSTOR 3072403.
14. Wills, Harold R.; Parks, Dean C. (June 2009). Connections between combinatorics of permutations and algorithms and geometry (PhD). Oregon State University. hdl:1957/11929.
15. Donchian, P. S.; Coxeter, H. S. M. (July 1935). "1142. An n-dimensional extension of Pythagoras' Theorem". The Mathematical Gazette. 19 (234): 206. doi:10.2307/3605876. JSTOR 3605876. S2CID 125391795.
16. Lee, John M. (2006). Introduction to Topological Manifolds. Springer. pp. 292–3. ISBN 978-0-387-22727-6.
17. Cornell, John (2002). Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data (third ed.). Wiley. ISBN 0-471-07916-2.
18. Vondran, Gary L. (April 1998). "Radial and Pruned Tetrahedral Interpolation Techniques" (PDF). HP Technical Report. HPL-98-95: 1–32.
References
• Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. ISBN 0-07-054235-X. (See chapter 10 for a simple review of topological properties.)
• Tanenbaum, Andrew S. (2003). "§2.5.3". Computer Networks (4th ed.). Prentice Hall. ISBN 0-13-066102-3.
• Devroye, Luc (1986). Non-Uniform Random Variate Generation. Springer. ISBN 0-387-96305-7. Archived from the original on 2009-05-05.
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). Dover. ISBN 0-486-61480-8.
• pp. 120–121, §7.2. see illustration 7-2A
• p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
• Weisstein, Eric W. "Simplex". MathWorld.
• Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge University Press. ISBN 978-1-107-39400-1. As PDF
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Heptadecagon
In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.
Regular heptadecagon
A regular heptadecagon
TypeRegular polygon
Edges and vertices17
Schläfli symbol{17}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D17), order 2×17
Internal angle (degrees)≈158.82°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
Regular heptadecagon
A regular heptadecagon is represented by the Schläfli symbol {17}.
Construction
As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19.[1] This proof represented the first progress in regular polygon construction in over 2000 years.[1] Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of $N$, the number of sides of the regular polygon, are distinct Fermat primes, which are of the form $F_{n}=2^{2^{n}}+1$ for some nonnegative integer $n$. Constructing a regular heptadecagon thus involves finding the cosine of $2\pi /17$ in terms of square roots. Gauss's book Disquisitiones Arithmeticae[2] gives this (in modern notation) as[3]
${\begin{aligned}\cos {\frac {2\pi }{17}}=&{\frac {1}{16}}\left({\sqrt {17}}-1+{\sqrt {34-2{\sqrt {17}}}}\right)\\&+{\frac {1}{8}}\left({\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\right).\\\end{aligned}}$
Constructions for the regular triangle, pentagon, pentadecagon, and polygons with 2h times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. (The only known Fermat primes are Fn for n = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)
The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893. The following method of construction uses Carlyle circles, as shown below. Based on the construction of the regular 17-gon, one can readily construct n-gons with n being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular n-gon with 2h times as many sides.
Another construction of the regular heptadecagon using straightedge and compass is the following:
T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in The Analyst in the year 1874:[5]
"To construct a regular polygon of seventeen sides in a circle. Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."
The following simple design comes from Herbert William Richmond from the year 1893:[6]
"LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N3 and N5; then if ordinates N3P3, N5P5 are drawn to the circle, the arcs AP3, AP5 will be 3/17 and 5/17 of the circumference."
• The point N3 is very close to the center point of Thales' theorem over AF.
The following construction is a variation of H. W. Richmond's construction.
The differences to the original:
• The circle k2 determines the point H instead of the bisector w3.
• The circle k4 around the point G' (reflection of the point G at m) yields the point N, which is no longer so close to M, for the construction of the tangent.
• Some names have been changed.
Another more recent construction is given by Callagy.[3]
Exact value of sin and cos of mπ/(17 × 2n)
If $A={\sqrt {2(17\pm {\sqrt {17}})}}$, $B=({\sqrt {17}}\pm 1)$ and $C=17\mp 4{\sqrt {17}}$ then, depending on any integer m
$\cos {\frac {m\pi }{17}}=\pm {\frac {(A\pm B)\pm 2{\sqrt {(A\mp B){\sqrt {C}}}}}{16}}$
$=\pm {\frac {{\sqrt {34\pm {\sqrt {68}}}}\pm ({\sqrt {17}}\pm 1)\pm 2{\sqrt {{\sqrt {34\pm {\sqrt {68}}}}\mp ({\sqrt {17}}\pm 1)}}{\sqrt {\sqrt {17\mp {\sqrt {272}}}}}}{16}}$
For example, if m = 1
$\cos {\frac {\pi }{17}}={\frac {{\sqrt {34-{\sqrt {68}}}}-{\sqrt {17}}+1+2{\sqrt {{\sqrt {34-{\sqrt {68}}}}+{\sqrt {17}}-1}}{\sqrt {\sqrt {17+{\sqrt {272}}}}}}{16}}$
Here are the expressions simplified into the following table.
Cos and Sin (m π / 17) in first quadrant, from which other quadrants are computible.
m16 cos (m π / 17)8 sin (m π / 17)
1$+1-{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}+{\sqrt {2720+{\sqrt {6284288}}}}}}$${\sqrt {34-{\sqrt {68}}-{\sqrt {136-{\sqrt {1088}}}}-{\sqrt {272+{\sqrt {39168}}-{\sqrt {43520+{\sqrt {1608777728}}}}}}}}$
2$-1+{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}-{\sqrt {2720+{\sqrt {6284288}}}}}}$${\sqrt {34-{\sqrt {68}}+{\sqrt {136-{\sqrt {1088}}}}-{\sqrt {272+{\sqrt {39168}}+{\sqrt {43520+{\sqrt {1608777728}}}}}}}}$
3$+1+{\sqrt {17}}+{\sqrt {34+{\sqrt {68}}}}+{\sqrt {68-{\sqrt {2448}}-{\sqrt {2720-{\sqrt {6284288}}}}}}$${\sqrt {34+{\sqrt {68}}-{\sqrt {136+{\sqrt {1088}}}}-{\sqrt {272-{\sqrt {39168}}+{\sqrt {43520-{\sqrt {1608777728}}}}}}}}$
4$-1+{\sqrt {17}}-{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}+{\sqrt {2720+{\sqrt {6284288}}}}}}$${\sqrt {34-{\sqrt {68}}-{\sqrt {136-{\sqrt {1088}}}}+{\sqrt {272+{\sqrt {39168}}-{\sqrt {43520+{\sqrt {1608777728}}}}}}}}$
5$+1+{\sqrt {17}}+{\sqrt {34+{\sqrt {68}}}}-{\sqrt {68-{\sqrt {2448}}-{\sqrt {2720-{\sqrt {6284288}}}}}}$${\sqrt {34+{\sqrt {68}}-{\sqrt {136+{\sqrt {1088}}}}+{\sqrt {272-{\sqrt {39168}}+{\sqrt {43520-{\sqrt {1608777728}}}}}}}}$
6$-1-{\sqrt {17}}+{\sqrt {34+{\sqrt {68}}}}+{\sqrt {68-{\sqrt {2448}}+{\sqrt {2720-{\sqrt {6284288}}}}}}$${\sqrt {34+{\sqrt {68}}+{\sqrt {136+{\sqrt {1088}}}}-{\sqrt {272-{\sqrt {39168}}-{\sqrt {43520-{\sqrt {1608777728}}}}}}}}$
7$+1+{\sqrt {17}}-{\sqrt {34+{\sqrt {68}}}}+{\sqrt {68-{\sqrt {2448}}+{\sqrt {2720-{\sqrt {6284288}}}}}}$${\sqrt {34+{\sqrt {68}}+{\sqrt {136+{\sqrt {1088}}}}+{\sqrt {272-{\sqrt {39168}}-{\sqrt {43520-{\sqrt {1608777728}}}}}}}}$
8$-1+{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}-{\sqrt {68+{\sqrt {2448}}-{\sqrt {2720+{\sqrt {6284288}}}}}}$${\sqrt {34-{\sqrt {68}}+{\sqrt {136-{\sqrt {1088}}}}+{\sqrt {272+{\sqrt {39168}}+{\sqrt {43520+{\sqrt {1608777728}}}}}}}}$
Therefore, applying induction with m=1 and starting with n=0:
$\cos {\frac {\pi }{17\times 2^{0}}}={\frac {1-{\sqrt {17}}+{\sqrt {34-{\sqrt {68}}}}+{\sqrt {68+{\sqrt {2448}}+{\sqrt {2720+{\sqrt {6284288}}}}}}}{16}}$
$\cos {\frac {\pi }{17\times 2^{n+1}}}={\frac {\sqrt {2+2\cos {\frac {\pi }{17\times 2^{n}}}}}{2}}$ and $\sin {\frac {\pi }{17\times 2^{n+1}}}={\frac {\sqrt {2-2\cos {\frac {\pi }{17\times 2^{n}}}}}{2}}.$
Symmetry
The regular heptadecagon has Dih17 symmetry, order 34. Since 17 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z17, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon. John Conway labels these by a letter and group order.[7] Full symmetry of the regular form is r34 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g17 subgroup has no degrees of freedom but can seen as directed edges.
Related polygons
Heptadecagrams
A heptadecagram is a 17-sided star polygon. There are seven regular forms given by Schläfli symbols: {17/2}, {17/3}, {17/4}, {17/5}, {17/6}, {17/7}, and {17/8}. Since 17 is a prime number, all of these are regular stars and not compound figures.
Picture
{17/2}
{17/3}
{17/4}
{17/5}
{17/6}
{17/7}
{17/8}
Interior angle ≈137.647° ≈116.471° ≈95.2941° ≈74.1176° ≈52.9412° ≈31.7647° ≈10.5882°
Petrie polygons
The regular heptadecagon is the Petrie polygon for one higher-dimensional regular convex polytope, projected in a skew orthogonal projection:
16-simplex (16D)
References
1. Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, Abstract Algebra and Famous Impossibilities, Springer, 1991, ISBN 0387976612, p. 178.
2. Carl Friedrich Gauss "Disquisitiones Arithmeticae" eod books2ebooks, p. 662 item 365.
3. Callagy, James J. "The central angle of the regular 17-gon", Mathematical Gazette 67, December 1983, 290–292.
4. Duane W. DeTemple "Carlyle Circles and the Lemoine Simplicity of Polygon Constructions" in The American Mathematical Monthly,Volume 98, Issuc 1 (Feb. 1991), 97–108. "4. Construction of the Regular Heptadecagon (17-gon)" pp. 101–104, , p.103, web.archive document, selected on 28 January 2017
5. Hendricks, J. E. (1874). "Answer to Mr. Heal's Query; T. P. Stowell of Rochester, N. Y." The Analyst: A Monthly Journal of Pure and Applied Mathematicus. 1: 94–95. Query, by W. E. Heal, Wheeling, Indiana p. 64; accessdate 30 April 2017
6. Herbert W. Richmond, description "A Construction for a regular polygon of seventeen side" illustration (Fig. 6), The Quarterly Journal of Pure and Applied Mathematics 26: pp. 206–207. Retrieved 4 December 2015
7. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
Further reading
• Dunham, William (September 1996). "1996—a triple anniversary". Math Horizons. 4: 8–13. doi:10.1080/10724117.1996.11974982. Retrieved 6 December 2009.
• Klein, Felix et al. Famous Problems and Other Monographs. – Describes the algebraic aspect, by Gauss.
External links
Wikimedia Commons has media related to 17-gons.
• Weisstein, Eric W. "Heptadecagon". MathWorld. Contains a description of the construction.
• "Constructing the Heptadecagon". MathPages.com.
• Heptadecagon trigonometric functions
• BBC video of New R&D center for SolarUK
• Archived at Ghostarchive and the Wayback Machine: Eisenbud, David. "The Amazing Heptadecagon (17-gon)" (Video). Brady Haran. Retrieved 2 March 2015.
• OEIS: A210644
Polygons (List)
Triangles
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By number
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11–20 sides
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>20 sides
• Icositrigon (23)
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Star polygons
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Classes
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History of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.
Mathematics
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• Number theory
• Geometry
• Algebra
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• Discrete mathematics
• Logic and set theory
• Probability
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The earliest mathematical texts available are from Mesopotamia and Egypt – Plimpton 322 (Babylonian c. 2000 – 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian c. 1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".[4] Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system and the first use of negative numbers.[6][7] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī.[8][9] Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations.[10] Contemporaneous with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation.[11] Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century.
Table of numerals
European (descended from the West Arabic) 0123456789
Arabic-Indic ٠١٢٣٤٥٦٧٨٩
Eastern Arabic-Indic (Persian and Urdu) ۰۱۲۳۴۵۶۷۸۹
Devanagari (Hindi) ०१२३४५६७८९
Chinese - Japanese 〇一二三四五六七八九
Tamil ௧௨௩௪௫௬௭௮௯
Prehistoric
The origins of mathematical thought lie in the concepts of number, patterns in nature, magnitude, and form.[12] Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. The idea of the "number" concept evolving gradually over time is supported by the existence of languages which preserve the distinction between "one", "two", and "many", but not of numbers larger than two.[12]
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and consists of a series of marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers[13] or a six-month lunar calendar.[14] Peter Rudman argues that the development of the concept of prime numbers could only have come about after the concept of division, which he dates to after 10,000 BC, with prime numbers probably not being understood until about 500 BC. He also writes that "no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10."[15] The Ishango bone, according to scholar Alexander Marshack, may have influenced the later development of mathematics in Egypt as, like some entries on the Ishango bone, Egyptian arithmetic also made use of multiplication by 2; this however, is disputed.[16]
Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.[17] All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.[18]
Babylonian
Main article: Babylonian mathematics
See also: Plimpton 322
Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (modern Iraq) from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity.[19] The majority of Babylonian mathematical work comes from two widely separated periods: The first few hundred years of the second millennium BC (Old Babylonian period), and the last few centuries of the first millennium BC (Seleucid period).[20] It is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics.
In contrast to the sparsity of sources in Egyptian mathematics, knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.[21] Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework.[22]
The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.[23] From around 2500 BC onward, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[24]
Babylonian mathematics were written using a sexagesimal (base-60) numeral system.[21] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is thought the sexagesimal system was initially used by Sumerian scribes because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30,[21] and for scribes (doling out the aforementioned grain allotments, recording weights of silver, etc) being able to easily calculate by hand was essential, and so a sexagesimal system is pragmatically easier to calculate by hand with; however, there is the possibility that using a sexagesimal system was an ethno-linguistic phenomenon (that might not ever be known), and not a mathematical/practical decision.[25] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the [[decimal]] system.The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation. The notational system of the Babylonians was the best of any civilization until the Renaissance,and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.[26] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.[20] By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.[20] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.[20]
Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.[27] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.[28] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.[29] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.[22]
Egyptian
Main article: Egyptian mathematics
Egyptian mathematics refers to mathematics written in the Egyptian language. From the Hellenistic period, Greek replaced Egyptian as the written language of Egyptian scholars. Mathematical study in Egypt later continued under the Arab Empire as part of Islamic mathematics, when Arabic became the written language of Egyptian scholars. Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.[30] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.[31]
The most extensive Egyptian mathematical text is the Rhind papyrus (sometimes also called the Ahmes Papyrus after its author), dated to c. 1650 BC but likely a copy of an older document from the Middle Kingdom of about 2000–1800 BC.[32] It is an instruction manual for students in arithmetic and geometry. In addition to giving area formulas and methods for multiplication, division and working with unit fractions, it also contains evidence of other mathematical knowledge,[33] including composite and prime numbers; arithmetic, geometric and harmonic means; and simplistic understandings of both the Sieve of Eratosthenes and perfect number theory (namely, that of the number 6).[34] It also shows how to solve first order linear equations[35] as well as arithmetic and geometric series.[36]
Another significant Egyptian mathematical text is the Moscow papyrus, also from the Middle Kingdom period, dated to c. 1890 BC.[37] It consists of what are today called word problems or story problems, which were apparently intended as entertainment. One problem is considered to be of particular importance because it gives a method for finding the volume of a frustum (truncated pyramid).
Finally, the Berlin Papyrus 6619 (c. 1800 BC) shows that ancient Egyptians could solve a second-order algebraic equation.[38]
Greek
Main article: Greek mathematics
Greek mathematics refers to the mathematics written in the Greek language from the time of Thales of Miletus (~600 BC) to the closure of the Academy of Athens in 529 AD.[39] Greek mathematicians lived in cities spread over the entire Eastern Mediterranean, from Italy to North Africa, but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics.[40]
Greek mathematics was much more sophisticated than the mathematics that had been developed by earlier cultures. All surviving records of pre-Greek mathematics show the use of inductive reasoning, that is, repeated observations used to establish rules of thumb. Greek mathematicians, by contrast, used deductive reasoning. The Greeks used logic to derive conclusions from definitions and axioms, and used mathematical rigor to prove them.[41]
Greek mathematics is thought to have begun with Thales of Miletus (c. 624–c.546 BC) and Pythagoras of Samos (c. 582–c. 507 BC). Although the extent of the influence is disputed, they were probably inspired by Egyptian and Babylonian mathematics. According to legend, Pythagoras traveled to Egypt to learn mathematics, geometry, and astronomy from Egyptian priests.
Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.[42] Pythagoras established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number".[43] It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem,[44] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers.[45][46] Although he was preceded by the Babylonians, Indians and the Chinese,[47] the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables, whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum).[48] The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica.[49]
Plato (428/427 BC – 348/347 BC) is important in the history of mathematics for inspiring and guiding others.[50] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came.[51] Plato also discussed the foundations of mathematics,[52] clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.[53] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.[51]
Eudoxus (408–c. 355 BC) developed the method of exhaustion, a precursor of modern integration[54] and a theory of ratios that avoided the problem of incommensurable magnitudes.[55] The former allowed the calculations of areas and volumes of curvilinear figures,[56] while the latter enabled subsequent geometers to make significant advances in geometry. Though he made no specific technical mathematical discoveries, Aristotle (384–c. 322 BC) contributed significantly to the development of mathematics by laying the foundations of logic.[57]
In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.[59] It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.[1] The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.[60] The Elements was known to all educated people in the West up through the middle of the 20th century and its contents are still taught in geometry classes today.[61] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,[60] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.[62]
Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity,[63] used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.[64] He also showed one could use the method of exhaustion to calculate the value of π with as much precision as desired, and obtained the most accurate value of π then known, 310/71 < π < 310/70.[65] He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid),[64] and an ingenious method of exponentiation for expressing very large numbers.[66] While he is also known for his contributions to physics and several advanced mechanical devices, Archimedes himself placed far greater value on the products of his thought and general mathematical principles.[67] He regarded as his greatest achievement his finding of the surface area and volume of a sphere, which he obtained by proving these are 2/3 the surface area and volume of a cylinder circumscribing the sphere.[68]
Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.[69] He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").[70] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.[71] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.[72]
Around the same time, Eratosthenes of Cyrene (c. 276–194 BC) devised the Sieve of Eratosthenes for finding prime numbers.[73] The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.[74] Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.[74] Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.[75] Heron of Alexandria (c. 10–70 AD) is credited with Heron's formula for finding the area of a scalene triangle and with being the first to recognize the possibility of negative numbers possessing square roots.[76] Menelaus of Alexandria (c. 100 AD) pioneered spherical trigonometry through Menelaus' theorem.[77] The most complete and influential trigonometric work of antiquity is the Almagest of Ptolemy (c. AD 90–168), a landmark astronomical treatise whose trigonometric tables would be used by astronomers for the next thousand years.[78] Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.[79]
Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.[80] During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis".[81] The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.[82] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).[83] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.[82]
Among the last great Greek mathematicians is Pappus of Alexandria (4th century AD). He is known for his hexagon theorem and centroid theorem, as well as the Pappus configuration and Pappus graph. His Collection is a major source of knowledge on Greek mathematics as most of it has survived.[84] Pappus is considered the last major innovator in Greek mathematics, with subsequent work consisting mostly of commentaries on earlier work.
The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415). She succeeded her father (Theon of Alexandria) as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed.[85] Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius.[86] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.[87] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.[88]
Roman
Further information: Roman abacus and Roman numerals
Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire, there were no noteworthy native Latin mathematicians in comparison.[89][90] Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surveyors and calculators were far more interested in applied mathematics than the theoretical mathematics and geometry that were prized by the Greeks.[91] It is unclear if the Romans first derived their numerical system directly from the Greek precedent or from Etruscan numerals used by the Etruscan civilization centered in what is now Tuscany, central Italy.[92]
Using calculation, Romans were adept at both instigating and detecting financial fraud, as well as managing taxes for the treasury.[93] Siculus Flaccus, one of the Roman gromatici (i.e. land surveyor), wrote the Categories of Fields, which aided Roman surveyors in measuring the surface areas of allotted lands and territories.[94] Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering, including the erection of architecture such as bridges, road-building, and preparation for military campaigns.[95] Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.[96][97]
The creation of the Roman calendar also necessitated basic mathematics. The first calendar allegedly dates back to 8th century BC during the Roman Kingdom and included 356 days plus a leap year every other year.[98] In contrast, the lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February.[99] This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle.[100] This calendar, which contained an error of 11 minutes and 14 seconds, was later corrected by the Gregorian calendar organized by Pope Gregory XIII (r. 1572–1585), virtually the same solar calendar used in modern times as the international standard calendar.[101]
At roughly the same time, the Han Chinese and the Romans both invented the wheeled odometer device for measuring distances traveled, the Roman model first described by the Roman civil engineer and architect Vitruvius (c. 80 BC – c. 15 BC).[102] The device was used at least until the reign of emperor Commodus (r. 177 – 192 AD), but its design seems to have been lost until experiments were made during the 15th century in Western Europe.[103] Perhaps relying on similar gear-work and technology found in the Antikythera mechanism, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m). With each revolution, a pin-and-axle device engaged a 400-tooth cogwheel that turned a second gear responsible for dropping pebbles into a box, each pebble representing one mile traversed.[104]
Chinese
Main article: Chinese mathematics
An analysis of early Chinese mathematics has demonstrated its unique development compared to other parts of the world, leading scholars to assume an entirely independent development.[105] The oldest extant mathematical text from China is the Zhoubi Suanjing (周髀算經), variously dated to between 1200 BC and 100 BC, though a date of about 300 BC during the Warring States Period appears reasonable.[106] However, the Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.[47]
Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.[107] Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.[108] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's Supplementary Notes on the Art of Figures.
The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.[109] It also defined the concepts of circumference, diameter, radius, and volume.[110]
In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles.[106] It created mathematical proof for the Pythagorean theorem,[111] and a mathematical formula for Gaussian elimination.[112] The treatise also provides values of π,[106] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,[113] as well as 3.162 by taking the square root of 10.[114][115] Liu Hui commented on the Nine Chapters in the 3rd century AD and gave a value of π accurate to 5 decimal places (i.e. 3.14159).[116][117] Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years.[116][118] He also established a method which would later be called Cavalieri's principle to find the volume of a sphere.[119]
The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.[116] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.[120] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).[120]
Even after European mathematics began to flourish during the Renaissance, European and Chinese mathematics were separate traditions, with significant Chinese mathematical output in decline from the 13th century onwards. Jesuit missionaries such as Matteo Ricci carried mathematical ideas back and forth between the two cultures from the 16th to 18th centuries, though at this point far more mathematical ideas were entering China than leaving.[120]
Japanese mathematics, Korean mathematics, and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere.[121] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644).[122] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.[123] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.[124]
Indian
Main article: Indian mathematics
See also: History of the Hindu–Arabic numeral system
Indian numerals in stone and copper inscriptions[125]
Ancient Brahmi numerals in a part of India
The earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature second phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.[126]
The oldest extant mathematical records from India are the Sulba Sutras (dated variously between the 8th century BC and the 2nd century AD),[127] appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.[128] As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.[127] The Sulba Sutras give methods for constructing a circle with approximately the same area as a given square, which imply several different approximations of the value of π.[129][130][lower-alpha 1] In addition, they compute the square root of 2 to several decimal places, list Pythagorean triples, and give a statement of the Pythagorean theorem.[130] All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.[127] It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.[127]
Pāṇini (c. 5th century BC) formulated the rules for Sanskrit grammar.[131] His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion.[132] Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system.[133][134] His discussion of the combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).[135]
The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence.[136] They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.[137] Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".[137]
Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.[138]It is in the Aryabhatiya that the decimal place-value system first appears. Several centuries later, the Muslim mathematician Abu Rayhan Biruni described the Aryabhatiya as a "mix of common pebbles and costly crystals".[139]
In the 7th century, Brahmagupta identified the Brahmagupta theorem, Brahmagupta's identity and Brahmagupta's formula, and for the first time, in Brahma-sphuta-siddhanta, he lucidly explained the use of zero as both a placeholder and decimal digit, and explained the Hindu–Arabic numeral system.[140] It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as Arabic numerals. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the Brahmi numerals. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.
In the 12th century, Bhāskara II[141] lived in southern India and wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, derivatives, the mean value theorem and the derivative of the sine function. To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics.[142]
In the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and obtained from it a transformed series, whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found the Madhava-Gregory series to determine the arctangent, the Madhava-Newton power series to determine sine and cosine and the Taylor approximation for sine and cosine functions.[143] In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the Yukti-bhāṣā.[144] [145] It has been argued that the advances of the Kerala school, which laid the foundations of the calculus, were transmitted to Europe in the 16th century[146] via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time and, as a result, directly influenced later European developments in analysis and calculus.[147] However, other scholars argue that the Kerala School did not formulate a systematic theory of differentiation and integration, and that there is not any direct evidence of their results being transmitted outside Kerala.[148][149][150][151]
Islamic empires
Main article: Mathematics in medieval Islam
See also: History of the Hindu–Arabic numeral system
The Islamic Empire established across the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of Greek in the Hellenistic world, Arabic was used as the written language of non-Arab scholars throughout the Islamic world at the time.[152]
In the 9th century, the mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[153] and he was the first to teach algebra in an elementary form and for its own sake.[154] He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[155] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[156]
In Egypt, Abu Kamil extended algebra to the set of irrational numbers, accepting square roots and fourth roots as solutions and coefficients to quadratic equations. He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions.[157] His works formed an important foundation for the development of algebra and influenced later mathematicians, such as al-Karaji and Fibonacci.
Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[158] The historian of mathematics, F. Woepcke,[159] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.[160]
In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations. He was also very influential in calendar reform.[161]
In the 13th century, Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.
Other achievements of Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals, the discovery of all the modern trigonometric functions besides the sine, al-Kindi's introduction of cryptanalysis and frequency analysis, the development of analytic geometry by Ibn al-Haytham, the beginning of algebraic geometry by Omar Khayyam and the development of an algebraic notation by al-Qalasādī.[162]
During the time of the Ottoman Empire and Safavid Empire from the 15th century, the development of Islamic mathematics became stagnant.
Maya
In the Pre-Columbian Americas, the Maya civilization that flourished in Mexico and Central America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.[163] Maya numerals used a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures.[163] The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy.[163] While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.[163]
Medieval European
Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage (in the Book of Wisdom) that God had ordered all things in measure, and number, and weight.[164]
Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.[165][166]
In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.[167][168] These and other new sources sparked a renewal of mathematics.
Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that)[169] which Fibonacci used as an unremarkable example.
The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.[170] One important contribution was development of mathematics of local motion.
Thomas Bradwardine proposed that speed (V) increases in arithmetic proportion as the ratio of force (F) to resistance (R) increases in geometric proportion. Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R).[171] Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.[172]
One of the 14th-century Oxford Calculators, William Heytesbury, lacking differential calculus and the concept of limits, proposed to measure instantaneous speed "by the path that would be described by [a body] if... it were moved uniformly at the same degree of speed with which it is moved in that given instant".[174]
Heytesbury and others mathematically determined the distance covered by a body undergoing uniformly accelerated motion (today solved by integration), stating that "a moving body uniformly acquiring or losing that increment [of speed] will traverse in some given time a [distance] completely equal to that which it would traverse if it were moving continuously through the same time with the mean degree [of speed]".[175]
Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.[176] In a later mathematical commentary on Euclid's Elements, Oresme made a more detailed general analysis in which he demonstrated that a body will acquire in each successive increment of time an increment of any quality that increases as the odd numbers. Since Euclid had demonstrated the sum of the odd numbers are the square numbers, the total quality acquired by the body increases as the square of the time.[177]
Renaissance
Further information: Mathematics and art
During the Renaissance, the development of mathematics and of accounting were intertwined.[178] While there is no direct relationship between algebra and accounting, the teaching of the subjects and the books published often intended for the children of merchants who were sent to reckoning schools (in Flanders and Germany) or abacus schools (known as abbaco in Italy), where they learned the skills useful for trade and commerce. There is probably no need for algebra in performing bookkeeping operations, but for complex bartering operations or the calculation of compound interest, a basic knowledge of arithmetic was mandatory and knowledge of algebra was very useful.
Piero della Francesca (c. 1415–1492) wrote books on solid geometry and linear perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De quinque corporibus regularibus (On the Five Regular Solids).[179][180][181]
Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494. It included a 27-page treatise on bookkeeping, "Particularis de Computis et Scripturis" (Italian: "Details of Calculation and Recording"). It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons.[182] In Summa Arithmetica, Pacioli introduced symbols for plus and minus for the first time in a printed book, symbols that became standard notation in Italian Renaissance mathematics. Summa Arithmetica was also the first known book printed in Italy to contain algebra. Pacioli obtained many of his ideas from Piero Della Francesca whom he plagiarized.
In Italy, during the first half of the 16th century, Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations.
Simon Stevin's book De Thiende ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation, which influenced all later work on the real number system.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Regiomontanus's table of sines and cosines was published in 1533.[183]
During the Renaissance the desire of artists to represent the natural world realistically, together with the rediscovered philosophy of the Greeks, led artists to study mathematics. They were also the engineers and architects of that time, and so had need of mathematics in any case. The art of painting in perspective, and the developments in geometry that involved, were studied intensely.[184]
Mathematics during the Scientific Revolution
17th century
The 17th century saw an unprecedented increase of mathematical and scientific ideas across Europe. Galileo observed the moons of Jupiter in orbit about that planet, using a telescope based Hans Lipperhey's. Tycho Brahe had gathered a large quantity of mathematical data describing the positions of the planets in the sky. By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion.[185] The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.
Building on earlier work by many predecessors, Isaac Newton discovered the laws of physics that explain Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, developed calculus and much of the calculus notation still in use today. He also refined the binary number system, which is the foundation of nearly all digital (electronic, solid-state, discrete logic) computers, including the Von Neumann architecture, which is the standard design paradigm, or "computer architecture", followed from the second half of the 20th century, and into the 21st. Leibniz has been called the "founder of computer science".[186]
Science and mathematics had become an international endeavor, which would soon spread over the entire world.[187]
In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence of Pierre de Fermat and Blaise Pascal. Pascal and Fermat set the groundwork for the investigations of probability theory and the corresponding rules of combinatorics in their discussions over a game of gambling. Pascal, with his wager, attempted to use the newly developing probability theory to argue for a life devoted to religion, on the grounds that even if the probability of success was small, the rewards were infinite. In some sense, this foreshadowed the development of utility theory in the 18th–19th century.
18th century
The most influential mathematician of the 18th century was arguably Leonhard Euler (1707–1783). His contributions range from founding the study of graph theory with the Seven Bridges of Königsberg problem to standardizing many modern mathematical terms and notations. For example, he named the square root of minus 1 with the symbol i, and he popularized the use of the Greek letter $\pi $ to stand for the ratio of a circle's circumference to its diameter. He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.
Other important European mathematicians of the 18th century included Joseph Louis Lagrange, who did pioneering work in number theory, algebra, differential calculus, and the calculus of variations, and Pierre-Simon Laplace, who, in the age of Napoleon, did important work on the foundations of celestial mechanics and on statistics.
Modern
19th century
Throughout the 19th century mathematics became increasingly abstract.[188] Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where uniqueness of parallels no longer holds. In this geometry the sum of angles in a triangle add up to less than 180°. Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the concept of a manifold, which generalizes the ideas of curves and surfaces.
The 19th century saw the beginning of a great deal of abstract algebra. Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the starting point of mathematical logic and has important applications in electrical engineering and computer science. Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion.
Also, for the first time, the limits of mathematics were explored. Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to trisect an arbitrary angle, to construct the side of a cube twice the volume of a given cube, nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks. On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers.
Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra. In the 20th century physicists and other scientists have seen group theory as the ideal way to study symmetry.
In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L.E.J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.
The 19th century saw the founding of a number of national mathematical societies: the London Mathematical Society in 1865, the Société Mathématique de France in 1872, the Circolo Matematico di Palermo in 1884, the Edinburgh Mathematical Society in 1883, and the American Mathematical Society in 1888. The first international, special-interest society, the Quaternion Society, was formed in 1899, in the context of a vector controversy.
In 1897, Kurt Hensel introduced p-adic numbers.
20th century
The 20th century saw mathematics become a major profession. By the end of the century, thousands of new Ph.D.s in mathematics were being awarded every year, and jobs were available in both teaching and industry.[189] An effort to catalogue the areas and applications of mathematics was undertaken in Klein's encyclopedia.
In a 1900 speech to the International Congress of Mathematicians, David Hilbert set out a list of 23 unsolved problems in mathematics. These problems, spanning many areas of mathematics, formed a central focus for much of 20th-century mathematics. Today, 10 have been solved, 7 are partially solved, and 2 are still open. The remaining 4 are too loosely formulated to be stated as solved or not.
Notable historical conjectures were finally proven. In 1976, Wolfgang Haken and Kenneth Appel proved the four color theorem, controversial at the time for the use of a computer to do so. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. Paul Cohen and Kurt Gödel proved that the continuum hypothesis is independent of (could neither be proved nor disproved from) the standard axioms of set theory. In 1998 Thomas Callister Hales proved the Kepler conjecture.
Mathematical collaborations of unprecedented size and scope took place. An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole. The resulting several dozen volumes has had a controversial influence on mathematical education.[190]
Differential geometry came into its own when Albert Einstein used it in general relativity. Entirely new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do, the concept of an abstract structure was itself abstracted and led to category theory. Grothendieck and Serre recast algebraic geometry using sheaf theory. Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s. Measure theory was developed in the late 19th and early 20th centuries. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. Knot theory greatly expanded. Quantum mechanics led to the development of functional analysis. Other new areas include Laurent Schwartz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals. Lie theory with its Lie groups and Lie algebras became one of the major areas of study.
Non-standard analysis, introduced by Abraham Robinson, rehabilitated the infinitesimal approach to calculus, which had fallen into disrepute in favour of the theory of limits, by extending the field of real numbers to the Hyperreal numbers which include infinitesimal and infinite quantities. An even larger number system, the surreal numbers were discovered by John Horton Conway in connection with combinatorial games.
The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with this: Alan Turing's computability theory; complexity theory; Derrick Henry Lehmer's use of ENIAC to further number theory and the Lucas-Lehmer test; Rózsa Péter's recursive function theory; Claude Shannon's information theory; signal processing; data analysis; optimization and other areas of operations research. In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory. The speed and data processing abilities of computers also enabled the handling of mathematical problems that were too time-consuming to deal with by pencil and paper calculations, leading to areas such as numerical analysis and symbolic computation. Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error-correcting codes, the Kalman filter from control theory and the RSA algorithm of public-key cryptography.
At the same time, deep insights were made about the limitations to mathematics. In 1929 and 1930, it was proved the truth or falsity of all statements formulated about the natural numbers plus either addition or multiplication (but not both), was decidable, i.e. could be determined by some algorithm. In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incompleteness theorems is that in any mathematical system that includes Peano arithmetic (including all of analysis and geometry), truth necessarily outruns proof, i.e. there are true statements that cannot be proved within the system. Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated.
One of the more colorful figures in 20th-century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved over 3000 theorems, including properties of highly composite numbers, the partition function and its asymptotics, and mock theta functions. He also made major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Paul Erdős published more papers than any other mathematician in history,[191] working with hundreds of collaborators. Mathematicians have a game equivalent to the Kevin Bacon Game, which leads to the Erdős number of a mathematician. This describes the "collaborative distance" between a person and Erdős, as measured by joint authorship of mathematical papers.
Emmy Noether has been described by many as the most important woman in the history of mathematics.[192] She studied the theories of rings, fields, and algebras.
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized areas in mathematics and the Mathematics Subject Classification was dozens of pages long.[193] More and more mathematical journals were published and, by the end of the century, the development of the World Wide Web led to online publishing.
21st century
See also: List of unsolved problems in mathematics § Problems solved since 1995
In 2000, the Clay Mathematics Institute announced the seven Millennium Prize Problems, and in 2003 the Poincaré conjecture was solved by Grigori Perelman (who declined to accept an award, as he was critical of the mathematics establishment).
Most mathematical journals now have online versions as well as print versions, and many online-only journals are launched. There is an increasing drive toward open access publishing, first popularized by arXiv.
Future
Main article: Future of mathematics
There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, and the volume of data being produced by science and industry, facilitated by computers, is expanding exponentially.
See also
• Archives of American Mathematics
• History of algebra
• History of arithmetic
• History of calculus
• History of combinatorics
• History of the function concept
• History of geometry
• History of logic
• History of mathematicians
• History of mathematical notation
• History of measurement
• History of numbers
• History of ancient numeral systems
• Prehistoric counting
• History of number theory
• History of statistics
• History of trigonometry
• History of writing numbers
• Kenneth O. May Prize
• List of important publications in mathematics
• Lists of mathematicians
• List of mathematics history topics
• Timeline of mathematics
Notes
1. The approximate values for π are 4 x (13/15)2 (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389)
1. (Boyer 1991, "Euclid of Alexandria" p. 119)
2. Friberg, J. (1981). "Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations", Historia Mathematica, 8, pp. 277–318.
3. Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity. pp. 1–191. ISBN 978-0-486-22332-2. PMID 14884919. {{cite book}}: |journal= ignored (help) Chap. IV "Egyptian Mathematics and Astronomy", pp. 71–96.
4. Turnbull (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0. S2CID 3994109.
5. Heath, Thomas L. (1963). A Manual of Greek Mathematics, Dover, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science."
6. Joseph, George Gheverghese (1991). The Crest of the Peacock: Non-European Roots of Mathematics. Penguin Books, London, pp. 140–48.
7. Ifrah, Georges (1986). Universalgeschichte der Zahlen. Campus, Frankfurt/New York, pp. 428–37.
8. Kaplan, Robert (1999). The Nothing That Is: A Natural History of Zero. Allen Lane/The Penguin Press, London.
9. "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
10. Juschkewitsch, A. P. (1964). Geschichte der Mathematik im Mittelalter. Teubner, Leipzig.
11. Eves, Howard (1990). History of Mathematics, 6th Edition, "After Pappus, Greek mathematics ceased to be a living study, ..." p. 185; "The Athenian school struggled on against growing opposition from Christians until the latter finally, in A.D. 529, obtained a decree from Emperor Justinian that closed the doors of the school forever." p. 186; "The period starting with the fall of the Roman Empire, in the middle of the fifth century, and extending into the eleventh century is known in Europe as the Dark Ages... Schooling became almost nonexistent." p. 258.
12. (Boyer 1991, "Origins" p. 3)
13. Williams, Scott W. (2005). "The Oldest Mathematical Object is in Swaziland". Mathematicians of the African Diaspora. SUNY Buffalo mathematics department. Retrieved 2006-05-06.
14. Marshack, Alexander (1991): The Roots of Civilization, Colonial Hill, Mount Kisco, NY.
15. Rudman, Peter Strom (2007). How Mathematics Happened: The First 50,000 Years. Prometheus Books. p. 64. ISBN 978-1-59102-477-4.
16. Marshack, A. (1972). The Roots of Civilization: the Cognitive Beginning of Man's First Art, Symbol and Notation. New York: McGraw-Hill.
17. Thom, Alexander; Archie Thom (1988). "The metrology and geometry of Megalithic Man", pp. 132–51 in Ruggles, C. L. N. (ed.), Records in Stone: Papers in memory of Alexander Thom. Cambridge University Press. ISBN 0-521-33381-4.
18. Damerow, Peter (1996). "The Development of Arithmetical Thinking: On the Role of Calculating Aids in Ancient Egyptian & Babylonian Arithmetic". Abstraction & Representation: Essays on the Cultural Evolution of Thinking (Boston Studies in the Philosophy & History of Science). Springer. ISBN 0792338162. Retrieved 2019-08-17.
19. (Boyer 1991, "Mesopotamia" p. 24)
20. (Boyer 1991, "Mesopotamia" p. 26)
21. (Boyer 1991, "Mesopotamia" p. 25)
22. (Boyer 1991, "Mesopotamia" p. 41)
23. Sharlach, Tonia (2006), "Calendars and Counting", The Sumerian World, Routledge, pp. 307–308, doi:10.4324/9780203096604.ch15, ISBN 978-0-203-09660-4, retrieved 2023-07-07
24. Melville, Duncan J. (2003). Third Millennium Chronology Archived 2018-07-07 at the Wayback Machine, Third Millennium Mathematics. St. Lawrence University.
25. Powell, M. (1976), "The Antecedents of Old Babylonian Place Notation and the Early History of Babylonian Mathematics" (PDF), Historia Mathematica, vol. 3, pp. 417–439, retrieved July 6, 2023
26. (Boyer 1991, "Mesopotamia" p. 27)
27. Aaboe, Asger (1998). Episodes from the Early History of Mathematics. New York: Random House. pp. 30–31.
28. (Boyer 1991, "Mesopotamia" p. 33)
29. (Boyer 1991, "Mesopotamia" p. 39)
30. Eglash, Ron (1999). African fractals : modern computing and indigenous design. New Brunswick, N.J.: Rutgers University Press. pp. 89, 141. ISBN 0813526140.
31. Eglash, R. (1995). "Fractal Geometry in African Material Culture". Symmetry: Culture and Science. 6–1: 174–177.
32. (Boyer 1991, "Egypt" p. 11)
33. Egyptian Unit Fractions at MathPages
34. Egyptian Unit Fractions
35. "Egyptian Papyri". www-history.mcs.st-andrews.ac.uk.
36. "Egyptian Algebra – Mathematicians of the African Diaspora". www.math.buffalo.edu.
37. (Boyer 1991, "Egypt" p. 19)
38. "Egyptian Mathematical Papyri – Mathematicians of the African Diaspora". www.math.buffalo.edu.
39. Eves, Howard (1990). An Introduction to the History of Mathematics, Saunders, ISBN 0-03-029558-0
40. (Boyer 1991, "The Age of Plato and Aristotle" p. 99)
41. Bernal, Martin (2000). "Animadversions on the Origins of Western Science", pp. 72–83 in Michael H. Shank, ed. The Scientific Enterprise in Antiquity and the Middle Ages. Chicago: University of Chicago Press, p. 75.
42. (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
43. (Boyer 1991, "Ionia and the Pythagoreans" p. 49)
44. Eves, Howard (1990). An Introduction to the History of Mathematics, Saunders, ISBN 0-03-029558-0.
45. Kurt Von Fritz (1945). "The Discovery of Incommensurability by Hippasus of Metapontum". The Annals of Mathematics.
46. Choike, James R. (1980). "The Pentagram and the Discovery of an Irrational Number". The Two-Year College Mathematics Journal. 11 (5): 312–316. doi:10.2307/3026893. JSTOR 3026893.
47. Qiu, Jane (7 January 2014). "Ancient times table hidden in Chinese bamboo strips". Nature. doi:10.1038/nature.2014.14482. S2CID 130132289. Retrieved 15 September 2014.
48. David E. Smith (1958), History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics, New York: Dover Publications (a reprint of the 1951 publication), ISBN 0-486-20429-4, pp. 58, 129.
49. Smith, David E. (1958). History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics, New York: Dover Publications (a reprint of the 1951 publication), ISBN 0-486-20429-4, p. 129.
50. (Boyer 1991, "The Age of Plato and Aristotle" p. 86)
51. (Boyer 1991, "The Age of Plato and Aristotle" p. 88)
52. Calian, George F. (2014). "One, Two, Three… A Discussion on the Generation of Numbers" (PDF). New Europe College. Archived from the original (PDF) on 2015-10-15.
53. (Boyer 1991, "The Age of Plato and Aristotle" p. 87)
54. (Boyer 1991, "The Age of Plato and Aristotle" p. 92)
55. (Boyer 1991, "The Age of Plato and Aristotle" p. 93)
56. (Boyer 1991, "The Age of Plato and Aristotle" p. 91)
57. (Boyer 1991, "The Age of Plato and Aristotle" p. 98)
58. Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. Retrieved 2008-09-26.
59. (Boyer 1991, "Euclid of Alexandria" p. 100)
60. (Boyer 1991, "Euclid of Alexandria" p. 104)
61. Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0 p. 141: "No work, except The Bible, has been more widely used...."
62. (Boyer 1991, "Euclid of Alexandria" p. 102)
63. (Boyer 1991, "Archimedes of Syracuse" p. 120)
64. (Boyer 1991, "Archimedes of Syracuse" p. 130)
65. (Boyer 1991, "Archimedes of Syracuse" p. 126)
66. (Boyer 1991, "Archimedes of Syracuse" p. 125)
67. (Boyer 1991, "Archimedes of Syracuse" p. 121)
68. (Boyer 1991, "Archimedes of Syracuse" p. 137)
69. (Boyer 1991, "Apollonius of Perga" p. 145)
70. (Boyer 1991, "Apollonius of Perga" p. 146)
71. (Boyer 1991, "Apollonius of Perga" p. 152)
72. (Boyer 1991, "Apollonius of Perga" p. 156)
73. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 161)
74. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 175)
75. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 162)
76. S.C. Roy. Complex numbers: lattice simulation and zeta function applications, p. 1 . Harwood Publishing, 2007, 131 pages. ISBN 1-904275-25-7
77. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 163)
78. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 164)
79. (Boyer 1991, "Greek Trigonometry and Mensuration" p. 168)
80. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178)
81. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 180)
82. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 181)
83. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 183)
84. (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 183–90)
85. "Internet History Sourcebooks Project". sourcebooks.fordham.edu.
86. (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 190–94)
87. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 193)
88. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 194)
89. (Goodman 2016, p. 119)
90. (Cuomo 2001, pp. 194, 204–06)
91. (Cuomo 2001, pp. 192–95)
92. (Goodman 2016, pp. 120–21)
93. (Cuomo 2001, p. 196)
94. (Cuomo 2001, pp. 207–08)
95. (Goodman 2016, pp. 119–20)
96. (Tang 2005, pp. 14–15, 45)
97. (Joyce 1979, p. 256)
98. (Gullberg 1997, p. 17)
99. (Gullberg 1997, pp. 17–18)
100. (Gullberg 1997, p. 18)
101. (Gullberg 1997, pp. 18–19)
102. (Needham & Wang 2000, pp. 281–85)
103. (Needham & Wang 2000, p. 285)
104. (Sleeswyk 1981, pp. 188–200)
105. (Boyer 1991, "China and India" p. 201)
106. (Boyer 1991, "China and India" p. 196)
107. Katz 2007, pp. 194–99
108. (Boyer 1991, "China and India" p. 198)
109. (Needham & Wang 1995, pp. 91–92)
110. (Needham & Wang 1995, p. 94)
111. (Needham & Wang 1995, p. 22)
112. (Straffin 1998, p. 164)
113. (Needham & Wang 1995, pp. 99–100)
114. (Berggren, Borwein & Borwein 2004, p. 27)
115. (de Crespigny 2007, p. 1050)
116. (Boyer 1991, "China and India" p. 202)
117. (Needham & Wang 1995, pp. 100–01)
118. (Berggren, Borwein & Borwein 2004, pp. 20, 24–26)
119. Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009). Calculus: Early Transcendentals (3 ed.). Jones & Bartlett Learning. p. xxvii. ISBN 978-0-7637-5995-7. Extract of p. 27
120. (Boyer 1991, "China and India" p. 205)
121. (Volkov 2009, pp. 153–56)
122. (Volkov 2009, pp. 154–55)
123. (Volkov 2009, pp. 156–57)
124. (Volkov 2009, p. 155)
125. Development Of Modern Numerals And Numeral Systems: The Hindu-Arabic system, Encyclopaedia Britannica, Quote: "The 1, 4, and 6 are found in the Ashoka inscriptions (3rd century BC); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st or 2nd century AD – all in forms that have considerable resemblance to today’s, 2 and 3 being well-recognized cursive derivations from the ancient = and ≡."
126. (Boyer 1991, "China and India" p. 206)
127. (Boyer 1991, "China and India" p. 207)
128. Puttaswamy, T.K. (2000). "The Accomplishments of Ancient Indian Mathematicians". In Selin, Helaine; D'Ambrosio, Ubiratan (eds.). Mathematics Across Cultures: The History of Non-western Mathematics. Springer. pp. 411–12. ISBN 978-1-4020-0260-1.
129. Kulkarni, R.P. (1978). "The Value of π known to Śulbasūtras" (PDF). Indian Journal of History of Science. 13 (1): 32–41. Archived from the original (PDF) on 2012-02-06.
130. Connor, J.J.; Robertson, E.F. "The Indian Sulbasutras". Univ. of St. Andrew, Scotland.
131. Bronkhorst, Johannes (2001). "Panini and Euclid: Reflections on Indian Geometry". Journal of Indian Philosophy. 29 (1–2): 43–80. doi:10.1023/A:1017506118885. S2CID 115779583.
132. Kadvany, John (2008-02-08). "Positional Value and Linguistic Recursion". Journal of Indian Philosophy. 35 (5–6): 487–520. CiteSeerX 10.1.1.565.2083. doi:10.1007/s10781-007-9025-5. ISSN 0022-1791. S2CID 52885600.
133. Sanchez, Julio; Canton, Maria P. (2007). Microcontroller programming : the microchip PIC. Boca Raton, Florida: CRC Press. p. 37. ISBN 978-0-8493-7189-9.
134. Anglin, W. S. and J. Lambek (1995). The Heritage of Thales, Springer, ISBN 0-387-94544-X
135. Hall, Rachel W. (2008). "Math for poets and drummers" (PDF). Math Horizons. 15 (3): 10–11. doi:10.1080/10724117.2008.11974752. S2CID 3637061.
136. (Boyer 1991, "China and India" p. 208)
137. (Boyer 1991, "China and India" p. 209)
138. (Boyer 1991, "China and India" p. 210)
139. (Boyer 1991, "China and India" p. 211)
140. Boyer (1991). "The Arabic Hegemony". History of Mathematics. Wiley. p. 226. ISBN 9780471543978. By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek.
141. Plofker 2009 182–207
142. Plofker 2009 pp. 197–98; George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, Indian Mathematics, pp. 118–30 in Companion History of the History and Philosophy of the Mathematical Sciences, ed. I. Grattan.Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126
143. Plofker 2009 pp. 217–53
144. Raju, C. K. (2001). "Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā" (PDF). Philosophy East & West. 51 (3): 325–362. doi:10.1353/pew.2001.0045. S2CID 170341845. Retrieved 2020-02-11.
145. Divakaran, P. P. (2007). The first textbook of calculus: Yukti-bhāṣā, Journal of Indian Philosophy 35, pp. 417–33.
146. C. K. Raju (2007). Cultural foundations of mathematics: the nature of mathematical proof and the transmission of the calculus from india to Europe in the 16th c. CE. Delhi: Pearson Longman.
147. Almeida, D. F.; J. K. John and A. Zadorozhnyy (2001). "Keralese mathematics: its possible transmission to Europe and the consequential educational implications". Journal of Natural Geometry. 20 (1): 77–104.{{cite journal}}: CS1 maint: multiple names: authors list (link)
148. Pingree, David (December 1992). "Hellenophilia versus the History of Science". Isis. 83 (4): 554–563. Bibcode:1992Isis...83..554P. doi:10.1086/356288. JSTOR 234257. S2CID 68570164. One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.
149. Bressoud, David (2002). "Was Calculus Invented in India?". College Mathematics Journal. 33 (1): 2–13. doi:10.2307/1558972. JSTOR 1558972.
150. Plofker, Kim (November 2001). "The 'Error' in the Indian "Taylor Series Approximation" to the Sine". Historia Mathematica. 28 (4): 293. doi:10.1006/hmat.2001.2331. It is not unusual to encounter in discussions of Indian mathematics such assertions as that 'the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)' [Joseph 1991, 300], or that 'we may consider Madhava to have been the founder of mathematical analysis' (Joseph 1991, 293), or that Bhaskara II may claim to be 'the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus' (Bag 1979, 294).... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)).... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian 'discovery of the principle of the differential calculus' somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential 'principle' was not generalized to arbitrary functions – in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here
151. Katz, Victor J. (June 1995). "Ideas of Calculus in Islam and India" (PDF). Mathematics Magazine. 68 (3): 163–74. doi:10.2307/2691411. JSTOR 2691411.
152. Abdel Haleem, Muhammad A. S. "The Semitic Languages", https://doi.org/10.1515/9783110251586.811, "Arabic became the language of scholarship in science and philosophy in the 9th century when the ‘translation movement’ saw concerted work on translations of Greek, Indian, Persian and Chinese, medical, philosophical and scientific texts", p. 811.
153. (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions."
154. Gandz and Saloman (1936), The sources of Khwarizmi's algebra, Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
155. (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
156. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 978-0-7923-2565-9. OCLC 29181926.
157. Sesiano, Jacques (1997). "Abū Kāmil". Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. pp. 4–5.
158. (Katz 1998, pp. 255–59)
159. F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
160. Katz, Victor J. (1995). "Ideas of Calculus in Islam and India". Mathematics Magazine. 68 (3): 163–74. doi:10.2307/2691411. JSTOR 2691411.
161. Alam, S (2015). "Mathematics for All and Forever" (PDF). Indian Institute of Social Reform & Research International Journal of Research.
162. O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics Archive, University of St Andrews
163. (Goodman 2016, p. 121)
164. Wisdom, 11:20
165. Caldwell, John (1981) "The De Institutione Arithmetica and the De Institutione Musica", pp. 135–54 in Margaret Gibson, ed., Boethius: His Life, Thought, and Influence, (Oxford: Basil Blackwell).
166. Folkerts, Menso (1970). "Boethius" Geometrie II, (Wiesbaden: Franz Steiner Verlag.
167. Marie-Thérèse d'Alverny, "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century, (Cambridge: Harvard University Press, 1982).
168. Guy Beaujouan, "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century, (Cambridge: Harvard University Press, 1982).
169. Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7
170. Grant, Edward and John E. Murdoch, eds. (1987). Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages. Cambridge: Cambridge University Press. ISBN 0-521-32260-X.
171. Clagett, Marshall (1961) The Science of Mechanics in the Middle Ages, (Madison: University of Wisconsin Press), pp. 421–40.
172. Murdoch, John E. (1969). "Mathesis in Philosophiam Scholasticam Introducta: The Rise and Development of the Application of Mathematics in Fourteenth Century Philosophy and Theology", in Arts libéraux et philosophie au Moyen Âge (Montréal: Institut d'Études Médiévales), pp. 224–27.
173. Pickover, Clifford A. (2009), The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, Inc., p. 104, ISBN 978-1-4027-5796-9, Nicole Oresme ... was the first to prove the divergence of the harmonic series (c. 1350). His results were lost for several centuries, and the result was proved again by Italian mathematician Pietro Mengoli in 1647 and by Swiss mathematician Johann Bernoulli in 1687.
174. Clagett, Marshall (1961). The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 210, 214–15, 236.
175. Clagett, Marshall (1961). The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, p. 284.
176. Clagett, Marshall (1961) The Science of Mechanics in the Middle Ages. Madison: University of Wisconsin Press, pp. 332–45, 382–91.
177. Nicole Oresme, "Questions on the Geometry of Euclid" Q. 14, pp. 560–65, in Marshall Clagett, ed., Nicole Oresme and the Medieval Geometry of Qualities and Motions. Madison: University of Wisconsin Press, 1968.
178. Heeffer, Albrecht: On the curious historical coincidence of algebra and double-entry bookkeeping, Foundations of the Formal Sciences, Ghent University, November 2009, p. 7
179. della Francesca, Piero. De Prospectiva Pingendi, ed. G. Nicco Fasola, 2 vols., Florence (1942).
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181. della Francesca, Piero. L'opera "De corporibus regularibus" di Pietro Franceschi detto della Francesca usurpata da Fra Luca Pacioli, ed. G. Mancini, Rome, (1916).
182. Alan Sangster, Greg Stoner & Patricia McCarthy: "The market for Luca Pacioli’s Summa Arithmetica" (Accounting, Business & Financial History Conference, Cardiff, September 2007) pp. 1–2
183. Grattan-Guinness, Ivor (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 978-0-393-32030-5.
184. Kline, Morris (1953). Mathematics in Western Culture. Great Britain: Pelican. pp. 150–51.
185. Struik, Dirk (1987). A Concise History of Mathematics (3rd. ed.). Courier Dover Publications. pp. 89. ISBN 978-0-486-60255-4.
186. "2021: 375th birthday of Leibniz, father of computer science". people.idsia.ch.
187. Eves, Howard, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0-03-029558-0, p. 379, "...the concepts of calculus...(are) so far reaching and have exercised such an impact on the modern world that it is perhaps correct to say that without some knowledge of them a person today can scarcely claim to be well educated."
188. Howard Eves, An Introduction to the History of Mathematics, 6th edition, 1990, "In the nineteenth century, mathematics underwent a great forward surge ... . The new mathematics began to free itself from its ties to mechanics and astronomy, and a purer outlook evolved." p. 493
189. Lori Thurgood; Mary J. Golladay; Susan T. Hill (June 2006). "U.S. Doctorates in the 20th Century" (PDF). nih.gov. Retrieved 5 April 2023.
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References
• de Crespigny, Rafe (2007), A Biographical Dictionary of Later Han to the Three Kingdoms (23–220 AD), Leiden: Koninklijke Brill, ISBN 978-90-04-15605-0.
• Berggren, Lennart; Borwein, Jonathan M.; Borwein, Peter B. (2004), Pi: A Source Book, New York: Springer, ISBN 978-0-387-20571-7
• Boyer, C.B. (1991) [1989], A History of Mathematics (2nd ed.), New York: Wiley, ISBN 978-0-471-54397-8
• Cuomo, Serafina (2001), Ancient Mathematics, London: Routledge, ISBN 978-0-415-16495-5
• Goodman, Michael, K.J. (2016), An introduction of the Early Development of Mathematics, Hoboken: Wiley, ISBN 978-1-119-10497-1{{citation}}: CS1 maint: multiple names: authors list (link)
• Gullberg, Jan (1997), Mathematics: From the Birth of Numbers, New York: W.W. Norton and Company, ISBN 978-0-393-04002-9
• Joyce, Hetty (July 1979), "Form, Function and Technique in the Pavements of Delos and Pompeii", American Journal of Archaeology, 83 (3): 253–63, doi:10.2307/505056, JSTOR 505056, S2CID 191394716.
• Katz, Victor J. (1998), A History of Mathematics: An Introduction (2nd ed.), Addison-Wesley, ISBN 978-0-321-01618-8
• Katz, Victor J. (2007), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, ISBN 978-0-691-11485-9
• Needham, Joseph; Wang, Ling (1995) [1959], Science and Civilization in China: Mathematics and the Sciences of the Heavens and the Earth, vol. 3, Cambridge: Cambridge University Press, ISBN 978-0-521-05801-8
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Further reading
General
• Aaboe, Asger (1964). Episodes from the Early History of Mathematics. New York: Random House.
• Bell, E. T. (1937). Men of Mathematics. Simon and Schuster.
• Burton, David M. (1997). The History of Mathematics: An Introduction. McGraw Hill.
• Grattan-Guinness, Ivor (2003). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. The Johns Hopkins University Press. ISBN 978-0-8018-7397-3.
• Kline, Morris. Mathematical Thought from Ancient to Modern Times.
• Struik, D. J. (1987). A Concise History of Mathematics, fourth revised edition. Dover Publications, New York.
Books on a specific period
• Gillings, Richard J. (1972). Mathematics in the Time of the Pharaohs. Cambridge, MA: MIT Press.
• Heath, Thomas Little (1921). A History of Greek Mathematics. Oxford, Claredon Press.
• van der Waerden, B. L. (1983). Geometry and Algebra in Ancient Civilizations, Springer, ISBN 0-387-12159-5.
Books on a specific topic
• Corry, Leo (2015), A Brief History of Numbers, Oxford University Press, ISBN 978-0198702597
• Hoffman, Paul (1998). The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hyperion. ISBN 0-7868-6362-5.
• Menninger, Karl W. (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 978-0-262-13040-0.
• Stigler, Stephen M. (1990). The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press. ISBN 978-0-674-40341-3.
External links
Wikiquote has quotations related to History of mathematics.
Documentaries
• BBC (2008). The Story of Maths.
• Renaissance Mathematics, BBC Radio 4 discussion with Robert Kaplan, Jim Bennett & Jackie Stedall (In Our Time, Jun 2, 2005)
Educational material
• MacTutor History of Mathematics archive (John J. O'Connor and Edmund F. Robertson; University of St Andrews, Scotland). An award-winning website containing detailed biographies on many historical and contemporary mathematicians, as well as information on notable curves and various topics in the history of mathematics.
• History of Mathematics Home Page (David E. Joyce; Clark University). Articles on various topics in the history of mathematics with an extensive bibliography.
• The History of Mathematics (David R. Wilkins; Trinity College, Dublin). Collections of material on the mathematics between the 17th and 19th century.
• Earliest Known Uses of Some of the Words of Mathematics (Jeff Miller). Contains information on the earliest known uses of terms used in mathematics.
• Earliest Uses of Various Mathematical Symbols (Jeff Miller). Contains information on the history of mathematical notations.
• Mathematical Words: Origins and Sources (John Aldrich, University of Southampton) Discusses the origins of the modern mathematical word stock.
• Biographies of Women Mathematicians (Larry Riddle; Agnes Scott College).
• Mathematicians of the African Diaspora (Scott W. Williams; University at Buffalo).
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| Wikipedia |
1 + 2 + 4 + 8 + ⋯
In mathematics, 1 + 2 + 4 + 8 + ⋯ is the infinite series whose terms are the successive powers of two. As a geometric series, it is characterized by its first term, 1, and its common ratio, 2. As a series of real numbers it diverges to infinity, so the sum of this series is infinity.
However, it can be manipulated to yield a number of mathematically interesting results. For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. For example, the Ramanujan summation of this series is −1, which is the limit of the series using the 2-adic metric.
Summation
The partial sums of $1+2+4+8+\cdots $ are $1,3,7,15,\ldots ;$ ;} since these diverge to infinity, so does the series.
$2^{0}+2^{1}+\cdots +2^{k}=2^{k+1}-1$
Therefore, any totally regular summation method gives a sum of infinity, including the Cesàro sum and Abel sum.[1] On the other hand, there is at least one generally useful method that sums $1+2+4+8+\cdots $ to the finite value of −1. The associated power series
$f(x)=1+2x+4x^{2}+8x^{3}+\cdots +2^{n}{}x^{n}+\cdots ={\frac {1}{1-2x}}$
has a radius of convergence around 0 of only ${\frac {1}{2}}$ so it does not converge at $x=1.$ Nonetheless, the so-defined function $f$ has a unique analytic continuation to the complex plane with the point $x={\frac {1}{2}}$ deleted, and it is given by the same rule $f(x)={\frac {1}{1-2x}}.$ Since $f(1)=-1,$ the original series $1+2+4+8+\cdots $ is said to be summable (E) to −1, and −1 is the (E) sum of the series. (The notation is due to G. H. Hardy in reference to Leonhard Euler's approach to divergent series).[2]
An almost identical approach (the one taken by Euler himself) is to consider the power series whose coefficients are all 1, that is,
$1+y+y^{2}+y^{3}+\cdots ={\frac {1}{1-y}}$
and plugging in $y=2.$ These two series are related by the substitution $y=2x.$
The fact that (E) summation assigns a finite value to $1+2+4+8+\cdots $ shows that the general method is not totally regular. On the other hand, it possesses some other desirable qualities for a summation method, including stability and linearity. These latter two axioms actually force the sum to be −1, since they make the following manipulation valid:
${\begin{array}{rcl}s&=&\displaystyle 1+2+4+8+16+\cdots \\&=&\displaystyle 1+2(1+2+4+8+\cdots )\\&=&\displaystyle 1+2s\end{array}}$
In a useful sense, $s=\infty $ is a root of the equation $s=1+2s.$ (For example, $\infty $ is one of the two fixed points of the Möbius transformation $z\mapsto 1+2z$ on the Riemann sphere). If some summation method is known to return an ordinary number for $s$; that is, not $\infty ,$ then it is easily determined. In this case $s$ may be subtracted from both sides of the equation, yielding $0=1+s,$ so $s=-1.$[3]
The above manipulation might be called on to produce −1 outside the context of a sufficiently powerful summation procedure. For the most well-known and straightforward sum concepts, including the fundamental convergent one, it is absurd that a series of positive terms could have a negative value. A similar phenomenon occurs with the divergent geometric series $1-1+1-1+\cdots $ (Grandi's series), where a series of integers appears to have the non-integer sum ${\frac {1}{2}}.$ These examples illustrate the potential danger in applying similar arguments to the series implied by such recurring decimals as $0.111\ldots $ and most notably $0.999\ldots $. The arguments are ultimately justified for these convergent series, implying that $0.111\ldots ={\frac {1}{9}}$ and $0.999\ldots =1,$ but the underlying proofs demand careful thinking about the interpretation of endless sums.[4]
It is also possible to view this series as convergent in a number system different from the real numbers, namely, the 2-adic numbers. As a series of 2-adic numbers this series converges to the same sum, −1, as was derived above by analytic continuation.[5]
See also
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 1 + 1 + 1 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Two's complement, a data convention for representing negative numbers where −1 is represented as if it were 1 + 2 + 4 + ⋯ + 2n−1.
Notes
1. Hardy p. 10
2. Hardy pp. 8, 10
3. The two roots of $s=1+2s$ are briefly touched on by Hardy p. 19.
4. Gardiner pp. 93–99; the argument on p. 95 for $1+2+4+8+\cdots $ is slightly different but has the same spirit.
5. Koblitz, Neal (1984). p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol. 58. Springer-Verlag. pp. chapter I, exercise 16, p. 20. ISBN 0-387-96017-1.
References
• Euler, Leonhard (1760). "De seriebus divergentibus". Novi Commentarii Academiae Scientiarum Petropolitanae. 5: 205–237.
• Gardiner, A. (2002) [1982]. Understanding infinity: the mathematics of infinite processes (Dover ed.). Dover. ISBN 0-486-42538-X.
• Hardy, G. H. (1949). Divergent Series. Clarendon Press. LCC QA295 .H29 1967.
Further reading
• Barbeau, E. J.; Leah, P. J. (May 1976). "Euler's 1760 paper on divergent series". Historia Mathematica. 3 (2): 141–160. doi:10.1016/0315-0860(76)90030-6.
• Ferraro, Giovanni (2002). "Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730". Annals of Science. 59 (2): 179–199. doi:10.1080/00033790010028179. S2CID 143992318.
• Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. doi:10.2307/2690371. JSTOR 2690371.
• Sandifer, Ed (June 2006). "Divergent series" (PDF). How Euler Did It. MAA Online. Archived from the original (PDF) on 2013-03-20. Retrieved 2007-02-17.
• Sierpińska, Anna (November 1987). "Humanities students and epistemological obstacles related to limits". Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986. JSTOR 3482354. S2CID 144880659.
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| Wikipedia |
1 − 2 + 3 − 4 + ⋯
In mathematics, 1 − 2 + 3 − 4 + ··· is an infinite series whose terms are the successive positive integers, given alternating signs. Using sigma summation notation the sum of the first m terms of the series can be expressed as
$\sum _{n=1}^{m}n(-1)^{n-1}.$
The infinite series diverges, meaning that its sequence of partial sums, (1, −1, 2, −2, 3, ...), does not tend towards any finite limit. Nonetheless, in the mid-18th century, Leonhard Euler wrote what he admitted to be a paradoxical equation:
$1-2+3-4+\cdots ={\frac {1}{4}}.$
A rigorous explanation of this equation would not arrive until much later. Starting in 1890, Ernesto Cesàro, Émile Borel and others investigated well-defined methods to assign generalized sums to divergent series—including new interpretations of Euler's attempts. Many of these summability methods easily assign to 1 − 2 + 3 − 4 + ... a "value" of 1/4. Cesàro summation is one of the few methods that do not sum 1 − 2 + 3 − 4 + ..., so the series is an example where a slightly stronger method, such as Abel summation, is required.
The series 1 − 2 + 3 − 4 + ... is closely related to Grandi's series 1 − 1 + 1 − 1 + .... Euler treated these two as special cases of the more general sequence 1 − 2n + 3n − 4n + ..., where n = 1 and n = 0 respectively. This line of research extended his work on the Basel problem and leading towards the functional equations of what are now known as the Dirichlet eta function and the Riemann zeta function.
Divergence
The series' terms (1, −2, 3, −4, ...) do not approach 0; therefore 1 − 2 + 3 − 4 + ... diverges by the term test. Divergence can also be shown directly from the definition: an infinite series converges if and only if the sequence of partial sums converges to limit, in which case that limit is the value of the infinite series. The partial sums of 1 − 2 + 3 − 4 + ... are:[1]
1,
1 − 2 = −1,
1 − 2 + 3 = 2,
1 − 2 + 3 − 4 = −2,
1 − 2 + 3 − 4 + 5 = 3,
1 − 2 + 3 − 4 + 5 − 6 = −3,
...
The sequence of partial sums shows that the series does not converge to a particular number: for any proposed limit x, there exists a point beyond which the subsequent partial sums are all outside the interval [x−1, x+1]), so 1 − 2 + 3 − 4 + ... diverges.
The partial sums include every integer exactly once—even 0 if one counts the empty partial sum—and thereby establishes the countability of the set $\mathbb {Z} $ of integers.[2]
Heuristics for summation
Stability and linearity
Since the terms 1, −2, 3, −4, 5, −6, ... follow a simple pattern, the series 1 − 2 + 3 − 4 + ... can be manipulated by shifting and term-by-term addition to yield a numerical value. If it can make sense to write s = 1 − 2 + 3 − 4 + ... for some ordinary number s, the following manipulations argue for s = 1⁄4:[3]
${\begin{array}{rclllll}4s&=&&(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )&{}+(1-2+3-4+\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+(1-2)+(3-4+5-6\cdots )\\&=&&(1-2+3-4+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}+1+(-2+3-4+5+\cdots )&{}-1+(3-4+5-6\cdots )\\&=&1+&(1-2+3-4+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(-2+3-4+5+\cdots )&{}+(3-4+5-6\cdots )\\&=&1+[&(1-2-2+3)&{}+(-2+3+3-4)&{}+(3-4-4+5)&{}+(-4+5+5-6)+\cdots ]\\&=&1+[&0+0+0+0+\cdots ]\\4s&=&1\end{array}}$
So $s={\frac {1}{4}}$.
Although 1 − 2 + 3 − 4 + ... does not have a sum in the usual sense, the equation s = 1 − 2 + 3 − 4 + ... = 1⁄4 can be supported as the most natural answer if such a sum is to be defined. A generalized definition of the "sum" of a divergent series is called a summation method or summability method. There are many different methods and it is desirable that they share some properties of ordinary summation. What the above manipulations actually prove is the following: Given any summability method that is linear and stable and sums the series 1 − 2 + 3 − 4 + ..., the sum it produces is 1⁄4.[4] Furthermore, since
${\begin{array}{rcllll}2s&=&&(1-2+3-4+\cdots )&+&(1-2+3-4+\cdots )\\&=&1+{}&(-2+3-4+\cdots )&{}+1-2&{}+(3-4+5\cdots )\\&=&0+{}&(-2+3)+(3-4)+(-4+5)+\cdots \\2s&=&&1-1+1-1\cdots \end{array}}$
such a method must also sum Grandi's series as 1 − 1 + 1 − 1 + ... = 1⁄2.[4]
Cauchy product
In 1891, Ernesto Cesàro expressed hope that divergent series would be rigorously brought into calculus, pointing out, "One already writes (1 − 1 + 1 − 1 + ...)2 = 1 − 2 + 3 − 4 + ... and asserts that both the sides are equal to 1⁄4."[5] For Cesàro, this equation was an application of a theorem he had published the previous year, which is the first theorem in the history of summable divergent series.[1] The details on his summation method are below; the central idea is that 1 − 2 + 3 − 4 + ... is the Cauchy product (discrete convolution) of 1 − 1 + 1 − 1 + ... with 1 − 1 + 1 − 1 + ....
The Cauchy product of two infinite series is defined even when both of them are divergent. In the case where an = bn = (−1)n, the terms of the Cauchy product are given by the finite diagonal sums
${\begin{array}{rcl}c_{n}&=&\displaystyle \sum _{k=0}^{n}a_{k}b_{n-k}=\sum _{k=0}^{n}(-1)^{k}(-1)^{n-k}\\[1em]&=&\displaystyle \sum _{k=0}^{n}(-1)^{n}=(-1)^{n}(n+1).\end{array}}$
The product series is then
$\sum _{n=0}^{\infty }(-1)^{n}(n+1)=1-2+3-4+\cdots .$
Thus a summation method that respects the Cauchy product of two series — and assigns to the series 1 − 1 + 1 − 1 + ... the sum 1/2 — will also assign to the series 1 − 2 + 3 − 4 + ... the sum 1/4. With the result of the previous section, this implies an equivalence between summability of 1 − 1 + 1 − 1 + ... and 1 − 2 + 3 − 4 + ... with methods that are linear, stable, and respect the Cauchy product.
Cesàro's theorem is a subtle example. The series 1 − 1 + 1 − 1 + ... is Cesàro-summable in the weakest sense, called (C, 1)-summable, while 1 − 2 + 3 − 4 + ... requires a stronger form of Cesàro's theorem,[6] being (C, 2)-summable. Since all forms of Cesàro's theorem are linear and stable,[7] the values of the sums are as calculated above.
Specific methods
Cesàro and Hölder
To find the (C, 1) Cesàro sum of 1 − 2 + 3 − 4 + ..., if it exists, one needs to compute the arithmetic means of the partial sums of the series. The partial sums are:
1, −1, 2, −2, 3, −3, ...,
and the arithmetic means of these partial sums are:
1, 0, 2⁄3, 0, 3⁄5, 0, 4⁄7, ....
This sequence of means does not converge, so 1 − 2 + 3 − 4 + ... is not Cesàro summable.
There are two well-known generalizations of Cesàro summation: the conceptually simpler of these is the sequence of (H, n) methods for natural numbers n. The (H, 1) sum is Cesàro summation, and higher methods repeat the computation of means. Above, the even means converge to 1⁄2, while the odd means are all equal to 0, so the means of the means converge to the average of 0 and 1⁄2, namely 1⁄4.[8] So 1 − 2 + 3 − 4 + ... is (H, 2) summable to 1⁄4.
The "H" stands for Otto Hölder, who first proved in 1882 what mathematicians now think of as the connection between Abel summation and (H, n) summation; 1 − 2 + 3 − 4 + ... was his first example.[9] The fact that 1⁄4 is the (H, 2) sum of 1 − 2 + 3 − 4 + ... guarantees that it is the Abel sum as well; this will also be proved directly below.
The other commonly formulated generalization of Cesàro summation is the sequence of (C, n) methods. It has been proven that (C, n) summation and (H, n) summation always give the same results, but they have different historical backgrounds. In 1887, Cesàro came close to stating the definition of (C, n) summation, but he gave only a few examples. In particular, he summed 1 − 2 + 3 − 4 + ..., to 1⁄4 by a method that may be rephrased as (C, n) but was not justified as such at the time. He formally defined the (C, n) methods in 1890 in order to state his theorem that the Cauchy product of a (C, n)-summable series and a (C, m)-summable series is (C, m + n + 1)-summable.[10]
Abel summation
In a 1749 report, Leonhard Euler admits that the series diverges but prepares to sum it anyway:
... when it is said that the sum of this series 1 − 2 + 3 − 4 + 5 − 6 etc. is 1⁄4, that must appear paradoxical. For by adding 100 terms of this series, we get −50, however, the sum of 101 terms gives +51, which is quite different from 1⁄4 and becomes still greater when one increases the number of terms. But I have already noticed at a previous time, that it is necessary to give to the word sum a more extended meaning ...[11]
Euler proposed a generalization of the word "sum" several times. In the case of 1 − 2 + 3 − 4 + ..., his ideas are similar to what is now known as Abel summation:
... it is no more doubtful that the sum of this series 1 − 2 + 3 − 4 + 5 etc. is 1⁄4; since it arises from the expansion of the formula 1⁄(1+1)2, whose value is incontestably 1⁄4. The idea becomes clearer by considering the general series 1 − 2x + 3x2 − 4x3 + 5x4 − 6x5 + &c. that arises while expanding the expression 1⁄(1+x)2, which this series is indeed equal to after we set x = 1.[12]
There are many ways to see that, at least for absolute values |x| < 1, Euler is right in that
$1-2x+3x^{2}-4x^{3}+\cdots ={\frac {1}{(1+x)^{2}}}.$
One can take the Taylor expansion of the right-hand side, or apply the formal long division process for polynomials. Starting from the left-hand side, one can follow the general heuristics above and try multiplying by (1 + x) twice or squaring the geometric series 1 − x + x2 − .... Euler also seems to suggest differentiating the latter series term by term.[13]
In the modern view, the generating function 1 − 2x + 3x2 − 4x3 + ... does not define a function at x = 1, so that value cannot simply be substituted into the resulting expression. Since the function is defined for all |x| < 1, one can still take the limit as x approaches 1, and this is the definition of the Abel sum:
$\lim _{x\rightarrow 1^{-}}\sum _{n=1}^{\infty }n(-x)^{n-1}=\lim _{x\rightarrow 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.$
Euler and Borel
Euler applied another technique to the series: the Euler transform, one of his own inventions. To compute the Euler transform, one begins with the sequence of positive terms that makes up the alternating series—in this case 1, 2, 3, 4, .... The first element of this sequence is labeled a0.
Next one needs the sequence of forward differences among 1, 2, 3, 4, ...; this is just 1, 1, 1, 1, .... The first element of this sequence is labeled Δa0. The Euler transform also depends on differences of differences, and higher iterations, but all the forward differences among 1, 1, 1, 1, ... are 0. The Euler transform of 1 − 2 + 3 − 4 + ... is then defined as
${\frac {1}{2}}a_{0}-{\frac {1}{4}}\Delta a_{0}+{\frac {1}{8}}\Delta ^{2}a_{0}-\cdots ={\frac {1}{2}}-{\frac {1}{4}}.$
In modern terminology, one says that 1 − 2 + 3 − 4 + ... is Euler summable to 1⁄4.
The Euler summability also implies Borel summability, with the same summation value, as it does in general.[14]
Separation of scales
Saichev and Woyczyński arrive at 1 − 2 + 3 − 4 + ... = 1⁄4 by applying only two physical principles: infinitesimal relaxation and separation of scales. To be precise, these principles lead them to define a broad family of "φ-summation methods", all of which sum the series to 1⁄4:
• If φ(x) is a function whose first and second derivatives are continuous and integrable over (0, ∞), such that φ(0) = 1 and the limits of φ(x) and xφ(x) at +∞ are both 0, then[15]
$\lim _{\delta \rightarrow 0}\sum _{m=0}^{\infty }(-1)^{m}(m+1)\varphi (\delta m)={\frac {1}{4}}.$
This result generalizes Abel summation, which is recovered by letting φ(x) = exp(−x). The general statement can be proved by pairing up the terms in the series over m and converting the expression into a Riemann integral. For the latter step, the corresponding proof for 1 − 1 + 1 − 1 + ... applies the mean value theorem, but here one needs the stronger Lagrange form of Taylor's theorem.
Generalization
The threefold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 3 + 6 − 10 + ..., the alternating series of triangular numbers; its Abel and Euler sum is 1⁄8.[16] The fourfold Cauchy product of 1 − 1 + 1 − 1 + ... is 1 − 4 + 10 − 20 + ..., the alternating series of tetrahedral numbers, whose Abel sum is 1⁄16.
Another generalization of 1 − 2 + 3 − 4 + ... in a slightly different direction is the series 1 − 2n + 3n − 4n + ... for other values of n. For positive integers n, these series have the following Abel sums:[17]
$1-2^{n}+3^{n}-\cdots ={\frac {2^{n+1}-1}{n+1}}B_{n+1}$
where Bn are the Bernoulli numbers. For even n, this reduces to
$1-2^{2k}+3^{2k}-\cdots =0,$
which can be interpreted as stating that negative even values of the Riemann zeta function are zero. This sum became an object of particular ridicule by Niels Henrik Abel in 1826:
Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes. Can one think of anything more appalling than to say that
0 = 1 − 22n + 32n − 42n + etc.
where n is a positive number. Here's something to laugh at, friends.[18]
Cesàro's teacher, Eugène Charles Catalan, also disparaged divergent series. Under Catalan's influence, Cesàro initially referred to the "conventional formulas" for 1 − 2n + 3n − 4n + ... as "absurd equalities", and in 1883 Cesàro expressed a typical view of the time that the formulas were false but still somehow formally useful. Finally, in his 1890 Sur la multiplication des séries, Cesàro took a modern approach starting from definitions.[19]
The series are also studied for non-integer values of n; these make up the Dirichlet eta function. Part of Euler's motivation for studying series related to 1 − 2 + 3 − 4 + ... was the functional equation of the eta function, which leads directly to the functional equation of the Riemann zeta function. Euler had already become famous for finding the values of these functions at positive even integers (including the Basel problem), and he was attempting to find the values at the positive odd integers (including Apéry's constant) as well, a problem that remains elusive today. The eta function in particular is easier to deal with by Euler's methods because its Dirichlet series is Abel summable everywhere; the zeta function's Dirichlet series is much harder to sum where it diverges.[20] For example, the counterpart of 1 − 2 + 3 − 4 + ... in the zeta function is the non-alternating series 1 + 2 + 3 + 4 + ..., which has deep applications in modern physics but requires much stronger methods to sum.
See also
• 1 + 2 + 3 + 4 + ⋯
• 1 + 1 + 1 + 1 + ⋯
• 1 + 2 + 4 + 8 + ·...
• 1 − 2 + 4 − 8 + ⋯
References
1. Hardy, p. 8
2. Beals, p. 23
3. Hardy (p. 6) presents this derivation in conjunction with evaluation of Grandi's series 1 − 1 + 1 − 1 + ....
4. Hardy, p. 6
5. Ferraro, p. 130.
6. Hardy, p. 3; Weidlich, pp. 52–55.
7. Alabdulmohsin 2018.
8. Hardy, p. 9. For the full details of the calculation, see Weidlich, pp. 17–18.
9. Ferraro, p. 118; Tucciarone, p. 10. Ferraro criticizes Tucciarone's explanation (p. 7) of how Hölder himself thought of the general result, but the two authors' explanations of Hölder's treatment of 1 − 2 + 3 − 4 + ... are similar.
10. Ferraro, pp. 123–128.
11. Euler et al., p. 2. Although the paper was written in 1749, it was not published until 1768.
12. Euler et al., pp. 3, 25.
13. For example, Lavine (p. 23) advocates long division but does not carry it out; Vretblad (p. 231) calculates the Cauchy product. Euler's advice is vague; see Euler et al., pp. 3, 26. John Baez even suggests a category-theoretic method involving multiply pointed sets and the quantum harmonic oscillator. Baez, John C. Euler's Proof That 1 + 2 + 3 + ... = −1/12 (PDF). Archived 2017-10-13 at the Wayback Machine math.ucr.edu (December 19, 2003). Retrieved on March 11, 2007.
14. Shawyer and Watson, p. 32
15. Saichev and Woyczyński, pp. 260–264.
16. Kline, p. 313.
17. Hardy, p. 3; Knopp, p. 491
18. Grattan-Guinness, p. 80. See Markushevich, p. 48, for a different translation from the original French; the tone remains the same.
19. Ferraro, pp. 120–128.
20. Euler et al., pp. 20–25.
Bibliography
• Alabdulmohsin, Ibrahim M. (2018). "Analytic summability theory". Summability Calculus. Springer International Publishing. pp. 65–91. doi:10.1007/978-3-319-74648-7_4. ISBN 978-3-319-74647-0.
• Beals, Richard (2004). Analysis: An Introduction. Cambridge UP. ISBN 978-0-521-60047-7.
• Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN 978-0-486-65973-2.
• Euler, Leonhard; Willis, Lucas; Osler, Thomas J. (2006). "Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series". The Euler Archive. Retrieved 2007-03-22. Originally published as Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". Mémoires de l'Académie des Sciences de Berlin. 17: 83–106.
• Ferraro, Giovanni (June 1999). "The First Modern Definition of the Sum of a Divergent Series: An Aspect of the Rise of 20th Century Mathematics". Archive for History of Exact Sciences. 54 (2): 101–135. doi:10.1007/s004070050036. S2CID 119766124.
• Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 978-0-262-07034-8.
• Hardy, G. H. (1949). Divergent Series. Clarendon Press. xvi+396. ISBN 978-0-8218-2649-2. LCCN 49005496. MR 0030620. OCLC 808787. 2nd Ed. published by Chelsea Pub. Co., 1991. LCCN 91-75377. ISBN 0-8284-0334-1.
• Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. CiteSeerX 10.1.1.639.6923. doi:10.2307/2690371. JSTOR 2690371.
• Knopp, Konrad (1990). Theory and Application of Infinite Series. New York: Dover Publications. ISBN 0486661652. LCCN 89071388.
• Lavine, Shaughan (1994). Understanding the Infinite. Harvard UP. ISBN 978-0-674-92096-5.
• Markusevič, Aleksej Ivanovič (1967). Series: fundamental concepts with historical exposition (English translation of 3rd revised edition (1961) in Russian ed.). Delhi, India: Hindustan Pub. Corp. p. 176. LCCN sa68017528. OCLC 729238507. Author also known as A. I. Markushevich and Alekseï Ivanovitch Markouchevitch. Also published in Boston, Mass by Heath with OCLC 474456247. Additionally, OCLC 208730, OCLC 487226828.
• Saichev, A. I.; Woyczyński, W. A. (1996). Distributions in the Physical and Engineering Sciences, Volume 1. Birkhaüser. ISBN 978-0-8176-3924-2.
• Shawyer, Bruce; Watson, Bruce (1994). Borel's Methods of Summability: Theory and Application. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York. ISBN 0-19-853585-6. MR 1320266.
• Tucciarone, John (January 1973). "The development of the theory of summable divergent series from 1880 to 1925". Archive for History of Exact Sciences. 10 (1–2): 1–40. doi:10.1007/BF00343405. S2CID 121888821.
• Vretblad, Anders (2003). Fourier Analysis and Its Applications. Springer. ISBN 978-0-387-00836-3.
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
Orders of magnitude (area)
This page is a progressive and labelled list of the SI area orders of magnitude, with certain examples appended to some list objects.
10−70 to 10−9 square metres
List of orders of magnitude for area 10−70 to 10−9 square metres
Factor (m2) Multiple Value Item
10−70 2.6×10−70 m2 Planck area, ${\frac {G\hbar }{c^{3}}}$[1]
10−60 1 square quectometre
10−54 1 square rontometre
10−52 100 rm2 1 shed[2]
10−48 1 square yoctometre (ym2) 1 ym2
10−43 100,000 ym2 1 femtobarn[3]
10−42 1 square zeptometre (zm2) 1 zm2
10−36 1 square attometre (am2) 1 am2
10−30 1 square femtometre (fm2) 1 fm2
10−29 66.52 fm2 Thomson cross-section of the electron[4]
10−28 100 fm2 1 barn, roughly the cross-sectional area of a uranium nucleus[5]
10−24 1 square picometre (pm2) 1 pm2
10−20 1 square angstrom (Å2) 10,000 pm2
10−19 100,000 pm2 Area of a lipid bilayer, per molecule[6]
75,000–260,000 pm2 Surface area of the 20 standard amino acids[7]
10−18 1 square nanometre (nm2) 1 nm2
10−16 100 nm2 Globular proteins: solvent-accessible surface area of a typical globular protein, having a typical molecular mass of ~35000 u (quite variable)[8]
10−14 17,000 nm2 Cross-sectional area of a nuclear pore complex in vertebrates[9]
10−12 1 square micrometre (μm2) 6 μm2 Surface area of an E. coli bacterium[10]
10−10 100 μm2 Surface area of a red blood cell[11]
10−9 6,000–110,000 μm2 Range of common LCD screen pixel sizes[12]
7,000 μm2 Area of a dot printed using 300 dots per inch resolution[13]
8,000 μm2 Cross-sectional area of a straight human hair that is 100 μm[14] in diameter[15]
10−8 to 10−1 square metres
List of orders of magnitude for areas 10−8 to 10−1 square metres
Factor (m2) Multiple Value Item
10−8 55,000 μm2 Size of a pixel on a typical modern computer display
10−7 2-400,000 μm2 Cross-sectional area of a mechanical pencil lead (0.5-0.7 mm in diameter)[16]
10−6 1 square millimetre (mm2) 1–2 mm2 Area of a human fovea[17]
2 mm2 Area of the head of a pin
10−5 30–50 mm2 Area of a 6–8 mm hole punched in a piece of paper by a hole punch[18]
10−4 1 square centimetre (cm2) 290 mm2 Area of one side of a U.S. penny[19][20]
500 mm2 Area of a typical postage stamp
10−3 1,100 mm2 Area of a human retina[21]
4,600 mm2 Area of the face of a credit card[22]
4,800 mm2 Largest side of a cigarette box
10−2 1 square decimetre (dm2) 10,000 mm2Index card (3 × 5 inches)[23]
60,000 mm2American letter paper (11 × 8.5 inches, "A" size)
62,370 mm2International A4 paper (210 × 297 mm)
92,903 mm21 square foot[24]
10−1 125,000 mm2International A3 paper (297 × 420 mm)
180,000 mm2Surface area of a basketball (diameter 24 cm)[25][26]
250,000 mm2International A2 paper (420 × 594 mm)
500,000 mm2International A1 paper (594 × 841 mm)
100 to 107 square metres
List of orders of magnitude for areas 100 to 107 square metres.
Factor (m2) Multiple Value Item
100 1 square metre 1 m2International A0 paper (841 × 1189 mm)
1.73 m2A number commonly used as the average body surface area of a human[27]
1–4 m2Area of the top of an office desk
101 10–20 m2 A parking space
70 m2 Approximate surface area of a human lung[28]
102 1 square decametre (dam2) 100 m2 One are (a)
162 m2 Size of a volleyball court (18 × 9 metres)[29]
202 m2 Floor area of a median suburban three-bedroom house in the US in 2010: 2,169 sq ft (201.5 m2)[30]
261 m2 Size of a tennis court[31]
437 m2 Size of an NBA/WNBA/NCAA basketball court[32]
103 1 kilo square meter k(m²) 1,000 m2Surface area of a modern stremma or dunam
1,250 m2Surface area of the water in an Olympic-size swimming pool[33]
4,047 m21 acre[34]
5,400 m2Size of an American football field[35][36]
7,140 m2Size of a typical football (soccer) field[37][38]
104 1 square hectometre (hm2) 10,000 m21 hectare (ha)[39]
17,000 m2Approximate area of a cricket field (theoretical limits: 6,402 m2 to 21,273 m2)[40]
22,100 m2Area of a Manhattan city block
53,000 m2Base of the Great Pyramid of Giza[41][42]
105 195,000 m2Irish National Botanic Gardens[43]
490,000 m2Vatican City[44]
600,000 m2Total floor area of the Pentagon[45]
887,800 m2AvtoVAZ main assembly building, Tolyatti, Russia (largest building by footprint)
106 1 mega square meter M(m²)
1 square kilometre (km2)
1.76 km2New Century Global Center, Chengdu, China (largest building by total floor area)
2 km2Monaco (country ranked 192nd by area)[46]
2.59 km21 square mile[47]
2.9 km2City of London (not all of modern London)[48]
107 59.5 km2Manhattan Island (land area)[49]
61 km2San Marino[50]
108 to 1014 square metres
Factor (m2) Multiple Value Item
108 105 km2Paris (inner city only)[51]
110 km2Walt Disney World[52]
272 km2Taipei City[53]
630 km2Toronto[54]
109 1 giga square meter G(m²) 1100 km2Hong Kong[55]
1290 km2Los Angeles, California, United States (city)[56]
1962 km2Jacksonville, Florida; largest city in the Continental US[57]
2188 km2Tokyo[58]
3,130 km2 Average area of an American county
5780 km2Administrative area of Bali[59]
8030 km2Community of Madrid, Spain
1010 11,000 km2Jamaica[60]
30,528 km2Belgium
68,870 km2Lake Victoria[61]
84,000 km2Austria[62]
1011 100,000 km2South Korea[63]
167,996 km2Jiuquan in China
232,000 km2Total area covered by underwater search for Malaysia Airlines Flight 370 (including both 2014-2017 and 2018 searches)
301,338 km2Italy[64]
357,000 km2Germany[65]
377,900 km2Japan[66]
510,000 km2Spain[67]
780,000 km2Turkey[68]
1012 1 tera square meter T(m²)
1 square megametre (Mm2)
1.0 Mm2Egypt (country ranked 29th by area)[69]
2 Mm2 Mexico
3.10 Mm2Sakha (Yakutia) Republic in Russia (largest subnational governing body)[70]
5 Mm2Largest extent of the Roman Empire[71][72]
7.74 Mm2Australia (country ranked 6th by area)[73]
8.5 Mm2Brazil
9.5 Mm2 China/ United States of America
1013 10 Mm2Canada (including water)[74]
14 Mm2Antarctica[75]
14 Mm2Arable land worldwide[76]
16.6 Mm2Surface area of Pluto[77]
17 Mm2Russia (country ranked 1st by area)[78]
30 Mm2Africa[79]
35.5 Mm2Largest extent of the British Empire[80]
38 Mm2Surface area of the Moon[81]
77 Mm2Atlantic Ocean[82]
1014 144 Mm2Surface area of Mars[83]
150 Mm2Land area of Earth[84]
156 Mm2Pacific Ocean[85]
360 Mm2Water area of Earth[84]
510 Mm2Total surface area of Earth[84]
1015 to 1026 square metres
List of orders of magnitude for areas 1015 to 1026 square metres.
Factor (m2) Multiple Value Item
1015 1 peta square meter P(m²) 1,000 Mm2Surface area of the white dwarf, Van Maanen's star
7,600 Mm2Surface area of Neptune[86]
1016 43,000 Mm2Surface area of Saturn[87]
61 000 Mm2Surface area of Jupiter,[88] the "surface" area of the spheroid (calculated from the mean radius as reported by NASA). The cross-sectional area of Jupiter, which is the same as the "circle" of Jupiter seen by an approaching spacecraft, is almost exactly one quarter the surface-area of the overall sphere, which in the case of Jupiter is approximately 1.535×1016 m2.
1017 2-600 000 Mm2Surface area of the brown dwarf CT Chamaeleontis B.
460,000 Mm2Area swept by the Moon's orbit of Earth
1018 1 square gigametre (Gm2) 6.1 Gm2Surface area of the Sun[89]
1019 30 Gm2Surface area of the star Vega
1020 100 Gm2
1021 1 zetta square meter Z(m²) 1 000 Gm2
1022 11 000 Gm2Area swept by Mercury's orbit around the Sun
37 000 Gm2Area swept by Venus' orbit around the Sun
71 000 Gm2Area swept by Earth's orbit around the Sun
1023 160 000 Gm2Area swept by Mars' orbit around the Sun
281 000 Gm2Surface area of a Dyson sphere with a radius of 1 AU
1024 1 yotta square meter (m²)
1 square terametre (Tm2)
1.9 Tm2Area swept by Jupiter's orbit around the Sun
6.4 Tm2Area swept by Saturn's orbit around the Sun
8.5 Tm2Surface area of the red supergiant star Betelgeuse
1025 24 Tm2Surface area of the hypergiant star VY Canis Majoris
26 Tm2Area swept by Uranus' orbit around the Sun
64 Tm2Area swept by Neptune's orbit around the Sun
1026 110 Tm2Area swept by Pluto's orbit around the Sun
1027 square metres and larger
List of orders of magnitude for areas 1027 square metres and larger.
Factor (m2) Multiple Value Item
1030 1 square petametre (Pm2)
1031 10 Pm2
1032 200 Pm2 Roughly the surface area of an Oort Cloud
300 Pm2 Roughly the surface area of a Bok globule
1033 1 000 Pm2
1034 30 000 Pm2 Roughly the surface area of The Bubble nebula
1035 100 000 Pm2
1036 1 square exametre (Em2)
...
1041 700 000 Em2 Roughly the area of Milky Way's galactic disk
1042 1 square zettametre (Zm2)
...
1048 1 square yottametre (Ym2)
1054 1 square ronnametre (Rm2) 2.4 Rm2 Surface area of the observable universe[90]
See also
• Orders of magnitude
• List of political and geographic subdivisions by total area
References
1. Calculated: square of the Planck length = (1.62e-35 m)^2 = 2.6e-70 m^2
2. Russ Rowlett (September 1, 2004). "Units: S". How Many? A Dictionary of Units of Measurement. University of North Carolina at Chapel Hill. Retrieved 2011-10-25.
3. "Femtobarn". CERN writing guidelines. CERN. Retrieved 2015-10-22.
4. Eric W. Weisstein. "Thomson Cross Section". Eric Weisstein's World of Science. Wolfram Research. Retrieved 2015-10-22.
5. "Other non-SI units". SI brochure. BIPM. Archived from the original on 2008-08-21. Retrieved 2011-10-25.
6. ""Rule of thumb" for the area per molecule in lipid bilayer". BioNumbers. Retrieved 2011-10-09.
7. "Individual Properties of the 20 Standard Amino Acids: Properties and Images". The Amino Acid Repository. Jena Library of Biological Macromolecules. Retrieved 2011-10-10.
8. Janin, J. E. L. (1979). "Surface and inside volumes in globular proteins". Nature. 277 (5696): 491–492. Bibcode:1979Natur.277..491J. doi:10.1038/277491a0. PMID 763335. S2CID 4338901.
9. "The Nuclear Pore Complex". UIUC Theoretical and Computational Biophysics Group. Retrieved 2011-10-14.
10. "E. coli Statistics". The CyberCell Database. Archived from the original on 2011-10-27. Retrieved 2011-09-11.
11. Marcelli, Gianluca; Parker, Kim H.; Winlove, C. Peter (2005). "Thermal Fluctuations of Red Blood Cell Membrane via a Constant-Area Particle-Dynamics Model". Biophysical Journal. 89 (4): 2473–2480. Bibcode:2005BpJ....89.2473M. doi:10.1529/biophysj.104.056168. PMC 1366746. PMID 16055528.
12. Calculated: Smallest and largest common pitches were 77 micrometers and 337 micrometers. (77e-6 m)^2 ~= 6e-9 m^2. (337e-6 m)^2 ~= 114e-9 m^2 ~= 110e-9 m^2
13. Calculated: (300 dots per inch / 2.54e-2 m/inch)^(-2) = 7.2e-9 m^2
14. "Hair Fiber Composition". Retrieved 2011-09-30.
15. Calculated: 100 μm in diameter => pi * ((1e-4 m)/2)**2 = 7.9e-9 m^2
16. Calculated: pi * (0.5mm/2)^2 = 2.0e-7 m^2 and pi * (0.7mm/2)^2 = 3.8e-7 m^2)
17. "Part XIII: Facts and Figures concerning the human retina". Webvision. University of Utah. Archived from the original on 2011-10-11. Retrieved 2011-09-28.
18. Calculated: ((6e-3 m)/2)**2 * pi = 2.8e-5 m^2 and ((8e-3 m)/2)**2 * pi = 5.0e-5 m^2
19. "Coin specifications". United States Mint. Retrieved 2011-12-28.
20. Calculated: area = pi * diameter^2 / 4 = 3.14 * (19.05e-3 m)^2 = 2.850e-4 m^2
21. Taylor, Enid; Jennings, Alan (1971). "Calculation of total retinal area". Br. J. Ophthalmol. 55 (4): 262–5. doi:10.1136/bjo.55.4.262. PMC 1208280. PMID 5572268.
22. "Credit Card Dimensions". Retrieved 2011-09-30.
23. Calculated: 3 inches * 5 inches * (2.54e-2 m/inch)^2 = 9.7e-3 m^2 ~= 0.01 m^2
24. Calculated: 1 foot * 1 foot * (0.3048 meters / foot)^2 = 0.092.90304 m^2
25. "Rules of the Game". USA Basketball. Archived from the original on 2011-10-27. Retrieved 2011-10-28.
26. Calculated: 29.5-29.75 inch circumference * 2.54 cm / in = 23.85-24.05 cm diameter => radius = 0.119-0.120 m => Area = 4 * pi * (0.119 m)^2 = 0.18 m^2
27. Sacco, Joseph J.; Botten, Joanne; Macbeth, Fergus; Bagust, Adrian; Clark, Peter (2010). "The Average Body Surface Area of Adult Cancer Patients in the UK: A Multicentre Retrospective Study". PLOS ONE. 5 (1): e8933. Bibcode:2010PLoSO...5.8933S. doi:10.1371/journal.pone.0008933. PMC 2812484. PMID 20126669.
28. Notter, Robert H. (2000). Lung surfactants: basic science and clinical applications. New York, N.Y: Marcel Dekker. p. 120. ISBN 0-8247-0401-0. Retrieved 2011-09-27.
29. "Section 1.1" (PDF). Official Volleyball Rules 2011-2012. FIVB. 2010. Retrieved 2011-10-27. The playing court is a rectangle measuring 18 x 9 m, surrounded by a free zone which is a minimum of 3 m wide on all sides.
30. "Median and Average Square Feet of Floor Area in New Single-Family Houses Completed by Location" (PDF). US Census Bureau. Retrieved 2011-09-26.
31. "Area of a Tennis Court". The Physics Factbook. Retrieved 2011-09-27.
32. Calculated: 4,700 sq ft * (0.3048 ft/m)2 = 436.644288 m2
33. Calculated: 50 m * 25 m = 1250 m^2
34. "General Tables of Units of Measurement" (PDF). NIST. Archived from the original (PDF) on 2006-11-26. Retrieved 2011-10-28. 4046.87
35. "What are the Dimensions of a Football Field". Dimensions Guide. Retrieved 2011-10-27.
36. Calculated: 360 feet * 160 feet * (0.3048 m/ft)^2 = 5351 m^2 ~= 5400 m^2
37. "How Big Is An Olympic Soccer Field?". LIVESTRONG.COM. Retrieved 2012-01-04. For the Olympics, fields are supposed to measure exactly 105 meters long and 68 meters wide
38. Calculated: 105 m * 68 m = 7140 m^2
39. "General Tables of Units of Measurement" (PDF). NIST. Archived from the original (PDF) on 2006-11-26. Retrieved 2011-10-28.
40. "AFL Ground Sizes | Passy's World of Mathematics". passyworldofmathematics.com. 11 September 2011. Retrieved 2016-11-12.
41. Greenberg, Ralph. "THE GREAT PYRAMID OF GIZA (Some Elegant Numerical Relationships)". Retrieved 2012-01-04. average length of the four sides is 230.364 meters
42. Calculated: 230.364 m^2 ~= 53068 m^2
43. Gartland, Fiona. "Valuable lead roofing stolen from Dublin bandstands". Archived from the original on 30 May 2018. Retrieved 29 May 2018.
44. "Holy See (Vatican City)". The World Factbook. Central Intelligence Agency. Retrieved 2011-10-28.
45. "The Pentagon - George Bergstrom". Great Buildings Online. Retrieved 2011-10-28. Floor area of 6.5 million square feet, 34 acres, 13.8 hectares, of which 3.7 million square feet are used for offices.
46. "Monaco". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-30.
47. Calculated: 1 mile * 1 mile * (1.61 km / mile)^2 = 2.59 km^2
48. "Jurisdictions: London". The International Finance Centre Portal. Retrieved 2011-10-28.
49. "New York -- Place and County Subdivision: Population, Housing Units, Area, and Density 2000". Census 2000 Summary File 1. US Census Bureau. Archived from the original on 2011-01-03. Retrieved 2011-10-28.
50. "San Marino". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-30.
51. "Comparateur de territoire: Commune de Paris (75056)". INSEE. Retrieved 2020-08-26.
52. "Walt Disney World Resort". Disney By The Numb3rs. Archived from the original on 2015-06-12. Retrieved 2011-10-28. 30,500 acres
53. "Appendix II Statistics". Taipei Yearbook 2010. Archived from the original on 2012-05-22. Retrieved 2011-10-28.
54. "Population and Dwelling Counts". 2001 Census. Statistics Canada. Retrieved 2011-10-28.
55. "Hong Kong". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
56. "California by Place: Los Angeles city". US Census. Archived from the original on 2020-02-12. Retrieved 2011-10-28. 498.29 square miles
57. "Cities with 100,000 or More Population in 2000 ranked by Land Area (square miles) /1, 2000 in Rank Order". U.S. Census Bureau, Administrative and Customer Services Division, Statistical Compendia Branch. March 16, 2004. Archived from the original on October 17, 2002. Retrieved 2010-10-26.
58. "OVERVIEW OF TOKYO". Tokyo Metropolitan Government. Archived from the original on 2011-11-08. Retrieved 2011-10-28.
59. "Kabupaten Klungkung : Data Agregat per Kecamatan" (PDF). Sp2010.bps.go.id. 2010. Retrieved 5 January 2018.
60. "Jamaica". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
61. "Lake Profile: Victoria". World Lakes. LakeNet. Retrieved 2011-10-28.
62. "Austria". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
63. "South Korea". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
64. "Italy". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
65. "Germany". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
66. "Japan". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
67. "Spain". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
68. "Turkey". The World Factbook. Central Intelligence Agency. Archived from the original on January 10, 2021. Retrieved 2011-09-29.
69. "Egypt". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
70. Rosstat (Russian Statistical Service), 2010 Archived 2012-10-18 at the Wayback Machine (xls). Retrieved 2012-06-15.
71. Turchin, Peter; Adams, Jonathan M.; Hall, Thomas D (December 2006). "East-West Orientation of Historical Empires". Journal of World-Systems Research. 12 (2): 222. ISSN 1076-156X. Retrieved 2016-09-16.
72. Taagepera, Rein (1979). "Size and Duration of Empires: Growth-Decline Curves, 600 B.C. to 600 A.D.". Social Science History. 3 (3/4): 125. doi:10.2307/1170959. JSTOR 1170959.
73. "Australia". The World Factbook. Central Intelligence Agency. Retrieved 2011-10-28.
74. "Canada". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
75. "Antarctica". The World Factbook. Central Intelligence Agency. Retrieved 2011-10-28.
76. "FAO Resources page". FAO.org. 2010.
77. "Pluto: By the Numbers". Solar System Exploration. NASA. Archived from the original on 2015-09-28. Retrieved 2015-12-11.
78. "Russia". The World Factbook. Central Intelligence Agency. Retrieved 2011-09-29.
79. "Map of Africa". Worldatlas.com. Retrieved 2012-01-04. 30,065,000 sq km
80. Rein Taagepera (September 1997). "Expansion and Contraction Patterns of Large Polities: Context for Russia". International Studies Quarterly. 41 (3): 502. doi:10.1111/0020-8833.00053. JSTOR 2600793.
81. "Earth's Moon: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2004-02-24. Retrieved 2011-09-29.
82. "The World Factbook: Atlantic Ocean". Central Intelligence Agency. 2011-03-24. Retrieved 2011-09-30.
83. "Mars: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2003-12-15. Retrieved 2011-09-29.
84. "The World Factbook: World". Central Intelligence Agency. 2011-08-31. Retrieved 2011-09-27.
85. "The World Factbook: Pacific Ocean". Central Intelligence Agency. 2011-11-17. Retrieved 2011-09-30.
86. "Neptune: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2003-12-15. Retrieved 2011-09-29.
87. "Saturn: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2004-02-24. Retrieved 2011-09-29.
88. "Jupiter: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2003-12-15. Retrieved 2011-09-29.
89. "Sun: Facts & Figures". Solar System Exploration. NASA. Archived from the original on 2011-07-03. Retrieved 2011-09-29.
90. "Wolfram|Alpha: Computational Knowledge Engine". www.wolframalpha.com. Retrieved 2016-03-01.
Orders of magnitude
Quantity
• Acceleration
• Angular momentum
• Area
• Bit rate
• Charge
• Computing
• Current
• Data
• Density
• Energy / Energy density
• Entropy
• Force
• Frequency
• Illuminance
• Length
• Luminance
• Magnetic field
• Mass
• Molarity
• Numbers
• Power
• Pressure
• Probability
• Radiation
• Sound pressure
• Specific heat capacity
• Speed
• Temperature
• Time
• Voltage
• Volume
See also
• Back-of-the-envelope calculation
• Best-selling electronic devices
• Fermi problem
• Powers of 10 and decades
• 10th
• 100th
• 1000000th
• Metric (SI) prefix
• Macroscopic scale
• Microscopic scale
Related
• Astronomical system of units
• Earth's location in the Universe
• Cosmic View (1957 book)
• To the Moon and Beyond (1964 film)
• Cosmic Zoom (1968 film)
• Powers of Ten (1968 and 1977 films)
• Cosmic Voyage (1996 documentary)
• Cosmic Eye (2012)
• Category
| Wikipedia |
Orders of magnitude (numbers)
This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.
Smaller than 10−100 (one googolth)
• Mathematics – random selections: Approximately 10−183,800 is a rough first estimate of the probability that a typing "monkey", or an English-illiterate typing robot, when placed in front of a typewriter, will type out William Shakespeare's play Hamlet as its first set of inputs, on the precondition it typed the needed number of characters.[1] However, demanding correct punctuation, capitalization, and spacing, the probability falls to around 10−360,783.[2]
• Computing: 2.2×10−78913 is approximately equal to the smallest non-zero value that can be represented by an octuple-precision IEEE floating-point value.
• 1×10−6176 is equal to the smallest non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value.
• 6.5×10−4966 is approximately equal to the smallest non-zero value that can be represented by a quadruple-precision IEEE floating-point value.
• 3.6×10−4951 is approximately equal to the smallest non-zero value that can be represented by an 80-bit x86 double-extended IEEE floating-point value.
• 1×10−398 is equal to the smallest non-zero value that can be represented by a double-precision IEEE decimal floating-point value.
• 4.9×10−324 is approximately equal to the smallest non-zero value that can be represented by a double-precision IEEE floating-point value.
• 1.5×10−157 is approximately equal to the probability that in a randomly selected group of 365 people, all of them will have different birthdays.[3]
• 1×10−101 is equal to the smallest non-zero value that can be represented by a single-precision IEEE decimal floating-point value.
10−100 to 10−30
• Mathematics: The chances of shuffling a standard 52-card deck in any specific order is around 1.24×10−68 (or exactly 1⁄52!)[4]
• Computing: The number 1.4×10−45 is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.
10−30
(0.000000000000000000000000000001; 1000−10; short scale: one nonillionth; long scale: one quintillionth)
ISO: quecto- (q)
• Mathematics: The probability in a game of bridge of all four players getting a complete suit each is approximately 4.47×10−28.[5]
10−27
(0.000000000000000000000000001; 1000−9; short scale: one octillionth; long scale: one quadrilliardth)
ISO: ronto- (r)
10−24
(0.000000000000000000000001; 1000−8; short scale: one septillionth; long scale: one quadrillionth)
ISO: yocto- (y)
10−21
(0.000000000000000000001; 1000−7; short scale: one sextillionth; long scale: one trilliardth)
ISO: zepto- (z)
• Mathematics: The probability of matching 20 numbers for 20 in a game of keno is approximately 2.83 × 10−19.
• Mathematics: The odds of a perfect bracket in the NCAA Division I men's basketball tournament are 1 in 263, approximately 1.08 × 10−19, if coin flips are used to predict the winners of the 63 matches.[6]
10−18
(0.000000000000000001; 1000−6; short scale: one quintillionth; long scale: one trillionth)
ISO: atto- (a)
• Mathematics: The probability of rolling snake eyes 10 times in a row on a pair of fair dice is about 2.74×10−16.
10−15
(0.000000000000001; 1000−5; short scale: one quadrillionth; long scale: one billiardth)
ISO: femto- (f)
• Mathematics: The Ramanujan constant, $e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots ,$ is an almost integer, differing from the nearest integer by approximately 7.5×10−13.
10−12
(0.000000000001; 1000−4; short scale: one trillionth; long scale: one billionth)
ISO: pico- (p)
• Mathematics: The probability in a game of bridge of one player getting a complete suit is approximately 2.52×10−11 (0.00000000252%).
• Biology: Human visual sensitivity to 1000 nm light is approximately 1.0×10−10 of its peak sensitivity at 555 nm.[7]
10−9
(0.000000001; 1000−3; short scale: one billionth; long scale: one milliardth)
ISO: nano- (n)
• Mathematics – Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the US Powerball lottery, with a single ticket, under the rules as of October 2015, are 292,201,338 to 1 against, for a probability of 3.422×10−9 (0.0000003422%).
• Mathematics – Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the Australian Powerball lottery, with a single ticket, under the rules as of April 2018, are 134,490,400 to 1 against, for a probability of 7.435×10−9 (0.0000007435%).
• Mathematics – Lottery: The odds of winning the Jackpot (matching the 6 main numbers) in the UK National Lottery, with a single ticket, under the rules as of August 2009, are 13,983,815 to 1 against, for a probability of 7.151×10−8 (0.000007151%).
10−6
(0.000001; 1000−2; long and short scales: one millionth)
ISO: micro- (μ)
Poker hands
HandChance
1. Royal flush0.00015%
2. Straight flush0.0014%
3. Four of a kind0.024%
4. Full house0.14%
5. Flush0.19%
6. Straight0.59%
7. Three of a kind2.1%
8. Two pairs4.8%
9. One pair42%
10. No pair50%
• Mathematics – Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5×10−6 (0.00015%).[8]
• Mathematics – Poker: The odds of being dealt a straight flush (other than a royal flush) in poker are 72,192 to 1 against, for a probability of 1.4×10−5 (0.0014%).
• Mathematics – Poker: The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4×10−4 (0.024%).
10−3
(0.001; 1000−1; one thousandth)
ISO: milli- (m)
• Mathematics – Poker: The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10−3 (0.14%).
• Mathematics – Poker: The odds of being dealt a flush in poker are 507.8 to 1 against, for a probability of 1.9 × 10−3 (0.19%).
• Mathematics – Poker: The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10−3 (0.39%).
• Physics: α = 0.007297352570(5), the fine-structure constant.
10−2
(0.01; one hundredth)
ISO: centi- (c)
• Mathematics – Lottery: The odds of winning any prize in the UK National Lottery, with a single ticket, under the rules as of 2003, are 54 to 1 against, for a probability of about 0.018 (1.8%).
• Mathematics – Poker: The odds of being dealt a three of a kind in poker are 46 to 1 against, for a probability of 0.021 (2.1%).
• Mathematics – Lottery: The odds of winning any prize in the Powerball, with a single ticket, under the rules as of 2015, are 24.87 to 1 against, for a probability of 0.0402 (4.02%).
• Mathematics – Poker: The odds of being dealt two pair in poker are 21 to 1 against, for a probability of 0.048 (4.8%).
10−1
(0.1; one tenth)
ISO: deci- (d)
• Legal history: 10% was widespread as the tax raised for income or produce in the ancient and medieval period; see tithe.
• Mathematics – Poker: The odds of being dealt only one pair in poker are about 5 to 2 against (2.37 to 1), for a probability of 0.42 (42%).
• Mathematics – Poker: The odds of being dealt no pair in poker are nearly 1 to 2, for a probability of about 0.5 (50%).
100
(1; one)
• Demography: The population of Monowi, an incorporated village in Nebraska, United States, was one in 2010.
• Religion: One is the number of gods in Judaism, Christianity, and Islam (monotheistic religions).
• Computing – Unicode: One character is assigned to the Lisu Supplement Unicode block, the fewest of any public-use Unicode block as of Unicode 15.0 (2022).
• Mathematics: √2 ≈ 1.414213562373095049, the ratio of the diagonal of a square to its side length.
• Mathematics: φ ≈ 1.618033988749894848, the golden ratio.
• Mathematics: √3 ≈ 1.732050807568877293, the ratio of the diagonal of a unit cube.
• Mathematics: the number system understood by most computers, the binary system, uses 2 digits: 0 and 1.
• Mathematics: √5 ≈ 2.236 067 9775, the correspondent to the diagonal of a rectangle whose side lengths are 1 and 2.
• Mathematics: √2 + 1 ≈ 2.414213562373095049, The ratio of smaller of the two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity.
• Mathematics: e ≈ 2.718281828459045087, the base of the natural logarithm.
• Mathematics: the number system understood by ternary computers, the ternary system, uses 3 digits: 0, 1, and 2.
• Religion: three manifestations of God in the Christian Trinity.
• Mathematics: π ≈ 3.141592653589793238, the ratio of a circle's circumference to its diameter.
• Religion: the Four Noble Truths in Buddhism.
• Biology: 7 ± 2, in cognitive science, George A. Miller's estimate of the number of objects that can be simultaneously held in human working memory.
• Music: 7 notes in a major or minor scale.
• Astronomy: 8 planets in the Solar System.
• Religion: the Noble Eightfold Path in Buddhism.
• Literature: 9 circles of Hell in the Inferno by Dante Alighieri.
101
(10; ten)
ISO: deca- (da)
• Demography: The population of Pesnopoy, a village in Bulgaria, was 10 in 2007.
• Human scale: There are 10 digits on a pair of human hands, and 10 toes on a pair of human feet.
• Mathematics: The number system used in everyday life, the decimal system, has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Religion: the Ten Commandments in the Abrahamic religions.
• Music: The number of notes (12) in a chromatic scale.
• Astrology: There are 12 zodiac signs, each one representing part of the annual path of the sun's movement across the night sky.
• Computing – Microsoft Windows: Twelve successive consumer versions of Windows NT have been released as of December 2021.
• Music: The number (15) of completed, numbered string quartets by each of Ludwig van Beethoven and Dmitri Shostakovich.
• Linguistics: The Finnish language has fifteen noun cases.
• Mathematics: The hexadecimal system, a common number system used in computer programming, uses 16 digits where the last 6 are usually represented by letters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
• Computing – Unicode: The minimum possible size of a Unicode block is 16 contiguous code points (i.e., U+abcde0 - U+abcdeF).
• Computing – UTF-16/Unicode: There are 17 addressable planes in UTF-16, and, thus, as Unicode is limited to the UTF-16 code space, 17 valid planes in Unicode.
• Science fiction: The 23 enigma plays a prominent role in the plot of The Illuminatus! Trilogy by Robert Shea and Robert Anton Wilson.
• Mathematics: eπ ≈ 23.140692633
• Music: a combined total of 24 major and minor keys, also the number of works in some musical cycles of J. S. Bach, Frédéric Chopin, Alexander Scriabin, and Dmitri Shostakovich.
• Alphabetic writing: There are 26 letters in the Latin-derived English alphabet (excluding letters found only in foreign loanwords).
• Science fiction: The number 42, in the novel The Hitchhiker's Guide to the Galaxy by Douglas Adams, is the Answer to the Ultimate Question of Life, the Universe, and Everything which is calculated by an enormous supercomputer over a period of 7.5 million years.
• Biology: Each human cell contains 46 chromosomes.
• Phonology: There are 47 phonemes in English phonology in Received Pronunciation.
• Syllabic writing: There are 49 letters in each of the two kana syllabaries (hiragana and katakana) used to represent Japanese (not counting letters representing sound patterns that have never occurred in Japanese).
• Chess: Either player in a chess game can claim a draw if 50 consecutive moves are made by each side without any captures or pawn moves.
• Demography: The population of Nassau Island, part of the Cook Islands, was around 78 in 2016.
• Syllabic writing: There are 85 letters in the modern version of the Cherokee syllabary.
• Music: There are 88 keys on a grand piano.
• Computing – ASCII: There are 95 printable characters in the ASCII character set.
102
(100; hundred)
ISO: hecto- (h)
• European history: Groupings of 100 homesteads were a common administrative unit in Northern Europe and Great Britain (see Hundred (county division)).
• Music: There are 104 numbered symphonies of Franz Josef Haydn.
• Religion: 108 is a sacred number in Hinduism.
• Chemistry: 118 chemical elements have been discovered or synthesized as of 2016.
• Computing – ASCII: There are 128 characters in the ASCII character set, including nonprintable control characters.
• Phonology: The Taa language is estimated to have between 130 and 164 distinct phonemes.
• Political Science: There were 193 member states of the United Nations as of 2011.
• Computing: A GIF image (or an 8-bit image) supports maximum 256 (=28) colors.
• Computing – Unicode: There are 327 different Unicode blocks as of Unicode 15.0 (2022).
• Aviation: 583 people died in the 1977 Tenerife airport disaster, the deadliest accident not caused by deliberate terrorist action in the history of civil aviation.
• Music: The highest number (626) in the Köchel catalogue of works of Wolfgang Amadeus Mozart.
• Demography: Vatican City, the least populous independent country, has an approximate population of 800 as of 2018.
103
(1000; thousand)
ISO: kilo- (k)
• Demography: The population of Ascension Island is 1,122.
• Music: 1,128: number of known extant works by Johann Sebastian Bach recognized in the Bach-Werke-Verzeichnis as of 2017.
• Typesetting: 2,000–3,000 letters on a typical typed page of text.
• Mathematics: 2,520 (5×7×8×9 or 23×32×5×7) is the least common multiple of every positive integer under (and including) 10.
• Terrorism: 2,996 persons (including 19 terrorists) died in the terrorist attacks of September 11, 2001.
• Biology: the DNA of the simplest viruses has 3,000 base pairs.[9]
• Military history: 4,200 (Republic) or 5,200 (Empire) was the standard size of a Roman legion.
• Linguistics: Estimates for the linguistic diversity of living human languages or dialects range between 5,000 and 10,000. (SIL Ethnologue in 2009 listed 6,909 known living languages.)
• Astronomy – Catalogues: There are 7,840 deep-sky objects in the NGC Catalogue from 1888.
• Lexicography: 8,674 unique words in the Hebrew Bible.
104
(10000; ten thousand or a myriad)
• Biology: Each neuron in the human brain is estimated to connect to 10,000 others.
• Demography: The population of Tuvalu was 10,544 in 2007.
• Lexicography: 14,500 unique English words occur in the King James Version of the Bible.
• Zoology: There are approximately 17,500 distinct butterfly species known.[10]
• Language: There are 20,000–40,000 distinct Chinese characters in more than occasional use.
• Biology: Each human being is estimated to have 20,000 coding genes.[11]
• Grammar: Each regular verb in Cherokee can have 21,262 inflected forms.
• War: 22,717 Union and Confederate soldiers were killed, wounded, or missing in the Battle of Antietam, the bloodiest single day of battle in American history.
• Computing – Unicode: 42,720 characters are encoded in CJK Unified Ideographs Extension B, the most of any single public-use Unicode block as of Unicode 15.0 (2022).
• Aviation: As of July 2021, 44,000+ airframes have been built of the Cessna 172, the most-produced aircraft in history.
• Computing - Fonts: The maximum possible number of glyphs in a TrueType or OpenType font is 65,535 (216-1), the largest number representable by the 16-bit unsigned integer used to record the total number of glyphs in the font.
• Computing – Unicode: A plane contains 65,536 (216) code points; this is also the maximum size of a Unicode block, and the total number of code points available in the obsolete UCS-2 encoding.
• Mathematics: 65,537 is the largest known Fermat prime.
• Memory: As of 2015, the largest number of decimal places of π that have been recited from memory is 70,030.[12]
105
(100000; one hundred thousand or a lakh).
• Demography: The population of Saint Vincent and the Grenadines was 100,982 in 2009.
• Biology – Strands of hair on a head: The average human head has about 100,000–150,000 strands of hair.
• Literature: approximately 100,000 verses (shlokas) in the Mahabharata.
• Computing – Unicode: 149,186 characters (including control characters) encoded in Unicode as of version 15.0 (2022).
• Language: 267,000 words in James Joyce's Ulysses.
• Computing – Unicode: 293,168 code points assigned to a Unicode block as of Unicode 15.0.
• Genocide: 300,000 people killed in the Rape of Nanking.
• Language – English words: The New Oxford Dictionary of English contains about 360,000 definitions for English words.
• Mathematics: 360,000 – The approximate number of entries in The On-Line Encyclopedia of Integer Sequences as of January 2023.[13]
• Biology – Plants: There are approximately 390,000 distinct plant species known, of which approximately 20% (or 78,000) are in risk of extinction.[14]
• Biology – Flowers: There are approximately 400,000 distinct flower species on Earth.[15]
• Literature: 564,000 words in War and Peace by Leo Tolstoy.
• Literature: 930,000 words in the King James Version of the Bible.
• Mathematics: There are 933,120 possible combinations on the Pyraminx.
• Computing – Unicode: There are 974,530 publicly-assignable code points (i.e., not surrogates, private-use code points, or noncharacters) in Unicode.
106
(1000000; 10002; long and short scales: one million)
ISO: mega- (M)
• Demography: The population of Riga, Latvia was 1,003,949 in 2004, according to Eurostat.
• Computing – UTF-8: There are 1,112,064 (220 + 216 - 211) valid UTF-8 sequences (excluding overlong sequences and sequences corresponding to code points used for UTF-16 surrogates or code points beyond U+10FFFF).
• Computing – UTF-16/Unicode: There are 1,114,112 (220 + 216) distinct values encodable in UTF-16, and, thus (as Unicode is currently limited to the UTF-16 code space), 1,114,112 valid code points in Unicode (1,112,064 scalar values and 2,048 surrogates).
• Ludology – Number of games: Approximately 1,181,019 video games have been created as of 2019.[16]
• Biology – Species: The World Resources Institute claims that approximately 1.4 million species have been named, out of an unknown number of total species (estimates range between 2 and 100 million species). Some scientists give 8.8 million species as an exact figure.
• Genocide: Approximately 800,000–1,500,000 (1.5 million) Armenians were killed in the Armenian genocide.
• Linguistics: The number of possible conjugations for each verb in the Archi language is 1,502,839.[17]
• Info: The freedb database of CD track listings has around 1,750,000 entries as of June 2005.
• War: 1,857,619 casualties at the Battle of Stalingrad.
• Computing – UTF-8: 2,164,864 (221 + 216 + 211 + 27) possible one- to four-byte UTF-8 sequences, if the restrictions on overlong sequences, surrogate code points, and code points beyond U+10FFFF are not adhered to. (Note that not all of these correspond to unique code points.)
• Mathematics – Playing cards: There are 2,598,960 different 5-card poker hands that can be dealt from a standard 52-card deck.
• Mathematics: There are 3,149,280 possible positions for the Skewb.
• Mathematics – Rubik's Cube: 3,674,160 is the number of combinations for the Pocket Cube (2×2×2 Rubik's Cube).
• Geography/Computing – Geographic places: The NIMA GEOnet Names Server contains approximately 3.88 million named geographic features outside the United States, with 5.34 million names. The USGS Geographic Names Information System claims to have almost 2 million physical and cultural geographic features within the United States.
• Computing - Supercomputer hardware: 4,981,760 processor cores in the final configuration of the Tianhe-2 supercomputer.
• Genocide: Approximately 5,100,000–6,200,000 Jews were killed in the Holocaust.
• Info – Web sites: As of August 20, 2023, the English Wikipedia contains approximately 6.7 million articles in the English language.
107
(10000000; a crore; long and short scales: ten million)
• Demography: The population of Haiti was 10,085,214 in 2010.
• Literature: 11,206,310 words in Devta by Mohiuddin Nawab, the longest continuously published story known in the history of literature.
• Genocide: An estimated 12 million persons shipped from Africa to the New World in the Atlantic slave trade.
• Mathematics: 12,988,816 is the number of domino tilings of an 8×8 checkerboard.
• Genocide/Famine: 15 million is an estimated lower bound for the death toll of the 1959–1961 Great Chinese Famine, the deadliest known famine in human history.
• War: 15 to 22 million casualties estimated as a result of World War I.
• Computing: 16,777,216 different colors can be generated using the hex code system in HTML (note that the trichromatic color vision of the human eye can only distinguish between about an estimated 1,000,000 different colors).[18]
• Science Fiction: In Isaac Asimov's Galactic Empire, in 22,500 CE, there are 25,000,000 different inhabited planets in the Galactic Empire, all inhabited by humans in Asimov's "human galaxy" scenario.
• Genocide/Famine: 55 million is an estimated upper bound for the death toll of the Great Chinese Famine.
• Literature: Wikipedia contains a total of around 61 million articles in 334 languages as of August 2023.
• War: 70 to 85 million casualties estimated as a result of World War II.
• Mathematics: 73,939,133 is the largest right-truncatable prime.
108
(100000000; long and short scales: one hundred million)
• Demography: The population of the Philippines was 100,981,437 in 2015.
• Internet – YouTube: The number of YouTube channels is estimated to be 113.9 million.[19]
• Info – Books: The British Library claims that it holds over 150 million items. The Library of Congress claims that it holds approximately 148 million items. See The Gutenberg Galaxy.
• Video gaming: As of 2020, approximately 200 million copies of Minecraft (the most-sold video game in history) have been sold.
• Mathematics: More than 215,000,000 mathematical constants are collected on the Plouffe's Inverter as of 2010.[20]
• Mathematics: 275,305,224 is the number of 5×5 normal magic squares, not counting rotations and reflections. This result was found in 1973 by Richard Schroeppel.
• Demography: The population of the United States was 328,239,523 in 2019.
• Mathematics: 358,833,097 stellations of the rhombic triacontahedron.
• Info – Web sites: As of November 2011, the Netcraft web survey estimates that there are 525,998,433 (526 million) distinct websites.
• Astronomy – Cataloged stars: The Guide Star Catalog II has entries on 998,402,801 distinct astronomical objects.
109
(1000000000; 10003; short scale: one billion; long scale: one thousand million, or one milliard)
ISO: giga- (G)
• Transportation – Cars: As of 2018, there are approximately 1.4 billion cars in the world, corresponding to around 18% of the human population.[21]
• Demographics – India 1,420,000,000 – approximate population of India in 2023.
• Demographics – Africa: The population of Africa reached 1,430,000,000 sometime in 2023.
• Demographics – China: 1,455,000,000 – approximate population of the People's Republic of China in 2023.
• Internet – Google: There are more than 1,500,000,000 active Gmail users globally.[22]
• Internet: Approximately 1,500,000,000 active users were on Facebook as of October 2015.[23]
• Computing – Computational limit of a 32-bit CPU: 2,147,483,647 is equal to 231−1, and as such is the largest number which can fit into a signed (two's complement) 32-bit integer on a computer.
• Computing – UTF-8: 2,147,483,648 (231) possible code points (U+0000 - U+7FFFFFFF) in the pre-2003 version of UTF-8 (including five- and six-byte sequences), before the UTF-8 code space was limited to the much smaller set of values encodable in UTF-16.
• Biology – base pairs in the genome: approximately 3.3×109 base pairs in the human genome.[11]
• Linguistics: 3,400,000,000 – the total number of speakers of Indo-European languages, of which 2,400,000,000 are native speakers; the other 1,000,000,000 speak Indo-European languages as a second language.
• Mathematics and computing: 4,294,967,295 (232 − 1), the product of the five known Fermat primes and the maximum value for a 32-bit unsigned integer in computing.
• Computing – IPv4: 4,294,967,296 (232) possible unique IP addresses.
• Computing: 4,294,967,296 – the number of bytes in 4 gibibytes; in computation, 32-bit computers can directly access 232 units (bytes) of address space, which leads directly to the 4-gigabyte limit on main memory.
• Mathematics: 4,294,967,297 is a Fermat number and semiprime. It is the smallest number of the form $2^{2^{n}}+1$ which is not a prime number.
• Demographics – world population: 8,300,000,000 – Estimated population for the world as of April 2023.[24]
1010
(10000000000; short scale: ten billion; long scale: ten thousand million, or ten milliard)
• Biology – bacteria in the human body: There are roughly 1010 bacteria in the human mouth.[25]
• Computing – web pages: approximately 5.6×1010 web pages indexed by Google as of 2010.
1011
(100000000000; short scale: one hundred billion; long scale: hundred thousand million, or hundred milliard)
• Astronomy: There are 100 billion planets located in the Milky Way.[26][27]
• Biology – Neurons in the brain: approximately (1±0.2) × 1011 neurons in the human brain.[28]
• Medicine: The United States Food and Drug Administration requires a minimum of 3 x 1011 (300 billion) platelets per apheresis unit.[29]
• Paleodemography – Number of humans that have ever lived: approximately (1.2±0.3) × 1011 live births of anatomically modern humans since the beginning of the Upper Paleolithic.[30]
• Astronomy – stars in our galaxy: of the order of 1011 stars in the Milky Way galaxy.[31]
1012
(1000000000000; 10004; short scale: one trillion; long scale: one billion)
ISO: tera- (T)
• Astronomy: Andromeda Galaxy, which is part of the same Local Group as our galaxy, contains about 1012 stars.
• Biology – Bacteria on the human body: The surface of the human body houses roughly 1012 bacteria.[25]
• Astronomy – Galaxies: A 2016 estimate says there are 2 × 1012 galaxies in the observable universe.[32]
• Biology – Blood cells in the human body: The average human body has 2.5 × 1012 red blood cells.
• Biology: An estimate says there were 3.04 × 1012 trees on Earth in 2015.[33]
• Marine biology: 3,500,000,000,000 (3.5 × 1012) – estimated population of fish in the ocean.
• Mathematics: 7,625,597,484,987 – a number that often appears when dealing with powers of 3. It can be expressed as $19683^{3}$, $27^{9}$, $3^{27}$, $3^{3^{3}}$ and 33 or when using Knuth's up-arrow notation it can be expressed as $3\uparrow \uparrow 3$ and $3\uparrow \uparrow \uparrow 2$.
• Astronomy: A light-year, as defined by the International Astronomical Union (IAU), is the distance that light travels in a vacuum in one year, which is equivalent to about 9.46 trillion kilometers (9.46×1012 km).
• Mathematics: 1013 – The approximate number of known non-trivial zeros of the Riemann zeta function as of 2004.[34]
• Mathematics – Known digits of π: As of March 2019, the number of known digits of π is 31,415,926,535,897 (the integer part of π×1013).[35]
• Biology – approximately 1014 synapses in the human brain.[36]
• Biology – Cells in the human body: The human body consists of roughly 1014 cells, of which only 1013 are human.[37][38] The remaining 90% non-human cells (though much smaller and constituting much less mass) are bacteria, which mostly reside in the gastrointestinal tract, although the skin is also covered in bacteria.
• Mathematics: The first case of exactly 18 prime numbers between multiples of 100 is 122,853,771,370,900 + n,[39] for n = 1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, 99.
• Cryptography: 150,738,274,937,250 configurations of the plug-board of the Enigma machine used by the Germans in WW2 to encode and decode messages by cipher.
• Computing – MAC-48: 281,474,976,710,656 (248) possible unique physical addresses.
• Mathematics: 953,467,954,114,363 is the largest known Motzkin prime.
1015
(1000000000000000; 10005; short scale: one quadrillion; long scale: one thousand billion, or one billiard)
ISO: peta- (P)
• Biology – Insects: 1,000,000,000,000,000 to 10,000,000,000,000,000 (1015 to 1016) – The estimated total number of ants on Earth alive at any one time (their biomass is approximately equal to the total biomass of the human species).[40]
• Computing: 9,007,199,254,740,992 (253) – number until which all integer values can exactly be represented in IEEE double precision floating-point format.
• Mathematics: 48,988,659,276,962,496 is the fifth taxicab number.
• Science Fiction: In Isaac Asimov's Galactic Empire, in what we call 22,500 CE, there are 25,000,000 different inhabited planets in the Galactic Empire, all inhabited by humans in Asimov's "human galaxy" scenario, each with an average population of 2,000,000,000, thus yielding a total Galactic Empire population of approximately 50,000,000,000,000,000.
• Science Fiction: There are approximately 1017 sentient beings in the Star Wars galaxy.
• Cryptography: There are 256 = 72,057,594,037,927,936 different possible keys in the obsolete 56-bit DES symmetric cipher.
1018
(1000000000000000000; 10006; short scale: one quintillion; long scale: one trillion)
ISO: exa- (E)
• Mathematics: The first case of exactly 19 prime numbers between multiples of 100 is 1,468,867,005,116,420,800 + n,[39] for n = 1, 3, 7, 9, 21, 31, 37, 39, 43, 49, 51, 63, 67, 69, 73, 79, 81, 87, 93.
• Mathematics: Goldbach's conjecture has been verified for all n ≤ 4×1018 by a project which computed all prime numbers up to that limit.[41]
• Computing – Manufacturing: An estimated 6×1018 transistors were produced worldwide in 2008.[42]
• Computing – Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22×1018) is equal to 263−1, and as such is the largest number which can fit into a signed (two's complement) 64-bit integer on a computer.
• Mathematics – NCAA basketball tournament: There are 9,223,372,036,854,775,808 (263) possible ways to enter the bracket.
• Mathematics – Bases: 9,439,829,801,208,141,318 (≈9.44×1018) is the 10th and (by conjecture) largest number with more than one digit that can be written from base 2 to base 18 using only the digits 0 to 9, meaning the digits for 10 to 17 are not needed in bases above 10.[43]
• Biology – Insects: It has been estimated that the insect population of the Earth is about 1019.[44]
• Mathematics – Answer to the wheat and chessboard problem: When doubling the grains of wheat on each successive square of a chessboard, beginning with one grain of wheat on the first square, the final number of grains of wheat on all 64 squares of the chessboard when added up is 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019).
• Mathematics – Legends: The Tower of Brahma legend tells about a Hindu temple containing a large room with three posts, on one of which are 64 golden discs, and the object of the mathematical game is for the Brahmins in this temple to move all of the discs to another pole so that they are in the same order, never placing a larger disc above a smaller disc, moving only one at a time. Using the simplest algorithm for moving the disks, it would take 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019) turns to complete the task (the same number as the wheat and chessboard problem above).[45]
• Computing – IPv6: 18,446,744,073,709,551,616 (264; ≈1.84×1019) possible unique /64 subnetworks.
• Mathematics – Rubik's Cube: There are 43,252,003,274,489,856,000 (≈4.33×1019) different positions of a 3×3×3 Rubik's Cube.
• Password strength: Usage of the 95-character set found on standard computer keyboards for a 10-character password yields a computationally intractable 59,873,693,923,837,890,625 (9510, approximately 5.99×1019) permutations.
• Economics: Hyperinflation in Zimbabwe estimated in February 2009 by some economists at 10 sextillion percent,[46] or a factor of 1020.
1021
(1000000000000000000000; 10007; short scale: one sextillion; long scale: one thousand trillion, or one trilliard)
ISO: zetta- (Z)
• Geo – Grains of sand: All the world's beaches combined have been estimated to hold roughly 1021 grains of sand.[47]
• Computing – Manufacturing: Intel predicted that there would be 1.2×1021 transistors in the world by 2015[48] and Forbes estimated that 2.9×1021 transistors had been shipped up to 2014.[49]
• Mathematics – Sudoku: There are 6,670,903,752,021,072,936,960 (≈6.7×1021) 9×9 sudoku grids.[50]
• Astronomy – Stars: 70 sextillion = 7×1022, the estimated number of stars within range of telescopes (as of 2003).[51]
• Astronomy – Stars: in the range of 1023 to 1024 stars in the observable universe.[52]
• Mathematics: 146,361,946,186,458,562,560,000 (≈1.5×1023) is the fifth unitary perfect number.
• Mathematics: 357,686,312,646,216,567,629,137 (≈3.6×1023) is the largest left-truncatable prime.
• Chemistry – Physics: The Avogadro constant (6.02214076×1023) is the number of constituents (e.g. atoms or molecules) in one mole of a substance, defined for convenience as expressing the order of magnitude separating the molecular from the macroscopic scale.
1024
(1000000000000000000000000; 10008; short scale: one septillion; long scale: one quadrillion)
ISO: yotta- (Y)
• Mathematics: 2,833,419,889,721,787,128,217,599 (≈2.8×1024) is the fifth Woodall prime.
• Mathematics: 3,608,528,850,368,400,786,036,725 (≈3.6×1024) is the largest polydivisible number.
• Mathematics: 286 = 77,371,252,455,336,267,181,195,264 is the largest known power of two not containing the digit '0' in its decimal representation.[53]
1027
(1000000000000000000000000000; 10009; short scale: one octillion; long scale: one thousand quadrillion, or one quadrilliard)
ISO: ronna- (R)
• Biology – Atoms in the human body: the average human body contains roughly 7×1027 atoms.[54]
• Mathematics – Poker: the number of unique combinations of hands and shared cards in a 10-player game of Texas hold 'em is approximately 2.117×1028.
1030
(1000000000000000000000000000000; 100010; short scale: one nonillion; long scale: one quintillion)
ISO: quetta- (Q)
• Biology – Bacterial cells on Earth: The number of bacterial cells on Earth is estimated at 5,000,000,000,000,000,000,000,000,000,000, or 5 × 1030.[55]
• Mathematics: 5,000,000,000,000,000,000,000,000,000,027 is the largest quasi-minimal prime.
• Mathematics: The number of partitions of 1000 is 24,061,467,864,032,622,473,692,149,727,991.[56]
• Mathematics: 368 = 278,128,389,443,693,511,257,285,776,231,761 is the largest known power of three not containing the digit '0' in its decimal representation.
• Mathematics: 2108 = 324,518,553,658,426,726,783,156,020,576,256 is the largest known power of two not containing the digit '9' in its decimal representation.[57]
1033
(1000000000000000000000000000000000; 100011; short scale: one decillion; long scale: one thousand quintillion, or one quintilliard)
• Mathematics – Alexander's Star: There are 72,431,714,252,715,638,411,621,302,272,000,000 (about 7.24×1034) different positions of Alexander's Star.
1036
(1000000000000000000000000000000000000; 100012; short scale: one undecillion; long scale: one sextillion)
• Mathematics: 227−1 − 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727 (≈1.7×1038) is the largest known double Mersenne prime.
• Computing: 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×1038), the theoretical maximum number of Internet addresses that can be allocated under the IPv6 addressing system, one more than the largest value that can be represented by a single-precision IEEE floating-point value, the total number of different Universally Unique Identifiers (UUIDs) that can be generated.
• Cryptography: 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×1038), the total number of different possible keys in the AES 128-bit key space (symmetric cipher).
1039
(1000000000000000000000000000000000000000; 100013; short scale: one duodecillion; long scale: one thousand sextillion, or one sextilliard)
• Cosmology: The Eddington–Dirac number is roughly 1040.
• Mathematics: 97# × 25 × 33 × 5 × 7 = 69,720,375,229,712,477,164,533,808,935,312,303,556,800 (≈6.97×1040) is the least common multiple of every integer from 1 to 100.
1042 to 10100
(1000000000000000000000000000000000000000000; 100014; short scale: one tredecillion; long scale: one septillion)
• Mathematics: 141×2141+1 = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833 (≈3.93×1044) is the second Cullen prime.
• Mathematics: There are 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (≈7.4×1045) possible permutations for the Rubik's Revenge (4×4×4 Rubik's Cube).
• Chess: 4.52×1046 is a proven upper bound for the number of chess positions allowed according to the rules of chess.[58]
• Geo: 1.33×1050 is the estimated number of atoms on Earth.
• Mathematics: 2168 = 374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856 is the largest known power of two which is not pandigital: There is no digit '2' in its decimal representation.[59]
• Mathematics: 3106 = 375,710,212,613,636,260,325,580,163,599,137,907,799,836,383,538,729 is the largest known power of three which is not pandigital: There is no digit '4'.[59]
• Mathematics: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (≈8.08×1053) is the order of the monster group.
• Cryptography: 2192 = 6,277,101,735,386,680,763,835,789,423,207,666,416,102,355,444,464,034,512,896 (6.27710174×1057), the total number of different possible keys in the Advanced Encryption Standard (AES) 192-bit key space (symmetric cipher).
• Cosmology: 8×1060 is roughly the number of Planck time intervals since the universe is theorised to have been created in the Big Bang 13.799 ± 0.021 billion years ago.[60]
• Cosmology: 1×1063 is Archimedes' estimate in The Sand Reckoner of the total number of grains of sand that could fit into the entire cosmos, the diameter of which he estimated in stadia to be what we call 2 light-years.
• Mathematics – Cards: 52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 (≈8.07×1067) – the number of ways to order the cards in a 52-card deck.
• Mathematics: There are ≈1.01×1068 possible combinations for the Megaminx.
• Mathematics: 1,808,422,353,177,349,564,546,512,035,512,530,001,279,481,259,854,248,860,454,348,989,451,026,887 (≈1.81×1072) – The largest known prime factor found by Lenstra elliptic-curve factorization (LECF) factorization as of 2010.[61]
• Mathematics: There are 282,870,942,277,741,856,536,180,333,107,150,328,293,127,731,985,672,134,721,536,000,000,000,000,000 (≈2.83×1074) possible permutations for the Professor's Cube (5×5×5 Rubik's Cube).
• Cryptography: 2256 = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936 (≈1.15792089×1077), the total number of different possible keys in the Advanced Encryption Standard (AES) 256-bit key space (symmetric cipher).
• Cosmology: Various sources estimate the total number of fundamental particles in the observable universe to be within the range of 1080 to 1085.[62][63] However, these estimates are generally regarded as guesswork. (Compare the Eddington number, the estimated total number of protons in the observable universe.)
• Computing: 9.999 999×1096 is equal to the largest value that can be represented in the IEEE decimal32 floating-point format.
• Computing: 69! (roughly 1.7112245×1098), is the highest factorial value that can be represented on a calculator with two digits for powers of ten without overflow.
• Mathematics: One googol, 1×10100, 1 followed by one hundred zeros, or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
10100 (one googol) to 101000
See also: googol
(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard)[64]
• Mathematics: There are 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000 000 000 000 000 000 000 (≈1.57×10116) distinguishable permutations of the V-Cube 6 (6×6×6 Rubik's Cube).
• Chess: Shannon number, 10120, a lower bound of the game-tree complexity of chess.
• Physics: 10120, discrepancy between the observed value of the cosmological constant and a naive estimate based on Quantum Field Theory and the Planck energy.
• Physics: 8×10120, ratio of the mass-energy in the observable universe to the energy of a photon with a wavelength the size of the observable universe.
• Mathematics: 19 568 584 333 460 072 587 245 340 037 736 278 982 017 213 829 337 604 336 734 362 294 738 647 777 395 483 196 097 971 852 999 259 921 329 236 506 842 360 439 300 (≈1.96×10121) is the period of primary pretenders.
• History – Religion: Asaṃkhyeya is a Buddhist name for the number 10140. It is listed in the Avatamsaka Sutra and metaphorically means "innumerable" in the Sanskrit language of ancient India.
• Xiangqi: 10150, an estimation of the game-tree complexity of xiangqi.
• Mathematics: There are 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 (≈1.95×10160) distinguishable permutations of the V-Cube 7 (7×7×7 Rubik's Cube).
• Go: There are 208 168 199 381 979 984 699 478 633 344 862 770 286 522 453 884 530 548 425 639 456 820 927 419 612 738 015 378 525 648 451 698 519 643 907 259 916 015 628 128 546 089 888 314 427 129 715 319 317 557 736 620 397 247 064 840 935 (≈2.08×10170) legal positions in the game of Go. See Go and mathematics.
• Economics: The annualized rate of the hyperinflation in Hungary in 1946 was estimated to be 2.9×10177%.[65] It was the most extreme case of hyperinflation ever recorded.
• Board games: 3.457×10181, number of ways to arrange the tiles in English Scrabble on a standard 15-by-15 Scrabble board.
• Physics: 10186, approximate number of Planck volumes in the observable universe.
• Shogi: 10226, an estimation of the game-tree complexity of shogi.
• Physics: 7×10245, approximate spacetime volume of the history of the observable universe in Planck units.[66]
• Computing: 1.797 693 134 862 315 807×10308 is approximately equal to the largest value that can be represented in the IEEE double precision floating-point format.
• Computing: (10 – 10−15)×10384 is equal to the largest value that can be represented in the IEEE decimal64 floating-point format.
• Mathematics: 997# × 31# × 25 × 34 × 54 × 7 = 7 128 865 274 665 093 053 166 384 155 714 272 920 668 358 861 885 893 040 452 001 991 154 324 087 581 111 499 476 444 151 913 871 586 911 717 817 019 575 256 512 980 264 067 621 009 251 465 871 004 305 131 072 686 268 143 200 196 609 974 862 745 937 188 343 705 015 434 452 523 739 745 298 963 145 674 982 128 236 956 232 823 794 011 068 809 262 317 708 861 979 540 791 247 754 558 049 326 475 737 829 923 352 751 796 735 248 042 463 638 051 137 034 331 214 781 746 850 878 453 485 678 021 888 075 373 249 921 995 672 056 932 029 099 390 891 687 487 672 697 950 931 603 520 000 (≈7.13×10432) is the least common multiple of every integer from 1 to 1000.
101000 to 1010100 (one googolplex)
See also: googolplex
• Mathematics: There are approximately 1.869×104099 distinguishable permutations of the world's largest Rubik's Cube (33×33×33).
• Computing: 1.189 731 495 357 231 765 05×104932 is approximately equal to the largest value that can be represented in the IEEE 80-bit x86 extended precision floating-point format.
• Computing: 1.189 731 495 357 231 765 085 759 326 628 007 0×104932 is approximately equal to the largest value that can be represented in the IEEE quadruple-precision floating-point format.
• Computing: (10 – 10−33)×106144 is equal to the largest value that can be represented in the IEEE decimal128 floating-point format.
• Computing: 1010,000 − 1 is equal to the largest value that can be represented in Windows Phone's calculator.
• Mathematics: 86562929 + 29298656 is the largest proven Leyland prime; with 30,008 digits as of April 2023.[67]
• Mathematics: approximately 7.76 × 10206,544 cattle in the smallest herd which satisfies the conditions of Archimedes's cattle problem.
• Mathematics: 2,618,163,402,417 × 21,290,000 − 1 is a 388,342-digit Sophie Germain prime; the largest known as of April 2023.[68]
• Mathematics: 2,996,863,034,895 × 21,290,000 ± 1 are 388,342-digit twin primes; the largest known as of April 2023.[69]
• Mathematics: 3,267,113# – 1 is a 1,418,398-digit primorial prime; the largest known as of April 2023.[70]
• Mathematics – Literature: Jorge Luis Borges' Library of Babel contains at least 251,312,000 ≈ 1.956 × 101,834,097 books (this is a lower bound).[71]
• Mathematics: 101,888,529 − 10944,264 – 1 is a 1,888,529-digit palindromic prime, the largest known as of April 2023.[72]
• Mathematics: 4 × 721,119,849 − 1 is the smallest prime of the form 4 × 72n − 1.[73]
• Mathematics: 422,429! + 1 is a 2,193,027-digit factorial prime; the largest known as of April 2023.[74]
• Mathematics: (215,135,397 + 1)/3 is a 4,556,209-digit Wagstaff probable prime, the largest known as of June 2021.
• Mathematics: 1,963,7361,048,576 + 1 is a 6,598,776-digit Generalized Fermat prime, the largest known as of April 2023.[75]
• Mathematics: (108,177,207 − 1)/9 is a 8,177,207-digit probable prime, the largest known as of 8 May 2021.[76]
• Mathematics: 10,223 × 231,172,165 + 1 is a 9,383,761-digit Proth prime, the largest known Proth prime[77] and non-Mersenne prime as of 2021.[78]
• Mathematics: 282,589,933 − 1 is a 24,862,048-digit Mersenne prime; the largest known prime of any kind as of 2020.[78]
• Mathematics: 282,589,932 × (282,589,933 − 1) is a 49,724,095-digit perfect number, the largest known as of 2020.[79]
• Mathematics – History: 108×1016, largest named number in Archimedes' Sand Reckoner.
• Mathematics: 10googol ($10^{10^{100}}$), a googolplex. A number 1 followed by 1 googol zeros. Carl Sagan has estimated that 1 googolplex, fully written out, would not fit in the observable universe because of its size, while also noting that one could also write the number as 1010100.[80]
Larger than 1010100
(One googolplex; 10googol; short scale: googolplex; long scale: googolplex)
• Mathematics – Literature: The number of different ways in which the books in Jorge Luis Borges' Library of Babel can be arranged is approximately $10^{10^{1,834,102}}$, the factorial of the number of books in the Library of Babel.
• Cosmology: In chaotic inflation theory, proposed by physicist Andrei Linde, our universe is one of many other universes with different physical constants that originated as part of our local section of the multiverse, owing to a vacuum that had not decayed to its ground state. According to Linde and Vanchurin, the total number of these universes is about $10^{10^{10,000,000}}$.[81]
• Mathematics: $10^{\,\!10^{10^{34}}}$, order of magnitude of an upper bound that occurred in a proof of Skewes (this was later estimated to be closer to 1.397 × 10316).
• Cosmology: The estimated number of Planck time units for quantum fluctuations and tunnelling to generate a new Big Bang is estimated to be $10^{10^{10^{56}}}$.
• Mathematics: $10^{\,\!10^{10^{100}}}$, a number in the googol family called a googolplexplex, googolplexian, or googolduplex. 1 followed by a googolplex zeros, or 10googolplex
• Cosmology: The uppermost estimate to the size of the entire universe is approximately $10^{10^{10^{122}}}$ times that of the observable universe.[82]
• Mathematics: $10^{\,\!10^{10^{963}}}$, order of magnitude of another upper bound in a proof of Skewes.
• Mathematics: $10^{\,\!10^{10^{10^{100}}}}$, a number in the googol family called a googolplexplexplex, googolplexianth, or googoltriplex. 1 followed by a googolduplex zeros, or 10googolduplex
• Mathematics: Steinhaus' mega lies between 10[4]257 and 10[4]258 (where a[n]b is hyperoperation).
• Mathematics: Moser's number, "2 in a mega-gon" in Steinhaus–Moser notation, is approximately equal to 10[10[4]257]10, the last four digits are ...1056.
• Mathematics: Graham's number, the last ten digits of which are ...2464195387, equals 3[3[3[...3[3[3[6]3+2]3+2]3...]3+2]3+2]3 with 64 levels of brackets. Arises as an upper bound solution to a problem in Ramsey theory. Representation in powers of 10 would be impractical (the number of 10s in the power tower $10^{\,\!10^{10^{...}}}$ would be virtually indistinguishable from the number itself).
• Mathematics: TREE(3): appears in relation to a theorem on trees in graph theory. Representation of the number is difficult, but one weak lower bound is AA(187196)(1), where A(n) is a version of the Ackermann function.
• Mathematics: SSCG(3): appears in relation to the Robertson–Seymour theorem. Known to be greater than TREE(3).
• Mathematics: Transcendental integers: a set of numbers defined in 2000 by Harvey Friedman, appears in proof theory.[83]
• Mathematics: Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest number to have ever been named.[84] It was originally defined in a "big number duel" at MIT on 26 January 2007.[85]
See also
• Conway chained arrow notation
• Encyclopedic size comparisons on Wikipedia
• Fast-growing hierarchy
• Indian numbering system
• Large numbers
• List of numbers
• Mathematical constant
• Names of large numbers
• Names of small numbers
• Power of 10
References
1. Charles Kittel and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 53. ISBN 978-0-7167-1088-2.
2. There are around 130,000 letters and 199,749 total characters in Hamlet; 26 letters ×2 for capitalization, 12 for punctuation characters = 64, 64199749 ≈ 10360,783.
3. Calculated: 365! / 365365 ≈ 1.455×10−157
4. Robert Matthews. "What are the odds of shuffling a deck of cards into the right order?". Science Focus. Retrieved December 10, 2018.
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13. Sloane, N. J. A. (ed.). "Sequence A360000". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-04-14.
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26. "How Many Planets are in the Milky Way? | Amount, Location & Key Facts". The Nine Planets. Retrieved 2020-11-28.
27. January 2013, Space com Staff 02 (2 January 2013). "100 Billion Alien Planets Fill Our Milky Way Galaxy: Study". Space.com. Retrieved 2020-11-28.
28. "there was, to our knowledge, no actual, direct estimate of numbers of cells or of neurons in the entire human brain to be cited until 2009. A reasonable approximation was provided by Williams and Herrup (1988), from the compilation of partial numbers in the literature. These authors estimated the number of neurons in the human brain at about 85 billion [...] With more recent estimates of 21–26 billion neurons in the cerebral cortex (Pelvig et al., 2008 ) and 101 billion neurons in the cerebellum (Andersen et al., 1992 ), however, the total number of neurons in the human brain would increase to over 120 billion neurons." Herculano-Houzel, Suzana (2009). "The human brain in numbers: a linearly scaled-up primate brain". Front. Hum. Neurosci. 3: 31. doi:10.3389/neuro.09.031.2009. PMC 2776484. PMID 19915731.
29. "Platelets dosing, indications, interactions, adverse effects, and more". reference.medscape.com. Retrieved 2022-10-31.
30. Kapitsa, Sergei P (1996). "The phenomenological theory of world population growth". Physics-Uspekhi. 39 (1): 57–71. Bibcode:1996PhyU...39...57K. doi:10.1070/pu1996v039n01abeh000127. S2CID 250877833. (citing the range of 80 to 150 billion, citing K. M. Weiss, Human Biology 56637, 1984, and N. Keyfitz, Applied Mathematical Demography, New York: Wiley, 1977). C. Haub, "How Many People Have Ever Lived on Earth?", Population Today 23.2), pp. 5–6, cited an estimate of 105 billion births since 50,000 BC, updated to 107 billion as of 2011 in Haub, Carl (October 2011). "How Many People Have Ever Lived on Earth?". Population Reference Bureau. Archived from the original on April 24, 2013. Retrieved April 29, 2013. (due to the high infant mortality in pre-modern times, close to half of this number would not have lived past infancy).
31. Elizabeth Howell, How Many Stars Are in the Milky Way? Archived 2016-05-28 at the Wayback Machine, Space.com, 21 May 2014 (citing estimates from 100 to 400 billion).
32. Hollis, Morgan (13 October 2016). "A universe of two trillion galaxies". The Royal Astronomical Society. Retrieved 9 November 2017.
33. Jonathan Amos (3 September 2015). "Earth's trees number 'three trillion'". BBC. Archived from the original on 6 June 2017.
34. Xavier Gourdon (October 2004). "Computation of zeros of the Zeta function". Archived from the original on 15 January 2011. Retrieved 2 November 2010.
35. Haruka Iwao, Emma (14 March 2019). "Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud". Archived from the original on 19 October 2019. Retrieved 12 April 2019.
36. Koch, Christof. Biophysics of computation: information processing in single neurons. Oxford university press, 2004.
37. Savage, D. C. (1977). "Microbial Ecology of the Gastrointestinal Tract". Annual Review of Microbiology. 31: 107–33. doi:10.1146/annurev.mi.31.100177.000543. PMID 334036.
38. Berg, R. (1996). "The indigenous gastrointestinal microflora". Trends in Microbiology. 4 (11): 430–5. doi:10.1016/0966-842X(96)10057-3. PMID 8950812.
39. Sloane, N. J. A. (ed.). "Sequence A186311 (Least century 100k to 100k+99 with exactly n primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-06-16.
40. Bert Holldobler and E.O. Wilson The Superorganism: The Beauty, Elegance, and Strangeness of Insect Societies New York:2009 W.W. Norton Page 5
41. Silva, Tomás Oliveira e. "Goldbach conjecture verification". Retrieved 11 April 2021.
42. "60th Birthday of Microelectronics Industry". Semiconductor Industry Association. 13 December 2007. Archived from the original on 13 October 2008. Retrieved 2 November 2010.
43. Sequence A131646 Archived 2011-09-01 at the Wayback Machine in The On-Line Encyclopedia of Integer Sequences
44. "Smithsonian Encyclopedia: Number of Insects Archived 2016-12-28 at the Wayback Machine". Prepared by the Department of Systematic Biology, Entomology Section, National Museum of Natural History, in cooperation with Public Inquiry Services, Smithsonian Institution. Accessed 27 December 2016. Facts about numbers of insects. Puts the number of individual insects on Earth at about 10 quintillion (1019).
45. Ivan Moscovich, 1000 playthinks: puzzles, paradoxes, illusions & games, Workman Pub., 2001 ISBN 0-7611-1826-8.
46. "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. Archived from the original on 1 March 2009. Retrieved 14 March 2009.
47. "To see the Universe in a Grain of Taranaki Sand". Archived from the original on 2012-06-30.
48. "Intel predicts 1,200 quintillion transistors in the world by 2015". Archived from the original on 2013-04-05.
49. "How Many Transistors Have Ever Shipped? – Forbes". Forbes. Archived from the original on 30 June 2015. Retrieved 1 September 2015.
50. "Sudoku enumeration". Archived from the original on 2006-10-06.
51. "Star count: ANU astronomer makes best yet". The Australian National University. 17 July 2003. Archived from the original on July 24, 2005. Retrieved 2 November 2010.
52. "Astronomers count the stars". BBC News. July 22, 2003. Archived from the original on August 13, 2006. Retrieved 2006-07-18. "trillions-of-earths-could-be-orbiting-300-sextillion-stars" van Dokkum, Pieter G.; Charlie Conroy (2010). "A substantial population of low-mass stars in luminous elliptical galaxies". Nature. 468 (7326): 940–942. arXiv:1009.5992. Bibcode:2010Natur.468..940V. doi:10.1038/nature09578. PMID 21124316. S2CID 205222998. "How many stars?" Archived 2013-01-22 at the Wayback Machine; see mass of the observable universe
53. (sequence A007377 in the OEIS)
54. "Questions and Answers – How many atoms are in the human body?". Archived from the original on 2003-10-06.
55. William B. Whitman; David C. Coleman; William J. Wiebe (1998). "Prokaryotes: The unseen majority". Proceedings of the National Academy of Sciences of the United States of America. 95 (12): 6578–6583. Bibcode:1998PNAS...95.6578W. doi:10.1073/pnas.95.12.6578. PMC 33863. PMID 9618454.
56. (sequence A070177 in the OEIS)
57. (sequence A035064 in the OEIS)
58. John Tromp (2010). "John's Chess Playground". Archived from the original on 2014-06-01.
59. Merickel, James G. (ed.). "Sequence A217379 (Numbers having non-pandigital power of record size (excludes multiples of 10).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-03-17.
60. Planck Collaboration (2016). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". Astronomy & Astrophysics. 594: A13. arXiv:1502.01589. Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830. S2CID 119262962.
61. Paul Zimmermann, "50 largest factors found by ECM Archived 2009-02-20 at the Wayback Machine".
62. Matthew Champion, "Re: How many atoms make up the universe?" Archived 2012-05-11 at the Wayback Machine, 1998
63. WMAP- Content of the Universe Archived 2016-07-26 at the Wayback Machine. Map.gsfc.nasa.gov (2010-04-16). Retrieved on 2011-05-01.
64. "Names of large and small numbers". bmanolov.free.fr. Miscellaneous pages by Borislav Manolov. Archived from the original on 2016-09-30.
65. Hanke, Steve; Krus, Nicholas. "Hyperinflation Table" (PDF). Retrieved 26 March 2021.
66. "Richard Eldridge".
67. Chris Caldwell, The Top Twenty: Elliptic Curve Primality Proof at The Prime Pages.
68. Chris Caldwell, The Top Twenty: Sophie Germain (p) at The Prime Pages.
69. Chris Caldwell, The Top Twenty: Twin at The Prime Pages.
70. Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
71. From the third paragraph of the story: "Each book contains 410 pages; each page, 40 lines; each line, about 80 black letters." That makes 410 x 40 x 80 = 1,312,000 characters. The fifth paragraph tells us that "there are 25 orthographic symbols" including spaces and punctuation. The magnitude of the resulting number is found by taking logarithms. However, this calculation only gives a lower bound on the number of books as it does not take into account variations in the titles – the narrator does not specify a limit on the number of characters on the spine. For further discussion of this, see Bloch, William Goldbloom. The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press: Oxford, 2008.
72. Chris Caldwell, The Top Twenty: Palindrome at The Prime Pages.
73. Gary Barnes, Riesel conjectures and proofs Archived 2021-04-12 at the Wayback Machine
74. Chris Caldwell, The Top Twenty: Factorial primes Archived 2013-04-10 at the Wayback Machine at The Prime Pages.
75. Chris Caldwell, The Top Twenty: Generalized Fermat Archived 2021-03-28 at the Wayback Machine at The Prime Pages.
76. PRP records
77. Chris Caldwell, The Top Twenty: Proth Archived 2020-11-24 at the Wayback Machine at The Prime Pages.
78. Chris Caldwell, The Top Twenty: Largest Known Primes at The Prime Pages.
79. Chris Caldwell, Mersenne Primes: History, Theorems and Lists at The Prime Pages.
80. asantos (15 December 2007). "Googol and Googolplex by Carl Sagan". Archived from the original on 2021-12-12 – via YouTube.
81. Zyga, Lisa "Physicists Calculate Number of Parallel Universes" Archived 2011-06-06 at the Wayback Machine, PhysOrg, 16 October 2009.
82. Don N. Page for Cornell University (2007). "Susskind's challenge to the Hartle–Hawking no-boundary proposal and possible resolutions". Journal of Cosmology and Astroparticle Physics. 2007 (1): 004. arXiv:hep-th/0610199. Bibcode:2007JCAP...01..004P. doi:10.1088/1475-7516/2007/01/004. S2CID 17403084.
83. H. Friedman, Enormous integers in real life (accessed 2021-02-06)
84. "CH. Rayo's Number". The Math Factor Podcast. Retrieved 24 March 2014.
85. Kerr, Josh (7 December 2013). "Name the biggest number contest". Archived from the original on 20 March 2016. Retrieved 27 March 2014.
External links
• Seth Lloyd's paper Computational capacity of the universe provides a number of interesting dimensionless quantities.
• Notable properties of specific numbers
• Clewett, James. "4,294,967,296 – The Internet is Full". Numberphile. Brady Haran. Archived from the original on 2013-05-24. Retrieved 2013-04-06.
Orders of magnitude
Quantity
• Acceleration
• Angular momentum
• Area
• Bit rate
• Charge
• Computing
• Current
• Data
• Density
• Energy / Energy density
• Entropy
• Force
• Frequency
• Illuminance
• Length
• Luminance
• Magnetic field
• Mass
• Molarity
• Numbers
• Power
• Pressure
• Probability
• Radiation
• Sound pressure
• Specific heat capacity
• Speed
• Temperature
• Time
• Voltage
• Volume
See also
• Back-of-the-envelope calculation
• Best-selling electronic devices
• Fermi problem
• Powers of 10 and decades
• 10th
• 100th
• 1000000th
• Metric (SI) prefix
• Macroscopic scale
• Microscopic scale
Related
• Astronomical system of units
• Earth's location in the Universe
• Cosmic View (1957 book)
• To the Moon and Beyond (1964 film)
• Cosmic Zoom (1968 film)
• Powers of Ten (1968 and 1977 films)
• Cosmic Voyage (1996 documentary)
• Cosmic Eye (2012)
• Category
Large numbers
Examples
in
numerical
order
• Thousand
• Ten thousand
• Hundred thousand
• Million
• Ten million
• Hundred million
• Billion
• Trillion
• Quadrillion
• Quintillion
• Sextillion
• Septillion
• Octillion
• Nonillion
• Decillion
• Eddington number
• Googol
• Shannon number
• Googolplex
• Skewes's number
• Moser's number
• Graham's number
• TREE(3)
• SSCG(3)
• BH(3)
• Rayo's number
• Transfinite numbers
Expression
methods
Notations
• Scientific notation
• Knuth's up-arrow notation
• Conway chained arrow notation
• Steinhaus–Moser notation
Operators
• Hyperoperation
• Tetration
• Pentation
• Ackermann function
• Grzegorczyk hierarchy
• Fast-growing hierarchy
Related
articles
(alphabetical
order)
• Busy beaver
• Extended real number line
• Indefinite and fictitious numbers
• Infinitesimal
• Largest known prime number
• List of numbers
• Long and short scales
• Number systems
• Number names
• Orders of magnitude
• Power of two
• Power of three
• Power of 10
• Sagan Unit
• Names
• History
| Wikipedia |
1000 (number)
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
← 999 1000 1001 →
• List of numbers
• Integers
← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinalone thousand
Ordinal1000th
(one thousandth)
Factorization23 × 53
Divisors1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
Greek numeral,Α´
Roman numeralM
Roman numeral (unicode)M, m, ↀ
Unicode symbol(s)ↀ
Greek prefixchilia
Latin prefixmilli
Binary11111010002
Ternary11010013
Senary43446
Octal17508
Duodecimal6B412
Hexadecimal3E816
Tamil௲
Chinese千
Punjabi੧੦੦੦
Look up thousand or 1000 in Wiktionary, the free dictionary.
A group of one thousand things is sometimes known, from Ancient Greek, as a chiliad.[1] A period of one thousand years may be known as a chiliad or, more often from Latin, as a millennium. The number 1000 is also sometimes described as a short thousand in medieval contexts where it is necessary to distinguish the Germanic concept of 1200 as a long thousand.
Notation
• The decimal representation for one thousand is
• 1000—a one followed by three zeros, in the general notation;
• 1 × 103—in engineering notation, which for this number coincides with:
• 1 × 103 exactly—in scientific normalized exponential notation;
• 1 E+3 exactly—in scientific E notation.
• The SI prefix for a thousand units is "kilo-", abbreviated to "k"—for instance, a kilogram or "kg" is a thousand grams. This is sometimes extended to non-SI contexts, such as "ka" (kiloannum) being used as a shorthand for periods of 1000 years. In computer science, however, "kilo-" is used more loosely to mean 2 to the 10th power (1024).
• In the SI writing style, a non-breaking space can be used as a thousands separator, i.e., to separate the digits of a number at every power of 1000.
• Multiples of thousands are occasionally represented by replacing their last three zeros with the letter "K" or "k": for instance, writing "$30k" for $30 000 or denoting the Y2K computer bug of the year 2000.
• A thousand units of currency, especially dollars or pounds, are colloquially called a grand. In the United States, this is sometimes abbreviated with a "G" suffix.
Properties
There are 168 prime numbers less than 1000.[2]
1000 is the 10th icositetragonal number, or 24-gonal number.[3]
1000 has a reduced totient value of 100, and totient of 400. It is equal to the sum of Euler's totient function over the first 57 integers, with 11 integers having a totient value of 1000.
1000 is the smallest number that generates three primes in the fastest way possible by concatenation of decremented numbers: (1,000,999), (1,000,999,998,997), and (1,000,999,998,997,996,995,994,993) are all prime.[4]
The 1000th prime number is 7919. It is a difference of 1 from the order of the smallest sporadic group: $|\mathrm {M} _{11}|$ = 7920.
Selected numbers in the range 1001–1999
1001 to 1099
1001 = sphenic number (7 × 11 × 13), pentagonal number, pentatope number
1002 = sphenic number, Mertens function zero, abundant number, number of partitions of 22
1003 = the product of some prime p and the pth prime, namely p = 17.
1004 = heptanacci number[5]
1005 = Mertens function zero, decagonal pyramidal number[6]
1006 = semiprime, product of two distinct isolated primes (2 and 503); unusual number; square-free number; number of compositions (ordered partitions) of 22 into squares; sum of two distinct pentatope numbers (5 and 1001); number of undirected Hamiltonian paths in 4 by 5 square grid graph;[7] record gap between twin primes;[8] number that is the sum of 7 positive 5th powers.[9] In decimal: equidigital number; when turned around, the number looks like a prime, 9001; its cube can be concatenated from other cubes, 1_0_1_8_1_0_8_216 ("_" indicates concatenation, 0 = 03, 1 = 13, 8 = 23, 216 = 63)[10]
1007 = number that is the sum of 8 positive 5th powers[11]
1008 = divisible by the number of primes below it
1009 = smallest four-digit prime, palindromic in bases 11, 15, 19, 24 and 28: (83811, 47415, 2F219, 1I124, 18128). It is also a Lucky prime and Chen prime.
1010 = 103 + 10,[12] Mertens function zero
1011 = the largest n such that 2n contains 101 and doesn't contain 11011, Harshad number in bases 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75 (and 202 other bases), number of partitions of 1 into reciprocals of positive integers <= 16 Egyptian fraction[13]
1012 = ternary number, (3210) quadruple triangular number (triangular number is 253),[14] number of partitions of 1 into reciprocals of positive integers <= 17 Egyptian fraction[13]
1013 = Sophie Germain prime,[15] centered square number,[16] Mertens function zero
1014 = 210-10,[17] Mertens function zero, sum of the nontriangular numbers between successive triangular numbers
1015 = square pyramidal number[18]
1016 = member of the Mian–Chowla sequence,[19] stella octangula number, number of surface points on a cube with edge-length 14[20]
1017 = generalized triacontagonal number[21]
1018 = Mertens function zero, 101816 + 1 is prime[22]
1019 = Sophie Germain prime,[15] safe prime,[23] Chen prime
1020 = polydivisible number
1021 = twin prime with 1019. It is also a Lucky prime.
1022 = Friedman number
1023 = sum of five consecutive primes (193 + 197 + 199 + 211 + 223);[24] the number of three-dimensional polycubes with 7 cells;[25] number of elements in a 9-simplex; highest number one can count to on one's fingers using binary; magic number used in Global Positioning System signals.
1024 = 322 = 45 = 210, the number of bytes in a kilobyte (in 1999, the IEC coined kibibyte to use for 1024 with kilobyte being 1000, but this convention has not been widely adopted). 1024 is the smallest 4-digit square and also a Friedman number.
1025 = Proth number 210 + 1; member of Moser–de Bruijn sequence, because its base-4 representation (1000014) contains only digits 0 and 1, or it's a sum of distinct powers of 4 (45 + 40); Jacobsthal-Lucas number; hypotenuse of primitive Pythagorean triangle
1026 = sum of two distinct powers of 2 (1024 + 2)
1027 = sum of the squares of the first eight primes; can be written from base 2 to base 18 using only the digits 0 to 9.
1028 = sum of totient function for first 58 integers; can be written from base 2 to base 18 using only the digits 0 to 9; number of primes <= 213.[26]
1029 = can be written from base 2 to base 18 using only the digits 0 to 9.
1030 = generalized heptagonal number
1031 = exponent and number of ones for the fifth base-10 repunit prime,[27] Sophie Germain prime,[15] super-prime, Chen prime
1032 = sum of two distinct powers of 2 (1024 + 8)
1033 = emirp, twin prime with 1031
1034 = sum of 12 positive 9th powers[28]
1035 = triangular number,[29] hexagonal number[30]
1036 = central polygonal number[31]
1037 = number in E-toothpick sequence[32]
1038 = even integer that is an unordered sum of two primes in exactly n ways[33]
1039 = prime of the form 8n+7,[34] number of partitions of 30 that do not contain 1 as a part,[35] Chen prime
1040 = 45 + 42: sum of distinct powers of 4.[36] The number of pieces that could be seen in a 6 × 6 × 6× 6 Rubik's Tesseract.
1041 = sum of 11 positive 5th powers[37]
1042 = sum of 12 positive 5th powers[38]
1043 = number whose sum of even digits and sum of odd digits are even[39]
1044 = sum of distinct powers of 4[36]
1045 = octagonal number[40]
1046 = coefficient of f(q) (3rd order mock theta function)[41]
1047 = number of ways to split a strict composition of n into contiguous subsequences that have the same sum[42]
1048 = number of partitions of n into squarefree parts[43]
1049 = Sophie Germain prime,[15] highly cototient number,[44] Chen prime
1050 = 10508 to decimal becomes a pronic number (55210),[45] number of parts in all partitions of 29 into distinct parts[46]
1051 = centered pentagonal number,[47] centered decagonal number
1052 = number that is the sum of 9 positive 6th powers[48]
1053 = triangular matchstick number[49]
1054 = centered triangular number[50]
1055 = number that is the sum of 12 positive 6th powers[51]
1056 = pronic number[52]
1057 = central polygonal number[53]
1058 = number that is the sum of 4 positive 5th powers,[54] area of a square with diagonal 46[55]
1059 = number n such that n4 is written in the form of a sum of four positive 4th powers[56]
1060 = sum of the first 25 primes
1061 = emirp, twin prime with 1063
1062 = number that is not the sum of two palindromes[57]
1063 = super-prime, sum of seven consecutive primes (137 + 139 + 149 + 151 + 157 + 163 + 167); near-wall-sun-sun prime[58]
1064 = sum of two positive cubes[59]
1065 = generalized duodecagonal[60]
1066 = number whose sum of their divisors is a square[61]
1067 = number of strict integer partitions of n in which are empty or have smallest part not dividing the other ones[62]
1068 = number that is the sum of 7 positive 5th powers,[9] total number of parts in all partitions of 15[63]
1069 = emirp[64]
1070 = number that is the sum of 9 positive 5th powers[65]
1071 = heptagonal number[66]
1072 = centered heptagonal number[67]
1073 = number that is the sum of 12 positive 5th powers[38]
1074 = number that is not the sum of two palindromes[57]
1075 = number non-sum of two palindromes[57]
1076 = number of strict trees weight n[68]
1077 = number where 7 outnumbers every other digit in the number[69]
1078 = Euler transform of negative integers[70]
1079 = every positive integer is the sum of at most 1079 tenth powers.
1080 = pentagonal number[71]
1081 = triangular number,[29] member of Padovan sequence[72]
1082 = central polygonal number[31]
1083 = three-quarter square,[73] number of partitions of 53 into prime parts
1084 = third spoke of a hexagonal spiral,[74] 108464 + 1 is prime
1085 = number of partitions of n into distinct parts > or = 2[75]
1086 = Smith number,[76] sum of totient function for first 59 integers
1087 = super-prime, cousin prime, lucky prime[77]
1088 = octo-triangular number, (triangular number result being 136)[78] sum of two distinct powers of 2, (1024 + 64)[79] number that is divisible by exactly seven primes with the inclusion of multiplicity[80]
1089 = 332, nonagonal number, centered octagonal number, first natural number whose digits in its decimal representation get reversed when multiplied by 9.[81]
1090 = sum of 5 positive 5th powers[82]
1091 = cousin prime and twin prime with 1093
1092 = divisible by the number of primes below it
1093 = the smallest Wieferich prime (the only other known Wieferich prime is 3511[83]), twin prime with 1091 and star number[84]
1094 = sum of 9 positive 5th powers,[65] 109464 + 1 is prime
1095 = sum of 10 positive 5th powers,[85] number that is not the sum of two palindromes
1096 = hendecagonal number,[86] number of strict solid partitions of 18[87]
1097 = emirp,[64] Chen prime
1098 = multiple of 9 containing digit 9 in its base-10 representation[88]
1099 = number where 9 outnumbers every other digit[89]
1100 to 1199
1100 = number of partitions of 61 into distinct squarefree parts[90]
1101 = pinwheel number[91]
1102 = sum of totient function for first 60 integers
1103 = Sophie Germain prime,[15] balanced prime[92]
1104 = Keith number[93]
1105 = 332 + 42 = 322 + 92 = 312 + 122 = 232 + 242, Carmichael number,[94] magic constant of n × n normal magic square and n-queens problem for n = 13, decagonal number,[95] centered square number,[16] Fermat pseudoprime[96]
1106 = number of regions into which the plane is divided when drawing 24 ellipses[97]
1107 = number of non-isomorphic strict T0 multiset partitions of weight 8[98]
1108 = number k such that k64 + 1 is prime
1109 = Friedlander-Iwaniec prime,[99] Chen prime
1110 = k such that 2k + 3 is prime[100]
1111 = 11 × 101, palindrome that is a product of two palindromic primes[101]
1112 = k such that 9k - 2 is a prime[102]
1113 = number of strict partions of 40[103]
1114 = number of ways to write 22 as an orderless product of orderless sums[104]
1115 = number of partitions of 27 into a prime number of parts[105]
1116 = divisible by the number of primes below it
1117 = number of diagonally symmetric polyominoes with 16 cells,[106] Chen prime
1118 = number of unimodular 2 × 2 matrices having all terms in {0,1,...,21}[107]
1119 = number of bipartite graphs with 9 nodes[108]
1120 = number k such that k64 + 1 is prime
1121 = number of squares between 342 and 344.[109]
1122 = pronic number,[52] divisible by the number of primes below it
1123 = balanced prime[92]
1124 = Leyland number[110]
1125 = Achilles number
1126 = number of 2 × 2 non-singular integer matrices with entries from {0, 1, 2, 3, 4, 5}[111]
1127 = maximal number of pieces that can be obtained by cutting an annulus with 46 cuts[112]
1128 = triangular number,[29] hexagonal number,[30] divisible by the number of primes below it
1129 = number of lattice points inside a circle of radius 19[113]
1130 = skiponacci number[114]
1131 = number of edges in the hexagonal triangle T(26)[115]
1132 = number of simple unlabeled graphs with 9 nodes of 2 colors whose components are complete graphs[116]
1133 = number of primitive subsequences of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}[117]
1134 = divisible by the number of primes below it, triangular matchstick number[49]
1135 = centered triangular number[118]
1136 = number of independent vertex sets and vertex covers in the 7-sunlet graph[119]
1137 = sum of values of vertices at level 5 of the hyperbolic Pascal pyramid[120]
1138 = recurring number in the works of George Lucas and his companies, beginning with his first feature film – THX 1138; particularly, a special code for Easter eggs on Star Wars DVDs.
1139 = wiener index of the windmill graph D(3,17)[121]
1140 = tetrahedral number[122]
1141 = 7-Knödel number[123]
1142 = n such that n32 + 1 is prime[124]
1143 = number of set partitions of 8 elements with 2 connectors[125]
1144 is not the sum of a pair of twin primes[126]
1145 = 5-Knödel number[127]
1146 is not the sum of a pair of twin primes[126]
1147 = 31 × 37 (a product of 2 successive primes)[128]
1148 is not the sum of a pair of twin primes[126]
1149 = a product of two palindromic primes[129]
1150 = number of 11-iamonds without bilateral symmetry.[130]
1151 = first prime following a prime gap of 22,[131] Chen prime
1152 = highly totient number,[132] 3-smooth number (27×32), area of a square with diagonal 48,[55] Achilles number
1153 = super-prime, Proth prime[133]
1154 = 2 × 242 + 2 = number of points on surface of tetrahedron with edgelength 24[134]
1155 = number of edges in the join of two cycle graphs, both of order 33[135]
1156 = 342, octahedral number,[136] centered pentagonal number,[47] centered hendecagonal number.[137]
1157 = smallest number that can be written as n^2+1 without any prime factors that can be written as a^2+1.[138]
1158 = number of points on surface of octahedron with edgelength 17[139]
1159 = member of the Mian–Chowla sequence,[19] a centered octahedral number[140]
1160 = octagonal number[141]
1161 = sum of the first 26 primes
1162 = pentagonal number,[71] sum of totient function for first 61 integers
1163 = smallest prime > 342.[142] See Legendre's conjecture. Chen prime.
1164 = number of chains of multisets that partition a normal multiset of weight 8, where a multiset is normal if it spans an initial interval of positive integers[143]
1165 = 5-Knödel number[127]
1166 = heptagonal pyramidal number[144]
1167 = number of rational numbers which can be constructed from the set of integers between 1 and 43[145]
1168 = antisigma(49)[146]
1169 = highly cototient number[44]
1170 = highest possible score in a National Academic Quiz Tournaments (NAQT) match
1171 = super-prime
1172 = number of subsets of first 14 integers that have a sum divisible by 14[147]
1173 = number of simple triangulation on a plane with 9 nodes[148]
1174 = number of widely totally strongly normal compositions of 16
1175 = maximal number of pieces that can be obtained by cutting an annulus with 47 cuts[112]
1176 = triangular number[29]
1177 = heptagonal number[66]
1178 = number of surface points on a cube with edge-length 15[20]
1179 = number of different permanents of binary 7*7 matrices[149]
1180 = smallest number of non-integral partitions into non-integral power >1000.[150]
1181 = smallest k over 1000 such that 8*10^k-49 is prime.[151]
1182 = number of necklaces possible with 14 beads of 2 colors (that cannot be turned over)[152]
1183 = pentagonal pyramidal number
1184 = amicable number with 1210[153]
1185 = number of partitions of 45 into pairwise relatively prime parts[154]
1186 = number of diagonally symmetric polyominoes with 15 cells,[106] number of partitions of 54 into prime parts
1187 = safe prime,[23] Stern prime,[155] balanced prime,[92] Chen prime
1188 = first 4 digit multiple of 18 to contain 18[156]
1189 = number of squares between 352 and 354.[109]
1190 = pronic number,[52] number of cards to build an 28-tier house of cards[157]
1191 = 352 - 35 + 1 = H35 (the 35th Hogben number)[158]
1192 = sum of totient function for first 62 integers
1193 = a number such that 41193 - 31193 is prime, Chen prime
1194 =number of permutations that can be reached with 8 moves of 2 bishops and 1 rook on a 3 × 3 chessboard[159]
1195 = smallest four digit number for which a−1(n) is an integer is a(n) is 2*a(n-1) - (-1)n[160]
1196 = $\sum _{k=1}^{38}\sigma (k)$[161]
1197 = pinwheel number[91]
1198 = centered heptagonal number[67]
1199 = area of the 20th conjoined trapezoid[162]
1200 to 1299
1200 = the long thousand, ten "long hundreds" of 120 each, the traditional reckoning of large numbers in Germanic languages, the number of households the Nielsen ratings sample,[163] number k such that k64 + 1 is prime
1201 = centered square number,[16] super-prime, centered decagonal number
1202 = number of regions the plane is divided into by 25 ellipses[97]
1203: first 4 digit number in the coordinating sequence for the (2,6,∞) tiling of the hyperbolic plane[164]
1204: magic constant of a 7 × 7 × 7 magic cube[165]
1205 = number of partitions of 28 such that the number of odd parts is a part[166]
1206 = 29-gonal number [167]
1207 = composite de Polignac number[168]
1208 = number of strict chains of divisors starting with the superprimorial A006939(3)[169]
1209 = The product of all ordered non-empty subsets of {3,1} if {a,b} is a||b: 1209=1*3*13*31
1210 = amicable number with 1184[170]
1211 = composite de Polignac number[168]
1212 = $\sum _{k=0}^{17}p(k)$, where $p$ is the number of partions of $k$[171]
1213 = emirp
1214 = sum of first 39 composite numbers[172]
1215 = number of edges in the hexagonal triangle T(27)[115]
1216 = nonagonal number[173]
1217 = super-prime, Proth prime[133]
1218 = triangular matchstick number[49]
1219 = Mertens function zero, centered triangular number[118]
1220 = Mertens function zero, number of binary vectors of length 16 containing no singletons[174]
1221 = product of the first two digit, and three digit repdigit
1222 = hexagonal pyramidal number
1223 = Sophie Germain prime,[15] balanced prime, 200th prime number[92]
1224 = number of edges in the join of two cycle graphs, both of order 34[135]
1225 = 352, square triangular number,[175] hexagonal number,[30] centered octagonal number[176]
1226 = number of rooted identity trees with 15 nodes [177]
1227 = smallest number representable as the sum of 3 triangular numbers in 27 ways[178]
1228 = sum of totient function for first 63 integers
1229 = Sophie Germain prime,[15] number of primes between 0 and 10000, emirp
1230 = the Mahonian number: T(9, 6)[179]
1231 = smallest mountain emirp, as 121, smallest mountain number is 11 × 11
1232 = number of labeled ordered set of partitions of a 7-set into odd parts[180]
1233 = 122 + 332
1234 = number of parts in all partitions of 30 into distinct parts,[46] smallest whole number containing all numbers from 1 to 4
1235 = excluding duplicates, contains the first four Fibbonacci numbers [181]
1236 = 617 + 619: sum of twin prime pair[182]
1237 = prime of the form 2p-1
1238 = number of partitions of 31 that do not contain 1 as a part[35]
1239 = toothpick number in 3D[183]
1240 = square pyramidal number[18]
1241 = centered cube number[184]
1242 = decagonal number[95]
1243 = composite de Polignac number[168]
1244 = number of complete partitions of 25[185]
1245 = Number of labeled spanning intersecting set-systems on 5 vertices.[186]
1246 = number of partitions of 38 such that no part occurs more than once[187]
1247 = pentagonal number[71]
1248 = the first four powers of 2 concatenated together
1249 = emirp, trimorphic number[188]
1250 = area of a square with diagonal 50[55]
1251 = 2 × 252 + 1 = number of different 2 × 2 determinants with integer entries from 0 to 25[189]
1252 = 2 × 252 + 2 = number of points on surface of tetrahedron with edgelength 25[134]
1253 = number of partitions of 23 with at least one distinct part[190]
1254 = number of partitions of 23 into relatively prime parts[191]
1255 = Mertens function zero, number of ways to write 23 as an orderless product of orderless sums,[104] number of partitions of 23[192]
1256 = Mertens function zero
1257 = number of lattice points inside a circle of radius 20[113]
1258 = Mertens function zero
1259 = highly cototient number[44]
1260 = highly composite number,[193] pronic number,[52] the smallest vampire number,[194] sum of totient function for first 64 integers, number of strict partions of 41[103] and appears twice in the Book of Revelation
1261 = star number,[84] Mertens function zero
1262 = maximal number of regions the plane is divided into by drawing 36 circles[195]
1263 = rounded total surface area of a regular tetrahedron with edge length 27[196]
1264 = sum of the first 27 primes
1265 = number of rooted trees with 43 vertices in which vertices at the same level have the same degree[197]
1266 = centered pentagonal number,[47] Mertens function zero
1267 = 7-Knödel number[123]
1268 = number of partitions of 37 into prime power parts[198]
1269 = least number of triangles of the Spiral of Theodorus to complete 11 revolutions[199]
1270 = 25 + 24×26 + 23×27,[200] Mertens function zero
1271 = sum of first 40 composite numbers[172]
1272 = sum of first 41 nonprimes[201]
1273 = 19 × 67 = 19 × prime(19)[202]
1274 = sum of the nontriangular numbers between successive triangular numbers
1275 = triangular number,[29] sum of the first 50 natural numbers
1276 = number of irredundant sets in the 25-cocktail party graph[203]
1277 = the start of a prime constellation of length 9 (a "prime nonuple")
1278 = number of Narayana's cows and calves after 20 years[204]
1279 = Mertens function zero, Mersenne prime exponent
1280 = Mertens function zero, number of parts in all compositions of 9[205]
1281 = octagonal number[141]
1282 = Mertens function zero, number of partitions of 46 into pairwise relatively prime parts[154]
1283 = safe prime[23]
1284 = 641 + 643: sum of twin prime pair[182]
1285 = Mertens function zero, number of free nonominoes, number of parallelogram polyominoes with 10 cells.[206]
1286 = number of inequivalent connected planar figures that can be formed from five 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree[207]
1287 = ${13 \choose 5}$[208]
1288 = heptagonal number[66]
1289 = Sophie Germain prime,[15] Mertens function zero
1290 = ${\frac {1289+1291}{2}}$, average of a twin prime pair[209]
1291 = largest prime < 64,[210] Mertens function zero
1292 = number such that phi(1292) = phi(sigma(1292)),[211] Mertens function zero
1293 = $\sum _{j=1}^{n}j\times prime(j)$[212]
1294 = rounded volume of a regular octahedron with edge length 14[213]
1295 = number of edges in the join of two cycle graphs, both of order 35[135]
1296 = 362 = 64, sum of the cubes of the first eight positive integers, the number of rectangles on a normal 8 × 8 chessboard, also the maximum font size allowed in Adobe InDesign
1297 = super-prime, Mertens function zero, pinwheel number[91]
1298 = number of partitions of 55 into prime parts
1299 = Mertens function zero, number of partitions of 52 such that the smallest part is greater than or equal to number of parts[214]
1300 to 1399
1300 = Sum of the first 4 fifth powers, mertens function zero, largest possible win margin in an NAQT match
1301 = centered square number,[16] Honaker prime,[215] number of trees with 13 unlabeled nodes[216]
1302 = Mertens function zero, number of edges in the hexagonal triangle T(28)[115]
1303 = prime of form 21n+1 and 31n+1[217] [218]
1304 = sum of 13046 and 1304 9 which is 328+976
1305 = triangular matchstick number[49]
1306 = Mertens function zero. In base 10, raising the digits of 1306 to powers of successive integers equals itself: 1306 = 11 + 32 + 03 + 64. 135, 175, 518, and 598 also have this property. Centered triangular number.[118]
1307 = safe prime[23]
1308 = sum of totient function for first 65 integers
1309 = the first sphenic number followed by two consecutive such number
1310 = smallest number in the middle of a set of three sphenic numbers
1311 = number of integer partitions of 32 with no part dividing all the others[219]
1312 = member of the Mian-Chowla sequence;[19]
1313 = sum of all parts of all partitions of 14 [220]
1314 = number of integer partitions of 41 whose distinct parts are connected[221]
1315 = 10^(2n+1)-7*10^n-1 is prime.[222]
1316 = Euler transformation of sigma(11)[223]
1317 = 1317 Only odd four digit number to divide the concatenation of all number up to itself in base 25[224]
1318 = Mertens function zero
1319 = safe prime[23]
1320 = 659 + 661: sum of twin prime pair[182]
1321 = Friedlander-Iwaniec prime[99]
1322 = area of the 21th conjoined trapezoid[162]
1323 = Achilles number
1324 = if D(n) is the nth representation of 1, 2 arranged lexicographically. 1324 is the first non-1 number which is D(D(x))[225]
1325 = Markov number,[226] centered tetrahedral number[227]
1326 = triangular number,[29] hexagonal number,[30] Mertens function zero
1327 = first prime followed by 33 consecutive composite numbers
1328 = sum of totient function for first 66 integers
1329 = Mertens function zero, sum of first 41 composite numbers[172]
1330 = tetrahedral number,[110] forms a Ruth–Aaron pair with 1331 under second definition
1331 = 113, centered heptagonal number,[67] forms a Ruth–Aaron pair with 1330 under second definition. This is the only non-trivial cube of the form x2 + x − 1, for x = 36.
1332 = pronic number[52]
1333 = 372 - 37 + 1 = H37 (the 37th Hogben number)[158]
1334 = maximal number of regions the plane is divided into by drawing 37 circles[195]
1335 = pentagonal number,[71] Mertens function zero
1336 = Mertens function zero
1337 = Used in the novel form of spelling called leet. Approximate melting point of gold in kelvins.
1338 = Mertens function zero
1339 = First 4 digit number to appear twice in the sequence of sum of cubes of primes dividing n[228]
1340 = k such that 5 × 2k - 1 is prime[229]
1341 = First mountain number with 2 jumps of more than one.
1342 = $\sum _{k=1}^{40}\sigma (k)$,[161] Mertens function zero
1343 = cropped hexagone[230]
1344 = 372 - 52, the only way to express 1344 as a difference of prime squares[231]
1345 = k such that k, k+1 and k+2 are products of two primes[232]
1346= number of locally disjointed rooted trees with 10 nodes[233]
1347 = concatenation of first 4 Lucas numbers [234]
1348 = number of ways to stack 22 pennies such that every penny is in a stack of one or two[235]
1349 = Stern-Jacobsthal number[236]
1350 = nonagonal number[173]
1351 = number of partitions of 28 into a prime number of parts[105]
1352 = number of surface points on a cube with edge-length 16,[20] Achilles number
1353 = 2 × 262 + 1 = number of different 2 × 2 determinants with integer entries from 0 to 26[189]
1354 = 2 × 262 + 2 = number of points on surface of tetrahedron with edgelength 26[134]
1356 is not the sum of a pair of twin primes[126]
1357 = number of nonnegative solutions to x2 + y2 ≤ 412[237]
1358 = rounded total surface area of a regular tetrahedron with edge length 28[196]
1360 = 372 - 32, the only way to express 1360 as a difference of prime squares[231]
1361 = first prime following a prime gap of 34,[131] centered decagonal number, Honaker prime[215]
1362 = number of achiral integer partitions of 48[238]
1363 = the number of ways to modify a circular arrangement of 14 objects by swapping one or more adjacent pairs[239]
1364 = Lucas number[240]
1365 = pentatope number[241]
1366 = Arima number, after Yoriyuki Arima who in 1769 constructed this sequence as the number of moves of the outer ring in the optimal solution for the Chinese Rings puzzle[242]
1367 = safe prime,[23] balanced prime, sum of three, nine, and eleven consecutive primes (449 + 457 + 461, 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 + 173, and 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151),[92]
1368 = number of edges in the join of two cycle graphs, both of order 36[135]
1369 = 372, centered octagonal number[176]
1370 = σ2(37): sum of squares of divisors of 37[243]
1371 = sum of the first 28 primes
1372 = Achilles number
1373 = number of lattice points inside a circle of radius 21[113]
1374 = number of unimodular 2 × 2 matrices having all terms in {0,1,...,23}[107]
1375 = decagonal pyramidal number[244]
1376 = primitive abundant number (abundant number all of whose proper divisors are deficient numbers)[245]
1377 = maximal number of pieces that can be obtained by cutting an annulus with 51 cuts[112]
1378 = triangular number[29]
1379 = magic constant of n × n normal magic square and n-queens problem for n = 14.
1380 = number of 8-step mappings with 4 inputs[246]
1381 = centered pentagonal number[47]Mertens function zero
1382 = first 4 digit tetrachi number [247]
1384 = $\sum _{k=1}^{41}\sigma (k)$[161]
1385 = up/down number[248]
1386 = octagonal pyramidal number[249]
1387 = 5th Fermat pseudoprime of base 2,[250] 22nd centered hexagonal number and the 19th decagonal number,[95] second Super-Poulet number.[251]
1389 = sum of first 42 composite numbers[172]
1391 = number of rational numbers which can be constructed from the set of integers between 1 and 47[145]
1392 = number of edges in the hexagonal triangle T(29)[115]
1393 = 7-Knödel number[123]
1394 = sum of totient function for first 67 integers
1395 = vampire number,[194] member of the Mian–Chowla sequence[19] triangular matchstick number[49]
1396 = centered triangular number[118]
1398 = number of integer partitions of 40 whose distinct parts are connected[221]
1400 to 1499
1400 = number of sum-free subsets of {1, ..., 15}[252]
1401 = pinwheel number[91]
1402 = number of integer partitions of 48 whose augmented differences are distinct[253]
1404 = heptagonal number[66]
1405 = 262 + 272, 72 + 82 + ... + 162, centered square number[16]
1406 = pronic number,[52] semi-meandric number[254]
1407 = 382 - 38 + 1 = H38 (the 38th Hogben number)[158]
1408 = maximal number of regions the plane is divided into by drawing 38 circles[195]
1409 = super-prime, Sophie Germain prime,[15] smallest number whose eighth power is the sum of 8 eighth powers, Proth prime[133]
1414 = smallest composite that when added to sum of prime factors reaches a prime after 27 iterations[255]
1415 = the Mahonian number: T(8, 8)[179]
1417 = number of partitions of 32 in which the number of parts divides 32[256]
1419 = Zeisel number[257]
1420 = Number of partitions of 56 into prime parts
1423 = 200 + 1223 and the 200th prime is 1223[258] Also Used as a Hate symbol
1424 = number of nonnegative solutions to x2 + y2 ≤ 422[237]
1425 = self-descriptive number in base 5
1426 = sum of totient function for first 68 integers, pentagonal number,[71] number of strict partions of 42[103]
1429 = number of partitions of 53 such that the smallest part is greater than or equal to number of parts[214]
1430 = Catalan number[259]
1431 = triangular number,[29] hexagonal number[30]
1432 = member of Padovan sequence[72]
1433 = super-prime, Honaker prime,[215] typical port used for remote connections to Microsoft SQL Server databases
1434 = rounded volume of a regular tetrahedron with edge length 23[260]
1435 = vampire number;[194] the standard railway gauge in millimetres, equivalent to 4 feet 8+1⁄2 inches (1.435 m)
1437 = smallest number of complexity 20: smallest number requiring 20 1's to build using +, * and ^[261]
1438 = k such that 5 × 2k - 1 is prime[229]
1439 = Sophie Germain prime,[15] safe prime[23]
1440 = a highly totient number[132] and a 481-gonal number. Also, the number of minutes in one day, the blocksize of a standard 3+1/2 floppy disk, and the horizontal resolution of WXGA(II) computer displays
1441 = star number[84]
1442 = number of parts in all partitions of 31 into distinct parts[46]
1443 = the sum of the second trio of three-digit permutable primes in decimal: 337, 373, and 733. Also the number of edges in the join of two cycle graphs, both of order 37[135]
1444 = 382, smallest pandigital number in Roman numerals
1446 = number of points on surface of octahedron with edgelength 19[139]
1447 = super-prime, happy number
1448 = number k such that phi(prime(k)) is a square[262]
1449 = Stella octangula number
1450 = σ2(34): sum of squares of divisors of 34[243]
1451 = Sophie Germain prime[15]
1452 = first Zagreb index of the complete graph K12[263]
1453 = Sexy prime with 1459
1454 = 3 × 222 + 2 = number of points on surface of square pyramid of side-length 22[264]
1455 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1457 = 2 × 272 − 1 = a twin square[266]
1458 = maximum determinant of an 11 by 11 matrix of zeroes and ones, 3-smooth number (2×36)
1459 = Sexy prime with 1453, sum of nine consecutive primes (139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181), pierpont prime
1460 = The number of years that would have to pass in the Julian calendar in order to accrue a full year's worth of leap days.
1461 = number of partitions of 38 into prime power parts[198]
1462 = (35 - 1) × (35 + 8) = the first Zagreb index of the wheel graph with 35 vertices[267]
1463 = total number of parts in all partitions of 16[63]
1464 = rounded total surface area of a regular icosahedron with edge length 13[268]
1465 = 5-Knödel number[127]
1469 = octahedral number,[136] highly cototient number[44]
1470 = pentagonal pyramidal number,[269] sum of totient function for first 69 integers
1471 = super-prime, centered heptagonal number[67]
1473 = cropped hexagone[230]
1476 = coreful perfect number[270]
1477 = 7-Knödel number[123]
1479 = number of planar partitions of 12[271]
1480 = sum of the first 29 primes
1481 = Sophie Germain prime[15]
1482 = pronic number,[52] number of unimodal compositions of 15 where the maximal part appears once[272]
1483 = 392 - 39 + 1 = H39 (the 39th Hogben number)[158]
1484 = maximal number of regions the plane is divided into by drawing 39 circles[195]
1485 = triangular number
1486 = number of strict solid partitions of 19[87]
1487 = safe prime[23]
1488 = triangular matchstick number[49]
1489 = centered triangular number[118]
1490 = tetranacci number[273]
1491 = nonagonal number,[173] Mertens function zero
1492 = Mertens function zero
1493 = Stern prime[155]
1494 = sum of totient function for first 70 integers
1496 = square pyramidal number[18]
1497 = skiponacci number[114]
1498 = number of flat partitions of 41[274]
1499 = Sophie Germain prime,[15] super-prime
1500 to 1599
1500 = hypotenuse in three different Pythagorean triangles[275]
1501 = centered pentagonal number[47]
1502 = number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most 47[276]
1503 = least number of triangles of the Spiral of Theodorus to complete 12 revolutions[199]
1504 = primitive abundant number (abundant number all of whose proper divisors are deficient numbers)[245]
1507 = number of partitions of 32 that do not contain 1 as a part[35]
1508 = heptagonal pyramidal number[144]
1509 = pinwheel number[91]
1510 = deficient number, odious number
1511 = Sophie Germain prime,[15] balanced prime[92]
1512 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1513 = centered square number[16]
1514 = sum of first 44 composite numbers[172]
1517 = number of lattice points inside a circle of radius 22[113]
1518 = Mertens function zero
1519 = Mertens function zero
1520 = pentagonal number,[71] Mertens function zero, forms a Ruth–Aaron pair with 1521 under second definition
1521 = 392, Mertens function zero, centered octagonal number,[176] forms a Ruth–Aaron pair with 1520 under second definition
1522 = k such that 5 × 2k - 1 is prime[229]
1523 = super-prime, Mertens function zero, safe prime,[23] member of the Mian–Chowla sequence[19]
1524 = Mertens function zero, k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1525 = heptagonal number,[66] Mertens function zero
1526 = number of conjugacy classes in the alternating group A27[277]
1527 = Mertens function zero
1528 = Mertens function zero, rounded total surface area of a regular octahedron with edge length 21[278]
1529 = composite de Polignac number[168]
1530 = vampire number[194]
1531 = prime number, centered decagonal number, Mertens function zero
1532 = Mertens function zero
1534 = number of achiral integer partitions of 50[238]
1535 = Thabit number
1536 = a common size of microplate, 3-smooth number (29×3), number of threshold functions of exactly 4 variables[279]
1537 = Keith number,[93] Mertens function zero
1538 = number of surface points on a cube with edge-length 17[20]
1539 = maximal number of pieces that can be obtained by cutting an annulus with 54 cuts[112]
1540 = triangular number, hexagonal number,[30] decagonal number,[95] tetrahedral number[110]
1541 = octagonal number[141]
1543 = Mertens function zero
1544 = Mertens function zero, number of partitions of integer partitions of 17 where all parts have the same length[280]
1546 = Mertens function zero
1547 = hexagonal pyramidal number
1548 = coreful perfect number[270]
1549 = de Polignac prime[281]
1552 = Number of partitions of 57 into prime parts
1556 = sum of the squares of the first nine primes
1557 = number of graphs with 8 nodes and 13 edges[282]
1558 = number k such that k64 + 1 is prime
1559 = Sophie Germain prime[15]
1560 = pronic number[52]
1561 = a centered octahedral number,[140] number of series-reduced trees with 19 nodes[283]
1562 = maximal number of regions the plane is divided into by drawing 40 circles[195]
1564 = sum of totient function for first 71 integers
1565 = ${\sqrt {1036^{2}+1173^{2}}}$ and $1036+1173=47^{2}$[284]
1566 = number k such that k64 + 1 is prime
1567 = number of partitions of 24 with at least one distinct part[190]
1568 = Achilles number[285]
1569 = 2 × 282 + 1 = number of different 2 × 2 determinants with integer entries from 0 to 28[189]
1570 = 2 × 282 + 2 = number of points on surface of tetrahedron with edgelength 28[134]
1571 = Honaker prime[215]
1572 = member of the Mian–Chowla sequence[19]
1575 = odd abundant number,[286] sum of the nontriangular numbers between successive triangular numbers, number of partitions of 24[192]
1578 = sum of first 45 composite numbers[172]
1579 = number of partitions of 54 such that the smallest part is greater than or equal to number of parts[214]
1580 = number of achiral integer partitions of 51[238]
1581 = number of edges in the hexagonal triangle T(31)[115]
1582 = a number such that the integer triangle [A070080(1582), A070081(1582), A070082(1582)] has an integer area[287]
1583 = Sophie Germain prime
1584 = triangular matchstick number[49]
1585 = Riordan number, centered triangular number[118]
1586 = area of the 23th conjoined trapezoid[162]
1588 = sum of totient function for first 72 integers
1589 = composite de Polignac number[168]
1590 = rounded volume of a regular icosahedron with edge length 9[288]
1591 = rounded volume of a regular octahedron with edge length 15[213]
1593 = sum of the first 30 primes
1594 = minimal cost of maximum height Huffman tree of size 17[289]
1595 = number of non-isomorphic set-systems of weight 10
1596 = triangular number
1597 = Fibonacci prime,[290] Markov prime,[226] super-prime, emirp
1598 = number of unimodular 2 × 2 matrices having all terms in {0,1,...,25}[107]
1599 = number of edges in the join of two cycle graphs, both of order 39[135]
1600 to 1699
1600 = 402, structured great rhombicosidodecahedral number,[291] repdigit in base 7 (44447), street number on Pennsylvania Avenue of the White House, length in meters of a common High School Track Event, perfect score on SAT (except from 2005 to 2015)
1601 = Sophie Germain prime, Proth prime,[133] the novel 1601 (Mark Twain)
1602 = number of points on surface of octahedron with edgelength 20[139]
1603 = number of partitions of 27 with nonnegative rank[292]
1606 = enneagonal pyramidal number[293]
1608 = $\sum _{k=1}^{44}\sigma (k)$[161]
1609 = cropped hexagonal number[230]
1610 = number of strict partions of 43[103]
1611 = number of rational numbers which can be constructed from the set of integers between 1 and 51[145]
1617 = pentagonal number[71]
1618 = centered heptagonal number[67]
1619 = palindromic prime in binary, safe prime[23]
1620 = 809 + 811: sum of twin prime pair[182]
1621 = super-prime, pinwheel number[91]
1624 = number of squares in the Aztec diamond of order 28[294]
1625 = centered square number[16]
1626 = centered pentagonal number[47]
1629 = rounded volume of a regular tetrahedron with edge length 24[260]
1630 = number k such that k^64 + 1 is prime
1632 = number of acute triangles made from the vertices of a regular 18-polygon[295]
1633 = star number[84]
1634 = Narcissistic number in base 10
1635 = number of partitions of 56 whose reciprocal sum is an integer[296]
1636 = number of nonnegative solutions to x2 + y2 ≤ 452[237]
1637 = prime island: least prime whose adjacent primes are exactly 30 apart[297]
1638 = harmonic divisor number,[298] 5 × 21638 - 1 is prime[229]
1639 = nonagonal number[173]
1640 = pronic number[52]
1641 = 412 - 41 + 1 = H41 (the 41st Hogben number)[158]
1642 = maximal number of regions the plane is divided into by drawing 41 circles[195]
1643 = sum of first 46 composite numbers[172]
1644 = 821 + 823: sum of twin prime pair[182]
1645 = number of 16-celled pseudo still lifes in Conway's Game of Life, up to rotation and reflection[299]
1646 = number of graphs with 8 nodes and 14 edges[282]
1647 and 1648 are both divisible by cubes[300]
1648 = number of partitions of 343 into distinct cubes[301]
1649 = highly cototient number,[44] Leyland number[110]
1650 = number of cards to build an 33-tier house of cards[157]
1651 = heptagonal number[66]
1652 = number of partitions of 29 into a prime number of parts[105]
1653 = triangular number, hexagonal number,[30] number of lattice points inside a circle of radius 23[113]
1654 = number of partitions of 42 into divisors of 42[302]
1655 = rounded volume of a regular dodecahedron with edge length 6[303]
1656 = 827 + 829: sum of twin prime pair[182]
1657 = cuban prime,[304] prime of the form 2p-1
1658 = smallest composite that when added to sum of prime factors reaches a prime after 25 iterations[255]
1659 = number of rational numbers which can be constructed from the set of integers between 1 and 52[145]
1660 = sum of totient function for first 73 integers
1661 = 11 × 151, palindrome that is a product of two palindromic primes[101]
1662 = number of partitions of 49 into pairwise relatively prime parts[154]
1663 = a prime number and 51663 - 41663 is a 1163-digit prime number[305]
1664 = k such that k, k+1 and k+2 are sums of 2 squares[306]
1665 = centered tetrahedral number[227]
1666 = largest efficient pandigital number in Roman numerals (each symbol occurs exactly once)
1667 = 228 + 1439 and the 228th prime is 1439[258]
1668 = number of partitions of 33 into parts all relatively prime to 33[307]
1669 = super-prime, smallest prime with a gap of exactly 24 to the next prime[308]
1670 = number of compositions of 12 such that at least two adjacent parts are equal[309]
1671 divides the sum of the first 1671 composite numbers[310]
1672 = 412 - 22, the only way to express 1672 as a difference of prime squares[231]
1673 = RMS number[311]
1674 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1675 = Kin number[312]
1676 = number of partitions of 34 into parts each of which is used a different number of times[313]
1677 = 412 - 32, the only way to express 1677 as a difference of prime squares[231]
1678 = n such that n32 + 1 is prime[124]
1679 = highly cototient number,[44] semiprime (23 × 73, see also Arecibo message), number of parts in all partitions of 32 into distinct parts[46]
1680 = highly composite number,[193] number of edges in the join of two cycle graphs, both of order 40[135]
1681 = 412, smallest number yielded by the formula n2 + n + 41 that is not a prime; centered octagonal number[176]
1682 = and 1683 is a member of a Ruth–Aaron pair (first definition)
1683 = triangular matchstick number[49]
1684 = centered triangular number[118]
1685 = 5-Knödel number[127]
1686 = $\sum _{k=1}^{45}\sigma (k)$[161]
1687 = 7-Knödel number[123]
1688 = number of finite connected sets of positive integers greater than one with least common multiple 72[314]
1689 = $9!!\sum _{k=0}^{4}{\frac {1}{2k+1}}$[315]
1690 = number of compositions of 14 into powers of 2[316]
1691 = the same upside down, which makes it a strobogrammatic number[317]
1692 = coreful perfect number[270]
1693 = smallest prime > 412.[142]
1694 = number of unimodular 2 × 2 matrices having all terms in {0,1,...,26}[107]
1695 = magic constant of n × n normal magic square and n-queens problem for n = 15. Number of partitions of 58 into prime parts
1696 = sum of totient function for first 74 integers
1697 = Friedlander-Iwaniec prime[99]
1698 = number of rooted trees with 47 vertices in which vertices at the same level have the same degree[197]
1699 = number of rooted trees with 48 vertices in which vertices at the same level have the same degree[197]
1700 to 1799
1700 = σ2(39): sum of squares of divisors of 39[243]
1701 = $\left\{{8 \atop 4}\right\}$, decagonal number, hull number of the U.S.S. Enterprise on Star Trek
1702 = palindromic in 3 consecutive bases: 89814, 78715, 6A616
1703 = 1703131131 / 1000077 and the divisors of 1703 are 1703, 131, 13 and 1[318]
1704 = sum of the squares of the parts in the partitions of 18 into two distinct parts[319]
1705 = tribonacci number[320]
1706 = 1 + 4 + 16 + 64 + 256 + 1024 + 256 + 64 + 16 + 4 + 1 sum of fifth row of triangle of powers of 4[321]
1707 = number of partitions of 30 in which the number of parts divides 30[256]
1708 = 22 × 7 × 61 a number whose product of prime indices 1 × 1 × 4 × 18 is divisible by its sum of prime factors 2 + 2 + 7 + 61[322]
1709 = first of a sequence of eight primes formed by adding 57 in the middle. 1709, 175709, 17575709, 1757575709, 175757575709, 17575757575709, 1757575757575709 and 175757575757575709 are all prime, but 17575757575757575709 = 232433 × 75616446785773
1710 = maximal number of pieces that can be obtained by cutting an annulus with 57 cuts[112]
1711 = triangular number, centered decagonal number
1712 = number of irredundant sets in the 29-cocktail party graph[203]
1713 = number of aperiodic rooted trees with 12 nodes[323]
1714 = number of regions formed by drawing the line segments connecting any two of the 18 perimeter points of an 3 × 6 grid of squares[324]
1715 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1716 = 857 + 859: sum of twin prime pair[182]
1717 = pentagonal number[71]
1718 = $\sum _{d|12}{\binom {12}{d}}$[325]
1719 = composite de Polignac number[168]
1720 = sum of the first 31 primes
1721 = twin prime; number of squares between 422 and 424.[109]
1722 = Giuga number,[326] pronic number[52]
1723 = super-prime
1724 = maximal number of regions the plane is divided into by drawing 42 circles[195]
1725 = 472 - 222 = (prime(15))2 - (nonprime(15))2[327]
1726 = number of partitions of 44 into distinct and relatively prime parts[328]
1727 = area of the 24th conjoined trapezoid[162]
1728 = the quantity expressed as 1000 in duodecimal, that is, the cube of twelve (called a great gross), and so, the number of cubic inches in a cubic foot, palindromic in base 11 (133111) and 23 (36323)
1729 = taxicab number, Carmichael number, Zeisel number, centered cube number, Hardy–Ramanujan number. In the decimal expansion of e the first time all 10 digits appear in sequence starts at the 1729th digit (or 1728th decimal place). In 1979 the rock musical Hair closed on Broadway in New York City after 1729 performances. Palindromic in bases 12, 32, 36.
1730 = 3 × 242 + 2 = number of points on surface of square pyramid of side-length 24[264]
1731 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1732 = $\sum _{k=0}^{5}{\binom {5}{k}}^{k}$[329]
1733 = Sophie Germain prime, palindromic in bases 3, 18, 19.
1734 = surface area of a cube of edge length 17[330]
1735 = number of partitions of 55 such that the smallest part is greater than or equal to number of parts[214]
1736 = sum of totient function for first 75 integers, number of surface points on a cube with edge-length 18[20]
1737 = pinwheel number[91]
1738 = number of achiral integer partitions of 52[238]
1739 = number of 1s in all partitions of 30 into odd parts[331]
1740 = number of squares in the Aztec diamond of order 29[294]
1741 = super-prime, centered square number[16]
1742 = number of regions the plane is divided into by 30 ellipses[97]
1743 = wiener index of the windmill graph D(3,21)[121]
1744 = k such that k, k+1 and k+2 are sums of 2 squares[306]
1745 = 5-Knödel number[127]
1746 = number of unit-distance graphs on 8 nodes[332]
1747 = balanced prime[92]
1748 = number of partitions of 55 into distinct parts in which the number of parts divides 55[333]
1749 = number of integer partitions of 33 with no part dividing all the others[219]
1750 = hypotenuse in three different Pythagorean triangles[275]
1751 = cropped hexagone[230]
1752 = 792 - 672, the only way to express 1752 as a difference of prime squares[231]
1753 = balanced prime[92]
1754 = k such that 5*2k - 1 is prime[229]
1755 = number of integer partitions of 50 whose augmented differences are distinct[253]
1756 = centered pentagonal number[47]
1757 = least number of triangles of the Spiral of Theodorus to complete 13 revolutions[199]
1758 = $\sum _{k=1}^{46}\sigma (k)$[161]
1759 = de Polignac prime[281]
1760 = the number of yards in a mile
1761 = k such that k, k+1 and k+2 are products of two primes[232]
1762 = number of binary sequences of length 12 and curling number 2[334]
1763 = number of edges in the join of two cycle graphs, both of order 41[135]
1764 = 422
1765 = number of stacks, or planar partitions of 15[335]
1766 = number of points on surface of octahedron with edgelength 21[139]
1767 = σ(282) = σ(352)[336]
1768 = number of nonequivalent dissections of an hendecagon into 8 polygons by nonintersecting diagonals up to rotation[337]
1769 = maximal number of pieces that can be obtained by cutting an annulus with 58 cuts[112]
1770 = triangular number, hexagonal number,[30] Seventeen Seventy, town in Australia
1771 = tetrahedral number[110]
1772 = centered heptagonal number,[67] sum of totient function for first 76 integers
1773 = number of words of length 5 over the alphabet {1,2,3,4,5} such that no two even numbers appear consecutively[338]
1774 = number of rooted identity trees with 15 nodes and 5 leaves[339]
1775 = $\sum _{1\leq i\leq 10}prime(i)\cdot (2\cdot i-1)$: sum of piles of first 10 primes[340]
1776 = square star number.[341] The number of pieces that could be seen in a 7 × 7 × 7× 7 Rubik's Tesseract.
1777 = smallest prime > 422.[142]
1778 = least k >= 1 such that the remainder when 6k is divided by k is 22[342]
1779 = number of achiral integer partitions of 53[238]
1780 = number of lattice paths from (0, 0) to (7, 7) using E (1, 0) and N (0, 1) as steps that horizontally cross the diagonal y = x with even many times[343]
1781 = the first 1781 digits of e form a prime[344]
1782 = heptagonal number[66]
1783 = de Polignac prime[281]
1784 = number of subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} such that every pair of distinct elements has a different quotient[345]
1785 = square pyramidal number,[18] triangular matchstick number[49]
1786 = centered triangular number[118]
1787 = super-prime, sum of eleven consecutive primes (137 + 139 + 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 + 191)
1788 = Euler transform of -1, -2, ..., -34[346]
1789 = number of wiggly sums adding to 17 (terms alternately increase and decrease or vice versa)[347]
1790 = number of partitions of 50 into pairwise relatively prime parts[154]
1791 = largest natural number that cannot be expressed as a sum of at most four hexagonal numbers.
1792 = Granville number
1793 = number of lattice points inside a circle of radius 24[113]
1794 = nonagonal number,[173] number of partitions of 33 that do not contain 1 as a part[35]
1795 = number of heptagons with perimeter 38[348]
1796 = k such that geometric mean of phi(k) and sigma(k) is an integer[265]
1797 = number k such that phi(prime(k)) is a square[262]
1798 = 2 × 29 × 31 = 102 × 111012 × 111112, which yield zero when the prime factors are xored together[349]
1799 = 2 × 302 − 1 = a twin square[266]
1800 to 1899
1800 = pentagonal pyramidal number,[269] Achilles number, also, in da Ponte's Don Giovanni, the number of women Don Giovanni had slept with so far when confronted by Donna Elvira, according to Leporello's tally
1801 = cuban prime, sum of five and nine consecutive primes (349 + 353 + 359 + 367 + 373 and 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227)[304]
1802 = 2 × 302 + 2 = number of points on surface of tetrahedron with edgelength 30,[134] number of partitions of 30 such that the number of odd parts is a part[166]
1803 = number of decahexes that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion)[350]
1804 = number k such that k^64 + 1 is prime
1805 = number of squares between 432 and 434.[109]
1806 = pronic number,[52] product of first four terms of Sylvester's sequence, primary pseudoperfect number,[351] only number for which n equals the denominator of the nth Bernoulli number,[352] Schröder number[353]
1807 = fifth term of Sylvester's sequence[354]
1808 = maximal number of regions the plane is divided into by drawing 43 circles[195]
1809 = sum of first 17 super-primes[355]
1810 = $\sum _{k=0}^{4}{\binom {4}{k}}^{4}$[356]
1811 = Sophie Germain prime
1812 = n such that n32 + 1 is prime[124]
1813 = number of polyominoes with 26 cells, symmetric about two orthogonal axes[357]
1814 = 1 + 6 + 36 + 216 + 1296 + 216 + 36 + 6 + 1 = sum of 4th row of triangle of powers of six[358]
1815 = polygonal chain number $\#(P_{2,1}^{3})$[359]
1816 = number of strict partions of 44[103]
1817 = total number of prime parts in all partitions of 20[360]
1818 = n such that n32 + 1 is prime[124]
1819 = sum of the first 32 primes, minus 32[361]
1820 = pentagonal number,[71] pentatope number,[241] number of compositions of 13 whose run-lengths are either weakly increasing or weakly decreasing[362]
1821 = member of the Mian–Chowla sequence[19]
1822 = number of integer partitions of 43 whose distinct parts are connected[221]
1823 = super-prime, safe prime[23]
1824 = 432 - 52, the only way to express 1824 as a difference of prime squares[231]
1825 = octagonal number[141]
1826 = decagonal pyramidal number[244]
1827 = vampire number[194]
1828 = meandric number, open meandric number, appears twice in the first 10 decimal digits of e
1829 = composite de Polignac number[168]
1830 = triangular number
1831 = smallest prime with a gap of exactly 16 to next prime (1847)[363]
1832 = sum of totient function for first 77 integers
1833 = number of atoms in a decahedron with 13 shells[364]
1834 = octahedral number,[136] sum of the cubes of the first five primes
1835 = absolute value of numerator of $D_{6}^{(5)}$[365]
1836 = factor by which a proton is more massive than an electron
1837 = star number[84]
1838 = number of unimodular 2 × 2 matrices having all terms in {0,1,...,27}[107]
1839 = $\lfloor {\sqrt[{3}]{13!}}\rfloor $[366]
1840 = 432 - 32, the only way to express 1840 as a difference of prime squares[231]
1841 = Mertens function zero
1842 = number of unlabeled rooted trees with 11 nodes[367]
1843 = Mertens function zero
1844 = Mertens function zero
1845 = Mertens function zero
1846 = sum of first 49 composite numbers[172]
1847 = super-prime
1848 = number of edges in the join of two cycle graphs, both of order 42[135]
1849 = 432, palindromic in base 6 (= 123216), centered octagonal number[176]
1850 = Number of partitions of 59 into prime parts
1851 = sum of the first 32 primes
1852 = number of quantales on 5 elements, up to isomorphism[368]
1853 = Mertens function zero
1854 = Mertens function zero
1855 = rencontres number: number of permutations of [7] with exactly one fixed point[369]
1856 = sum of totient function for first 78 integers
1857 = Mertens function zero, pinwheel number[91]
1858 = number of 14-carbon alkanes C14H30 ignoring stereoisomers[370]
1859 = composite de Polignac number[168]
1860 = number of squares in the Aztec diamond of order 30[371]
1861 = centered square number,[16] Mertens function zero
1862 = Mertens function zero, forms a Ruth–Aaron pair with 1863 under second definition
1863 = Mertens function zero, forms a Ruth–Aaron pair with 1862 under second definition
1864 = Mertens function zero, ${\frac {1864!-2}{2}}$ is a prime[372]
1865 = 123456: Largest senary metadrome (number with digits in strict ascending order in base 6)[373]
1866 = Mertens function zero, number of plane partitions of 16 with at most two rows[374]
1867 = prime de Polignac number[281]
1868 = smallest number of complexity 21: smallest number requiring 21 1's to build using +, * and ^[261]
1869 = Hultman number: SH(7, 4)[375]
1870 = decagonal number[95]
1871 = the first prime of the 2 consecutive twin prime pairs: (1871, 1873) and (1877, 1879)[376]
1872 = first Zagreb index of the complete graph K13[263]
1873 = number of Narayana's cows and calves after 21 years[204]
1874 = area of the 25th conjoined trapezoid[162]
1875 = 502 - 252
1876 = number k such that k^64 + 1 is prime
1877 = number of partitions of 39 where 39 divides the product of the parts[377]
1878 = n such that n32 + 1 is prime[124]
1879 = a prime with square index[378]
1880 = the 10th element of the self convolution of Lucas numbers[379]
1881 = tricapped prism number[380]
1882 = number of linearly separable Boolean functions in 4 variables[381]
1883 = number of conjugacy classes in the alternating group A28[277]
1884 = k such that 5*2k - 1 is prime[229]
1885 = Zeisel number[257]
1886 = number of partitions of 64 into fourth powers[382]
1887 = number of edges in the hexagonal triangle T(34)[115]
1888 = primitive abundant number (abundant number all of whose proper divisors are deficient numbers)[245]
1889 = Sophie Germain prime, highly cototient number[44]
1890 = triangular matchstick number[49]
1891 = triangular number, hexagonal number,[30] centered pentagonal number,[47] centered triangular number[118]
1892 = pronic number[52]
1893 = 442 - 44 + 1 = H44 (the 44th Hogben number)[158]
1894 = maximal number of regions the plane is divided into by drawing 44 circles[195]
1895 = Stern-Jacobsthal number[236]
1896 = member of the Mian-Chowla sequence[19]
1897 = member of Padovan sequence,[72] number of triangle-free graphs on 9 vertices[383]
1898 = smallest multiple of n whose digits sum to 26[384]
1899 = cropped hexagone[230]
1900 to 1999
1900 = number of primes <= 214.[26] Also 1900 (film) or Novecento, 1976 movie. 1900 was the year Thorold Gosset introduced his list of semiregular polytopes; it is also the year Max Brückner published his study of polyhedral models, including stellations of the icosahedron, such as the novel final stellation of the icosahedron.
1901 = Sophie Germain prime, centered decagonal number
1902 = number of symmetric plane partitions of 27[385]
1903 = generalized catalan number[386]
1904 = number of flat partitions of 43[274]
1905 = Fermat pseudoprime[96]
1906 = number n such that 3n - 8 is prime[387]
1907 = safe prime,[23] balanced prime[92]
1908 = coreful perfect number[270]
1909 = hyperperfect number[388]
1910 = number of compositions of 13 having exactly one fixed point[389]
1911 = heptagonal pyramidal number[144]
1912 = size of 6th maximum raising after one blind in pot-limit poker[390]
1913 = super-prime, Honaker prime[215]
1914 = number of bipartite partitions of 12 white objects and 3 black ones[391]
1915 = number of nonisomorphic semigroups of order 5[392]
1916 = sum of first 50 composite numbers[172]
1917 = number of partitions of 51 into pairwise relatively prime parts[154]
1918 = heptagonal number[66]
1919 = smallest number with reciprocal of period length 36 in base 10[393]
1920 = sum of the nontriangular numbers between successive triangular numbers
1921 = 4-dimensional centered cube number[394]
1922 = Area of a square with diagonal 62[55]
1923 = 2 × 312 + 1 = number of different 2 X 2 determinants with integer entries from 0 to 31[189]
1924 = 2 × 312 + 2 = number of points on surface of tetrahedron with edgelength 31[134]
1925 = number of ways to write 24 as an orderless product of orderless sums[104]
1926 = pentagonal number[71]
1927 = 211 - 112[395]
1928 = number of distinct values taken by 2^2^...^2 (with 13 2's and parentheses inserted in all possible ways)[396]
1929 = Mertens function zero, number of integer partitions of 42 whose distinct parts are connected[221]
1930 = number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most 53[276]
1931 = Sophie Germain prime
1932 = number of partitions of 40 into prime power parts[198]
1933 = centered heptagonal number,[67] Honaker prime[215]
1934 = sum of totient function for first 79 integers
1935 = number of edges in the join of two cycle graphs, both of order 43[135]
1936 = 442, 18-gonal number,[397] 324-gonal number.
1937 = number of chiral n-ominoes in 12-space, one cell labeled[398]
1938 = Mertens function zero, number of points on surface of octahedron with edgelength 22[139]
1939 = 7-Knödel number[123]
1940 = the Mahonian number: T(8, 9)[179]
1941 = maximal number of regions obtained by joining 16 points around a circle by straight lines[399]
1942 = number k for which 10k + 1, 10k + 3, 10k + 7, 10k + 9 and 10k + 13 are primes[400]
1943 = largest number not the sum of distinct tetradecagonal numbers[401]
1944 = 3-smooth number (23×35), Achilles number[285]
1945 = number of partitions of 25 into relatively prime parts such that multiplicities of parts are also relatively prime[402]
1946 = number of surface points on a cube with edge-length 19[20]
1947 = k such that 5·2k + 1 is a prime factor of a Fermat number 22m + 1 for some m[403]
1948 = number of strict solid partitions of 20[87]
1949 = smallest prime > 442.[142]
1950 = $1\cdot 2\cdot 3+4\cdot 5\cdot 6+7\cdot 8\cdot 9+10\cdot 11\cdot 12$,[404] largest number not the sum of distinct pentadecagonal numbers[401]
1951 = cuban prime[304]
1952 = number of covers of {1, 2, 3, 4}[405]
1953 = triangular number
1956 = number of sum-free subsets of {1, ..., 16}[252]
1955 = number of partitions of 25 with at least one distinct part[190]
1956 = nonagonal number[173]
1957 = $\sum _{k=0}^{6}{\frac {6!}{k!}}$ = total number of ordered k-tuples (k=0,1,2,3,4,5,6) of distinct elements from an 6-element set[406]
1958 = number of partitions of 25[192]
1959 = Heptanacci-Lucas number[407]
1960 = number of parts in all partitions of 33 into distinct parts[46]
1961 = number of lattice points inside a circle of radius 25[113]
1962 = number of edges in the join of the complete graph K36 and the cycle graph C36[408]
1963! - 1 is prime[409]
1964 = number of linear forests of planted planar trees with 8 nodes[410]
1965 = total number of parts in all partitions of 17[63]
1966 = sum of totient function for first 80 integers
1967 = least edge-length of a square dissectable into at least 30 squares in the Mrs. Perkins's quilt problem[411]
σ(1968) = σ(1967) + σ(1966)[412]
1969 = Only value less than four million for which a "mod-ification" of the standard Ackermann Function does not stabilize[413]
1970 = number of compositions of two types of 9 having no even parts[414]
1971 = $3^{7}-6^{3}$[415]
1972 = n such that ${\frac {n^{37}-1}{n-1}}$ is prime[416]
1973 = Sophie Germain prime, Leonardo prime
1974 = number of binary vectors of length 17 containing no singletons[174]
1975 = number of partitions of 28 with nonnegative rank[292]
1976 = octagonal number[141]
1977 = number of non-isomorphic multiset partitions of weight 9 with no singletons[417]
1978 = n such that n | (3n + 5)[418]
1979 = number of squares between 452 and 454.[109]
1980 = pronic number[52]
1981 = pinwheel number[91]
1982 = maximal number of regions the plane is divided into by drawing 45 circles[195]
1983 = skiponacci number[114]
1984 = 11111000000 in binary, see also: 1984 (disambiguation)
1985 = centered square number[16]
1986 = number of ways to write 25 as an orderless product of orderless sums[104]
1987 = 300th prime number
1988 = sum of the first 33 primes
1989 = number of 9-step mappings with 4 inputs[246]
1990 = Stella octangula number
1991 = 11 × 181, the 46th Gullwing number,[419] palindromic composite number with only palindromic prime factors[420]
1992 = number of nonisomorphic sets of nonempty subsets of a 4-set[421]
1993 = a number with the property that 41993 - 31993 is prime,[422] number of partitions of 30 into a prime number of parts[105]
1994 = Glaisher's function W(37)[423]
1995 = number of unlabeled graphs on 9 vertices with independence number 6[424]
1996 = a number with the property that (1996! + 3)/3 is prime[425]
1997 = $\sum _{k=1}^{21}{k\cdot \phi (k)}$[426]
1998 = triangular matchstick number[49]
1999 = centered triangular number[427] number of regular forms in a myriagram.
Prime numbers
There are 135 prime numbers between 1000 and 2000:[428][429]
1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999
References
Wikimedia Commons has media related to 1000 (number).
1. "chiliad". Merriam-Webster. Archived from the original on 25 March 2022.
2. Caldwell, Chris K (2021). "The First 1,000 Primes". PrimePages. University of Tennessee at Martin.
3. Sloane, N. J. A. (ed.). "Sequence A051876 (24-gonal numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 30 November 2022.
4. "1000". Prime Curious!. Archived from the original on 25 March 2022.
5. "Sloane's A122189 : Heptanacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 7 October 2016. Retrieved 13 July 2017.
6. Sloane, N. J. A. (ed.). "Sequence A007585 (10-gonal (or decagonal) pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
7. Sloane, N. J. A. (ed.). "Sequence A332307 (Array read by antidiagonals: T(m,n) is the number of (undirected) Hamiltonian paths in the m X n grid graph)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 January 2023.
8. Sloane, N. J. A. (ed.). "Sequence A036063 (Increasing gaps among twin primes: size)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 January 2023.
9. "A003352 - Oeis".
10. Sloane, N. J. A. (ed.). "Sequence A061341 (A061341 Numbers not ending in 0 whose cubes are concatenations of other cubes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 January 2023.
11. "A003353 - Oeis".
12. Sloane, N. J. A. (ed.). "Sequence A034262 (n^3 + n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
13. Sloane, N. J. A. (ed.). "Sequence A020473 (Egyptian fractions: number of partitions of 1 into reciprocals of positive integers < n+1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
14. "A046092 - Oeis".
15. "Sloane's A005384 : Sophie Germain primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 11 June 2016. Retrieved 12 June 2016.
16. "Sloane's A001844 : Centered square numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 11 June 2016. Retrieved 12 June 2016.
17. Sloane, N. J. A. (ed.). "Sequence A000325 (2^n - n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
18. "Sloane's A000330 : Square pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
19. "Sloane's A005282 : Mian-Chowla sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 17 May 2016. Retrieved 12 June 2016.
20. Sloane, N. J. A. (ed.). "Sequence A005897 (6*n^2 + 2 for n > 0)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
21. "A316729 - Oeis".
22. Sloane, N. J. A. (ed.). "Sequence A006313 (Numbers n such that n^16 + 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 24 May 2022.
23. "Sloane's A005385 : Safe primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
24. Sloane, N. J. A. (ed.). "Sequence A034964 (Sums of five consecutive primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 November 2022.
25. Sloane, N. J. A. (ed.). "Sequence A000162 (Number of 3-dimensional polyominoes (or polycubes) with n cells.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 November 2022.
26. Sloane, N. J. A. (ed.). "Sequence A007053 (Number of primes < 2^n+1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
27. Sloane, N. J. A. (ed.). "Sequence A004023 (Indices of prime repunits: numbers n such that 11...111 (with n 1's)... is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 25 February 2023.
28. "A004801 - Oeis".
29. "Sloane's A000217 : Triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 5 April 2016. Retrieved 12 June 2016.
30. "Sloane's A000384 : Hexagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 17 April 2016. Retrieved 12 June 2016.
31. "A000124 - Oeis".
32. "A161328 - Oeis".
33. "A023036 - Oeis".
34. "A007522 - Oeis".
35. Sloane, N. J. A. (ed.). "Sequence A002865 (Number of partitions of n that do not contain 1 as a part)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
36. "A000695 - Oeis".
37. "A003356 - Oeis".
38. "A003357 - Oeis".
39. "A036301 - Oeis".
40. "A000567 - Oeis".
41. "A000025 - Oeis".
42. "A336130 - Oeis".
43. "A073576 - Oeis".
44. "Sloane's A100827 : Highly cototient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
45. "Base converter | number conversion".
46. Sloane, N. J. A. (ed.). "Sequence A015723 (Number of parts in all partitions of n into distinct parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
47. "Sloane's A005891 : Centered pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
48. "A003365 - Oeis".
49. Sloane, N. J. A. (ed.). "Sequence A045943 (Triangular matchstick numbers: 3*n*(n+1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
50. "A005448 - Oeis".
51. "A003368 - Oeis".
52. "Sloane's A002378 : Oblong (or promic, pronic, or heteromecic) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 9 June 2016. Retrieved 12 June 2016.
53. "A002061 - Oeis".
54. "A003349 - Oeis".
55. Sloane, N. J. A. (ed.). "Sequence A001105 (2*n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
56. "A003294 - Oeis".
57. "A035137 - Oeis".
58. "A347565: Primes p such that A241014(A000720(p)) is +1 or -1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 25 March 2022. Retrieved 19 January 2022.
59. "A003325 - Oeis".
60. "A195162 - Oeis".
61. "A006532 - Oeis".
62. "A341450 - Oeis".
63. Sloane, N. J. A. (ed.). "Sequence A006128 (Total number of parts in all partitions of n. Also, sum of largest parts of all partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
64. "A006567 - Oeis".
65. "A003354 - Oeis".
66. "Sloane's A000566 : Heptagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 11 June 2016. Retrieved 12 June 2016.
67. "Sloane's A069099 : Centered heptagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 9 June 2016. Retrieved 12 June 2016.
68. "A273873 - Oeis".
69. "A292457 - Oeis".
70. "A073592 - Oeis".
71. "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
72. "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 10 June 2016. Retrieved 12 June 2016.
73. "A077043 - Oeis".
74. "A056107 - Oeis".
75. "A025147 - Oeis".
76. "Sloane's A006753 : Smith numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 9 June 2016. Retrieved 12 June 2016.
77. "Sloane's A031157 : Numbers that are both lucky and prime". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 4 March 2016. Retrieved 12 June 2016.
78. "A033996 - Oeis".
79. "A018900 - Oeis".
80. "A046308 - Oeis".
81. "Sloane's A001232 : Numbers n such that 9*n = (n written backwards)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 17 October 2015. Retrieved 14 June 2016.
82. "A003350 - Oeis".
83. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 163
84. "Sloane's A003154 : Centered 12-gonal numbers. Also star numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on 11 June 2016. Retrieved 12 June 2016.
85. "A003355 - Oeis".
86. "A051682 - Oeis".
87. Sloane, N. J. A. (ed.). "Sequence A323657 (Number of strict solid partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
88. "A121029 - Oeis".
89. "A292449 - Oeis".
90. Sloane, N. J. A. (ed.). "Sequence A087188 (number of partitions of n into distinct squarefree parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
91. Sloane, N. J. A. (ed.). "Sequence A059993 (Pinwheel numbers: 2*n^2 + 6*n + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
92. "Sloane's A006562 : Balanced primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
93. "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
94. "Sloane's A002997 : Carmichael numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
95. "Sloane's A001107 : 10-gonal (or decagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
96. Sloane, N. J. A. (ed.). "Sequence A001567 (Fermat pseudoprimes to base 2, also called Sarrus numbers or Poulet numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
97. Sloane, N. J. A. (ed.). "Sequence A051890 (2*(n^2 - n + 1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
98. Sloane, N. J. A. (ed.). "Sequence A319560 (Number of non-isomorphic strict T_0 multiset partitions of weight n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
99. Sloane, N. J. A. (ed.). "Sequence A028916 (Friedlander-Iwaniec primes: Primes of form a^2 + b^4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
100. Sloane, N. J. A. (ed.). "Sequence A057732 (Numbers k such that 2^k + 3 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
101. Sloane, N. J. A. (ed.). "Sequence A046376 (Palindromes with exactly 2 palindromic prime factors (counted with multiplicity), and no other prime factors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
102. Sloane, N. J. A. (ed.). "Sequence A128455 (Numbers k such that 9^k - 2 is a prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
103. Sloane, N. J. A. (ed.). "Sequence A000009 (Expansion of Product_{m > 0} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
104. Sloane, N. J. A. (ed.). "Sequence A318949 (Number of ways to write n as an orderless product of orderless sums)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
105. Sloane, N. J. A. (ed.). "Sequence A038499 (Number of partitions of n into a prime number of parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
106. Sloane, N. J. A. (ed.). "Sequence A006748 (Number of diagonally symmetric polyominoes with n cells)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
107. Sloane, N. J. A. (ed.). "Sequence A210000 (Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.}
108. Sloane, N. J. A. (ed.). "Sequence A033995 (Number of bipartite graphs with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
109. Sloane, N. J. A. (ed.). "Sequence A028387 (n + (n+1)^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
110. "Sloane's A076980 : Leyland numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
111. Sloane, N. J. A. (ed.). "Sequence A062801 (Number of 2 X 2 non-singular integer matrices with entries from {0,...,n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.}
112. Sloane, N. J. A. (ed.). "Sequence A000096 (n*(n+3)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
113. Sloane, N. J. A. (ed.). "Sequence A000328". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
114. Sloane, N. J. A. (ed.). "Sequence A001608 (Perrin sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
115. Sloane, N. J. A. (ed.). "Sequence A140091 (3*n*(n + 3)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
116. Sloane, N. J. A. (ed.). "Sequence A005380". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
117. Sloane, N. J. A. (ed.). "Sequence A051026 (Number of primitive subsequences of 1, 2, ..., n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
118. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers: 3n(n-1)/2 + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
119. Sloane, N. J. A. (ed.). "Sequence A080040 (2*a(n-1) + 2*a(n-2) for n > 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
120. Sloane, N. J. A. (ed.). "Sequence A264237 (Sum of values of vertices at level n of the hyperbolic Pascal pyramid)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
121. Sloane, N. J. A. (ed.). "Sequence A033991 (n*(4*n-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
122. "Sloane's A000292 : Tetrahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
123. Sloane, N. J. A. (ed.). "Sequence A208155 (7-Knödel numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
124. Sloane, N. J. A. (ed.). "Sequence A006315 (Numbers n such that n^32 + 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
125. Sloane, N. J. A. (ed.). "Sequence A185982 (Triangle read by rows: number of set partitions of n elements with k connectors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
126. Sloane, N. J. A. (ed.). "Sequence A007534 (Even numbers that are not the sum of a pair of twin primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
127. Sloane, N. J. A. (ed.). "Sequence A050993 (5-Knödel numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
128. Sloane, N. J. A. (ed.). "Sequence A006094 (Products of 2 successive primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
129. Sloane, N. J. A. (ed.). "Sequence A046368 (Products of two palindromic primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
130. "1150 (number)". The encyclopedia of numbers.
131. "Sloane's A000101 : Increasing gaps between primes (upper end)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 10 July 2016.
132. "Sloane's A097942 : Highly totient numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
133. "Sloane's A080076 : Proth primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
134. Sloane, N. J. A. (ed.). "Sequence A005893 (Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
135. Sloane, N. J. A. (ed.). "Sequence n*(n+2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
136. "Sloane's A005900 : Octahedral numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
137. "Sloane's A069125 : a(n) = (11*n^2 - 11*n + 2)/2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
138. "1157 (number)". The encyclopedia of numbers.
139. Sloane, N. J. A. (ed.). "Sequence A005899 (Number of points on surface of octahedron)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
140. Sloane, N. J. A. (ed.). "Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
141. Sloane, N. J. A. (ed.). "Sequence A000567 (Octagonal numbers: n*(3*n-2). Also called star numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
142. Sloane, N. J. A. (ed.). "Sequence A007491 (Smallest prime > n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
143. Sloane, N. J. A. (ed.). "Sequence A055887 (Number of ordered partitions of partitions)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
144. Sloane, N. J. A. (ed.). "Sequence A002413 (Heptagonal (or 7-gonal) pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
145. Sloane, N. J. A. (ed.). "Sequence A018805". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
146. Sloane, N. J. A. (ed.). "Sequence A024816 (Antisigma(n): Sum of the numbers less than n that do not divide n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
147. "A063776 - OEIS". oeis.org.
148. "A000256 - OEIS". oeis.org.
149. "1179 (number)". The encyclopedia of numbers.
150. "A000339 - OEIS". oeis.org.
151. "A271269 - OEIS". oeis.org.
152. "A000031 - OEIS". oeis.org.
153. Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 61. ISBN 978-1-84800-000-1.
154. Sloane, N. J. A. (ed.). "Sequence A051424 (Number of partitions of n into pairwise relatively prime parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
155. "Sloane's A042978 : Stern primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
156. "A121038 - OEIS". oeis.org.
157. Sloane, N. J. A. (ed.). "Sequence A005449 (Second pentagonal numbers: n*(3*n + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
158. Sloane, N. J. A. (ed.). "Sequence A002061 (Central polygonal numbers: n^2 - n + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
159. "A175654 - OEIS". oeis.org.
160. oeis.org/A062092
161. Sloane, N. J. A. (ed.). "Sequence A024916 (Sum_1^n sigma(k))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
162. >Sloane, N. J. A. (ed.). "Sequence A080663 (3*n^2 - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
163. Meehan, Eileen R., Why TV is not our fault: television programming, viewers, and who's really in control Lanham, MD: Rowman & Littlefield, 2005
164. "A265070 - OEIS". oeis.org.
165. "1204 (number)". The encyclopedia of numbers.
166. Sloane, N. J. A. (ed.). "Sequence A240574 (Number of partitions of n such that the number of odd parts is a part)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
167. "A303815 - OEIS". oeis.org.
168. Sloane, N. J. A. (ed.). "Sequence A098237 (Composite de Polignac numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
169. Sloane, N. J. A. (ed.). "Sequence A337070 (Number of strict chains of divisors starting with the superprimorial A006939(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
170. Higgins, ibid.
171. Sloane, N. J. A. (ed.). "Sequence A000070 (Sum_{0..n} A000041(k))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
172. Sloane, N. J. A. (ed.). "Sequence A053767 (Sum of first n composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
173. "Sloane's A001106 : 9-gonal (or enneagonal or nonagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
174. Sloane, N. J. A. (ed.). "Sequence A006355 (Number of binary vectors of length n containing no singletons)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
175. "Sloane's A001110 : Square triangular numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
176. "Sloane's A016754 : Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
177. "A004111 - OEIS". oeis.org.
178. "A061262 - OEIS". oeis.org.
179. Sloane, N. J. A. (ed.). "Sequence A008302 (Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product{0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
180. "A006154 - OEIS". oeis.org.
181. "A000045 - OEIS". oeis.org.
182. Sloane, N. J. A. (ed.). "Sequence A054735 (Sums of twin prime pairs)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
183. "A160160 - OEIS". oeis.org.
184. "Sloane's A005898 : Centered cube numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
185. Sloane, N. J. A. (ed.). "Sequence A126796 (Number of complete partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
186. oeis.org/A305843
187. "A007690 - OEIS". oeis.org.
188. "Sloane's A033819 : Trimorphic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
189. Sloane, N. J. A. (ed.). "Sequence A058331 (2*n^2 + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
190. Sloane, N. J. A. (ed.). "Sequence A144300 (Number of partitions of n minus number of divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
191. Sloane, N. J. A. (ed.). "Sequence A000837 (Number of partitions of n into relatively prime parts. Also aperiodic partitions.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
192. Sloane, N. J. A. (ed.). "Sequence A000041 (a(n) is the number of partitions of n (the partition numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
193. "Sloane's A002182 : Highly composite numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
194. "Sloane's A014575 : Vampire numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
195. Sloane, N. J. A. (ed.). "Sequence A014206 (n^2 + n + 2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
196. Sloane, N. J. A. (ed.). "Sequence A070169 (Rounded total surface area of a regular tetrahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
197. Sloane, N. J. A. (ed.). "Sequence A003238 (Number of rooted trees with n vertices in which vertices at the same level have the same degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
198. Sloane, N. J. A. (ed.). "Sequence A023894 (Number of partitions of n into prime power parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
199. Sloane, N. J. A. (ed.). "Sequence A072895 (Least k for the Theodorus spiral to complete n revolutions)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
200. Sloane, N. J. A. (ed.). "Sequence A100040 (2*n^2 + n - 5)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
201. Sloane, N. J. A. (ed.). "Sequence A051349 (Sum of first n nonprimes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
202. Sloane, N. J. A. (ed.). "Sequence A033286 (n * prime(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
203. Sloane, N. J. A. (ed.). "Sequence A084849 (1 + n + 2*n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
204. Sloane, N. J. A. (ed.). "Sequence A000930 (Narayana's cows sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
205. Sloane, N. J. A. (ed.). "Sequence A001792 ((n+2)*2^(n-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
206. Sloane, N. J. A. (ed.). "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
207. Sloane, N. J. A. (ed.). "Sequence A216492 (Number of inequivalent connected planar figures that can be formed from n 1 X 2 rectangles (or dominoes) such that each pair of touching rectangles shares exactly one edge, of length 1, and the adjacency graph of the rectangles is a tree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
208. Sloane, N. J. A. (ed.). "Sequence A007318 (Pascal's triangle read by rows)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
209. Sloane, N. J. A. (ed.). "Sequence A014574 (Average of twin prime pairs)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
210. Sloane, N. J. A. (ed.). "Sequence A173831 (Largest prime < n^4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
211. Sloane, N. J. A. (ed.). "Sequence A006872 (Numbers k such that phi(k) equals phi(sigma(k)))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
212. Sloane, N. J. A. (ed.). "Sequence A014285 (Sum_{1..n} j*prime(j))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
213. Sloane, N. J. A. (ed.). "Sequence A071400 (Rounded volume of a regular octahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
214. Sloane, N. J. A. (ed.). "Sequence A003114 (Number of partitions of n into parts 5k+1 or 5k+4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
215. Sloane, N. J. A. (ed.). "Sequence A033548 (Honaker primes: primes P(k) such that sum of digits of P(k) equals sum of digits of k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
216. Sloane, N. J. A. (ed.). "Sequence A000055 (Number of trees with n unlabeled nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
217. "A124826 - OEIS". oeis.org.
218. "A142005 - OEIS". oeis.org.
219. Sloane, N. J. A. (ed.). "Sequence A338470 (Number of integer partitions of n with no part dividing all the others)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
220. "A066186 - OEIS". oeis.org.
221. Sloane, N. J. A. (ed.). "Sequence A304716 (Number of integer partitions of n whose distinct parts are connected)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
222. "A115073 - OEIS". oeis.org.
223. "A061256 - OEIS". oeis.org.
224. "A061954 - OEIS". oeis.org.
225. "A030299 - OEIS". oeis.org.
226. "Sloane's A002559 : Markoff (or Markov) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
227. Sloane, N. J. A. (ed.). "Sequence A005894 (Centered tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
228. "A005064 - OEIS". oeis.org.
229. Sloane, N. J. A. (ed.). "Sequence A001770 (Numbers k such that 5*2^k - 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
230. Sloane, N. J. A. (ed.). "Sequence A144391 (3*n^2 + n - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
231. Sloane, N. J. A. (ed.). "Sequence A090781 (Numbers that can be expressed as the difference of the squares of primes in just one distinct way)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
232. Sloane, N. J. A. (ed.). "Sequence A056809 (Numbers k such that k, k+1 and k+2 are products of two primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
233. "A316473 - OEIS". oeis.org.
234. "A000032 - OEIS". oeis.org.
235. "1348 (number)". The encyclopedia of numbers.
236. Sloane, N. J. A. (ed.). "Sequence A101624 (Stern-Jacobsthal number)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
237. Sloane, N. J. A. (ed.). "Sequence A000603". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
238. Sloane, N. J. A. (ed.). "Sequence A330224 (Number of achiral integer partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
239. Sloane, N. J. A. (ed.). "Sequence A001610 (a(n-1) + a(n-2) + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
240. Sloane, N. J. A. (ed.). "Sequence A000032 (Lucas numbers: L(n-1) + L(n-2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
241. "Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
242. Sloane, N. J. A. (ed.). "Sequence A005578 (Arima sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
243. Sloane, N. J. A. (ed.). "Sequence A001157 (sigma_2(n): sum of squares of divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
244. Sloane, N. J. A. (ed.). "Sequence A007585 (10-gonal (or decagonal) pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
245. Sloane, N. J. A. (ed.). "Sequence A071395 (Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
246. Sloane, N. J. A. (ed.). "Sequence A005945 (Number of n-step mappings with 4 inputs)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
247. "A001631 - OEIS". oeis.org. Retrieved 25 June 2023.
248. Sloane, N. J. A. (ed.). "Sequence A000111 (Euler or up/down numbers: e.g.f. sec(x) + tan(x))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
249. Sloane, N. J. A. (ed.). "Sequence A002414 (Octagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
250. "Sloane's A001567 : Fermat pseudoprimes to base 2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
251. "Sloane's A050217 : Super-Poulet numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
252. Sloane, N. J. A. (ed.). "Sequence A007865 (Number of sum-free subsets of {1, ..., n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.}
253. Sloane, N. J. A. (ed.). "Sequence A325349 (Number of integer partitions of n whose augmented differences are distinct)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
254. "Sloane's A000682 : Semimeanders". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
255. Sloane, N. J. A. (ed.). "Sequence A050710 (Smallest composite that when added to sum of prime factors reaches a prime after n iterations)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
256. Sloane, N. J. A. (ed.). "Sequence A067538 (Number of partitions of n in which the number of parts divides n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
257. "Sloane's A051015 : Zeisel numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
258. Sloane, N. J. A. (ed.). "Sequence A061068 (Primes which are the sum of a prime and its subscript)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
259. "Sloane's A000108 : Catalan numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
260. Sloane, N. J. A. (ed.). "Sequence A071399 (Rounded volume of a regular tetrahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
261. Sloane, N. J. A. (ed.). "Sequence A003037 (Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
262. Sloane, N. J. A. (ed.). "Sequence A062325 (Numbers k for which phi(prime(k)) is a square)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
263. Sloane, N. J. A. (ed.). "Sequence A011379 (n^2*(n+1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
264. Sloane, N. J. A. (ed.). "Sequence A005918 (Number of points on surface of square pyramid: 3*n^2 + 2 (n>0))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
265. Sloane, N. J. A. (ed.). "Sequence A011257 (Geometric mean of phi(n) and sigma(n) is an integer)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
266. Sloane, N. J. A. (ed.). "Sequence A056220 (2*n^2 - 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
267. Sloane, N. J. A. (ed.). "Sequence A028569 (n*(n + 9))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
268. Sloane, N. J. A. (ed.). "Sequence A071398 (Rounded total surface area of a regular icosahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
269. "Sloane's A002411 : Pentagonal pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
270. Sloane, N. J. A. (ed.). "Sequence A307958 (Coreful perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
271. Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
272. Sloane, N. J. A. (ed.). "Sequence A006330 (Number of corners, or planar partitions of n with only one row and one column)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
273. "Sloane's A000078 : Tetranacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
274. Sloane, N. J. A. (ed.). "Sequence A034296 (Number of flat partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
275. Sloane, N. J. A. (ed.). "Sequence A084647 (Hypotenuses for which there exist exactly 3 distinct integer triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
276. Sloane, N. J. A. (ed.). "Sequence A002071 (Number of pairs of consecutive integers x, x+1 such that all prime factors of both x and x+1 are at most the n-th prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
277. Sloane, N. J. A. (ed.). "Sequence A000702 (number of conjugacy classes in the alternating group A_n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
278. Sloane, N. J. A. (ed.). "Sequence A071396 (Rounded total surface area of a regular octahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
279. Sloane, N. J. A. (ed.). "Sequence A000615 (Threshold functions of exactly n variables)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
280. Sloane, N. J. A. (ed.). "Sequence A319066 (Number of partitions of integer partitions of n where all parts have the same length)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
281. Sloane, N. J. A. (ed.). "Sequence A065381 (Primes not of the form p + 2^k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
282. Sloane, N. J. A. (ed.). "Sequence A008406 (Triangle T(n,k) read by rows, giving number of graphs with n nodes and k edges))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
283. Sloane, N. J. A. (ed.). "Sequence A000014 (Number of series-reduced trees with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
284. Sloane, N. J. A. (ed.). "Sequence A088319 (Ordered hypotenuses of primitive Pythagorean triangles having legs that add up to a square)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
285. Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
286. "Sloane's A005231 : Odd abundant numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
287. Sloane, N. J. A. (ed.). "Sequence A070142 (Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer area)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
288. Sloane, N. J. A. (ed.). "Sequence A071402 (Rounded volume of a regular icosahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
289. Sloane, N. J. A. (ed.). "Sequence A006327 (Fibonacci(n) - 3. Number of total preorders)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
290. "Sloane's A000045 : Fibonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
291. Sloane, N. J. A. (ed.). "Sequence A100145 (Structured great rhombicosidodecahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
292. Sloane, N. J. A. (ed.). "Sequence A064174 (Number of partitions of n with nonnegative rank)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
293. Sloane, N. J. A. (ed.). "Sequence A007584 (9-gonal (or enneagonal) pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
294. Sloane, N. J. A. (ed.). "Sequence A046092 (4 times triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
295. Sloane, N. J. A. (ed.). "Sequence A007290 (2*binomial(n,3))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
296. Sloane, N. J. A. (ed.). "Sequence A058360 (Number of partitions of n whose reciprocal sum is an integer)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
297. Sloane, N. J. A. (ed.). "Sequence A046931 (Prime islands: least prime whose adjacent primes are exactly 2n apart)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
298. "Sloane's A001599 : Harmonic or Ore numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
299. Sloane, N. J. A. (ed.). "Sequence A056613 (Number of n-celled pseudo still lifes in Conway's Game of Life, up to rotation and reflection)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
300. Sloane, N. J. A. (ed.). "Sequence A068140 (Smaller of two consecutive numbers each divisible by a cube greater than one)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
301. Sloane, N. J. A. (ed.). "Sequence A030272 (Number of partitions of n^3 into distinct cubes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
302. Sloane, N. J. A. (ed.). "Sequence A018818 (Number of partitions of n into divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
303. Sloane, N. J. A. (ed.). "Sequence A071401 (Rounded volume of a regular dodecahedron with edge length n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
304. "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
305. Sloane, N. J. A. (ed.). "Sequence A059802 (Numbers k such that 5^k - 4^k is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
306. Sloane, N. J. A. (ed.). "Sequence A082982 (Numbers k such that k, k+1 and k+2 are sums of 2 squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
307. Sloane, N. J. A. (ed.). "Sequence A057562 (Number of partitions of n into parts all relatively prime to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
308. Sloane, N. J. A. (ed.). "Sequence A000230 (smallest prime p such that there is a gap of exactly 2n between p and next prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
309. Sloane, N. J. A. (ed.). "Sequence A261983 (Number of compositions of n such that at least two adjacent parts are equal)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
310. Sloane, N. J. A. (ed.). "Sequence A053781 (Numbers k that divide the sum of the first k composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
311. Sloane, N. J. A. (ed.). "Sequence A140480 (RMS numbers: numbers n such that root mean square of divisors of n is an integer)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
312. Sloane, N. J. A. (ed.). "Sequence A023108 (Positive integers which apparently never result in a palindrome under repeated applications of the function A056964(x))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
313. Sloane, N. J. A. (ed.). "Sequence A098859 (Number of partitions of n into parts each of which is used a different number of times)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
314. Sloane, N. J. A. (ed.). "Sequence A286518 (Number of finite connected sets of positive integers greater than one with least common multiple n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
315. Sloane, N. J. A. (ed.). "Sequence A004041 (Scaled sums of odd reciprocals: (2*n + 1)!!*(Sum_{0..n} 1/(2*k + 1)))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
316. {{cite OEIS<A023359|Number of compositions (ordered partitions) of n into powers of 2}}
317. Sloane, N. J. A. (ed.). "Sequence A000787 (Strobogrammatic numbers: the same upside down)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
318. Sloane, N. J. A. (ed.). "Sequence A224930 (Numbers n such that n divides the concatenation of all divisors in descending order)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
319. Sloane, N. J. A. (ed.). "Sequence A294286 (Sum of the squares of the parts in the partitions of n into two distinct parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
320. "Sloane's A000073 : Tribonacci numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
321. Sloane, N. J. A. (ed.). "Sequence A020989 ((5*4^n - 2)/3)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
322. Sloane, N. J. A. (ed.). "Sequence A331378 (Numbers whose product of prime indices is divisible by their sum of prime factors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
323. Sloane, N. J. A. (ed.). "Sequence A301700 (Number of aperiodic rooted trees with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
324. Sloane, N. J. A. (ed.). "Sequence A331452 (number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
325. Sloane, N. J. A. (ed.). "Sequence A056045 ("Sum_{d divides n}(binomial(n,d))")". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
326. "Sloane's A007850 : Giuga numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
327. Sloane, N. J. A. (ed.). "Sequence A161757 ((prime(n))^2 - (nonprime(n))^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
328. Sloane, N. J. A. (ed.). "Sequence A078374 (Number of partitions of n into distinct and relatively prime parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
329. Sloane, N. J. A. (ed.). "Sequence A167008 (Sum_{0..n} C(n,k)^k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
330. Sloane, N. J. A. (ed.). "Sequence A033581 (6*n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
331. Sloane, N. J. A. (ed.). "Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
332. Sloane, N. J. A. (ed.). "Sequence A350507 (Number of (not necessarily connected) unit-distance graphs on n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
333. Sloane, N. J. A. (ed.). "Sequence A102627 (Number of partitions of n into distinct parts in which the number of parts divides n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
334. Sloane, N. J. A. (ed.). "Sequence A216955 (number of binary sequences of length n and curling number k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
335. Sloane, N. J. A. (ed.). "Sequence A001523 (Number of stacks, or planar partitions of n; also weakly unimodal compositions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
336. Sloane, N. J. A. (ed.). "Sequence A065764 (Sum of divisors of square numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
337. Sloane, N. J. A. (ed.). "Sequence A220881 (Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
338. Sloane, N. J. A. (ed.). "Sequence A154964 (3*a(n-1) + 6*a(n-2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
339. Sloane, N. J. A. (ed.). "Sequence A055327 (Triangle of rooted identity trees with n nodes and k leaves)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
340. Sloane, N. J. A. (ed.). "Sequence A316322 (Sum of piles of first n primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
341. Sloane, N. J. A. (ed.). "Sequence A045944 (Rhombic matchstick numbers: n*(3*n+2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
342. Sloane, N. J. A. (ed.). "Sequence A127816 (least k such that the remainder when 6^k is divided by k is n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
343. Sloane, N. J. A. (ed.). "Sequence A005317 ((2^n + C(2*n,n))/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
344. Sloane, N. J. A. (ed.). "Sequence A064118 (Numbers k such that the first k digits of e form a prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
345. Sloane, N. J. A. (ed.). "Sequence A325860 (Number of subsets of {1..n} such that every pair of distinct elements has a different quotient)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
346. Sloane, N. J. A. (ed.). "Sequence A073592 (Euler transform of negative integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
347. Sloane, N. J. A. (ed.). "Sequence A025047 (Alternating compositions, i.e., compositions with alternating increases and decreases, starting with either an increase or a decrease)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
348. Sloane, N. J. A. (ed.). "Sequence A288253 (Number of heptagons that can be formed with perimeter n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
349. Sloane, N. J. A. (ed.). "Sequence A235488 (Squarefree numbers which yield zero when their prime factors are xored together)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
350. Sloane, N. J. A. (ed.). "Sequence A075213 (Number of polyhexes with n cells that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
351. "Sloane's A054377 : Primary pseudoperfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
352. Kellner, Bernard C.; 'The equation denom(Bn) = n has only one solution'
353. Sloane, N. J. A. (ed.). "Sequence A006318 (Large Schröder numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2016.
354. "Sloane's A000058 : Sylvester's sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
355. Sloane, N. J. A. (ed.). "Sequence A083186 (Sum of first n primes whose indices are primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
356. Sloane, N. J. A. (ed.). "Sequence A005260 (Sum_{0..n} binomial(n,k)^4)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
357. Sloane, N. J. A. (ed.). "Sequence A056877 (Number of polyominoes with n cells, symmetric about two orthogonal axes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
358. Sloane, N. J. A. (ed.). "Sequence A061801 ((7*6^n - 2)/5)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
359. Sloane, N. J. A. (ed.). "Sequence A152927 (Number of sets (in the Hausdorff metric geometry) at each location between two sets defining a polygonal configuration consisting of k 4-gonal polygonal components chained with string components of length 1 as k varies)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
360. Sloane, N. J. A. (ed.). "Sequence A037032 (Total number of prime parts in all partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
361. Sloane, N. J. A. (ed.). "Sequence A101301 (The sum of the first n primes, minus n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
362. Sloane, N. J. A. (ed.). "Sequence A332835 (Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2 June 2022.
363. Sloane, N. J. A. (ed.). "Sequence A000230 (smallest prime p such that there is a gap of exactly 2n between p and next prime, or -1 if no such prime exists)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
364. Sloane, N. J. A. (ed.). "Sequence A004068 (Number of atoms in a decahedron with n shells)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
365. Sloane, N. J. A. (ed.). "Sequence A001905 (From higher-order Bernoulli numbers: absolute value of numerator of D-number D2n(2n-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
366. Sloane, N. J. A. (ed.). "Sequence A214083 (floor(n!^(1/3)))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
367. Sloane, N. J. A. (ed.). "Sequence A000081 (Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
368. Sloane, N. J. A. (ed.). "Sequence A354493 (Number of quantales on n elements, up to isomorphism)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
369. Sloane, N. J. A. (ed.). "Sequence A000240 (Rencontres numbers: number of permutations of [n] with exactly one fixed point)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
370. Sloane, N. J. A. (ed.). "Sequence A000602 (Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
371. ""Aztec Diamond"". Retrieved 20 September 2022.
372. Sloane, N. J. A. (ed.). "Sequence A082671 (Numbers n such that (n!-2)/2 is a prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
373. Sloane, N. J. A. (ed.). "Sequence A023811 (Largest metadrome (number with digits in strict ascending order) in base n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
374. Sloane, N. J. A. (ed.). "Sequence A000990 (Number of plane partitions of n with at most two rows)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
375. Sloane, N. J. A. (ed.). "Sequence A164652 (Hultman numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
376. Sloane, N. J. A. (ed.). "Sequence A007530 (Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
377. Sloane, N. J. A. (ed.). "Sequence A057568 (Number of partitions of n where n divides the product of the parts)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
378. Sloane, N. J. A. (ed.). "Sequence A011757 (prime(n^2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
379. Sloane, N. J. A. (ed.). "Sequence A004799 (Self convolution of Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
380. Sloane, N. J. A. (ed.). "Sequence A005920 (Tricapped prism numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
381. Sloane, N. J. A. (ed.). "Sequence A000609 (Number of threshold functions of n or fewer variables)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
382. Sloane, N. J. A. (ed.). "Sequence A259793 (Number of partitions of n^4 into fourth powers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
383. Sloane, N. J. A. (ed.). "Sequence A006785 (Number of triangle-free graphs on n vertices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
384. Sloane, N. J. A. (ed.). "Sequence A002998 (Smallest multiple of n whose digits sum to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
385. Sloane, N. J. A. (ed.). "Sequence A005987 (Number of symmetric plane partitions of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
386. Sloane, N. J. A. (ed.). "Sequence A023431 (Generalized Catalan Numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
387. Sloane, N. J. A. (ed.). "Sequence A217135 (Numbers n such that 3^n - 8 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
388. "Sloane's A034897 : Hyperperfect numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
389. Sloane, N. J. A. (ed.). "Sequence A240736 (Number of compositions of n having exactly one fixed point)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
390. Sloane, N. J. A. (ed.). "Sequence A007070 (4*a(n-1) - 2*a(n-2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
391. Sloane, N. J. A. (ed.). "Sequence A000412 (Number of bipartite partitions of n white objects and 3 black ones)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
392. Sloane, N. J. A. (ed.). "Sequence A027851 (Number of nonisomorphic semigroups of order n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
393. Sloane, N. J. A. (ed.). "Sequence A003060 (Smallest number with reciprocal of period length n in decimal (base 10))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
394. Sloane, N. J. A. (ed.). "Sequence A008514 (4-dimensional centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
395. Sloane, N. J. A. (ed.). "Sequence A024012 (2^n - n^2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
396. Sloane, N. J. A. (ed.). "Sequence A002845 (Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
397. "Sloane's A051870 : 18-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 12 June 2016.
398. Sloane, N. J. A. (ed.). "Sequence A045648 (Number of chiral n-ominoes in (n-1)-space, one cell labeled)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
399. Sloane, N. J. A. (ed.). "Sequence A000127 (Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
400. Sloane, N. J. A. (ed.). "Sequence A178084 (Numbers k for which 10k + 1, 10k + 3, 10k + 7, 10k + 9 and 10k + 13 are primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
401. Sloane, N. J. A. (ed.). "Sequence A007419 (Largest number not the sum of distinct n-th-order polygonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
402. Sloane, N. J. A. (ed.). "Sequence A100953 (Number of partitions of n into relatively prime parts such that multiplicities of parts are also relatively prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
403. Sloane, N. J. A. (ed.). "Sequence A226366 (Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
404. Sloane, N. J. A. (ed.). "Sequence A319014 (1*2*3 + 4*5*6 + 7*8*9 + 10*11*12 + 13*14*15 + 16*17*18 + ... + (up to n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
405. Sloane, N. J. A. (ed.). "Sequence A055621 (Number of covers of an unlabeled n-set)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
406. Sloane, N. J. A. (ed.). "Sequence A000522 (Total number of ordered k-tuples of distinct elements from an n-element set)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
407. Sloane, N. J. A. (ed.). "Sequence A104621 (Heptanacci-Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
408. Sloane, N. J. A. (ed.). "Sequence A005449 (Second pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
409. Sloane, N. J. A. (ed.). "Sequence A002982 (Numbers n such that n! - 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
410. Sloane, N. J. A. (ed.). "Sequence A030238 (Backwards shallow diagonal sums of Catalan triangle A009766)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
411. Sloane, N. J. A. (ed.). "Sequence A089046 (Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
412. Sloane, N. J. A. (ed.). "Sequence A065900 (Numbers n such that sigma(n) equals sigma(n-1) + sigma(n-2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
413. Jon Froemke & Jerrold W. Grossman (February 1993). "A Mod-n Ackermann Function, or What's So Special About 1969?". The American Mathematical Monthly. Mathematical Association of America. 100 (2): 180–183. doi:10.2307/2323780. JSTOR 2323780.
414. Sloane, N. J. A. (ed.). "Sequence A052542 (2*a(n-1) + a(n-2))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
415. Sloane, N. J. A. (ed.). "Sequence A024069 (6^n - n^7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
416. Sloane, N. J. A. (ed.). "Sequence A217076 (Numbers n such that (n^37-1)/(n-1) is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
417. Sloane, N. J. A. (ed.). "Sequence A302545 (Number of non-isomorphic multiset partitions of weight n with no singletons)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
418. Sloane, N. J. A. (ed.). "Sequence A277288 (Positive integers n such that n divides (3^n + 5))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
419. Sloane, N. J. A. (ed.). "Sequence A187220 (Gullwing sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
420. Sloane, N. J. A. (ed.). "Sequence A046351 (Palindromic composite numbers with only palindromic prime factors)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
421. Sloane, N. J. A. (ed.). "Sequence A000612 (Number of P-equivalence classes of switching functions of n or fewer variables, divided by 2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
422. OEIS: A059801
423. Sloane, N. J. A. (ed.). "Sequence A002470 (Glaisher's function W(n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
424. Sloane, N. J. A. (ed.). "Sequence A263341 (Triangle read by rows: T(n,k) is the number of unlabeled graphs on n vertices with independence number k)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
425. Sloane, N. J. A. (ed.). "Sequence A089085 (Numbers k such that (k! + 3)/3 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
426. Sloane, N. J. A. (ed.). "Sequence A011755 (Sum_{1..n} k*phi(k))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
427. Sloane, N. J. A. (ed.). "Sequence A005448 (Centered triangular numbers: 3n(n-1)/2 + 1)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.,
428. Sloane, N. J. A. (ed.). "Sequence A038823 (Number of primes between n*1000 and (n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
429. Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and Swinnerton-Dyer Conjecture". wstein.org. Retrieved 6 February 2021.
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| Wikipedia |
0.999...
In mathematics, 0.999... (also written as 0.9 or 0..9) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence.[1] This number is equal to 1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" 1 – rather, "0.999..." and "1" represent exactly the same number.
There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.
More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.
Elementary proof
There is an elementary proof of the equation 0.999... = 1, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by Stillwell (1994, p. 42), is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1.
Intuitive explanation
If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.
More precisely, the distance from 0.9 to 1 is 0.1 = 1/10, the distance from 0.99 to 1 is 0.01 = 1/102, and so on. The distance to 1 from the nth point (the one with n 9s after the decimal point) is 1/10n.
Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than 1/10n for every integer n. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10n for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so 1 = 0.999....
Discussion on completeness
Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.
Rigorous proof
The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number 0.999...9, with n nines after the decimal point, is denoted 0.(9)n. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. As 1/10n = 0.0...01, with n digits after the decimal point, the addition rule for decimal numbers implies
$0.(9)_{n}+1/10^{n}=1,$
and
$0.(9)_{n}<1,$
for every positive integer n.
One has to show that 1 is the smallest number that is no less than all 0.(9)n. For this, it suffices to prove that, if a number x is not larger than 1 and no less than all 0.(9)n, then x = 1. So let x such that
$0.(9)_{n}\leq x\leq 1,$
for every positive integer n. Therefore,
$1-1\leq 1-x\leq 1-0.(9)_{n}.$
which, using basic arithmetic and the first equality established above, simplifies to
$0\leq 1-x\leq 1/10^{n}.$
This implies that the difference between 1 and x is less than the inverse of any positive integer. Thus this difference must be zero, and, thus x = 1; that is
$0.999\ldots =1.$
This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all 0.(9)n. (This is implied by the fact that 0.(9)n ≤ x < 1 implies 0.(9)n–1 ≤ 2x – 1 < x < 1).
Algebraic arguments
Many algebraic arguments have been provided, which suggest that $1=0.999\ldots $ They are not mathematical proofs since they are typically based on the fact that the rules for adding and multiplying finite decimals extend to infinite decimals. This is true, but the proof is essentially the same as the proof of $1=0.999\ldots $ So, all these arguments are essentially circular reasoning.
Nevertheless, the matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. Byers (2007, p. 39) discusses the argument that, in elementary school, one is taught that 1⁄3=0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1=0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. Richman (1999) discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.
Byers also presents the following argument.
${\begin{aligned}x&=0.999\ldots \\10x&=9.999\ldots &&{\text{by multiplying by }}10\\10x&=9+0.999\ldots &&{\text{by splitting off integer part}}\\10x&=9+x&&{\text{by definition of }}x\\9x&=9&&{\text{by subtracting }}x\\x&=1&&{\text{by dividing by }}9\end{aligned}}$
Students who did not accept the first argument sometimes accept the second argument, but, in Byers's opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.
The same argument is also given by Richman (1999), who notes that skeptics may question whether x is cancellable – that is, whether it makes sense to subtract x from both sides.
Analytic proofs
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form
$b_{0}.b_{1}b_{2}b_{3}b_{4}b_{5}\dots .$
The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.
Infinite series and sequences
Further information: Decimal representation
A common development of decimal expansions is to define them as sums of infinite series. In general:
$b_{0}.b_{1}b_{2}b_{3}b_{4}\ldots =b_{0}+b_{1}\left({\tfrac {1}{10}}\right)+b_{2}\left({\tfrac {1}{10}}\right)^{2}+b_{3}\left({\tfrac {1}{10}}\right)^{3}+b_{4}\left({\tfrac {1}{10}}\right)^{4}+\cdots .$
For 0.999... one can apply the convergence theorem concerning geometric series:[2]
If $|r|<1$ then $ar+ar^{2}+ar^{3}+\cdots ={\frac {ar}{1-r}}.$
Since 0.999... is such a sum with a = 9 and common ratio r = 1⁄10, the theorem makes short work of the question:
$0.999\ldots =9\left({\tfrac {1}{10}}\right)+9\left({\tfrac {1}{10}}\right)^{2}+9\left({\tfrac {1}{10}}\right)^{3}+\cdots ={\frac {9\left({\tfrac {1}{10}}\right)}{1-{\tfrac {1}{10}}}}=1.$
This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.[3]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999...[4] A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.[5]
A sequence (x0, x1, x2, ...) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:[6]
$0.999\ldots \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}\ =\lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)=1-\lim _{n\to \infty }{\frac {1}{10^{n}}}=1\,-\,0=1.$
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 1⁄10n → 0 as n → ∞, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".[7] Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.
Nested intervals and least upper bounds
Further information: Nested intervals
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.
If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b0, b1, b2, b3, ..., and one writes
$x=b_{0}.b_{1}b_{2}b_{3}\ldots $
In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.[8]
One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b0.b1b2b3... is defined to be the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.[9]
The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b0.b1b2b3... to be the least upper bound of the set of approximants {b0, b0.b1, b0.b1b2, ...}.[10] One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.[11]
Proofs from the construction of the real numbers
Further information: Construction of the real numbers
Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.
The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.[12]
Dedekind cuts
Further information: Dedekind cut
In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x.[13] In particular, the real number 1 is the set of all rational numbers that are less than 1.[14] Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form[15]
$1-{\frac {1}{10^{n}}}=0.(9)_{n}=0.\underbrace {99\ldots 9} _{n{\text{ nines}}}.$
Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as
${\frac {a}{b}}<1,$
with b > 0 and b > a. This implies
$1-{\frac {a}{b}}={\frac {b-a}{b}}\geq {\frac {1}{b}}>{\frac {1}{10^{b}}},$
and thus
${\frac {a}{b}}<1-{\frac {1}{10^{b}}}.$
and since
$1-{\frac {1}{10^{b}}}=0.(9)_{b}<0.999\ldots $
by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1.
The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.[16] The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in Mathematics Magazine,[17] which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.[18] Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate. He also notes that typically the definitions allow { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."[19] A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1⁄3 has no representation; see "Alternative number systems" below.
Cauchy sequences
Further information: Cauchy sequence
Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)[20]
If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xn − yn) has the limit 0. Truncations of the decimal number b0.b1b2b3... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.[21] Thus in this formalism the task is to show that the sequence of rational numbers
$\left(1-0,1-{9 \over 10},1-{99 \over 100},\dots \right)=\left(1,{1 \over 10},{1 \over 100},\dots \right)$
has the limit 0. Considering the nth term of the sequence, for n ∈ $\mathbb {N} $, it must therefore be shown that
$\lim _{n\rightarrow \infty }{\frac {1}{10^{n}}}=0.$
This limit is plain[22] if one understands the definition of limit. So again 0.999... = 1.
The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.[16] The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.[23]
Infinite decimal representation
Further information: Construction of the real numbers § Stevin's construction
Commonly in secondary schools' mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version.[24] With this construction of the reals, all proofs of the statement "1 = 0.999..." can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.
Dense order
Further information: Dense order
One of the notions that can resolve the issue is the requirement that real numbers are densely ordered. Students are taking for granted that $0.99999...$ is before $1$ while this kind of intuitive ordering is better defined as purely lexicographical.
"... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1)."[25]
Dense order requires that there is a third real value strictly between $0.99999...$ and $1$, but there is none: we cannot change a single digit in either of the two to obtain such a number. If $0.99999...$ and $1$ are to represent real numbers they have to be equal. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal.
Generalizations
The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.[26]
Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111... equals 1, and in base 3 (the ternary numeral system) 0.222... equals 1. In general, any terminating base b expression has a counterpart with repeated trailing digits equal to b − 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.[27]
Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000.... This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik–Loreti constant q = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the Thue–Morse sequence, which does not repeat.[28]
A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example:[29]
• In the balanced ternary system, 1⁄2 = 0.111... = 1.111....
• In the reverse factorial number system (using bases 2!,3!,4!,... for positions after the decimal point), 1 = 1.000... = 0.1234....
Impossibility of unique representation
That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered lexicographically. Indeed, the following two properties account for the difficulty:
• If an interval of the real numbers is partitioned into two non-empty parts L, R, such that every element of L is (strictly) less than every element of R, then either L contains a largest element or R contains a smallest element, but not both.
• The collection of infinite strings of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts L, R, such that every element of L is less than every element of R, while L contains a largest element and R contains a smallest element. Indeed, it suffices to take two finite prefixes (initial substrings) p1, p2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for L the set of all strings in the collection whose corresponding prefix is at most p1, and for R the remainder, the strings in the collection whose corresponding prefix is at least p2. Then L has a largest element, starting with p1 and choosing the largest available symbol in all following positions, while R has a smallest element obtained by following p2 by the smallest symbol in all positions.
The first point follows from basic properties of the real numbers: L has a supremum and R has an infimum, which are easily seen to be equal; being a real number it either lies in R or in L, but not both since L and R are supposed to be disjoint. The second point generalizes the 0.999.../1.000... pair obtained for p1 = "0", p2 = "1". In fact one need not use the same alphabet for all positions (so that for instance mixed radix systems can be included) or consider the full collection of possible strings; the only important points are that at each position a finite set of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an order preserving map from the collection of strings to an interval of the real numbers cannot be a bijection: either some numbers do not correspond to any string, or some of them correspond to more than one string.
Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary point-set topology"; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.[30]
Applications
One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:
• 1⁄7 = 0.142857 and 142 + 857 = 999.
• 1⁄73 = 0.01369863 and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If it can be proved that if a decimal of the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.[31] Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.[32]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:
• A point in the unit interval lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.
The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.[33]
Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.[34] A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.[35]
Skepticism in education
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
• Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.[36]
• Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".[37]
• Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.[38]
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999...
Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'".[39]
The elementary argument of multiplying 0.333... = 1⁄3 by 3 can convince reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.[40] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a supremum definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.[41] Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.
Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."[42]
As part of Ed Dubinsky's APOS theory of mathematical learning, he and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.[43]
Cultural phenomenon
With the rise of the Internet, debates about 0.999... have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999... is described as a "popular sport", and it is one of the questions answered in its FAQ.[44] The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.
A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense.[45]
A Slate article reports that the concept of 0.999... is "hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums".[46] In the same vein, the question of 0.999... proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company issued a "press release" on April Fools' Day 2004 that it is 1:
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.[47]
Two proofs are then offered, based on limits and multiplication by 10.
0.999... features also in mathematical jokes, such as:[48]
Q: How many mathematicians does it take to screw in a lightbulb?
A: 0.999999....
In alternative number systems
Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999... = 1 is a convention as well:
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.[49]
Infinitesimals
Main article: Infinitesimal
Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[50] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[51] Lightstone shows how to associate to each number a sequence of digits,
$0.d_{1}d_{2}d_{3}\dots ;\dots d_{\infty -1}d_{\infty }d_{\infty +1}\dots ,$ ;\dots d_{\infty -1}d_{\infty }d_{\infty +1}\dots ,}
indexed by the hypernatural numbers. While he does not directly discuss 0.999..., he shows the real number 1⁄3 is represented by 0.333...;...333... which is a consequence of the transfer principle. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.
The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number uH=0.999...;...999000..., with last digit 9 at infinite hypernatural rank H, satisfies a strict inequality uH < 1. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be
${\underset {H}{0.\underbrace {999\ldots } }}\;=1\;-\;{\frac {1}{10^{H}}}.$[52]
All such interpretations of "0.999..." are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....[53] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999... < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[54][55]
Hackenbush
Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1⁄3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number 1⁄ω, where ω is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.[56]
This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to 3/4, but the first representation corresponds to the binary tree path LRLRLLL... while the second corresponds to the different path LRLLRRR....
Revisiting subtraction
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999... < 1.
First, Richman defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999... + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.[57]
In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction d he allows both the cut (−∞, d) and the "principal cut" (−∞, d]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut D, but there is "a sort of negative infinitesimal," 0−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + x = 1" has no solution.[58]
p-adic numbers
Main article: p-adic number
When asked about 0.999..., novices often believe there should be a "final 9", believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the final 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "final 9" in 0.999....[59] However, there is a system that contains an infinite string of 9s including a last 9.
The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to pn, than it is to 1. The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, and there are no infinitesimals.
In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.[60] Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
$\ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\cdots ={\frac {9}{1-10}}=-1.$[61]
(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = ...999 then 10x = ...990, so 10x = x − 9, hence x = −1 again.[60]
As a final extension, since 0.999... = 1 (in the reals) and ...999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols"[62] one may add the two equations and arrive at ...999.999... = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true in the doubly infinite decimal expansion of the 10-adic solenoid, with eventually repeating left ends to represent the real numbers[63] and eventually repeating right ends to represent the 10-adic numbers.
Related questions
• Zeno's paradoxes, particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.[64]
• Division by zero occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has a "point at infinity". Here, it makes sense to define 1⁄0 to be infinity;[65] and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.[66]
• Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.[67] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the sign and magnitude or ones' complement formats, or floating point numbers as specified by the IEEE floating-point standard).[68][69]
See also
• Finitism
• Informal mathematics
• Limit (mathematics)
• Series (mathematics)
Notes
1. This definition is equivalent to the definition of decimal numbers as the limits of their summed components, which, in the case of 0.999..., is the limit of the sequence (0.9, 0.99, 0.999, ...). The equivalence is due to bounded increasing sequences having their limit always equal to their least upper bound.
2. Rudin p. 61, Theorem 3.26; J. Stewart p. 706
3. Euler p. 170
4. Grattan-Guinness p. 69; Bonnycastle p. 177
5. For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31
6. The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) Thomas' Calculus: Early Transcendentals 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).
7. Davies p. 175; Smith and Harrington p. 115
8. Beals p. 22; I. Stewart p. 34
9. Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46
10. Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27
11. Apostol p. 12
12. The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30
13. Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number x can be named by giving an infinite set of rationals, namely all the rationals less than x. We will in effect define x to be the set of rationals smaller than x. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."
14. Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".
15. Richman p. 399
16. O'Connor, J. J.; Robertson, E. F. (October 2005). "History topic: The real numbers: Stevin to Hilbert". MacTutor History of Mathematics. Archived from the original on 29 September 2007. Retrieved 30 August 2006.
17. Fred Richman (December 1999). "Is 0.999... = 1?". Mathematics Magazine. Mathematical Association of America. pp. 396–400. Archived from the original on 11 December 2014. Retrieved 28 October 2014.
18. Richman
19. Richman pp. 398–399
20. Griffiths & Hilton §24.2 "Sequences" p. 386
21. Griffiths & Hilton pp. 388, 393
22. Griffiths & Hilton p. 395
23. Griffiths & Hilton pp.viii, 395
24. Liangpan Li (March 2011). "A new approach to the real numbers". arXiv:1101.1800 [math.CA].
25. Alan Schoenfeld, Derek Holton (2002). Holton, Derek; Artigue, Michèle; Kirchgräber, Urs; Hillel, Joel; Niss, Mogens; Schoenfeld, Alan (eds.). The Teaching and Learning of Mathematics at University Level. New ICMI Study Series. Vol. 7. Springer, Dordrecht. p. 212. doi:10.1007/0-306-47231-7. ISBN 978-0-306-47231-2.
26. Petkovšek p. 408
27. Protter and Morrey p. 503; Bartle and Sherbert p. 61
28. Komornik and Loreti p. 636
29. Kempner p. 611; Petkovšek p. 409
30. Petkovšek pp. 410–411
31. Leavitt 1984 p. 301
32. Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98
33. Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.
34. Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.
35. Rudin p. 50, Pugh p. 98
36. Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."
37. Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221
38. Tall and Schwarzenberger p. 6; Tall 2000 p. 221
39. Tall 2000 p. 221
40. Tall 1976 pp. 10–14
41. Pinto and Tall p. 5, Edwards and Ward pp. 416–417
42. Mazur pp. 137–141
43. Dubinsky et al. pp. 261–262
44. As observed by Richman (p. 396). de Vreught, Hans (1994). "sci.math FAQ: Why is 0.9999... = 1?". Archived from the original on 29 September 2007. Retrieved 29 June 2006.
45. Adams, Cecil (11 July 2003). "An infinite question: Why doesn't .999~ = 1?". The Straight Dope. Chicago Reader. Archived from the original on 15 August 2006. Retrieved 6 September 2006.
46. Ellenberg, Jordan (6 June 2014). "Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?". Slate.
47. "Blizzard Entertainment Announces .999~ (Repeating) = 1" (Press release). Blizzard Entertainment. 1 April 2004. Archived from the original on 4 November 2009. Retrieved 16 November 2009.
48. Renteln and Dundes, p. 27
49. Gowers p. 60
50. For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis.
51. Lightstone pp. 245–247
52. Katz & Katz 2010
53. Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.
54. Katz & Katz (2010b)
55. R. Ely (2010)
56. Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1⁄3 and touch on 1⁄ω. The game for 0.111...2 follows directly from Berlekamp's Rule.
57. Richman pp. 397–399
58. Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.
59. Gardiner p. 98; Gowers p. 60
60. Fjelstad p. 11
61. Fjelstad pp. 14–15
62. DeSua p. 901
63. DeSua pp. 902–903
64. Wallace p. 51, Maor p. 17
65. See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57
66. Maor p. 54
67. Munkres p. 34, Exercise 1(c)
68. Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2e ed.). W. H. Freeman. p. 462. ISBN 978-0-7167-1088-2.
69. "Floating point types". MSDN C# Language Specification. Archived from the original on 24 August 2006. Retrieved 29 August 2006.
References
• Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). "4.1 Cantor Sets". Chaos: An introduction to dynamical systems. Springer. ISBN 978-0-387-94677-1.
This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
• Apostol, Tom M. (1974). Mathematical Analysis (2e ed.). Addison-Wesley. ISBN 978-0-201-00288-1.
A transition from calculus to advanced analysis, Mathematical analysis is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)
• Baldwin, Michael; Norton, Anderson (2012). "Does 0.999... Really Equal 1?". The Mathematics Educator. 21 (2): 58–67.
• Bartle, R. G.; Sherbert, D. R. (1982). Introduction to Real Analysis. Wiley. ISBN 978-0-471-05944-8.
This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
• Beals, Richard (2004). Analysis: An Introduction. Cambridge UP. ISBN 978-0-521-60047-7.
• Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 978-0-12-091101-1.
• Berz, Martin (1992). Automatic Differentiation as Nonarchimedean Analysis. Computer Arithmetic and Enclosure Methods. Elsevier. pp. 439–450. CiteSeerX 10.1.1.31.3019.
• Beswick, Kim (2004). "Why Does 0.999... = 1?: A Perennial Question and Number Sense". Australian Mathematics Teacher. 60 (4): 7–9.
• Bunch, Bryan H. (1982). Mathematical Fallacies and Paradoxes. Van Nostrand Reinhold. ISBN 978-0-442-24905-2.
This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
• Burrell, Brian (1998). Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster. ISBN 978-0-87779-621-3.
• Byers, William (2007). How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton UP. ISBN 978-0-691-12738-5.
• Conway, John B. (1978) [1973]. Functions of One Complex Variable I (2e ed.). Springer-Verlag. ISBN 978-0-387-90328-6.
This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
• Davies, Charles (1846). The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications. A.S. Barnes. p. 175. Retrieved 4 July 2011.
• DeSua, Frank C. (November 1960). "A System Isomorphic to the Reals". The American Mathematical Monthly. 67 (9): 900–903. doi:10.2307/2309468. JSTOR 2309468.
• Dubinsky, Ed; Weller, Kirk; McDonald, Michael; Brown, Anne (2005). "Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2". Educational Studies in Mathematics. 60 (2): 253–266. doi:10.1007/s10649-005-0473-0. S2CID 45937062.
• Edwards, Barbara; Ward, Michael (May 2004). "Surprises from mathematics education research: Student (mis)use of mathematical definitions" (PDF). The American Mathematical Monthly. 111 (5): 411–425. CiteSeerX 10.1.1.453.7466. doi:10.2307/4145268. JSTOR 4145268. Archived from the original (PDF) on 22 July 2011. Retrieved 4 July 2011.
• Enderton, Herbert B. (1977). Elements of Set Theory. Elsevier. ISBN 978-0-12-238440-0.
An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
• Euler, Leonhard (1822) [1770]. Elements of Algebra. John Hewlett and Francis Horner, English translators (3rd English ed.). Orme Longman. p. 170. ISBN 978-0-387-96014-2. Retrieved 4 July 2011.
• Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285.
• Gardiner, Anthony (2003) [1982]. Understanding Infinity: The Mathematics of Infinite Processes. Dover. ISBN 978-0-486-42538-2.
• Gowers, Timothy (2002). Mathematics: A Very Short Introduction. Oxford UP. ISBN 978-0-19-285361-5.
• Grattan-Guinness, Ivor (1970). The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT Press. ISBN 978-0-262-07034-8.
• Griffiths, H. B.; Hilton, P. J. (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. London: Van Nostrand Reinhold. ISBN 978-0-442-02863-3. LCC QA37.2 G75.
This book grew out of a course for Birmingham-area grammar school mathematics teachers. The course was intended to convey a university-level perspective on school mathematics, and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of ideal theory, which is not reproduced here. (pp. vii, xiv)
• Katz, K.; Katz, M. (2010a). "When is .999... less than 1?". The Montana Mathematics Enthusiast. 7 (1): 3–30. arXiv:1007.3018. Bibcode:2010arXiv1007.3018U. doi:10.54870/1551-3440.1381. S2CID 11544878. Archived from the original on 20 July 2011. Retrieved 4 July 2011.
• Katz, Karin Usadi; Katz, Mikhail G. (2010b). "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era". Educational Studies in Mathematics. 74 (3): 259. arXiv:1003.1501. Bibcode:2010arXiv1003.1501K. doi:10.1007/s10649-010-9239-4. S2CID 115168622.
• Kempner, A. J. (December 1936). "Anormal Systems of Numeration". The American Mathematical Monthly. 43 (10): 610–617. doi:10.2307/2300532. JSTOR 2300532.
• Komornik, Vilmos; Loreti, Paola (1998). "Unique Developments in Non-Integer Bases". The American Mathematical Monthly. 105 (7): 636–639. doi:10.2307/2589246. JSTOR 2589246.
• Leavitt, W. G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly. 74 (6): 669–673. doi:10.2307/2314251. JSTOR 2314251.
• Leavitt, W. G. (September 1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. doi:10.2307/2686394. JSTOR 2686394.
• Lightstone, A. H. (March 1972). "Infinitesimals". The American Mathematical Monthly. 79 (3): 242–251. doi:10.2307/2316619. JSTOR 2316619.
• Mankiewicz, Richard (2000). The Story of Mathematics. Cassell. ISBN 978-0-304-35473-3.
Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
• Maor, Eli (1987). To Infinity and Beyond: A Cultural History of the Infinite. Birkhäuser. ISBN 978-3-7643-3325-6.
A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
• Mazur, Joseph (2005). Euclid in the Rainforest: Discovering Universal Truths in Logic and Math. Pearson: Pi Press. ISBN 978-0-13-147994-4.
• Munkres, James R. (2000) [1975]. Topology (2e ed.). Prentice-Hall. ISBN 978-0-13-181629-9.
Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres's treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
• Núñez, Rafael (2006). "Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics". 18 Unconventional Essays on the Nature of Mathematics. Springer. pp. 160–181. ISBN 978-0-387-25717-4. Archived from the original on 18 July 2011. Retrieved 4 July 2011.
• Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 978-0-387-94108-0.
• Peressini, Anthony; Peressini, Dominic (2007). "Philosophy of Mathematics and Mathematics Education". In van Kerkhove, Bart; van Bendegem, Jean Paul (eds.). Perspectives on Mathematical Practices. Logic, Epistemology, and the Unity of Science. Vol. 5. Springer. ISBN 978-1-4020-5033-6.
• Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.
• Pinto, Márcia; Tall, David (2001). PME25: Following students' development in a traditional university analysis course (PDF). pp. v4: 57–64. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.
• Protter, M. H.; Morrey, Charles B. Jr. (1991). A First Course in Real Analysis (2e ed.). Springer. ISBN 978-0-387-97437-8.
This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nondecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
• Pugh, Charles Chapman (2001). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.
While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
• Renteln, Paul; Dundes, Alan (January 2005). "Foolproof: A Sampling of Mathematical Folk Humor" (PDF). Notices of the AMS. 52 (1): 24–34. Archived from the original (PDF) on 25 February 2009. Retrieved 3 May 2009.
• Richman, Fred (December 1999). "Is 0.999... = 1?". Mathematics Magazine. 72 (5): 396–400. doi:10.2307/2690798. JSTOR 2690798. Free HTML preprint: Richman, Fred (June 1999). "Is 0.999... = 1?". Archived from the original on 2 September 2006. Retrieved 23 August 2006. Note: the journal article contains material and wording not found in the preprint.
• Robinson, Abraham (1996). Non-standard Analysis (Revised ed.). Princeton University Press. ISBN 978-0-691-04490-3. JSTOR j.ctt1cx3vb6.
• Rosenlicht, Maxwell (1985). Introduction to Analysis. Dover. ISBN 978-0-486-65038-8. This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
• Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.
A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
• Shrader-Frechette, Maurice (March 1978). "Complementary Rational Numbers". Mathematics Magazine. 51 (2): 90–98. doi:10.2307/2690144. JSTOR 2690144.
• Smith, Charles; Harrington, Charles (1895). Arithmetic for Schools. Macmillan. p. 115. ISBN 978-0-665-54808-6. Retrieved 4 July 2011.
• Sohrab, Houshang (2003). Basic Real Analysis. Birkhäuser. ISBN 978-0-8176-4211-2.
• Starbird, M.; Starbird, T. (March 1992). "Required Redundancy in the Representation of Reals". Proceedings of the American Mathematical Society. 114 (3): 769–774. doi:10.1090/S0002-9939-1992-1086343-5. JSTOR 2159403.
• Stewart, Ian (1977). The Foundations of Mathematics. Oxford UP. ISBN 978-0-19-853165-4.
• Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6.
• Stewart, James (1999). Calculus: Early transcendentals (4e ed.). Brooks/Cole. ISBN 978-0-534-36298-0.
This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
• Stillwell, John (1994), Elements of Algebra: Geometry, Numbers, Equations, Springer, ISBN 9783540942900
• Tall, David; Schwarzenberger, R. L. E. (1978). "Conflicts in the Learning of Real Numbers and Limits" (PDF). Mathematics Teaching. 82: 44–49. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.
• Tall, David (1977). "Conflicts and Catastrophes in the Learning of Mathematics" (PDF). Mathematical Education for Teaching. 2 (4): 2–18. Archived from the original (PDF) on 26 March 2009. Retrieved 3 May 2009.
• Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology" (PDF). Mathematics Education Research Journal. 12 (3): 210–230. Bibcode:2000MEdRJ..12..196T. doi:10.1007/BF03217085. S2CID 143438975. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.
• von Mangoldt, Hans (1911). "Reihenzahlen". Einführung in die höhere Mathematik (in German) (1st ed.). Leipzig: Verlag von S. Hirzel.
• Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 978-0-393-00338-3.
Further reading
• Burkov, S. E. (1987). "One-dimensional model of the quasicrystalline alloy". Journal of Statistical Physics. 47 (3/4): 409–438. Bibcode:1987JSP....47..409B. doi:10.1007/BF01007518. S2CID 120281766.
• Burn, Bob (March 1997). "81.15 A Case of Conflict". The Mathematical Gazette. 81 (490): 109–112. doi:10.2307/3618786. JSTOR 3618786. S2CID 187823601.
• Calvert, J. B.; Tuttle, E. R.; Martin, Michael S.; Warren, Peter (February 1981). "The Age of Newton: An Intensive Interdisciplinary Course". The History Teacher. 14 (2): 167–190. doi:10.2307/493261. JSTOR 493261.
• Choi, Younggi; Do, Jonghoon (November 2005). "Equality Involved in 0.999... and (-8)1/3". For the Learning of Mathematics. 25 (3): 13–15, 36. JSTOR 40248503.
• Choong, K. Y.; Daykin, D. E.; Rathbone, C. R. (April 1971). "Rational Approximations to π". Mathematics of Computation. 25 (114): 387–392. doi:10.2307/2004936. JSTOR 2004936.
• Edwards, B. (1997). "An undergraduate student's understanding and use of mathematical definitions in real analysis". In Dossey, J.; Swafford, J.O.; Parmentier, M.; Dossey, A.E. (eds.). Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Vol. 1. Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education. pp. 17–22.
• Eisenmann, Petr (2008). "Why is it not true that 0.999... < 1?" (PDF). The Teaching of Mathematics. 11 (1): 35–40. Retrieved 4 July 2011.
• Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals". Journal for Research in Mathematics Education. 41 (2): 117–146. doi:10.5951/jresematheduc.41.2.0117.
This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for 0.999... falling short of 1 by an infinitesimal 0.000...1.
• Ferrini-Mundy, J.; Graham, K. (1994). Kaput, J.; Dubinsky, E. (eds.). "Research in calculus learning: Understanding of limits, derivatives and integrals". MAA Notes: Research Issues in Undergraduate Mathematics Learning. 33: 31–45.
• Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". arXiv:math.NT/0605182.
• Gardiner, Tony (June 1985). "Infinite processes in elementary mathematics: How much should we tell the children?". The Mathematical Gazette. 69 (448): 77–87. doi:10.2307/3616921. JSTOR 3616921. S2CID 125222118.
• Monaghan, John (December 1988). "Real Mathematics: One Aspect of the Future of A-Level". The Mathematical Gazette. 72 (462): 276–281. doi:10.2307/3619940. JSTOR 3619940. S2CID 125825964.
• Navarro, Maria Angeles; Carreras, Pedro Pérez (2010). "A Socratic methodological proposal for the study of the equality 0.999...=1" (PDF). The Teaching of Mathematics. 13 (1): 17–34. Retrieved 4 July 2011.
• Przenioslo, Malgorzata (March 2004). "Images of the limit of function formed in the course of mathematical studies at the university". Educational Studies in Mathematics. 55 (1–3): 103–132. doi:10.1023/B:EDUC.0000017667.70982.05. S2CID 120453706.
• Sandefur, James T. (February 1996). "Using Self-Similarity to Find Length, Area, and Dimension". The American Mathematical Monthly. 103 (2): 107–120. doi:10.2307/2975103. JSTOR 2975103.
• Sierpińska, Anna (November 1987). "Humanities students and epistemological obstacles related to limits". Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986. JSTOR 3482354. S2CID 144880659.
• Szydlik, Jennifer Earles (May 2000). "Mathematical Beliefs and Conceptual Understanding of the Limit of a Function". Journal for Research in Mathematics Education. 31 (3): 258–276. doi:10.2307/749807. JSTOR 749807.
• Tall, David O. (2009). "Dynamic mathematics and the blending of knowledge structures in the calculus". ZDM Mathematics Education. 41 (4): 481–492. doi:10.1007/s11858-009-0192-6. S2CID 14289039.
• Tall, David O. (May 1981). "Intuitions of infinity". Mathematics in School. 10 (3): 30–33. JSTOR 30214290.
External links
Wikimedia Commons has media related to 0.999….
• .999999... = 1? from cut-the-knot
• Why does 0.9999... = 1 ?
• Proof of the equality based on arithmetic
• David Tall's research on mathematics cognition
• What is so wrong with thinking of real numbers as infinite decimals?
• Theorem 0.999... on Metamath
Real numbers
• 0.999...
• Absolute difference
• Cantor set
• Cantor–Dedekind axiom
• Completeness
• Construction
• Decidability of first-order theories
• Extended real number line
• Gregory number
• Irrational number
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| Wikipedia |
1 − 1 + 2 − 6 + 24 − 120 + ⋯
In mathematics,
$\sum _{k=0}^{\infty }(-1)^{k}k!$
For a related alternating partial sum of factorials, see Alternating factorial.
is a divergent series, first considered by Euler, that sums the factorials of the natural numbers with alternating signs. Despite being divergent, it can be assigned a value of approximately 0.596347 by Borel summation.
Euler and Borel summation
This series was first considered by Euler, who applied summability methods to assign a finite value to the series.[1] The series is a sum of factorials that are alternately added or subtracted. One way to assign a value to this divergent series is by using Borel summation, where one formally writes
$\sum _{k=0}^{\infty }(-1)^{k}k!=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{\infty }x^{k}e^{-x}\,dx.$
If summation and integration are interchanged (ignoring that neither side converges), one obtains:
$\sum _{k=0}^{\infty }(-1)^{k}k!=\int _{0}^{\infty }\left[\sum _{k=0}^{\infty }(-x)^{k}\right]e^{-x}\,dx.$
The summation in the square brackets converges when $|x|<1$, and for those values equals ${\tfrac {1}{1+x}}$. The analytic continuation of ${\tfrac {1}{1+x}}$ to all positive real $x$ leads to a convergent integral for the summation:
${\begin{aligned}\sum _{k=0}^{\infty }(-1)^{k}k!&=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\\[4pt]&=eE_{1}(1)\approx 0.596\,347\,362\,323\,194\,074\,341\,078\,499\,369\ldots \end{aligned}}$
where E1(z) is the exponential integral. This is by definition the Borel sum of the series.
Connection to differential equations
Consider the coupled system of differential equations
${\dot {x}}(t)=x(t)-y(t),\qquad {\dot {y}}(t)=-y(t)^{2}$
where dots denote derivatives with respect to t.
The solution with stable equilibrium at (x,y) = (0,0) as t → ∞ has y(t) = 1/t, and substituting it into the first equation gives a formal series solution
$x(t)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(n-1)!}{t^{n}}}$
Observe x(1) is precisely Euler's series.
On the other hand, the system of differential equations has a solution
$x(t)=e^{t}\int _{t}^{\infty }{\frac {e^{-u}}{u}}\,du.$
By successively integrating by parts, the formal power series is recovered as an asymptotic approximation to this expression for x(t). Euler argues (more or less) that since the formal series and the integral both describe the same solution to the differential equations, they should equal each other at $t=1$, giving
$\sum _{n=1}^{\infty }(-1)^{n+1}(n-1)!=e\int _{1}^{\infty }{\frac {e^{-u}}{u}}\,du.$
See also
• Alternating factorial
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's)
• 1 + 2 + 3 + 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 − 2 + 4 − 8 + ⋯
References
1. Euler, L. (1760). "De seriebus divergentibus" [On divergent series]. Novi Commentarii Academiae Scientiarum Petropolitanae (5): 205–237. arXiv:1202.1506. Bibcode:2012arXiv1202.1506E.
Further reading
• Kline, Morris (November 1983), "Euler and Infinite Series", Mathematics Magazine, 56 (5): 307–313, CiteSeerX 10.1.1.639.6923, doi:10.2307/2690371, JSTOR 2690371
• Kozlov, V. V. (2007), "Euler and mathematical methods in mechanics" (PDF), Russian Mathematical Surveys, 62 (4): 639–661, Bibcode:2007RuMaS..62..639K, doi:10.1070/rm2007v062n04abeh004427, S2CID 250892576
• Leah, P. J.; Barbeau, E. J. (May 1976), "Euler's 1760 paper on divergent series", Historia Mathematica, 3 (2): 141–160, doi:10.1016/0315-0860(76)90030-6
Sequences and series
Integer sequences
Basic
• Arithmetic progression
• Geometric progression
• Harmonic progression
• Square number
• Cubic number
• Factorial
• Powers of two
• Powers of three
• Powers of 10
Advanced (list)
• Complete sequence
• Fibonacci sequence
• Figurate number
• Heptagonal number
• Hexagonal number
• Lucas number
• Pell number
• Pentagonal number
• Polygonal number
• Triangular number
Properties of sequences
• Cauchy sequence
• Monotonic function
• Periodic sequence
Properties of series
Series
• Alternating
• Convergent
• Divergent
• Telescoping
Convergence
• Absolute
• Conditional
• Uniform
Explicit series
Convergent
• 1/2 − 1/4 + 1/8 − 1/16 + ⋯
• 1/2 + 1/4 + 1/8 + 1/16 + ⋯
• 1/4 + 1/16 + 1/64 + 1/256 + ⋯
• 1 + 1/2s + 1/3s + ... (Riemann zeta function)
Divergent
• 1 + 1 + 1 + 1 + ⋯
• 1 − 1 + 1 − 1 + ⋯ (Grandi's series)
• 1 + 2 + 3 + 4 + ⋯
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 4 + 8 + ⋯
• 1 − 2 + 4 − 8 + ⋯
• Infinite arithmetic series
• 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
• 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
• 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
• Taylor series
• Power series
• Formal power series
• Laurent series
• Puiseux series
• Dirichlet series
• Trigonometric series
• Fourier series
• Generating series
Hypergeometric series
• Generalized hypergeometric series
• Hypergeometric function of a matrix argument
• Lauricella hypergeometric series
• Modular hypergeometric series
• Riemann's differential equation
• Theta hypergeometric series
• Category
| Wikipedia |
Hundredth
In arithmetic, a hundredth is a single part of something that has been divided equally into a hundred parts. For example, a hundredth of 675 is 6.75. In this manner it is used with the prefix "centi" such as in centimeter.
A hundredth is the reciprocal of 100.
A hundredth is written as a decimal fraction as 0.01, and as a vulgar fraction as 1/100.
“Hundredth” is also the ordinal number that follows “ninety-ninth” and precedes “hundred and first.” It is written as 100th.
See also
• Basis point
• Cent (currency)
• Cent (music)
• Hundredth is an American rock band from Myrtle Beach, South Carolina, that formed in 2008.
• Order of magnitude (numbers)
• Orders of magnitude
• Percentage
• Point (gemstone)
Preceded by
Tenth
Decimal
orders of magnitude
Succeeded by
Thousandth
Orders of magnitude
Quantity
• Acceleration
• Angular momentum
• Area
• Bit rate
• Charge
• Computing
• Current
• Data
• Density
• Energy / Energy density
• Entropy
• Force
• Frequency
• Illuminance
• Length
• Luminance
• Magnetic field
• Mass
• Molarity
• Numbers
• Power
• Pressure
• Probability
• Radiation
• Sound pressure
• Specific heat capacity
• Speed
• Temperature
• Time
• Voltage
• Volume
See also
• Back-of-the-envelope calculation
• Best-selling electronic devices
• Fermi problem
• Powers of 10 and decades
• 10th
• 100th
• 1000000th
• Metric (SI) prefix
• Macroscopic scale
• Microscopic scale
Related
• Astronomical system of units
• Earth's location in the Universe
• Cosmic View (1957 book)
• To the Moon and Beyond (1964 film)
• Cosmic Zoom (1968 film)
• Powers of Ten (1968 and 1977 films)
• Cosmic Voyage (1996 documentary)
• Cosmic Eye (2012)
• Category
| Wikipedia |
Millionth
One millionth is equal to 0.000 001, or 1 x 10−6 in scientific notation. It is the reciprocal of a million, and can be also written as 1⁄1,000,000.[1] Units using this fraction can be indicated using the prefix "micro-" from Greek, meaning "small".[2] Numbers of this quantity are expressed in terms of μ (the Greek letter mu).[3]
"Millionth" can also mean the ordinal number that comes after the nine hundred, ninety-nine thousand, nine hundred, ninety-ninth and before the million and first.[4]
See also
• International System of Units
• Micro-
• International Map of the World
• Order of magnitude (numbers)
• Order of magnitude
• Parts-per notation
• Per mille
References
1. Laidler, Keith J. (30 December 2011), Science and Sensibility: The Elegant Logic of the Universe, Prometheus Books, p. 15, ISBN 9781615927036.
2. The American Heritage Guide to Contemporary Usage and Style, Houghton Mifflin Company, 2005, p. 300, ISBN 9780618604999.
3. Handley, Brett; Coon, Craig; Marshall, David (2011), Principles of Engineering, Cengage Learning, p. 212, ISBN 9781435428362.
4. Hurford, James R. (1994), Grammar: A Student's Guide, Cambridge University Press, p. 146, ISBN 9780521456272.
Preceded by
Thousandth
Decimal
orders of magnitude
Succeeded by
Billionth
Orders of magnitude
Quantity
• Acceleration
• Angular momentum
• Area
• Bit rate
• Charge
• Computing
• Current
• Data
• Density
• Energy / Energy density
• Entropy
• Force
• Frequency
• Illuminance
• Length
• Luminance
• Magnetic field
• Mass
• Molarity
• Numbers
• Power
• Pressure
• Probability
• Radiation
• Sound pressure
• Specific heat capacity
• Speed
• Temperature
• Time
• Voltage
• Volume
See also
• Back-of-the-envelope calculation
• Best-selling electronic devices
• Fermi problem
• Powers of 10 and decades
• 10th
• 100th
• 1000000th
• Metric (SI) prefix
• Macroscopic scale
• Microscopic scale
Related
• Astronomical system of units
• Earth's location in the Universe
• Cosmic View (1957 book)
• To the Moon and Beyond (1964 film)
• Cosmic Zoom (1968 film)
• Powers of Ten (1968 and 1977 films)
• Cosmic Voyage (1996 documentary)
• Cosmic Eye (2012)
• Category
| Wikipedia |
Subsets and Splits