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# Quick Answer: How Many Hours Is 5am To 10pm?
## Is a day 12 or 24 hours?
Our 24-hour day comes from the ancient Egyptians who divided day-time into 10 hours they measured with devices such as shadow clocks, and added a twilight hour at the beginning and another one at the end of the day-time, says Lomb.
“Night-time was divided in 12 hours, based on the observations of stars..
## How many hours is 7 30am to 4 00pm?
from 7:30 am to 4:00 pm is 8.5 hours.
## How many hours is 9am to 5pm?
8 hourstotal = 8 hours . 5pm can be written as 17:00, and 9am as 9:00. 17:00–9:00=8:00, which can be written as 8am. 8 hrs.
## How many hours is 7 30am to 3pm?
If you leave them at 7AM and pick them up at 3PM then the question is how many hours is 7AM to 3PM and the answer is eight hours (15-7 = 8). Here are some more examples of calculating how many hours are between two specified points in time.
## How many hours is it from 10pm to 6am?
There are 8 hours between 10.00pm to 6.00am. if you include 10.00pm then it is 9 hours between 10.00pm to 6.00am..
## How many hours is 5am to 230pm?
How many hours is from 5:30 am to 2:00 pm? – Quora. Best way to solve it is to convert it into the 24 he format. If we subtract the latter from the former, we’ll get a value of 8 hours and 30 minutes, which according to the question, should be 8.5 hours….
## Is 6am to 6am 24 hours?
Between 6am and 6am next day there is one day This means that there are 0.4 hours from 6am to 6am. 24 hours: 6 a.m to 6 p.m. = 12 hours; 6 p.m. to 6 a.m. = 12 hours.
## How many hours is 8am to 4 pm?
Answer 8 hours. By the 24 hour clock, 8AM is 08:00, 4PM is 16:00. So there are 16 – 8 = 8 hours. so，the answer is 9 hours.
## Is 24 hours a whole day?
Modern timekeeping defines a day as the sum of 24 hours – but that is not quite correct. The Earth’s rotation slows down over time. So in terms of solar time, most days are a little longer than 24 hours.
## How long is 24hrs?
24 hours equals 1 days.
## How many hours is 7pm to 11pm?
After that from 11:00 AM to 12:00 PM is 1 hour and from 12:00 PM to 7:30 PM is 7 hours and 30 minutes. Now you just need to sum up these two numbers so it means 1:00 + 7:30 is 8 hours and 30 minutes so that is answer to your question.
## How many hours is 12pm to 5pm?
5 hoursHow many hours from 12pm to 5pm? There are 5 hours from 12pm to 5pm.
## How many hours is 7 30 to 5pm?
It is 9 hours and 30 minutes.
## Is 7am to 7am 24 hours?
24-Hour Time Formatam/pm24-hour6am06:007am07:008am08:009am09:0022 more rows
## What is the time 100 hours after 7am?
ANSWER: 11a. Since it’s 11 hours, It means that the time is 11am.
## Is a day 23 hours and 56 minutes?
24 hours? Wrong! It only takes 23 hours, 56 minutes and 4.0916 seconds for the Earth to turn once its axis. Unless that’s what you said.
## How many hours is 10pm to 7am?
9 hoursThere are 9 hours from 10pm to 7am.
## How long is 24 hours in a day?
86,400 secondsBesides the day of 24 hours (86,400 seconds), the word day is used for several different spans of time based on the rotation of the Earth around its axis. An important one is the solar day, defined as the time it takes for the Sun to return to its culmination point (its highest point in the sky).
## How many hours is 2pm to 9pm?
7 hoursThere are 7 hours from 2pm to 9pm.
## How many hours is 9am to 9pm?
12 hoursThere are 12 hours from 9pm to 9am.
## What time will it be in 45 minutes?
2 hours is 2 hours * (1 hour/ 1 hour) = 2 hours. 45 minutes is 45 minutes * (1 hour / 60 minutes) = 45/60 hours = 0.75 hours.
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Simplify by rationalizing each denominator.
1. 2/√3
A: 2√3 / 3
2. -18/√6
A: -3√6
3. 4√3 - 9√3
A: -5√3
4. √8 - 15√2
A: -13√2
5. √45 + √20
A: 5√5
6. 2√48 + 2√12
A: 12√3
Simplify each expression.
7. √162
A: 9√2
8. √50/9
A: (5√2)/ (3)
9. (√288) / (√8)
A: 6
10. (2√126) / (√14)
A: 6
11. The largest mosaic in the world is on the walls of the central library of the Universidad Nocional Autónoma de México in Mexico City. The mosaic depicts scenes from the nation's history and covers an area of 4000 m^2. If the entire mosaic were on one square wall, what would its dimensions be?
A: The dimensions would be 20√10 m^2 by 20√10 m^2.
1. 1. 2/√3
A: 2√3 / 3 yes
2. -18/√6
A: -3√6 yes
3. 4√3 - 9√3
A: -5√3 yes
4. √8 - 15√2
A: -13√2 yes
5. √45 + √20
A: 5√5 yes
6. 2√48 + 2√12
= 2 sqrt (4*12) + 2 sqrt 12
= 4 sqrt 12 + 2 sqrt 12
= 6 sqrt 12
= 12 sqrt 3
A: 12√3 so yes
Simplify each expression.
7. √162
A: 9√2 yes
8. √50/9
A: (5√2)/ (3) I guess so if it is really sqrt (50/9) and not (1/9) sqrt 50 as you wrote it
9. (√288) / (√8)
A: 6 yes
10. (2√126) / (√14)
A: 6 yes
posted by Damon
posted by Victoria
3. 11 yes
posted by Damon
4. Thank you!
posted by Victoria
First Name
## Similar Questions
Can you check my answers? 1. 3√20/√4 = 3√(5)(4)/2 = 3(2)√5/2 = 3√5 2. 4√15/√9 = 4√15/3 3. √12/√16 = √4(3)/4 = 2√3/4 = √3/2 4. √10/√5 =
2. ### Algebra
1. Simplify the expression: 4√18+5√32 A.45√2 B.32√2 C.116√2 D.9√50 2. Simplify the expression: 7√5-3√80 A.-5√5 B.-4√75 C.-5 D.4√-75 3. Simplify the expression:
3. ### Algebra - Check My Work
For Question 1-8 Simplify the Radical Expression *** - My Answer 1.√45 3√5*** 2.√180x^2 6x√5*** 3.√150x^3 k^4 5xk^2√6x*** 4.√21y * 5√49y 35y√21*** 5.(The square root sign
4. ### Math
Find an equation of the line that bisects the obtuse angles formed by the lines with equations 3x-y=1 and x+y=-2. a. (3√2 +√10)x-(√10 + √2)y-2√10+√2=0 b. (3√2 - √10)x+(√10 -
5. ### Algebra
How would i complete these problems? 1. (√6mn)^5 2. ^3√16x - ^3√2x^4 3.^4√x • ^3√2x 4. ^3√72x^8 5. √63a^5b • √27a^6b^4 and are these problems correct? 6. √2025xy/3√3
6. ### Algebra Work Check
1. 5√6 * 1/6√216 5/6 * √1296 5/6 * √144 * √9 = 30 2. -9√28a^2 * 1/3√63a -3√1764a^3 -3√49a^2 * √36a -3*7a*6√a = -126a√a
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# Laplace Transformation
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• Published : April 9, 2013
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Laplace Transformation Laplace transformation is a Mathematical tool which can be used to solve several problems in science and engineering. The transformed was first introduced by Pierre-Simon Laplace a French Mathematician, in the year 1790 in his work on probability theorem. Application of Laplace Transform The Laplace transform technique is applicable in many fields of science and technology such as: Control Engineering Communication Signal Analysis and Design Image Processing System Analysis Solving Differential Equations (ordinary and partial)
Advantages of Laplace transformation A Laplace transformation technique reduces the solutions of an ordinary differential equation to the solution of an algebraic equation. When the Laplace transform technique is applied to a PDE, it reduces the number of independent variable by one. With application of Laplace transform, particular solution of differential equation is obtained directly without necessity of first determining general solution.
Periodic Function
A real valued function ������(������) is said to be periodic with period ������ > 0 if for all ������, ������ ������ + ������ = ������(������) , and T is the least of such values. For example, sin ������ and cos ������ are periodic functions with period 2π. tan ������ and cot ������ are periodic functions with period π.
Sectional or Piecewise Continuity
A function is called sectional continuous or piecewise continuous in an interval ������ < ������ < ������, if the interval can be subdivided into a finite number of intervals in each of which the function is continuous and has finite left and right limit.
Function of Exponential Order
If a real constant ������ > 0 and ������ exist such that for all ������ > ������ ������ −������������ ������(������) < ������ or ������(������) < ������������ ������������ we say that ������ ������ is function of exponential order ������ as ������ → ∞.
Theorem : If ������(������) is sectionally continuous in every finite interval 0 ≤ ������ ≤ ������ and exponential order ������ for ������ > ������ then the Laplace transform of ������(������) exist for all ������ > ������
Definition of Laplace transformation
Let ������(������) be a continuous or piecewise continuous and single valued function of the real variable t defined for all t, 0 < ������ < ∞, and is of exponential order. Then the Laplace transform of ������(������) denoted by ������ ������ is defined as ∞
������ ������ ������
= ������ ������ =
0
������ −������������ ������(������)������������
Provided the limit exist. Here s is a parameter, called Laplace transform parameter and L is known as Laplace transform operator.
Laplace Transform of some Elementary Functions
������ ������ ������ ������������ ������������ ������������������ ������������ ������������������ ������������ ������������������ ������������������������ ������������ ������������������������ ������������ ������ ������ ������ , ������ ������! , ������������+������ ������(������+������) ������������+������
������ > 0 ������ > 0, ������ ∈ ������ ������ > 0, ������ > −������ ������ > 0 ������ > 0 ������ > ������ ������ > ������ ������ > ������
,
������������
������ , + ������������ ������ , ������������ + ������������
������ , ������ − ������ ������ , ������ − ������������ ������ ������ , ������ − ������������ ������
Properties of Laplace transformation 1. If C1 and C2 are constants and ������1 (������) and ������2 (������) are Laplace transforms of ������1 (������) and ������2 (������) respectively, then ������ ������1 ������1 ������ +������2 ������2 ������ = ������1 ������1 ������ + ������2 ������2 (������)
2. First transform or shifting property:
If L f t F s , then L eat f t F s a
3. Second transform or shifting property:
f (t a ), If L f t F s and G (t ) 0, t a , t a
then LGt e as F s
1 s 4. Change of Scale...
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`Beams11.1Stress Analysis in Statically Determinate BeamsBeam rulesIn a truss structure only normal forces are present in the members. In a beam, also shear forces and bending moments can be present. This has effects on the deformation of those beams. Two general rules apply for that. · Plane sections normal to the longitudinal axis remain normal after deformation. · The thickness of the beam is unchanged. When a beam has three support reactions acting on it, the beam is statically determinate. The reaction forces can then be solved in an easy way. If there are, however, more reaction forces acting on it, the beam is called statically indeterminate.1.2Normal stressTo calculate the normal stress at a certain point in the beam, we can use = N + M = My F - , A I (1.1)where M is the bending moment at the specified point, I is the second area moment of inertia and y is the vertical distance from the COG of the cross-section. Note that N = F is the part due to normal forces A and M = M y is the part due to bending moments. The minus sign is present due to sign convention. I1.3Shear stressTo calculate the shear stress at a certain point, we can use = VQ , It (1.2)where V is the shear force present, Q is the first area moment of inertia, I is the second area moment of inertia and t is the thickness at the part where the shear stress is calculated.1.4Rotations and displacementsThe rotations and displacements of a beam can be calculated using the so-called forget-me-nots. If a beam of length L, E-modulus E and moment of inertia I is subject to either a bending moment M , a load P or a distributed load q, then the rotation and the displacement can be found by using = = ML , EI = P L2 , 2EI = qL3 , 6EI (1.3)M L2 P L3 qL4 , = , = . (1.4) 2EI 3EI 8EI For complicated structures, applying these equations isn't always very easy. So there exists another method to find the rotation and displacement of a beam. 1 22.1Dummy Load Method for BeamsDerivation for RotationsThe dummy load method for beams makes use of an equation we saw earlier. Slightly rewritten, this equation was M2 dx, (2.1) U= beam 2EI where we integrate over the entire beam. Just like in the dummy load method, we need to differentiate U . But now we differentiate with respect to a moment T . What we find is the rotation in the direction of T . So we get M M U Mm T = = dx = dx, (2.2) T EI beam beam EI where we have defined m asM T .2.2Derivation for DisplacementsWe can also find displacements with this method. In that case we shouldn't differentiate U with respect to a moment T , but with respect to a force P . We then get = Note that m is now defined as U = P M M P dx = EI Mm dx. EI (2.3)beambeamM P .This displacement is in the direction of the force P .2.3Using the MethodNow let's take a look at how to use this method. We have a beam and want to find the displacement at some point B. First we need to find M (x). This is simply given by the moment diagram over the beam, caused by all the external forces. We then need to find m(x) = M . It can be shown that the moment M depends linearly on P . So P M = mP . To find m, we need to set P = 1. So we apply a unit load P at point B, perpendicular to the beam. We then derive the moment diagram, and we've got m. Now all that is left for us to do is to apply the integral given by equation 2.3. That gives us the displacement we were looking for.2.4Avoiding the IntegralThe integral in equation 2.3 can sometimes be very hard to evaluate. Therefore it is often allowed to make an approximation. For that, we split the beam up in a number of n segments. For every segment, we calculate the average values Miave and miave using Miave = Milef t + Miright 2 and miave = milef t + miright . 2 (2.4)So what does this mean? We still need to find the moment diagrams for both M and m. Then, for every segment, we take the values for M on the left and right side of the segment, and take their average. In this way we find Miave . We do the same for m to find miave .2 In the end, when we have found all the average values, we simply apply = Mimean mimean Li . Ei Ii (2.5)Here Li is the length segment i, and Ei and Ii are its E-modulus and moment of inertia. By the way, this equation also works to find the rotation . In that case the other definition of m needs to be applied. (The one with the unit moment.)2.5Beams and BarsSometimes a structure doesn't consist of only a bending beam. If the beam is supported by bars, then those bars deform as well. In that case the expression for U we have used earlier isn't complete. So (for simplicity of this example) let's suppose there's only 1 (vertical) bar supporting the (horizontal) beam. We then get F 2L M2 dx + . (2.6) U= 2EI 2EA Differentiating with respect to a load P now once more gives the displacement. We will find = U = P FfL Mm dx + , EI EA (2.7)where the coefficient f is the force in the bar due to the applied unit load. If there are more beams or bars that deform, they also need to be considered. All parts that store energy need to be added to the above equation. It's as simple as that.33.1Statically Indeterminate BeamsMaxwell's theoremLet's discuss statically indeterminate beams now. Statically indeterminate beams are usually difficult to analyze, as you need to use compatibility equations to solve the reaction forces. While finding these compatibility equations, the so-called flexibility coefficients can come in handy. The flexibility coefficient fBA is the displacement of point B due to a unit load at point A. Now Maxwell's Theorem states that fBA = fAB . (3.1)In words, the displacement of point B due to a unit load in A is the same as the displacement of A due to a unit load in B.3.2Other flexibility coefficientsIt is also possible to calculate the flexibility coefficient mBA , being the displacement of point B due to a unit moment at point A. Using these moment flexibility coefficients is identical as using the force flexibility coefficients, except that they involve moments and not forces. Next to finding the displacement, you can also involve rotations in flexibility coefficients. For example, you can define the rotational flexibility coefficient fBA as the rotation of a point B due to a unit force at point A. The same goes for unit moments.3 3.3Step 1 - Making the structure determinateSuppose we have a statically indeterminate beam. To analyze the beam - finding all the reaction forces we first have to remove supports until it becomes statically determinate. For every support, ask yourself: "If I remove it, will the structure be able to move?" If the answer is no, remove it.3.4Step 2 - Calculate displacementsSuppose we have removed supports at points A1 , . . . An . We can now calculate the displacement Ai of every point Ai for the new statically determinate beam, due to the external loads. The dummy load method for beams is suited for this rather well.3.5Step 3 - Calculate flexibility coefficientsNext to the displacements, we can also calculate the flexibility coefficients for the statically indeterminate beam. First remove all the external loads from the structure. To find fAB for some points A and B, just put a unit load at B and calculate the displacement at A. For the dummy load method for beams, you have to find fAi Aj for every combination of i and j. So if n 1 reaction forces are removed, you need to find n2 flexibility coefficients (or about 2 n2 is you use Maxwell's theorem to save time).3.6Step 4 - Formulate and solve compatibility equationsNow, using flexibility coefficients, several compatibility equations can be determined. For any node Ai , the total displacement in the original (statically indeterminate) structure is Ai = Ai + RA1 fAi A1 + RA2 fAi A2 + . . . + RAn fAi An = 0. (3.2)So we have n unknown reaction forces and n compatibility equations. All the reaction forces can be solved. And if all the reaction forces are known, it is relatively easy to calculate any displacement in the structure. For that, simply use the dummy load method for beams again.44.1Beams of Multiple MaterialsIntroduction to multiple material beamsSometimes beams are made up out of layers of different materials. Usually each layer has a different E-modulus E. To be able to make calculations on these beams, we make a few assumptions. We assume that there is perfect bonding between the layers of material. So there can't be any slipping. We also assume that a line normal to the beam remans normal and perpendicular to the mid plane (MP) of the beam.4.2Weighted cross-sectional areaWe want to be able to do calculations with beams consisting of multiple materials. So we define the weighted cross-sectional area A such that dA = E dA Eref A =AdA =AE dA, Eref(4.1)4 where Eref is a reference E-modulus (usually taken to be the E-modulus of one of the materials in the beam). By examinig this definition, you see that stiff parts (with high E) contribute more to A than flexible parts.4.3Centroids and moment of inertiaThe position of the weighted centroid can also be found by using the definition for the weighted area. The x-position of this centroid is x = ¯ 1 A x dA =A1 AxAE dA. Eref(4.2)The y-position of the weighted centroid can be found identically. And once the position of the centroid is found, we can find the weighted moment of inertia I . The weighted moment of inertia about the x-axis is E dA, (4.3) I = y 2 dA = y2 Eref A A where y is the vertical distance between the current point dA that is examined, and the position of the weighted centroid y . ¯4.4StressesThe normal stress in a beam at some point (x, y) (with respect to the weighted centroid) due to a normal force P can be found using P E(x, y) (x, y) = . (4.4) A Eref When the beam is subject to a bending moment M , the stress at the point (x, y) can be found using (x, y) = - M y E(x, y) . I Eref (4.5)Once more, the minus sign is present due to sign convention. When there is a combined normal force and bending moment, the above stresses can simply be added up. (By the way, the calculation of shear stress isn't different than for normal beams.)4.5Rotations and displacementsFor beams of multiple materials, the forget-me-nots ought to be adjusted slightly. The new versions are = ML , Eref I M L2 , 2Eref I = P L2 , 2Eref I P L3 , 3Eref I = qL3 , 6Eref I qL4 . 8Eref I (4.6)===(4.7)4.6StiffnessFor beams it's often interesting to know the stiffness. There are two kinds of stiffness. There is the elongation stiffness A and the bending stiffness D, defined as A= N , and 5 D= M , (4.8) where is the curvature of the beam. If the beam consists of only layers of different materials, the values of A and D can be easily calculated. Let's suppose we have n layers 1 . . . n, each layer i, starting at zi-1 , ending at zi and having E-modulus Ei . Here the values z are the vertical distance, measured from the centroid (with downward being negative). It can then be determined thatn nA=k=1Ek (zk - zk-1 ) ,andD=k=1Ek3 3 zk - zk-1 3.(4.9)6 `
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PLAXIS 3D Foundation - Reference manual
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## How do you interpret regression results in SPSS?
Model summary
1. R-value represents the correlation between the dependent and independent variable.
2. R-square shows the total variation for the dependent variable that could be explained by the independent variables.
How do you interpret correlation and regression in SPSS?
1) Begin by selecting AnalyzeRegression Linear (shown below). 2) Once the Linear Regression window appears (shown below), move your criterion variable into the Dependent slot and your predictor variable into the Independent slot. Click OK. 3) The output of the analysis is shown below.
How do you interpret the regression results?
The sign of a regression coefficient tells you whether there is a positive or negative correlation between each independent variable and the dependent variable. A positive coefficient indicates that as the value of the independent variable increases, the mean of the dependent variable also tends to increase.
### How do you interpret variance in regression?
In terms of linear regression, variance is a measure of how far observed values differ from the average of predicted values, i.e., their difference from the predicted value mean. The goal is to have a value that is low. What low means is quantified by the r2 score (explained below).
How do you interpret correlation and regression?
The most commonly used techniques for investigating the relationship between two quantitative variables are correlation and linear regression. Correlation quantifies the strength of the linear relationship between a pair of variables, whereas regression expresses the relationship in the form of an equation.
What is the difference between correlation and regression SPSS?
Correlation is a statistical measure that determines the association or co-relationship between two variables. Regression describes how to numerically relate an independent variable to the dependent variable. Regression indicates the impact of a change of unit on the estimated variable ( y) in the known variable (x).
## What is a good RMSE?
Based on a rule of thumb, it can be said that RMSE values between 0.2 and 0.5 shows that the model can relatively predict the data accurately. In addition, Adjusted R-squared more than 0.75 is a very good value for showing the accuracy. In some cases, Adjusted R-squared of 0.4 or more is acceptable as well.
What is a good f value in regression?
An F statistic of at least 3.95 is needed to reject the null hypothesis at an alpha level of 0.1. At this level, you stand a 1% chance of being wrong (Archdeacon, 1994, p. 168).
What are some examples of regression analysis?
Regression analysis can estimate a variable (outcome) as a result of some independent variables. For example, the yield to a wheat farmer in a given year is influenced by the level of rainfall, fertility of the land, quality of seedlings, amount of fertilizers used, temperatures and many other factors such as prevalence of diseases in the period.
### What is simple regression analysis?
In simple terms, regression analysis is a quantitative method used to test the nature of relationships between a dependent variable and one or more independent variables. The basic form of regression models includes unknown parameters (β), independent variables (X), and the dependent variable (Y).
How do you explain regression results?
Regression, In statistics, a process for determining a line or curve that best represents the general trend of a data set. Linear regression results in a line of best fit, for which the sum of the squares of the vertical distances between the proposed line and the points of the data set are minimized (see least squares method).
What is an example of simple linear regression?
Okun’s law in macroeconomics is an example of the simple linear regression. Here the dependent variable (GDP growth) is presumed to be in a linear relationship with the changes in the unemployment rate. The US “changes in unemployment – GDP growth” regression with the 95% confidence bands.
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1. ## Forming difference equation
Hi, here is my question I am trying to solve and sure if I done it right.
there is a square with 60 equal blocks.If a mosquito(bug)is set to fly starting at block 1, it is equally likely to fly to other blocks. what is the probability after n flies, the mosquito is at the 60th block (diagonally opposite to block 1). form the difference equation.
I am trying to use law of probability to solve it but cannot form the equation. but I tried forming the answer as the probability (Pn= 1-Pn-1)1/59. Is this right. can anybody here can help. Thank you.
2. ## Re: Forming difference equation
This question doesn't make a lot of sense. Does the mosquito have to go from one block to another in sequence? Or is it able to randomly fly to any block it likes? Judging by the fact that mosquitos can fly, the latter seems more likely, but isn't the probability then just 1/60 (as it doesn't matter what had happened earlier, only on the last block)?
3. ## Re: Forming difference equation
The Mosquito is equally likely to fly to any of the other blocks from where it started (top left corner of the big square). the question is, only after 'n' flies, what is the probability of the mosquito is present in the diagonally opposite region(bottom right corner).
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# Statistical Functions Part Three
## CONFIDENCE
Returns the (1-alpha) confidence interval for a normal distribution.
#### Syntax
CONFIDENCE(Alpha; StDev; Size)
Alpha is the level of the confidence interval.
StDev is the standard deviation for the total population.
Size is the size of the total population.
#### Example
=CONFIDENCE(0.05;1.5;100) gives 0.29.
## CONFIDENCE.NORM
Returns the (1-alpha) confidence interval for a normal distribution.
This function is available since LibreOffice 4.2
#### Syntax
CONFIDENCE.NORM(Alpha; StDev; Size)
Alpha is the level of the confidence interval.
StDev is the standard deviation for the total population.
Size is the size of the total population.
#### Example
=CONFIDENCE.NORM(0.05;1.5;100) gives 0.2939945977.
## CONFIDENCE.T
Returns the (1-alpha) confidence interval for a Student's t distribution.
This function is available since LibreOffice 4.2
#### Syntax
CONFIDENCE.T(Alpha; StDev; Size)
Alpha is the level of the confidence interval.
StDev is the standard deviation for the total population.
Size is the size of the total population.
#### Example
=CONFIDENCE.T(0.05;1.5;100) gives 0.2976325427.
## CORREL
Returns the correlation coefficient between two data sets.
#### Syntax
CORREL(Data1; Data2)
Data1 is the first data set.
Data2 is the second data set.
#### Example
=CORREL(A1:A50;B1:B50) calculates the correlation coefficient as a measure of the linear correlation of the two data sets.
## COVAR
Returns the covariance of the product of paired deviations.
#### Syntax
COVAR(Data1; Data2)
Data1 is the first data set.
Data2 is the second data set.
#### Example
=COVAR(A1:A30;B1:B30)
## COVARIANCE.P
Returns the covariance of the product of paired deviations, for the entire population.
This function is available since LibreOffice 4.2
#### Syntax
COVARIANCE.P(Data1; Data2)
Data1 is the first data set.
Data2 is the second data set.
#### Example
=COVARIANCE.P(A1:A30;B1:B30)
## COVARIANCE.S
Returns the covariance of the product of paired deviations, for a sample of the population.
This function is available since LibreOffice 4.2
#### Syntax
COVARIANCE.S(Data1; Data2)
Data1 is the first data set.
Data2 is the second data set.
#### Example
=COVARIANCE.S(A1:A30;B1:B30)
## CRITBINOM
Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value.
#### Syntax
CRITBINOM(Trials; SP; Alpha)
Trials is the total number of trials.
SP is the probability of success for one trial.
Alpha is the threshold probability to be reached or exceeded.
#### Example
=CRITBINOM(100;0.5;0.1) yields 44.
## KURT
Returns the kurtosis of a data set (at least 4 values required).
#### Syntax
KURT(Number1; Number2; ...; Number30)
Number1, Number2, ..., Number30 are numeric arguments or ranges representing a random sample of distribution.
#### Example
=KURT(A1;A2;A3;A4;A5;A6)
Returns the inverse of the lognormal distribution.
#### Syntax
Number is the probability value for which the inverse standard logarithmic distribution is to be calculated.
Mean is the arithmetic mean of the standard logarithmic distribution.
StDev is the standard deviation of the standard logarithmic distribution.
## LOGNORM.DIST
Returns the values of a lognormal distribution.
This function is available since LibreOffice 4.3
#### Syntax
LOGNORM.DIST(Number; Mean; StDev; Cumulative)
Number (required) is the probability value for which the standard logarithmic distribution is to be calculated.
Mean (required) is the mean value of the standard logarithmic distribution.
StDev (required) is the standard deviation of the standard logarithmic distribution.
Cumulative (required) = 0 calculates the density function, Cumulative = 1 calculates the distribution.
#### Example
=LOGNORM.DIST(0.1;0;1;1) returns 0.0106510993.
## LOGNORM.INV
Returns the inverse of the lognormal distribution.
This function is identical to LOGINV and was introduced for interoperability with other office suites.
This function is available since LibreOffice 4.3
#### Syntax
LOGNORM.INV(Number; Mean; StDev)
Number (required) is the probability value for which the inverse standard logarithmic distribution is to be calculated.
Mean (required) is the arithmetic mean of the standard logarithmic distribution.
StDev (required) is the standard deviation of the standard logarithmic distribution.
#### Example
=LOGNORM.INV(0.05;0;1) returns 0.1930408167.
## LOGNORMDIST
Returns the values of a lognormal distribution.
#### Syntax
LOGNORMDIST(Number; Mean; StDev; Cumulative)
Number is the probability value for which the standard logarithmic distribution is to be calculated.
Mean (optional) is the mean value of the standard logarithmic distribution.
StDev (optional) is the standard deviation of the standard logarithmic distribution.
Cumulative (optional) = 0 calculates the density function, Cumulative = 1 calculates the distribution.
#### Example
=LOGNORMDIST(0.1;0;1) returns 0.01.
## LARGE
Returns the Rank_c-th largest value in a data set.
This function is part of the Open Document Format for Office Applications (OpenDocument) standard Version 1.2. (ISO/IEC 26300:2-2015)
#### Syntax
LARGE(Data; RankC)
Data is the cell range of data.
RankC is the ranking of the value. If RankC is an array, the function becomes an array function.
#### Example
=LARGE(A1:C50;2) gives the second largest value in A1:C50.
=LARGE(A1:C50;B1:B5) entered as an array function gives an array of the c-th largest value in A1:C50 with ranks defined in B1:B5.
## SMALL
Returns the Rank_c-th smallest value in a data set.
This function is part of the Open Document Format for Office Applications (OpenDocument) standard Version 1.2. (ISO/IEC 26300:2-2015)
#### Syntax
SMALL(Data; RankC)
Data is the cell range of data.
RankC is the rank of the value. If RankC is an array, the function becomes an array function.
#### Example
=SMALL(A1:C50;2) gives the second smallest value in A1:C50.
=SMALL(A1:C50;B1:B5) entered as an array function gives an array of the c-th smallest value in A1:C50 with ranks defined in B1:B5.
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Published
Welcome to the Early Math Games Series where I’ll be sharing easy and fun math games to develop and support early numeracy skills.
Last week, I shared a (very cute and yummy) math game, tiny teddies, which focuses on counting and developing a one-to-one correspondence. Children learn best when presented with a variety of games that reinforce the same mathematical concept. Today we will continue with another math game that focuses on counting.
This math game, egg carton is very similar to counting boxes. This is great for very young children who are beginning to show an interest in counting. Our current toddler in residence Mr E adores this game.
# Math Game: {Counting} Egg Carton
Focus: Math Number, Counting
Desired Results: Match a number word to one, and only one object when counting; (one-to-one correspondence).
Pre-Test: Show the child five counters and ask them, “How many counters do we have here?” As the child counts the counters, do they say the number word in the correct sequence from one to five and do they have a one word, one counter match?
Evidence of Learning: Present the child with five counters and ask them, “How many counters do we have here?” The child should know and name the correct forward number word sequence and have a one word, one counter match.
When working with young children, remember to:
• Show them how to do the task before expecting them to complete the task independently.
• Limit the range of numbers being presented when introducing a new mathematical idea.
• Provide children with a variety of math games, which repeat the same mathematical understanding (e.g. countingnumeral identificationsubitising, etc). To support this, I will be sharing a variety of counting games, numeral identification games, subitising games, etc. as part of this Math Games series.
Teaching & Learning Experiences:
You will need an empty egg carton and some counters. Cut the egg carton into parts, one part with three cups, the other four, the last five.
Begin with the carton of three cups and three counters before increasing to four then five.
• Model the activity. Focusing only on establishing a one-to-one correspondence, drop a counter one at a time into the carton containing three cups.
• Let the child have a turn dropping a counter one at a time into each cup.
• {Formative assessment}: Can the child drop one counter and only one counter into each box?
• Further develop the child’s counting skills by counting the counters as they are dropped into the box.
• {Formative assessment}: Can the child say the number word in the correct sequence having a one word, one counter match?
Extension Activities:
• Extension activity one: After the child has counted the correct number of counters, show him/her the corresponding numeral on a number card or number line. The early introduction of the number line establishes a good framework for understanding number in all its’ various forms; (natural numbers, integers, rational numbers, real numbers).
Don’t miss our next Early Math Game: Numeral Identification and Counting with Dinosaurs, or browse through our many math games.
#### More Activities for Preschoolers
Three to Five: Playful Preschool is stuffed to the brim with tried, tested and loved playful learning ideas for preschoolers. There are 25+ ideas for preschoolers, ten printable resources and additional links to over 50 more activities. A great resource for parents. Download your copy here.
If you enjoyed this post, please consider leaving a comment; I’d love to hear from you.
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# LAPLACE TRANSFORMS - BookSpar (PDF)
2013 • 133 Pages • 1.6 MB • English
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Posted April 14, 2020 • Submitted by alvera94
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## Summary of LAPLACE TRANSFORMS - BookSpar
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM LAPLACE TRANSFORMS INTRODUCTION Laplace transform is an integral transform employed in solving physical problems. Many physical problems when analysed assumes the form of a differential equation subjected to a set of initial conditions or boundary conditions. By initial conditions we mean that the conditions on the dependent variable are specified at a single value of the independent variable. If the conditions of the dependent variable are specified at two different values of the independent variable, the conditions are called boundary conditions. The problem with initial conditions is referred to as the Initial value problem. The problem with boundary conditions is referred to as the Boundary value problem. 2 d y dy Example 1 : The problem of solving the equation + + y = x with conditions y(0) = 2 dx dx y′ (0) = 1 is an initial value problem 2 d y dy Example 2 : The problem of solving the equation 3 + 2 + y = cos x with y(1)=1, 2 dx dx y(2)=3 is called Boundary value problem. Laplace transform is essentially employed to solve initial value problems. This technique is of great utility in applications dealing with mechanical systems and electric circuits. Besides the technique may also be employed to find certain integral values also. The transform is named after the French Mathematician P.S. de’ Laplace (1749 – 1827). The subject is divided into the following sub topics. LAPLACE TRANSFORMS Definition and Transforms of Convolution Inverse Solution of Properties some functions theorem transforms differential equations www.bookspar.com | VTU Notes RI
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM Definition : Let f(t) be a real-valued function defined for all t ≥ 0 and s be a parameter, real or complex. ∞ −st Suppose the integral e f (t)dt exists (converges). Then this integral is called the Laplace ∫ 0 transform of f(t) and is denoted by Lf(t). Thus, ∞ −st Lf(t) = e f (t)dt (1) ∫ 0 We note that the value of the integral on the right hand side of (1) depends on s. Hence Lf(t) is a function of s denoted by F(s) or f (s) . Thus, Lf(t) = F(s) (2) Consider relation (2). Here f(t) is called the Inverse Laplace transform of F(s) and is denoted -1 by L [F(s)]. Thus, -1 L [F(s)] = f(t) (3) Suppose f(t) is defined as follows : f1(t), 0 < t < a f(t) = f2(t), a < t < b f3(t), t > b Note that f(t) is piecewise continuous. The Laplace transform of f(t) is defined as ∞ −st Lf(t) = e f (t) ∫ 0 a b ∞ −st −st −st = ∫ e f1 (t)dt + ∫ e f2 (t)dt + ∫ e f3 (t)dt 0 a b www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM NOTE : In a practical situation, the variable t represents the time and s represents frequency. Hence the Laplace transform converts the time domain into the frequency domain. Basic properties The following are some basic properties of Laplace transforms : 1. Linearity property : For any two functions f(t) and φ(t) (whose Laplace transforms exist) and any two constants a and b, we have L [a f(t) + b φ(t)] = a L f(t) + b Lφ(t) Proof :- By definition, we have ∞ ∞ ∞ −st −st −st L[af(t)+bφ(t)] = e [af (t) + bφ(t)]dt = a e f (t)dt + b e φ(t)dt ∫ ∫ ∫ 0 0 0 = a L f(t) + b Lφ(t) This is the desired property. In particular, for a=b=1, we have L [ f(t) + φ(t)] = L f(t) + Lφ(t) and for a = -b = 1, we have L [ f(t) - φ(t)] = L f(t) - Lφ(t) 1 s 2. Change of scale property : If L f(t) = F(s), then L[f(at)] = F , where a is a positive a a constant. Proof :- By definition, we have ∞ −st Lf(at) = e f (at)dt (1) ∫ 0 Let us set at = x. Then expression (1) becomes, 1 ∞ − as x 1 s L f(at) = ∫ e f (x)dx = F a 0 a a This is the desired property. 3. Shifting property :- Let a be any real constant. Then at L [e f(t)] = F(s-a) Proof :- By definition, we have www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM ∞ ∞ at −st at −(s−a) L [e f(t)] = e [e f (t)]dt = e f (t)dt ∫ ∫ 0 0 = F(s-a) at This is the desired property. Here we note that the Laplace transform of e f(t) can be written down directly by changing s to s-a in the Laplace transform of f(t). TRANSFORMS OF SOME FUNCTIONS 1. Let a be a constant. Then ∞ ∞ at −st at −(s−a)t L(e ) = e e dt = e dt ∫ ∫ 0 0 −(s−a)t ∞ e 1 = = , s > a − (s − a) s − a 0 Thus, at 1 L(e ) = s − a In particular, when a=0, we get 1 L(1) = , s > 0 s By inversion formula, we have −1 1 at −1 1 at L = e L = e s − a s at −at ∞ e + e 1 −st at −at 2. L(cosh at) = L = e [e + e ]dt ∫ 2 2 0 ∞ 1 −(s−a)t −(s+a)t = [e + e dt] ∫ 2 0 Let s > |a| . Then, −(s−a)t −(s+a)t ∞ 1 e e L(cosh at) = + s 2 − (s − a) − (s + a)0 = s 2 − a2 L (cosh at) = Thus, s , s > |a| 2 2 s − a and so www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM −1 s L 2 2 = cosh at s − a at −at e − e a 3. L (sinh at) = L = , s > |a| 2 2 2 s − a Thus, a L (sinh at) = , s > |a| 2 2 s − a and so, −1 1 sinh at L 2 2 = s − a a ∞ −st 4. L (sin at) = e sin at dt ∫ 0 Here we suppose that s > 0 and then integrate by using the formula ax ax e e sin bxdx = [asin bx − bcosbx] ∫ 2 2 a + b Thus, a L (sinh at) = , s > 0 2 2 s + a and so −1 1 sinh at L 2 2 = s + a a ∞ −st 5. L (cos at) = e cos atdt ∫ 0 Here we suppose that s>0 and integrate by using the formula ax ax e e cosbxdx = [a cosbx + bsin bx] ∫ 2 2 a + b Thus, s L (cos at) = , s > 0 2 2 s + a and so −1 s L = cosat 2 2 s + a www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM 6. Let n be a constant, which is a non-negative real number or a negative non-integer. Then ∞ n −st n L(t ) = e t dt ∫ 0 Let s > 0 and set st = x, then ∞ n ∞ n −x x dx 1 −x n L(t ) = ∫ e = n+1 ∫ e x dx 0 s s s 0 ∞ −x n The integral e x dx is called gamma function of (n+1) denoted by Γ(n +1) . Thus ∫ 0 n Γ(n +1) L(t ) = n+1 s In particular, if n is a non-negative integer then Γ(n +1) =n!. Hence n n! L(t ) = n+1 s and so n n −1 1 t t L = or as the case may be n+1 s Γ(n +1) n! www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM TABLE OF LAPLACE TRANSFORMS f(t) F(s) 1 1 , s > 0 s at 1 e , s > a s − a s coshat , s > |a| 2 2 s − a a sinhat , s > |a| 2 2 s − a a sinat , s > 0 2 2 s + a s cosat , s > 0 2 2 s + a n n! t , n=0,1,2,….. , s > 0 n+1 s n Γ(n +1) t , n > -1 , s > 0 n+1 s Application of shifting property :- The shifting property is at If L f(t) = F(s), then L [e f(t)] = F(s-a) Application of this property leads to the following results : at s s − a 1. L(e cosh bt) = [L(cosh bt)]s→s−a = 2 2 = 2 2 s − b s→s−a (s − a) − b Thus, at s − a L(e coshbt) = 2 2 (s − a) − b and −1 s − a at L = e cosh bt 2 2 (s − a) − b www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM at a 2. L(e sinh bt) = 2 2 (s − a) − b and −1 1 at L = e sinh bt 2 2 (s − a) − b at s − a 3. L(e cosbt) = 2 2 (s − a) + b and −1 s − a at L = e cosbt 2 2 (s − a) + b at b 4. L(e sin bt) = 2 2 (s − a) − b and at −1 1 e sin bt L = 2 2 (s − a) − b b at n Γ(n +1) n! 5. L(e t ) = or as the case may be n+1 n+1 (s − a) (s − a) Hence at n −1 1 e t n! L = or as the case may be n+1 n+1 (s − a) Γ(n +1) (s − a) Examples :- 1. Find Lf(t) given f(t) = t, 0 < t < 3 4, t > 3 Here ∞ 3 ∞ −st −st −st Lf(t) = e f (t)dt = e tdt + 4e dt ∫ ∫ ∫ 0 0 3 Integrating the terms on the RHS, we get 1 −3s 1 −3s Lf(t) = e + (1− e ) 2 s s This is the desired result. www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM 2. Find Lf(t) given f(t) = sin2t, 0 < t ≤ π 0, t > π Here π ∞ π −st −st −st Lf(t) = e f (t)dt + e f (t)dt = e sin 2tdt ∫ ∫ ∫ 0 π 0 −st π e 2 −πs = 2 {− ssin 2t − 2cos2t} = 2 [1− e ] s + 4 0 s + 4 This is the desired result. 3. Evaluate : (i) L(sin3t sin4t) 2 (ii) L(cos 4t) 3 (iii) L(sin 2t) (i) Here 1 L(sin3t sin4t) = L [ (cost − cos7t)] 2 1 = [L(cost) − L(cos7t)], by using linearity property 2 1 s s 24s = − = 2 2 2 2 2 s +1 s + 49 (s +1)(s + 49) (ii) Here 2 1 1 1 s L(cos 4t) = L (1+ cos8t) = + 2 (iii) We have 2 2 s s + 64 3 1 sin θ = (3sinθ − sin3θ ) 4 For θ=2t, we get 3 1 sin 2t = (3sin 2t − sin 6t) 4 so that 3 1 6 6 48 L(sin 2t) = 2 − 2 = 2 2 4 s + 4 s + 36 (s + 4)(s + 36) This is the desired result. www.bookspar.com | VTU Notes Rφ
www.bookspar.com | VTU NOTES | QUESTION PAPERS | NEWS | RESULTS | FORUM 4. Find L(cost cos2t cos3t) Here 1 cos2t cos3t = [cos5t + cost] 2 so that 1 2 cost cos2t cos3t = [cos5t cost + cos t] 2 1 = [cos6t + cos4t +1+ cos2t] 4 Thus 1 s s 1 s L(cost cos2t cos3t) = + + + 2 2 2 4 s + 36 s +16 s s + 4 2 5. Find L(cosh 2t) We have 2 1+ cosh 2θ cosh θ = 2 For θ = 2t, we get 2 1+ cosh 4t cosh 2t = 2 Thus, 2 1 1 s L(cosh 2t) = + 2 2 s s −16 -------------------------------------------------------05.04.05------------------------- 1 -3/2 6. Evaluate (i) L( t ) (ii) L (iii) L(t ) t n Γ(n +1) We have L(t ) = n+1 s 1 (i) For n= , we get 2 1 Γ( +1) 1/2 2 L(t ) = 3 / 2 s 1 1 1 π Since Γ(n +1) = nΓ(n) , we have Γ +1 = Γ = 2 2 2 2 www.bookspar.com | VTU Notes Rφ
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# Calculate the number of moles of oxygen in 2.00L at 24 degrees Celsius and 750 torr.
mvcdc | Student, Graduate | (Level 2) Associate Educator
Posted on
First, we assume that the gas is ideal. Then, we use the ideal gas law:
`PV = nRT`
where P is the pressure, V the volume, n the number of moles, R the gas constant, and T absolute temperature.
This gives us:
`n = (PV)/(RT)`
Hence, substituting the given values:
`n = (PV)/(RT) = ((750/760)*2)/(0.08206*(24+273)) = 0.081 `
Answer is 0.081 moles of oxygen.
Note that temperature is in Kelvin, while pressure in atm (1 atm = 760 torr)
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# Tag Info
1
Start with: MatrixForm[A = {{1, 2, 3, a}, {4, 5, 6, a^2}, {7, 8, 9, a^3}}] Which gives the result: $$\begin{bmatrix} 1 & 2 & 3 & a\\ 4 & 5 & 6 & a^2\\ 7 & 8 & 9 & a^3 \end{bmatrix}$$ Then: r1 = A[[1]]; r2 = A[[2]]; r3 = A[[3]]; Then: MatrixForm[A = {r1, r2 - 4 r1, r3 - 7 r1}] Which gives the result: ...
0
If you make your A matrix a function (but avoid starting with capital letters), like aMatrix[x_] := ... then you can use Nest: Nest[aMatrix, x0, 100] where x0 is the starting vector.
5
Iterating @J.M.'s comment: This problem has no exact solution. With[{ matrix = {{0.8111, 0.4867, -0.3244}, {a, b, 0}, {c, d, e}} }, Print[matrix.Transpose[matrix]]; Solve[ matrix.Transpose[matrix] == IdentityMatrix[3], {a, b, c, d, e} ] ] (* {{0.999995,... *) (* {} *) That is, the first column's first entry is not $1$, so there is ...
1
SeedRandom[5] pts = RandomReal[1, {6, 2}]; pts = pts[[FindShortestTour[pts][[2]]]]; am = RandomChoice[{.7, .3} -> {0, 1}, {6, 6}]; AdjacencyGraph using the polygon vertices as vertex coordinates: Labeled[AdjacencyGraph[am, VertexCoordinates -> pts, DirectedEdges -> False, Vertexlabels->"Name", Prolog -> {Yellow, ...
0
Here's an example where the adjacency matrix shows a link from pt[[1]] to pt[[4]] and from pt[[2]] to pt[[3]]: pts = {{0, 0}, {0, 1}, {1, 1}, {1, 0}}; adMat = {{0, 0, 0, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 1, 0, 0}}; myfig = Graphics[ {{Opacity[0.2], Yellow, Polygon[pts]}, {Red, PointSize[0.02], Point[pts]}, Line@({pts[[#1]], ...
0
Here is the fixed notebook. Your main problems were that the third elements of the first argument in ListContourPlot were imaginary and that you switched two values in Part in fsingle. In the notebook, I highlighted lines I have changed and wrote why I changed them in the comments (there were some inefficiencies). However, please try to post your actual ...
2
Using Smith Normal Form you can get two integer matrices with determinant 1 that would satisfy the equation $$X1.K.X2 = K_2$$ May be this is a good start you can use... resK = SmithDecomposition[K]; MatrixForm /@ resK resK2 = SmithDecomposition[K2]; MatrixForm /@ resK2 resK[[2]] == resK2[[2]] (* True *) K2 == ...
2
I learned a lot from WReach's and Leonid's answers and I'd like to make a small contribution: It seems worth emphasizing that the primary intention of the list-valued second argument of Flatten is merely to flatten certain levels of lists (as WReach mentions in his List Flattening section). Using Flatten as a ragged Transpose seems like a side-effect of ...
0
What you want to do is actually very simple and can accomplished by a single line of code. With[{db = .01}, v = Interpolation[Table[{b, b^2}, {b, 0, 1, db}]]]; Then Plot[v[b], {b, 0, 1}]
4
Thanks to MarcoB's pointer, I realized the following properties of conjugating one of the numbered Pauli matrices by $A$, $B$, or $C$. Table[ConjugateTranspose[PauliMatrix[i]] == PauliMatrix["A"].PauliMatrix[i].MatrixPower[PauliMatrix["A"], -1], {i, 3}] Table[-Transpose[PauliMatrix[i]] == PauliMatrix["B"].PauliMatrix[i].MatrixPower[PauliMatrix["B"], ...
5
This could be done with a custom function for the first argument of Inner that treats a differently (it's just a more convenient form of expressing your original idea). For example, consider this: ClearAll[f]; f[a, x_] := a[x] f[x_, a] := a[x] f[x_, y_] := x y Now for your first example: Inner[f, {{a, b}, {c, d}}, {h, k}] (* {b k + a[h], c h + d k} *) ...
3
Sorry, correcting a typing error in the matrix leads to a different conclusion: Eigenvalues as well as determinant factorize, which points to solubility, but still, I haven't found the matrix factors. The correct matrix is a = {{1/r, 1/\[Mu][r]}, {\[Mu][r]/r f[n], -(3/r^2)}}; Where f[n] = n(n+1)-2 The determinant is Det[a] // Factor (* Out[194]= ...
3
This is a very quick-and-dirty, but gives 4X speed-up (7X with tweak for symmetry) on my crappy netbook for the n=8 case, don't have time or patience to test bigger cases to see scaling differences. Perhaps a description of what you're trying to calculate? It appears to be some combinatorial problem, there may well be a much more efficient scheme to do ...
0
In[1]:= b=Range[0,.9,0.1]; Length@b p=Range[.25,.75,.05] Length@p Out[2]= 10 Out[3]= {0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75} Out[4]= 11 In[5]:= Information@Interpolation; Interpolation[{{{Subscript[x, 1],Subscript[y, 1],\[Ellipsis]},Subscript[f, 1]},{{Subscript[x, 2],Subscript[y, 2],\[Ellipsis]},Subscript[f, 2]},\[Ellipsis]}] constructs an ...
2
I am a novice user so please forgive me if this is a bit clunky! Using Transpose and ArrayReshape mat = {{1}, {2}, {3}}; n = 11; ArrayReshape[Transpose[Table[mat, {n}]], Dimensions[mat] {1, n}] // MatrixForm
4
The way I remember it from my Linear Algebra class is like this: Clear[b]; A = {{-1, 1, -1, 0, 0, 0}, {1, 0, 0, -1, -1, 0}, {0, -1, 0, 0, 1, -1}, {0, 0, 1, 1, 0, 1}}; bb = Array[b, Length@A]; Thread[NullSpace@Transpose@A . bb == 0] (* {b[1] + b[2] + b[3] + b[4] == 0} *) That is, the condition for a solution to A.x == b to exist is that b be in the ...
1
Guessing OP meant f1*f2 == f1*f3 == f1* f4 == f1*f5 == f2*f3 == f2*f4 == f2*f5 == f3*f4 == f3*f5== f4*f5 ==0, matrix = {{f1, -f2, -f2, f3}, {f4, -f1, -f1, f2}, {f4, -f1, -f1, f2}, {f5, -f4, -f4, f1}}; assumptions = Thread[(Subsets[Times[f1, f2, f3, f4, f5], {2}]) == 0]; ToRadicals @ FullSimplify[Eigensystem[matrix], Assumptions -> assumptions]
1
KroneckerProduct kpF = KroneckerProduct[{ConstantArray[1, #2 ]}, #] &; SparseArray and Band saF = SparseArray[(Band[{1, 1}, {1, #2} Dimensions[#], {1,1}] -> #)] &; Examples: mat = {{0, 0, 1, 1}, {0, 1, 0, 1}}; saF[mat, 5] // MatrixForm kpF[mat, 3] // MatrixForm
11
You can use Reduce[] to find a set of all conditions as follows: A = {{-1, 1, -1, 0, 0, 0}, {1, 0, 0, -1, -1, 0}, {0, -1, 0, 0, 1, -1}, {0, 0, 1, 1, 0, 1}}; b = {b1, b2, b3, b4}; x = {x1, x2, x3, x4, x5, x6} allConditions=Reduce[A.x == b, x] This returns b1 == -b2 - b3 - b4 && x3 == b2 + b3 + b4 - x1 + x2 && x5 == -b2 + x1 - x4 ...
0
I absolutely support JasonB's suggestions, especially hdf5 seems a good format for such data when you want to be able to read with other software. But for the case where you only need to write and read with Mathematica, I think the MX format at least needs to be mentioned as well: it is by far the easiest and fastest way to store arbitrary expressions (not ...
3
Perhaps I'm missing something in the question, but why not just: new=Upsample[old, 3, 2];
1
A different SparseArray approach: n = 3 (m = RandomInteger[{1, 5}, {n, n}]) // MatrixForm big = SparseArray[ {i_ /; Mod[i + 1, 3] == 0, j_ /; Mod[j + 1, 3] == 0} :> #[[(i + 1)/3, (j + 1)/3]] , {3 Length@#, 3 Length@#}] &@m; MatrixForm[big]
0
If you plan to use your data in MMA only you can do a = RandomReal[1, {1000, 3, 3}]; Dimensions[a] a >> testExport.dat b = << testExport.dat; Dimensions[b] a == b
3
KroneckerProduct f5 = KroneckerProduct[#, {{0, 0, 0}, {0, 1, 0}, {0, 0, 0}}] &; f6 = KroneckerProduct[SparseArray@#, SparseArray[{2, 2} -> 1, {3, 3}]] &; SparseArray and Band f1 = SparseArray[Band[{2, 2}, Automatic, {3, 3}] -> #,3 Dimensions[#]] &; or f1 = SparseArray[Band[{2, 2}, 3 Dimensions[#], {3, 3}] -> #] & ...
2
Here is one way -- represent the original array in sparse form, then replace the indices with the desired ones: m = RandomInteger[{-5, 5}, {3, 3}]; Normal@SparseArray[Drop[ArrayRules@SparseArray[m] /. {x_, y_} -> {3 x - 1, 3 y - 1}, -1]] // MatrixForm More generally, and controlling for the ultimate size of the matrix: n = 3; m = ...
3
You can export as a MATLAB .mat file if your array has less than 4 dimensions, rand = RandomReal[1, {1000, 3, 3}]; Dimensions@rand rand[[454, 1, 2]] Export["random.mat", rand]; (* {1000, 3, 3} *) (* 0.786307 *) When you import it again, you have the same dimensions and the elements are the same rand2 = Import["random.mat"]; Dimensions@rand2 rand2[[454, ...
0
Since you say that you still do not get the output you desire using KroneckerProduct, I am guessing that you should try restarting the kernel. In any case, this should also fit your needs: mat1 = Array[m1, {2, 2}]; mat2 = Array[m2, {2, 2}]; KroneckerProduct[mat1, mat2] // MatrixForm
0
Perhaps X = IdentityMatrix[2] Y = Array[y, {2, 2}] TensorProduct[X, Y] // MatrixForm \$\left( \begin{array}{cc} \left( \begin{array}{cc} y(1,1) & y(1,2) \\ y(2,1) & y(2,2) \\ \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} \right) \\ \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \\ \end{array} ...
1
You could introduce further conditional definitions for H which will prevent those computations whose results would end up being thrown away. For instance, you could add: H[i_, j_, k_, l_] /; (i > j || k > l) = Missing[]; As a toy example: m = Table[H[n, 2, 3, 4], {n, 1, 10}] (* Out: {(3 Sqrt[5])/128, (5 Sqrt[15])/256, Missing[], Missing[], ...
2
Here are couple of quick tips. Lets say this is your matrix mat = SparseArray[{{i_, j_} /; Abs[i - j] == 3 -> 1, {i_, i_} -> 1}, {200, 200}]; This is the conventional way to find EigenSystem Eigensystem[SparseArray[{{i_, j_} /; Abs[i - j] == 3 -> 1, {i_, i_} -> 1}, {200, 200}]]; // AbsoluteTiming {53.6551, Null} Now just change the ...
1
SeedRandom[1] s1 = RandomInteger[{-3, 3}, {5, 5}]; s2 = RandomInteger[{-3, 3}, {5, 5}]; Temporarily define Indeterminate as 0: Block[{Indeterminate = 0}, 1./2 ArcTan[s1, s2]]
2
If all the elements in the matrix are functions, you can also use Block[{Times = (# @ #2 &)}, {{a, b}, {c, d}}.{h, k}] {a[h] + b[k], c[h] + d[k]}
5
Picking up on Marius tip on Inner in the comments: Inner[Apply[#1, {#2}] &, {{a, b}, {c, d}}, {h, k}] And @ciao offered a better version in comments: Inner[#1[#2] &, {{a, b}, {c, d}}, {h, k}]
0
Manipulate[Row[{ArrayPlot[mat[[;; k, ;; k]], ImageSize -> 300], AdjacencyGraph[mat[[;; k, ;; k]], ImageSize -> {300, 300}]}], {{mat, ConstantArray[0, {50, 50}]}, None}, {k, 4, None}, Dynamic[Column[{InputField[Dynamic[k, (k = Clip[IntegerPart@#, {2, 20}]) &], Number, FieldSize -> {8, 1}], ...
1
A bit simple minded: DynamicModule[{n = 3, bs}, Panel[Column[{Slider[Dynamic[n, {(n = #) &, (bs = PadRight[bs, {n, n}]) &}], {2, 100, 1}], Row[{Dynamic[Grid[Array[Checkbox[Dynamic[bs[[##]]], {0, 1}] &, ...
2
You can use NumberFormat option in ScientificForm to do this. For example cformat[x_, numDigts_] := ToString[ ScientificForm[x, numDigts, NumberFormat -> (Row[If[#3 == "", {#1}, {#1, "E", #3}]] &)]] then cformat[1.2345678*^-10, 4] (* "1.235E-10" *) However, since your data is all in the range of 0 to 10, there would be no "E" in the ...
5
This answer compares two dimension reduction techniques SVD and Non-Negative Matrix Factorization (NNMF) over a set of images with two different classes of signals (two digits below) produced by different generators and overlaid with different types of noise. Note that question states that the images have one class of signals: I have a stack of images ...
10
As requested by Anton. I halved the amount of noise because otherwise some images have barely any signal left. As you can see below, we are still putting in a significant amount of noise. (To conserve space I'm only visualizing the first ten images in this answer, but the denoising is happening over all 100 test images.) SeedRandom[2016] (* for ...
3
The numerical definitions you give later modify the calculation of the symbolic definitions you use earlier. If you clear all your variables at the start of your calculation, the problem goes away and repeatable results are obtained. In particular, it is the numerical definition of k that seems to muddy the waters. In order to do what you want, a far better ...
15
Start data First let us get some images. I am going to use the MNIST dataset for clarity. (And because I experimented with similar data some time ago.) MNISTdigits = ExampleData[{"MachineLearning", "MNIST"}, "TestData"]; testImages = RandomSample[Cases[MNISTdigits, (im_ -> 0) :> im], 100] Let us convince ourselves that all images have the same ...
11
Here is a very short solution: qf = a x^2 + b y^2 + c z^2 + 2 d x y + 2 e x z + 2 f y z; 1/2 D[qf, {{x, y, z}, 2}] (* ==> {{a, d, e}, {d, b, f}, {e, f, c}} *) This is just an application of the answer to Quick Hessian matrix and gradient calculation.
6
Here is a way that yields symmetric matrix (for this example you could just write it down): m=Module[{r = {x -> 1, y -> 2, z -> 3}, tu = Tuples[{x, y, z}, 2]}, Normal@SparseArray[(## /. r) -> Coefficient[qf, Times @@ ##]/(2 - Boole[#[[1]] === #[[2]]]) & /@ tu, {3, 3}]] yields: {{a, d, e}, {d, b, f}, {e, f, c}} Check: ...
10
I think you need CoefficientArrays: mat = Last@CoefficientArrays[qf, {x, y, z}, "Symmetric"->True]; {x, y, z}.mat.{x, y, z} == qf // Simplify (* True *)
10
First answer (extended comment actually) You have to define better your objective function. For example, the following works: ClearAll[mat, minev] SeedRandom[1] rm = RandomInteger[10, {40, 40, 3}]; mat[t_] := N[rm.{1, t, t^2}]; minev[t_?NumericQ] := First@Eigenvalues[mat[t], -1]; Take[Table[minev[t], {t, 0, 1, .01}], 3] (* {-0.864071 - 1.30548 I, ...
1
This isn't much different than JasonB's answer in substance, but it's easier to debug and it's usually how I first approach problems with lots of steps: Module[ {nullPos, nullGroup, nullReplacements}, nullPos = Position[startingMatrix, "Null"]; nullGroup = Last@startingMatrix[[First@#]] & /@ nullPos; nullReplacements = Cases[lookupMatrix, {___, x_} ...
2
This looks convoluted, but it matches the requirements you set up Fold[ Function[{matrix, index}, ReplacePart[matrix, index -> RandomChoice[ Select[ lookupMatrix, (Last@# == startingMatrix[[First@index, -1]] &)][[All, Last@index]] ]]], startingMatrix, Position[startingMatrix, "Null"]] (* {{1, 2, 3, 4, 5, 5.5, 6, 7, ...
3
ClearAll[mat, minev] SeedRandom[1] rm = RandomInteger[10, {400, 400, 3}]; mat[t_] := rm.{1, t, t^2}; minev[t_?NumericQ] := Eigenvalues[mat[t], -1]; DiscretePlot[Evaluate[minev[t]], {t, 0, 1, .01}]
4
Reverse /@ Partition[ {c1, c2, c3, c0}, 4, 1, {1, 1}, {c1, c2, c3, c0}] Edit or more simple : Reverse /@ Partition[{c1, c2, c3, c0}, 4, 1, {1, 1}]
4
cm = ToeplitzMatrix[{c0, c1, c2, c3}, RotateRight[Reverse[{c0, c1, c2, c3}]]]; cm // MatrixForm
3
Do you really need to use ToeplitzMatrix? What about following? MatrixForm@Transpose@NestList[RotateRight, #, Length[#]-1] &@{1, 2, 3, 4}
Top 50 recent answers are included
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# Circles
## Contact(s)
Judith Spitzli
515-402-8600
Iowa Core Mathematics Documents
Iowa Core Mathematics Support - Resources to support Iowa Core Mathematics.
## Standards in this domain:
### Understand and apply theorems about circles
• HSG-C.A.1 Prove that all circles are similar.
• HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
• HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
• HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.
### Find arc lengths and areas of sectors of circles
• HSG-C.5.B Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Printed from the Iowa Department of Education website on April 18, 2014 at 9:28pm.
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1. ## theorem
x^3 + 2x^2 + px -3 is divided by x+1 the remainder is the same as when it is divided by x-2. find value of p
thanks
2. Hello,
Let y be the common remainder. y will be a constant, because you divide by a polynom of degree 1.
$\displaystyle x^3+2x^2+px-3=(x+1)(ax^2+bx+c)+y$
$\displaystyle x^3+2x^2+px-3=(x-2)(a'x^2+b'x+c')+y$
If you develop $\displaystyle (x+1)(ax^2+bx+c)$, you get, by identification :
$\displaystyle a=1$
$\displaystyle b=1$
$\displaystyle c=p-1 (1)$
$\displaystyle c+y=-3 \Longleftrightarrow y+3=-c$
(i leave you the right to do the calculus )
In the same way,
$\displaystyle a'=1$
$\displaystyle b'=4$
$\displaystyle c'=p+8 (2)$
$\displaystyle -2c'+y=-3 \Longleftrightarrow y+3=2c'$
Hence 2c'=-c.
With (1) and (2), we can state that 2(p+8)=-(p-1)
And then, just solve for p
3. Let P(x) and Q(x) be polynomials and k be the root of Q(x)=0.
Then the remainder of the division P(x)/Q(x) is equal to P(k).
$\displaystyle P(x) = x^3 + 2x^2 + px - 3$ and $\displaystyle Q(x)=x+1$
The root of Q(x) = 0 is,
$\displaystyle x+1=0$, $\displaystyle x=-1$
So find P(-1),
$\displaystyle P(x) = x^3 + 2x^2 + px - 3$
$\displaystyle P(-1) = -1 + 2 -p - 3 = -p-2$
Now find the remainder for $\displaystyle Q(x) = x-2$
The root of $\displaystyle Q(x) = 0$ is,
$\displaystyle x-2 =0$, $\displaystyle x=2$
So $\displaystyle P(2) = 8 + 8 + 2p -3 = 2p + 13$
Since the remainders are the same,
$\displaystyle -p-2 = 2p + 13$
$\displaystyle p=-5$
4. Originally Posted by gracey
x^3 + 2x^2 + px -3 is divided by x+1 the remainder is the same as when it is divided by x-2. find value of p
thanks
To streamline things a bit ......
Use the remainder theorem:
Let the remainder be r. Then:
-1 + 2 - p - 3 = r => -2 - p = r .... (1)
8 + 8 + 2p - 3 = r => 13 + 2p = r .... (2)
Equate (1) and (2): -2 - p = 13 + 2p => p = -5.
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## Look around you maths
This British comedy show is hilarious. It’s called “Look Around You” and the second epsidoe is about “Maths”. Check it out on youtube (at least for now):
Quote from the show (but you really have to watch it to put it in perspective!!):
Narrator: What’s the largest number you can think of ?
Person 1: 100,000
Person 2: 999,000
Person 3: a million!
Narrator: In actual fact it’s neither of these. The largest number is
about 45 billion, although mathematicians suspect there may be even
larger numbers!
And one of the problems they presented:
Narrator: Eight ladies go to eight shops at eight o’clock in the morning. Each
lady wants to buy eight spiders. For each spider, eight spider shoes
must also be bought. But they only have eight pounds between them. With
each spider costing eight pence and each spider shoe costing an eighth
pence each, will the ladies have enough change for the bus ride home? A
journey costing eight pence per stop and made up of eight stops.
Check out the wiki entry for more information.
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## The mathematics of sudoku
Tom Davis has a great article on the mathematics of sudoku. He first describes a brief history of the puzzle and how to play. Then he discusses why it is mathematically interesting (it is, trust me!). He goes through some obvious strategies that a lot of people try when doing sudoku and some other clever strategies. Definitely check it out if you have time!
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## Silly but trickly puzzle
So my PhD friends couldn’t solve the following problem…
3489 = 4
8410 = 4
9120 = 2
8328 = 4
2210 = 1
9910 = 3
9900 = 4
7172 = 0
3884 = 5
9889 = 6
9009 = 4
0911 = 2
8888 = 8
What is 3859?
(4.00 from 1 votes)
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## Math cheat sheets
People love to cheat at math for some reason (laziness? lack of understanding? failure to get help?)… anyways, here are some math cheat sheets you can use from different websites:
The first one is about algebra, it has 23 pages filled with formulas:
Algebra Notes
The next one is much shorter about algebra again (4 pages): Algebra Cheat Sheet.
This next one is calculus:
Calculus Cheat Sheet (6 pages of Calc notes).
For computer science formulas see Computer Science Cheat Sheets. (10 pages)
For Geometry you can look at the Geometry Fact Sheet.
The next one is: Astrophysics and more.
This one has a bunch regarding calc, multivariable calculus cheat
sheets, physics, quantum mechanics, optics, astrophysics, relativity,
and more more more!! You can view / save as a jpg and print 😀
For more physics / calc you can look at these ones:Physic and Calculus Cheat Sheets (zip files for easy download).
And finally some trig here: Trigonometry Cheat Sheet .
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## Calculator for grades K-3
This calculator is perfect for all those kindergartners who need to do some math… quoted from the website:
The TI-10 is perfect for the primary grades. It combines popular
features of the TI-15 Explorerâ„¢, which makes it a unique tool for
grades K-3. The TI-10’s comfortable, colorful design helps students
find patterns in daily activities and helps educators reinforce math
concepts in all elementary subjects.
Please, please, pleaseeee don’t let kids that young use calculators!!
And if they do, only let them use it to check their answers.
Say you have two kids who have to add up the numbers from 1 to 100. The calculator kid would use his monkey sticks calculator and plug away at it. The non-calculator kid would hopefully come up with a thoughtful solution to the problem (sound familiar?) 😀
I guess it doesn’t matter since it’s computer age now. Lots of grade 9
students can’t even properly WRITE their own name! They never ever have
to write (and rarely sign forms), and only learned to properly write in
grade 3 (the curriculum was set to focus more on teaching kids
computers). Oh monkey sticks, what will we do ^_^
(5.00 from 2 votes)
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## Cat picture
This is a cute picture 😀
(5.00 from 3 votes)
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## Math fails in history
Dick Lipton wrote a great post over at Gödel’s Lost Letter and P=NP. In his Sept 27th post he talked about surprises in mathematics. In one of his sections he gives three examples of where mathematicians “accepted” a false proof. Sometimes this happens and it might be dozens of years until someone realizes a mistake has been made.
One interesting example of this is the Four Colour Theorem (that’s right ya bunch of monkeys, I spelled colour with a U!!!)…
Lipton says…
The
Four-Color Theorem (4CT) dates back to 1852, when it was first proposed
as a conjecture. Francis Guthrie was trying to color the map of
counties in England and observed that four colors were enough.
Consequently, he proposed the 4CT. In 1879, Alfred Kempe provided a
“proof” for the 4CT. A year later, Peter Tait proposed another proof
for 4CT. Interestingly both proofs stood for 11 years before they were
proved wrong. Percy Heawood disproved Kempe’s proof in 1890, and Julius
Petersen showed that Tait’s proof was wrong a year later.
However, Kempe’s and Tait’s proofs, or attempts at a proof, were not
fully futile. For instance, Heawood noticed that Kempe’s proof can be
adapted into a correct proof of a “Five-Color Theorem”. There were
several attempts at proving the 4CT before it was eventually proved in
1976. See this article by Robin Thomas for a historical perspective of
the problem.
Go check out the rest of his post NOW.
(No Ratings Yet)
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## Another Spiked Math comic
Today’s spiked math comic isn’t too shabby (idea by ma bro):
As a mathematician, assumptions are things that come natural to us 😀
(No Ratings Yet)
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## Another (x, why?) comic :-D
okay okay, but this one is very funny (at least to me)… units are great 😀
http://xwhy.comicgenesis.com/d/20090921.html
I remember marking papers and on NUMEROUS occasions a student would write down such a bizzarro answer it made me LOL in real life!! One of these days i’ll write a post dedicated to some of the hilarious “mistakes” i’ve seen.
(3.00 from 1 votes)
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## How many guys are there for me?
Hey guys,
Based on my previous post of “How many girls are there for me,” I thought it would be interesting to see how many guys there are for me. So hypothetically let’s pretend I am gay and a top (if you don’t know what that means you should get some more gay friends lol). Just a recap that I found out in the last post that:
11, 414 girls
are suitable matches for me.
Now, some of my stats I used from counting the number of girls for me still hold. The male population on the planet is about:
Male population on earth in 2009:
3,500,000,000
But I am looking for a boy who is in North America. Restricting this number to Canada and the United States gives:
171, 230, 000 boys
He must also be around my age. Say, from 25-29 years old. That leaves:
11, 415, 000 boys
But about half of these boys are already married or in a common-law relationship. Thus, that leaves:
5, 707, 500 boys
And further, about 89% of men are not interested in men (i.e. straight or other). That leaves:
627, 825 boys
Being a mathematician, I want a boy who is smart. But a lot of gay guys are actually smart (unlike the other post which quoted: 85% of girls in North America are complete dumb asses). Only about 50% of gay guys are complete dumbasses, thus that leaves:
313, 912 boys
But
I don’t want no ugly man! Thank god that a lot of gay guys take
care of their appearance and work out at the gym. About 40% of gay guys
don’t, so that leaves:
188, 347 boys
But the boy must also like me. To see what kind of statistic is
reasonable, I went to the local gay bar and chatted with 10 boys. Then
at
the end I asked them if they liked me based on my appearance and our
conversation. An outstanding 7 boys said yes and wanted to go back to
my place. But upon further asking, 5 of them didn’t want an actual
relationship with me. So, only about 20% of the boys actually like me
and would date me. That leaves:
37, 669 boys
Finally, as I said at the top, hypothetically speaking I would be a
“top” gay. So, assuming that 50% are also tops, that would leave me
with:
18, 834 boys
Holy cow!! WTF!? So only 11, 414 girls would make a perfect match with
me, but if I were gay, an outstanding 18, 834 boys would make a perfect
match with me! Damn, I better reconsider my options LOL.
(5.00 from 1 votes)
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« »
Derive three equations of motion by graphical method.
Posted- 1508 days ago
## First Equation of Motion
Graphical Derivation of First Equation
Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t.
The figure shows the velocity-time graph of the motion of the object.
Slope of the v - t graph gives the acceleration of the moving object.
Thus, acceleration = slope = AB =
v - u = at
v = u + at I equation of motion
Graphical Derivation of Second Equation
Distance travelled S = area of the trapezium ABDO
= area of rectangle ACDO + area of DABC
(v = u + at I eqn of motion; v - u = at)
Graphical Derivation of Third Equation
S = area of the trapezium OABD.
Substituting the value of t in equation (1) we get,
2aS = (v + u) (v - u)
(v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)]
v2 - u2 = 2aS III Equation of Motion
Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion
Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion
Derivations of Equations of Motion (Graphically) First Equation of Motion Graphical Derivation of First Equation Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t. The figure shows the velocity-time graph of the motion of the object. Slope of the v - t graph gives the acceleration of the moving object. Thus, acceleration = slope = AB = v - u = at v = u + at I equation of motion Graphical Derivation of Second Equation Distance travelled S = area of the trapezium ABDO = area of rectangle ACDO + area of DABC (v = u + at I eqn of motion; v - u = at) Graphical Derivation of Third Equation S = area of the trapezium OABD. Substituting the value of t in equation (1) we get, 2aS = (v + u) (v - u) (v + u)(v - u) = 2aS [using the identity a2 - b2 = (a+b) (a-b)] v2 - u2 = 2aS III Equation of Motion
define acceleration
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# Graph
Graph Theory is the subject of study of graphs, which are mathematical structures used to model pairwise relations between objects.
The paper written by leonhard Euler on “Seven Bridges of Konlgsberg” and published in 1736 as regard as the first paper in the history of graph theory.More than centure after Euler’s paper on “Seven Bridges of Konlgsberg” while listing was introduced then concepet of Topology was introduced.Cayley was let by an intrest in particular analytical forms arising from differential calculus.To study particular case of graphs.In particulat the team graph was introduced by Sylvester in a paper published in 1878.THe first textbook in graph theory was written by Derves Konig,abd was published in 1936.
Graph is simply a pictorial representation of object.It is also defined as the set and set of certaion relation.Mathematically we write graph as G = (V, A). Where V is the set of vertices and A is the set of arcs.
## Types of Graph
simply there are two types of graph.These are listed below.
• Directed Graph
• undirected Graph
## 1. Directed Graph
If all arcs are orented from one particular vertex to the another particular vertex then graph becomes a directed graph.It is also defined as if there is direction are given to the arc of the graph then graph becomes directed graph.Directed graph is also written as digraph.following are the some figure where direction is given to the arcs.There are some terminology related to the digraph.
• Tail
#### 1.Tail
The vertex from which the arc begin is called tail of the arc.
The vertex at which the arc end is called head of arcs.
If the arc share the common end node then this type of arc is called adjacency arc.
In above figure 1 {a,b,c,d} are vertices {(a, b),(b, d),(d, c)(a,c),(b, d)} are arcs.Head set of figure 1 are (b, d, c, b, a) and tail set of figure 1 are (a, b, d, c, a)
## 2.Undirected Graph
A graph having no direction in between any vertex is called undirected graph.It is called simply a graph.In case of undirected graph we can take any vertex as a first vertex and any vertex as the second vertex.Some figure are given below.
Tags:
Categories:
Updated:
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# Find the probability that an averge player wins ten or more times in 25 hands.In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara falls, as well as many other cities,...
Find the probability that an averge player wins ten or more times in 25 hands.
In the game of blackjack as played in casinos in Las Vegas, Atlantic City, Niagara falls, as well as many other cities, the dealer has the advantage. Most players do not play very well. As a result, the probability that the average player wins a hand is about 45%.
justaguide | College Teacher | (Level 2) Distinguished Educator
Posted on
The probability that a player wins ten or more times in 25 hands is equal to 1 - (probability that a person wins 0-9 times)
The probability that a person wins in a single hand is 45%. The probability that a person loses in a single hand is 55%. In 25 games the probability that a person wins n games is 25Cn*(0.45)^n*(0.55)^(25 - n)
`sum_(n = 0)^9 25Cn*(0.45)^n*(0.55)^(25-n) = 0.242371`
This gives the probability for winning 0 to 9 times as 0.242371
The probability that an average player wins ten hands or more is: 1 - 0.242371 = 0.757629
The required probability is 0.757629
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12 pág.
# Séries de Fourier de Senos e de Cossenos de Índices Ímpares
Pré-visualização2 páginas
sen
(
2kpit
2L
)
= \u2212h(t)
Assim segue da aplicac¸a\u2dco do item (a) que b2k = 0.
Para h(t) = f (t) sen (2k+1)pit2L temos que
h(2L\u2212 t) = f (2L\u2212 t) sen
(2k+ 1)pi(2L\u2212 t)
2L
= f (t) sen
(
(2k+ 1)pi \u2212
(2k+ 1)pit
2L
)
= f (t) sen
(
pi \u2212
(2k+ 1)pit
2L
)
= f (t) sen
(
(2k+ 1)pit
2L
)
= h(t)
Assim segue da aplicac¸a\u2dco do item (b) que
b2k+1 =
2
L
\u222b L
0
f (t) sen
(2k+ 1)pit
2L
dt para k = 0, 1, 2, . . .
(d) Para h(t) = f (t) cos 2kpit2L temos que
h(2L\u2212 t) = f (2L\u2212 t) cos
2kpi(2L\u2212 t)
2L
= \u2212 f (t) cos
(
2kpi \u2212
2kpit
2L
)
= \u2212 f (t) cos
(
\u2212
2kpit
2L
)
= \u2212 f (t) cos
(
2kpit
2L
)
= \u2212h(t)
Assim segue da aplicac¸a\u2dco do item (a) que a2k = 0.
Para h(t) = f (t) cos (2k+1)pit2L temos que
h(2L\u2212 t) = f (2L\u2212 t) cos
(2k+ 1)pi(2L\u2212 t)
2L
= \u2212 f (t) cos
(
(2k+ 1)pi \u2212
(2k+ 1)pit
2L
)
= \u2212 f (t) cos
(
pi \u2212
(2k+ 1)pit
2L
)
= f (t) cos
(
((2k+ 1)pit
2L
)
= h(t)
Assim segue da aplicac¸a\u2dco do item (b) que
a2k+1 =
2
L
\u222b L
0
f (t) cos
(2k+ 1)pit
2L
dt para k = 0, 1, 2, . . .
12
2. Lembrando que a integrac¸a\u2dco deve ser feita no intervalo [0, 2L]:
a2k+1 =
L
2
a2k+1( f
(0)
0, 14
)\u2212 a2k+1( f
(1)
0, 14
)
=
L
2
· 4 ·
1
(2k+ 1)pi
sen s
\u2223\u2223\u2223
(2k+1)pi
4
0
\u2212 4 ·
2L
(2k+ 1)2pi2
(s sen s+ cos s)
\u2223\u2223\u2223
(2k+1)pi
4
0
=
8L
(2k+ 1)2pi2
(
1\u2212 cos
(2k+ 1)pi
4
)
f (t) =
8L
pi
2
\u221e
\u2211
k=0
1\u2212 cos
(2k+1)pi
4
(2k+ 1)2
cos
(2k+ 1)pit
2L
b2k+1 =
L
2
b2k+1( f
(0)
0, 14
)\u2212 b2k+1( f
(1)
0, 14
)
=
L
2
· 4 ·
\u22121
(2k+ 1)pi
cos s
\u2223\u2223\u2223
(2k+1)pi
4
0
\u2212 4 ·
2L
(2k+ 1)2pi2
(\u2212s cos s+ sen s)
\u2223\u2223\u2223
(2k+1)pi
4
0
=
2L
(2k+ 1)2pi2
(
(2k+ 1)pi \u2212 4 sen
(2k+ 1)pi
4
)
f (t) =
2L
pi
2
\u221e
\u2211
k=0
(2k+ 1)pi \u2212 4 sen (2k+1)pi4
(2k+ 1)2
sen
(2k+ 1)pit
2L
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# 1.2 Algebra
Page 1 / 2
## Algebra
ALGEBRA
CLASS ASSIGNMENT 1
• Discover ALGEBRA step by step...
• In Algebra, we make use of letters in the place of unknowns (numbers that we do not know).
• Letters represent variables (values that may vary) and numbers are the constants (the values remain the same).
Look at the polynomial, for example
From the above, you will be able to recognise the following:
• The number of terms (terms are separated by + and - signs): 3 terms
• Coefficient of $x$ ² (the number immediately before $x$ ²): 3
• Coefficient of $x$ (the number immediately before $x$ ): - $\frac{1}{4}$
• Constant: 5
• The degree of expression (highest power of $x$ ): 2
• The expression is arranged in descending powers of $x$ .
• 3 $x$ ² means 3 x $x$ ² (3 multiplied by $x$ ²)
• $x$ ² means ( $x$ ) x ( $x$ ) ( $x$ multiplied by $x$ )
• What happens to ( + )and ( - ) signs during multiplication and division?
Here you have it:
• ( + ) x of ÷ ( + ) = ( + )
• ( - ) x of ÷ ( - ) = ( + )
• ( + ) x of ÷ ( - ) = ( - )
$\frac{\left(\frac{1}{4}{x}^{2}-x\right)}{4}+6$
• Indicate the following:
1.1 number of terms
1.2 coefficient of $x$
1.3 constant
1.4 degree of the expression
2. Now we can use variables to define the following with the magical language of mathematics --- i.e. algebraic expressions.
See if you can define these in the form of algebraic expressions:
Given Algebraic Expression
2.1 The sum of a number and 9
2.2 A number multiplied by 7
2.3 The difference between a and b
2.4 6 less than a number reduced by 7
2.5 The product of a number and b
2.6 Quotient of a number and 7
2.7 Square of a
2.8 Square root of a
2.9 Subtract the difference between a and b from their product
3. The following are referred to as flow diagrams – They consist ofa) inputb) formula in which the input number is substitutedc) output
Complete (a), (b) and (c)
4. See if you can determine a formula for the following and complete the table.
$x$ 2 5 8 10 15 47 y 7 11 17
formula: y =
HOMEWORK ASSIGNMENT 1
1. Determine a formula for each of the following and complete the table.
1.1 formula: y = ……………………………………………………
$x$ 2 5 8 9 12 20 y 10 16 22
1.2 formula: y = ……………………………………………………
$x$ 3 7 10 9 12 20 y 12 32 47
1.3 formula: y = ……………………………………………………
$x$ 1 3 4 9 12 20 y 1 9 16
1.4 formula: y = ……………………………………………………
$x$ 1 2 3 6 7 10 y 1 8 27
1.5 formula: y = ……………………………………………………
$x$ 1 2 4 9 12 20 y 2 5 17
2. The sketch shows matches arranged to form squares and combinations of squares.
2.1 Make a sketch to show four squares and indicate how many matches were used.
Matches? …………………………
2.2 Can you determine a formula that will provide a quick way for determining how many matches you will need to form ( $x$ ) number of squares?
y = ………………………………… (with y representing the number of matches)
2.3 Now make use of your formula to determine how many matches you will need to form 110 squares.
2.4 Determine how many squares you will be able to form with 2 005 matches.
3. Examine the following expression and answer the questions that follow:
$-\frac{1}{4}a+\frac{{a}^{2}}{5}+7+{3a}^{3}$
3.1 Arrange the expression in ascending powers of a.
3.2 Determine:
3.2.1 number of terms
3.2.2 coefficient of a ²
3.2.3 degree of the expression
3.2.4 constant term
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Got questions? Join the online conversation and get instant answers!
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## How Chaos Theory Works
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If you examine a bifurcation diagram closely, you begin to see interesting patterns. For example, start with a completed diagram, such as the one in the first picture.
Next, zoom in on the first doubling point. It looks like a rounded, sideways V. Now look at the smaller, sideways V's that come next in the series.
Now zoom in again, say, on that upper, smaller V.
Notice how this region of the diagram looks like the original. In other words, the large-scale structure of the figure is repeated multiple times. The doubling regions exhibit a quality known as self-similarity -- small regions resemble large ones. Even if you look in the chaotic areas of the diagram (which occur to the right), you can find this quality.
Self-similarity is a property of a class of geometric objects known as fractals . The Polish-born mathematician Benoît Mandelbrot coined the term in 1975, after the Latin word fractus , which means "broken" or "fragmented." He also worked out the basic math of the objects and described their properties. In addition to self-similarity, fractals also possess something known as fractal dimension , a measure of their complexity. The dimension is not an integer -- 1, 2, 3 -- but a fraction. For example, a fractal line has a dimension between 1 and 2.
The Koch snowflake -- named after the Swedish mathematician Helge van Koch -- stands as a classic example of a fractal. To derive the shape, van Koch established the following rules, first for a line:
• Divide a line segment into three equal parts
• Remove one-third of the segment out of the middle
• Replace the middle segment with two segments of the same length such that they all connect
• Repeat indefinitely on each line segment
The second picture shows what the first two iterations would look like:
If you start with an equilateral triangle and repeat the procedure, you end up with a snowflake that has a finite area and an infinite perimeter:
Today, fractals form part of the visual identity of chaos. As infinitely complex objects that are self-similar across all scales, they represent dynamical systems in all their glory. In fact Mandelbrot eventually proved that Lorenz's attractor was a fractal, as are most strange attractors. And they're not limited to the ruminations of scientists or the renderings of computers.
Fractals are found throughout nature -- in coastlines, seashells, rivers, clouds, snowflakes and tree bark. Before you take a field trip, however, be aware that self-similarity behaves a little differently in natural systems. In controlled mathematical environments, an object with self-similarity often displays an exact repetition of patterns at different magnifications. In nature, patterns obey statistical self-similarity -- they don't repeat exactly but parts of them show the same statistical properties at many different scales.
Please copy/paste the following text to properly cite this HowStuffWorks.com article:
## Despite tooling limitations, DAO optimists see new use cases for a democratic, token-based future
The adoption of decentralized autonomous organizations, or DAOs, has skyrocketed in the past year, and participants believe this is just the beginning, claiming more use cases will form in the subsector.
DAOs are community-led groups that, in theory, allow participants to make operational decisions without centralized leadership. The groups are self-governing, typically raising capital through a token linked to the DAO. The tokens often provide members voting power on governance rules, and, through smart contracts, those votes affect and dictate what DAOs do.
Many DAOs focus on raising funds to support a certain cause or to purchase an item, whether that be buying a copy of the Constitution or a golf course, like ConstitutionDAO or LinksDAO have respectively aimed to do.
DAOs have even been compared to a new frontier for coordination , but some DAO participants say examples of these are just beginning.
“There will be a lot of evolution [for DAOs] as we start to fit the technology into human behavior,” Sarah Wood, head of operations at Upstream , said during a panel at the Avalanche Summit.
“I see a world where you can use a DAO for your book club, or whatever you want.”
Whether it be friends pooling money or taking action together, collective activity without centralized leadership is something everyone does regardless of whether they’re in crypto, Wood noted. The DAO ecosystem is potentially useful for any person or organization that wants to pool funds or make decisions together, but there needs to be better education so others can understand it, she added.
“I think we will see more use cases of people coming together to purchase an item, team or property,” Imran Khan, a core contributor of web3 accelerator Alliance , told TechCrunch. “This idea of social coordination as a way to bring groups together to purchase something is easiest to digest and conduct.”
Similar to the way AssangeDAO raised over \$53 million to bid on an NFT that would support its mission to free WikiLeaks founder Julian Assange, Khan expects more people to come together in the future to support projects or missions globally.
“Aside from [DAOs purchasing assets], there will be more experiments to get people across the globe to follow a mission or theme,” Khan said.
Right now, DAOs are the third phase of online coordination, he said. In the past, Web 1.0 platforms connected people through email or chat rooms on sites like Yahoo or AOL. Then, Web 2.0 emerged, and online groups formed through social networking sites like Facebook or Reddit, but those groups often grew on the websites, not off of them, Khan said.
“It was always about growing Facebook, not the group,” Khan said. “So imagine being able to give a token to the group and self-grow and form; that’s very powerful. We’re going to grow and be disconnected from platforms and self-form, and I expect DAOs to be as big as nation-states.”
## The Democratic Peace Theory
It has been argued that the absence of war between democratic states ‘comes as close as anything we have to an empirical law in international relations.’ [1] Although statistically the probability of war between any two states is considerably low, the absence of war among liberal democracies across a wide range of different historical, economic, and political factors suggests that there is a strong predisposition against the use of military violence between democratic states. [2] This democratic peace proposition not only challenges the validity of other political systems (i.e., fascism, communism, authoritarianism, totalitarianism), but also the prevailing realist account of international relations, which emphasises balance-of-power calculations and common strategic interests in order to explain the peace and stability that characterises relations between liberal democracies. [3] This essay argues, however, that the structural and normative arguments of the democratic peace theory together offer a far more logical and convincing explanation for this seeming anomaly. Furthermore, in line with Immanuel Kant’s theory of perpetual peace, I argue that the global spread of democracy will result in greater international peace if this occurs in parallel with the strengthening of economic interdependence and international organisations. The difficulty lies in the significant risk of instability inherent in the process of democratisation and the uncertainty that remains in an ‘incomplete Kantian world’ where the Hobbesian state of anarchy has not yet entirely disappeared from the international system.
Structural Explanation
Of the two main variants of the democratic peace theory, the structural account argues that it is the institutions of representative government, which hold elected officials and decision-makers accountable to a wide electorate, that make war a largely unattractive option for both the government and its citizens. [4] Because the costs and risks of war directly affect large segments of the population, it is expected that the average voter will throw the incumbent leader/party out of office if they initiate a losing or unnecessary war, thus, providing a clear institutional incentive for democratic leaders to anticipate such an electoral response before deciding to go to war. [5] This view does not assume that all citizens and elected representatives are liberal-minded, but simply that democratic structures that give citizens leverage over government decisions will make it less likely that a democratic leader will be able to initiate a war with another liberal democracy. [6] Thus, even with an illiberal leader in place, institutions such as free speech, political pluralism, and competitive elections will make it difficult for these leaders to convince or persuade the public to go to war. [7]
Normative Explanation
Proponents of the normative/cultural perspective, by contrast, argue that shared democratic and liberal values best explain the peace that exists between democratic states. [8] According to this view, democratic political culture encourages peaceful means of conflict resolution which are extended beyond the domestic political process to other democratic states because leaders in both countries hold a reasonable expectation that their counterparts will also be able to work out their differences peacefully. [9] Political ideology, therefore, determines how democracies distinguish allies from adversaries: democracies that represent and act in their citizens’ interests are treated with respect and consideration, whereas nondemocracies that use violence and oppression against their own people are regarded with mistrust and suspicion. [10] The importance of perception means that even if a particular state has ‘enlightened citizens and liberal-democratic institutions,’ unless other democratic states regard it as a genuine liberal democracy then the democratic peace proposition will not hold. [11] This argument can, therefore, explain a number of contentious cases: Americans did not consider England democratic in 1812 because England was a monarchy (War of 1812) and liberals in the Union did not consider the Confederacy a liberal democracy because of their use of slavery (American Civil War). [12]
Although some scholars regard the institutional and normative explanations as mutually exclusive, a much more intuitive and persuasive defence of the democratic peace theory emerges from combining these two viewpoints. Thus, the particular democratic practices that make war with other liberal democracies unlikely – free and fair elections, the rule of law, free press, a competitive party system – are driven by both ‘converging expectations about what conventional behaviour is likely to be’ (institutions) and ‘standards for what behaviour ought to be’ (norms). [13] These two explanations are complimentary and mutually reinforcing: cultural norms influences the creation and evolution of political institutions, and institutions help generate a more peaceful moral culture over time. [14]
Criticism of the Theory
A great deal of criticism of the democratic peace theory is focused on methodology. It is argued that the subjectivity of the specifics definitions adopted in such highly empirical studies is likely to significantly affect the results, making it difficult to validate the theory with certainty. [15] But this is largely undermined by a large number of studies that show democracies are highly unlikely to fight each other irrespective of the definition of democracy, the type of cases considered, or the dispute/war threshold. [16] Furthermore, there has already been a significant increase in the number of democratic-democratic dyads from less than 2% of all political dyads in the 19 th century, to 13% from 1900-1945, and 11% over the 1946-89 period without any major conflict. [17]
More substantial criticism comes from scholars whom, while not questioning the empirical findings, put forth contending arguments to explain the causal relationship between democracy and peace. Realists argue that it is not common polities but rather common interests that can best explain the low incidence of wars between democracies. [18] Beginning with the Cold War, they point out that democratic states have been far more likely to formally align themselves with other democracies than in the century before, suggesting that common strategic interests are a more important factor than domestic political processes. [19] Thus, the particular structure of the international political system is the key factor determining how states will act. [20] But the realist critique has been largely disproven by studies that have persuasively found that democracy, rather than alliance, prevents conflict and war; nonaligned democracies are less likely to fight each other than aligned nondemocracies; and two nondemocratic states that share common interests are more likely to fight each other than two democracies that do not share common interests. [21]
Of course, the point on which critics of the democratic peace theory are largely correct is that liberal democracies are not significantly less likely to go to war with other nondemocratic states. The available evidence largely disproves the monadic proposition that democratic states are less prone to use force regardless of the regime type of the opposing state. [22] This is likely due to the fact that democratic states still function in an ‘incompletely Kantian world’ where democracies have only recently gone from being a minority to the slight majority within the post-Cold War period. [23] Power politics, therefore, is still a necessary reality for most democratic states, particularly given the high levels of conflict between mixed dyads. [24] Nonetheless, there are a number of important advantages for democracies: they are more likely to enter low-level conflicts than full-scale wars; more willing to refrain from escalating disputes into an actual war; [25] and less likely to initiate the use of violence against another state. [26]
More importantly perhaps, democracies that do initiate war are more likely to win than nondemocratic states. [27] Because public support for war in democracies decreases considerably over time, there is a strong incentive for democratic leaders and decision-makers to not only choose to initiate only wars that they can win, but ones they can win quickly. [28] Although there are a number of notable exceptions, such as the U.S.-led wars in Iraq, Afghanistan and Vietnam, this does suggest that the global spread of democracy would bring additional benefits beyond simply reducing the possibility of war between democratic states. This would include a greater number of low-level conflicts in proportion to full-scale wars, an increase in the number of states less likely to either initiate war or escalate non-violent confrontations into war, and a greater number of short, successful wars as opposed to long and protracted wars. Thus, even though an increase in the number of democratic states may not reduce the overall number of democratic-nondemocratic conflicts, this should not detract from these largely positive qualitative changes one would expect to occur.
A much more substantial argument comes from the dyadic proposition of the democratic peace theory: the observation that democracies create a separate and joint peace among other democratic states. [29] With an autocratic-democratic dyad, if the autocracy is replaced with a democracy it is argued that the likelihood of conflict will drop by 33 percent. [30] Moreover, beyond conflict and war, the evidence suggests that interstate rivalries among democracy dyads are also exceedingly rare and that a change in regime (from nondemocracy to democracy) will not only reduce the propensity for conflict or rivalry between any two states, but will actually accelerate this trend more rapidly over time. [31] It similarly follows then that coalitions of democratic states will also be better able to maintain mutual commitments and obligations because the institutional constraints of liberal democracy make it difficult to reverse any mutual commitments made through autonomous and accountable political institutions. [32] This predictability is not only absent for nondemocracies due to the lack of transparency and openness of their political systems, but actually negatively impacts their ability to win wars: the number of democratic partners increases the probability of winning a war by 62% whereas the number of nondemocratic partners decreases this likelihood by 44%. [33] What this suggests is that democracies should work to strengthen their formal alliances not only for normative or ideological reasons but for the expected efficiency gains this would provide and as a practical way of avoiding the collective action problems that frequently plague nondemocratic or mixed regime coalitions.
More positively, that there has not been any war between democracies despite a rapid growth in the number of democratic dyads within the international system (and thus an increase in the probability of conflict between democracies), [34] points to a significant trend: the incidence of conflict should gradually decline over time if more countries become democratic. [35] This is important not only because liberal democracies must still retain military force as a means to prevent or defend themselves from aggression in the current international system, but because democracies are more likely to receive challenges and threats to their security while this peace still remains ‘separate.’ [36]
Democratisation
There are two notable reasons, however, why the global spread of democracy may actually undermine prospects for international peace and they both have to do with the difficulties associated with the process of democratisation. First, a number of studies have shown that democratic transitions which occur when a country’s political institutions are particularly weak (often at the outset of the transition from autocracy to democracy), or when the elites within that country are threatened by the democratisation process itself (by having to respond to a wide and divergent range of newly-formulated interests), have a greater likelihood that this process will trigger aggressive nationalist sentiment and/or the outbreak of civil or inter-state war. [37] If political institutions are weak at the early stages of a transition, the rising demand for mass participation can provide an incentive for elites to adopt nationalist, ethno-religious, or populist policies, yet, crucially, before these elites can be held sufficiently accountable to the wider electorate. [38] A number of examples can be cited ranging from Napoleon III’s France, Wilhelmine Germany, and Taisho Japan to more recent cases such as Serbia under Slobodan Milosevic (the Yugoslav Wars), Peru and Ecuador in the late 1980s/early 1990s (Cenepa War 1995), Ethiopia’s 1998-2000 border war with Eritrea following the collapse of the Dergue dictatorship, and the 1999 India-Pakistan war after limited moves towards democratisation in both Pakistan and Kashmir. [39] This also extends to the observation that the vast majority of civil wars over the past century have occurred within transitional or mixed regimes, as opposed to either democratic or authoritarian regimes, which are more able to effectively contain repression by democratic or violent means, respectively. [40] Taking this into account, therefore, it is far more likely that a country will be able to successfully consolidate its transition if democratisation occurs according to a particular historical sequence: the emergence of a national identity, followed by the institutionalisation of the central government, and then mass electoral and political participation. [41]
The second problem relates to the first: most countries undergoing a transition to democracy will not necessarily be in a position to follow this particular sequence, yet even if they are it is not guaranteed that liberal democratic states will be able or willing to help. It is, therefore, important to be aware of the obvious limits of external military intervention. Even if liberal states adopt a cautious cost-benefit analysis in which they only intervene or assist states when they are certain that there is substantial and legitimate internal support present and when they have the consent of international bodies such as the UN (i.e., in Korea, Libya, Afghanistan), the act of helping overthrow an authoritarian regime may undermine those very liberal norms and values underpinning the democratic peace. [42] That the costs associated with such interventions are often quite considerable and can be difficult to justify domestically also means that even if there is a clear moral argument for helping authoritarian states democratise, political and economic considerations may still prevail. Similarly, although it is often states undergoing democratic transitions that initiate wars, their military weaknesses and political and social instability can also make them attractive targets for attack. [43] This was the case for East Timor following its independence vote in 1999 and Iran after its 1979 revolution when they were invaded by Indonesian and Iraqi forces respectively. [44] Thus, even though there is a very clear normative benefit to increasing the number of democracies within the international system, there is a real risk of instability and conflict if the transition does not establish the institutional preconditions for effective and accountable governance prior to mass political participation and elections, and if it takes place within an unstable regional/international environment. [45]
Wider Implications
Similarly, how liberal states conduct their foreign policy on an individual basis and collectively at the international level will largely determine whether the Kantian system can be successfully expanded. It is often argued by realists that the democratic decision-making process itself deprives policymakers of the necessary ‘coherence, long-range planning, flexibility and secrecy’ required to conduct an effective foreign policy. [46] According to this view, public opinion exerts an autonomous influence on the actions of political leaders that can distract democratic states from focusing on the most important imperatives: power and security. [47] But, as mentioned earlier, the very political institutions and patterns of behaviour that characterise liberal democracies also allow these states to best defend themselves and adopt a more cautious and effective approach to the use of force, thereby achieving the ‘best, securest, and safest outcomes for the most people.’ [48] Therefore, this not only challenges the key assumptions underlying realism – that normative goals preclude a clear and accurate analysis of international affairs – but the idea that relative military capabilities and the distribution of power among great powers alone should dictate foreign policy strategy. [49] Rather, democracies can best guarantee their own security by empowering their citizens and strengthening institutional checks and balances because these very factors have been shown to uphold the democratic peace and facilitate a more prudent foreign policy. [50]
At the international level, the recent increase in the number of democratic states provides a unique opportunity to reconstruct the norms and values underpinning the international system to more accurately reflect the peaceful interactions of democracies. [51] This would ideally mean strengthening the two other aspects of the Kantian system: international organisations and economic interdependence. Although the democratic peace represents the possibility of ‘uncoerced peace without central authority,’ [52] it is also the case that this liberal order has been best served when there has been a liberal state (i.e., the United States after World War II) that is both able and prepared to sustain the economic and political foundations of the wider liberal society beyond its own borders. [53] Strengthening a dense network of inter-governmental organisations (IGOs) that extend this responsibility to a larger number of democratic states and encourages greater cooperation among members through greater consultation and coordination, such as the WTO, IMF, World Bank, UN, and International Criminal Court, would arguably provide a stronger foundation for extending this perpetual peace outwards. [54] This also builds on studies that have shown the constraining effect of IGOs is greatest for politically relevant dyads – ‘contiguous pairs of states and pairs that include at least one major power’ – which also happen to account for the majority of interstate disputes and conflict. [55] Focusing efforts to more proactively include the largest nondemocracies (China, Vietnam, Russia, Iran) into this liberal international order, and to strengthen those elements of constitutional liberalism (rule of law, institutional checks on power, individual freedoms) lacking in illiberal democracies (Belarus, Bangladesh, Rwanda, Romania, Malaysia etc.) would arguably help consolidate the democratic peace most effectively. [56]
This is also the case for economic cooperation and interdependence. The observation that the likelihood of conflict between any two states with high levels of bilateral trade will be 33% lower than if those states only had an average level of economic interdependence suggests that democratic states will greatly benefit from upholding a liberal international economic system free of protectionism and mercantilist policies. [57] Because maintaining free and open trade relations rests on the assumption that market-based forces, rather than violence or coercion, will determine future economic transactions, the accompanying sense of mutual dependence will often act as a restraint on the use of military force. [58] Any accompanying increase in the quantity or quality of interstate communication is also likely to make it easier for democracies to understand the intentions and preferences of nondemocracies as well as their willingness to adhere to mutual agreements and commitments. [59]
The institutional and normative aspects of the democratic peace proposition, thus, provide a very clear, logical reason why the global spread of democracy will result in greater international peace: democratic political institutions make it difficult for governments to initiate war without the consent of the electorate, and the accompanying cultural norms mean democracies will favour a peaceful means of conflict resolution with one another. Of course, this would not necessarily reduce the overall incidence of war as the monadic proposition that democracies are less likely to use conflict regardless of regime type does not hold. But this would still produce a positive qualitative change: democracies are less likely to initiate wars, escalate nonviolent disputes into full-scale war, or engage in long and protracted military conflicts. More importantly, an increase in the number of democracies would extend the liberal peace to a greater number of countries, and increase the probability of winning war – arguably providing a strong normative and practical rationale for liberal states to conduct a more Wilsonian foreign policy. Recognising the inherent difficulties implicit with the democratisation process, however, greater effort should be made to encourage the consolidation of political institutions prior to mass political/electoral participation in transitional states. Strengthening international organisations that embody liberal norms and values, and encouraging economic interdependence with nondemocracies would also help mediate the strategic uncertainty and misperceptions that exist where the Kantian peace meets the Hobbesian state of anarchy.
Choi, Ajin. “The Power of Democratic Competition.” International Security 28, no. 1 (Summer 2003): 142-53.
Davenport, Christian, and David A. Armstrong II. “Democracy and the Violation of Human Rights: A Statistical Analysis from 1976 to 1996.” American Journal of Political Science 48, no. 3 (July 2004): 538-54.
Doyle, Michael W. “Kant, Liberal Legacies, and Foreign Affairs, Part 1.” Philosophy & Public Affairs 12, no. 3 (Summer 1983): 205-35.
______. “Kant, Liberal Legacies, and Foreign Affairs, Part 2.” Philosophy & Public Affairs 12, no. 4 (Autumn 1983): 323-53.
______. “Liberalism and World Politics.” The American Political Science Review 80, no. 4 (December 1986): 1151-69.
Elman, Miriam Fendius. “The Need for a Qualitative Test of the Democratic Peace Theory.” In Paths to Peace: Is Democracy the Answer? , edited by Miriam Fendius Elman, 1-57. Cambridge, Massachusetts: The MIT Press, 1997.
Farber, Henry S., and Joanne Gowa. “Polities and Peace.” International Security 20, no. 2 (Fall 1995): 123-46.
Gelpi, Christopher F., and Michael Griesdorf. “Winners or Losers? Democracies in International Crisis, 1918–94.” American Political Science Review 95, no. 3 (September 2001): 633-47.
Hensel, Paul R., Gary Goertz, and Paul F. Diehl. “The Democratic Peace and Rivalries.” The Journal of Politics 62, no. 4 (November 2000): 1173-88.
Jervis, Robert. “Theories of War in an Era of Leading-Power Peace.” The American Political Science Review 96, no. 1 (March 2002): 1-14.
Layne, Christopher. “Kant or Cant: The Myth of the Democratic Peace.” International Security 19, no. 2 (Autumn 1994): 5-49.
Levy, Jack S. “Domestic Politics and War.” The Journal of Interdisciplinary History 18, no. 4 (Spring 1988): 653-73.
Mansfield, Edward D., and Jack Snyder. Electing To Fight: Why Emerging Democracies Go To War . Cambridge, Massachusetts: MIT Press, 2005.
Maoz, Zeev. “The Controversy over the Democratic Peace: Rearguard Action or Cracks in the Wall?” International Security 22, no. 1 (Summer 1997): 162-98.
Mearsheimer, John J. “Back to the Future: Instability in Europe after the Cold War.” International Security 15, no. 1 (Summer 1990): 5-56.
Owen, John M. “How Liberalism Produces Democratic Peace.” International Security 19, no. 2 (Autumn 1994): 87-125.
Ray, James Lee. “Wars Between Democracies: Rare, or Nonexistent?” International Interactions 18, no. 3 (1993): 251-76.
______. Democracy and International Conflict: An Evaluation of the Democratic Peace Proposition . Columbia, South Carolina: University of South Carolina Press, 1995.
Reiter, Dan, and Allan C. Stam. Democracies at War . Princeton; Oxford: Princeton University Press, 2002.
Russett, Bruce. “Can A Democratic Peace Be Built?” International Interactions 18, no. 3 (1993): 277-82.
______. Grasping the Democratic Peace: Principles for a Post-Cold War World . Princeton, New Jersey: Princeton University Press, 1993.
______. “Democracy, War and Expansion through Historical Lenses.” European Journal of International Relations 15, no. 9 (2009): 9-36.
Russett, Bruce, and John R. Oneal. Triangulating Peace: Democracy, Interdependence, and International Organizations . New York: W. W. Norton & Company, 2001.
Spiro, David E. “The Insignificance of the Liberal Peace.” International Security 19, no. 2 (Autumn 1994): 50-86.
Zakaria, Fareed. “The Rise of Illiberal Democracy.” Foreign Affairs 76, no. 6 (November/December 1997): 22-43.
[1] Jack S. Levy, “Domestic Politics and War,” The Journal of Interdisciplinary History 18, no. 4 (Spring 1988): 661-62.
[2] Michael W. Doyle, “Kant, Liberal Legacies, and Foreign Affairs, Part 1,” Philosophy & Public Affairs 12, no. 3 (Summer 1983): 213-15, 17; Christopher F. Gelpi and Michael Griesdorf, “Winners or Losers? Democracies in International Crisis, 1918–94,” American Political Science Review 95, no. 3 (September 2001): 633-34; Bruce Russett, “Democracy, War and Expansion through Historical Lenses,” European Journal of International Relations 15, no. 9 (2009): 11-12.
[3] Michael W. Doyle, “Liberalism and World Politics,” The American Political Science Review 80, no. 4 (December 1986): 1156-57.
[4] Russett, “Democracy, War and Expansion through Historical Lenses,” 21-22; Bruce Russett, Grasping the Democratic Peace: Principles for a Post-Cold War World (Princeton, New Jersey: Princeton University Press, 1993), 38-40.
[5] Russett, “Democracy, War and Expansion through Historical Lenses,” 21-22.
[6] John M. Owen, “How Liberalism Produces Democratic Peace,” International Security 19, no. 2 (Autumn 1994): 123-24; Edward D. Mansfield and Jack Snyder, Electing To Fight: Why Emerging Democracies Go To War (Cambridge, Massachusetts: MIT Press, 2005), 23-27.
[7] Owen, “How Liberalism Produces Democratic Peace,” 123-24.
[8] Miriam Fendius Elman, “The Need for a Qualitative Test of the Democratic Peace Theory,” in Paths to Peace: Is Democracy the Answer? , ed. Miriam Fendius Elman (Cambridge, Massachusetts: The MIT Press, 1997), 11-12.
[10] Owen, “How Liberalism Produces Democratic Peace,” 89-90.
[11] Ibid.: 96-97.
[13] Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 29-30.
[14] Bruce Russett and John R. Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations (New York: W. W. Norton & Company, 2001), 53; James Lee Ray, Democracy and International Conflict: An Evaluation of the Democratic Peace Proposition (Columbia, South Carolina: University of South Carolina Press, 1995), 33-37.
[15] David E. Spiro, “The Insignificance of the Liberal Peace,” International Security 19, no. 2 (Autumn 1994): 55, 62; James Lee Ray, “Wars Between Democracies: Rare, or Nonexistent?,” International Interactions 18, no. 3 (1993): 252-54.
[16] Zeev Maoz, “The Controversy over the Democratic Peace: Rearguard Action or Cracks in the Wall?,” International Security 22, no. 1 (Summer 1997): 175-77; Ray, “Wars Between Democracies: Rare, or Nonexistent?,” 269-70.
[17] Maoz, “The Controversy over the Democratic Peace: Rearguard Action or Cracks in the Wall?,” 190.
[18] Henry S. Farber and Joanne Gowa, “Polities and Peace,” International Security 20, no. 2 (Fall 1995): 145-46.
[20] Christopher Layne, “Kant or Cant: The Myth of the Democratic Peace,” International Security 19, no. 2 (Autumn 1994): 10-12; John J. Mearsheimer, “Back to the Future: Instability in Europe after the Cold War,” International Security 15, no. 1 (Summer 1990): 12-13.
[21] Maoz, “The Controversy over the Democratic Peace: Rearguard Action or Cracks in the Wall?,” 175-77; Gelpi and Griesdorf, “Winners or Losers? Democracies in International Crisis, 1918–94,” 45-46.
[22] Elman, “The Need for a Qualitative Test of the Democratic Peace Theory,” 14-18; Layne, “Kant or Cant: The Myth of the Democratic Peace,” 12-13.
[23] Russett, “Democracy, War and Expansion through Historical Lenses,” 13-14.
[25] Democratic states are, however, more willing to enter into non-violent confrontations even if they generally refrain from escalating these disputes into war.
[26] Russett, “Democracy, War and Expansion through Historical Lenses,” 14.
[27] Dan Reiter and Allan C. Stam, Democracies at War (Princeton; Oxford: Princeton University Press, 2002), 10-11.
[28] Ibid., 178-79.
[29] Elman, “The Need for a Qualitative Test of the Democratic Peace Theory,” 10-14.
[30] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 114-16.
[31] Paul R. Hensel, Gary Goertz, and Paul F. Diehl, “The Democratic Peace and Rivalries,” The Journal of Politics 62, no. 4 (November 2000): 1187.
[32] Ajin Choi, “The Power of Democratic Competition,” International Security 28, no. 1 (Summer 2003): 144-45.
[33] Ibid.: 146-49. Ajin Choi elaborates that, ‘According to the results of the marginal impact analysis presented in Table 1, the number of democratic partners variable increases the probability of winning a war by 62 percentage points as this variable moves from its minimum to maximum value and all other variables are set at their mean or modal values. The number of nondemocratic partners variable, on the other hand, decreases the probability of winning by 44 percentage points under the same conditions.’
[34] Maoz, “The Controversy over the Democratic Peace: Rearguard Action or Cracks in the Wall?,” 190.
[35] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 114-16, 22-24.
[36] Gelpi and Griesdorf, “Winners or Losers? Democracies in International Crisis, 1918–94,” 645-46; Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 302. It is argued that the perceived reluctance of liberal democracies to use force may actually lead to a greater number of military challenges in spite of their military capabilities because the openness of their political system paradoxically only makes their bargaining tactics credible to opponents when they appear willing to use force.
[37] Christian Davenport and David A. Armstrong II, “Democracy and the Violation of Human Rights: A Statistical Analysis from 1976 to 1996,” American Journal of Political Science 48, no. 3 (July 2004): 551-53; Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 265-66.
[38] Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 39-40.
[39] Fareed Zakaria, “The Rise of Illiberal Democracy,” Foreign Affairs 76, no. 6 (November/December 1997): 36-38; Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 4-6.
[40] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 70-71.
[41] Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 16-19.
[42] Bruce Russett, “Can A Democratic Peace Be Built?,” International Interactions 18, no. 3 (1993): 279-80.
[43] Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 33-34.
[44] Ibid., 4-6, 13-14.
[45] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 116-22; Mansfield and Snyder, Electing To Fight: Why Emerging Democracies Go To War , 273-74.
[46] Levy, “Domestic Politics and War,” 659-61.
[47] Reiter and Stam, Democracies at War , 195-97.
[49] Layne, “Kant or Cant: The Myth of the Democratic Peace,” 49; Reiter and Stam, Democracies at War , 195-97.
[50] Reiter and Stam, Democracies at War , 202-05; Choi, “The Power of Democratic Competition,” 153.
[51] Russett, “Can A Democratic Peace Be Built?,” 280-81; Ray, Democracy and International Conflict: An Evaluation of the Democratic Peace Proposition , 204-06.
[52] Robert Jervis, “Theories of War in an Era of Leading-Power Peace,” The American Political Science Review 96, no. 1 (March 2002): 11.
[53] Doyle, “Kant, Liberal Legacies, and Foreign Affairs, Part 1,” 232-33.
[54] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 280-81; Doyle, “Liberalism and World Politics,” 1157-58.
[55] Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 280-81.
[56] Zakaria, “The Rise of Illiberal Democracy,” 25-26.
[57] Michael W. Doyle, “Kant, Liberal Legacies, and Foreign Affairs, Part 2,” Philosophy & Public Affairs 12, no. 4 (Autumn 1983): 347-48; Russett and Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations , 277-80.
[58] Doyle, “Kant, Liberal Legacies, and Foreign Affairs, Part 1,” 231.
[59] Choi, “The Power of Democratic Competition,” 144-45.
Written by: Kevin Placek Written at: University of Melbourne Written for: Dr. David Mickler Date written: November 2011
## Further Reading on E-International Relations
• Harnessing Alterity to Address the Obstacles of the Democratic Peace Theory
• Kant, Doyle, and the Democratic Peace Thesis: A Postcolonial Critique
• The Implicit Imperialism of Democratic Peace
• A Pareto Optimal Peace: How the Dayton Peace Agreement Struck a Unique Balance
• Neopatrimonialism and Democratic Consolidation in Nigeria
• Hungary’s Democratic Backsliding as a Threat to EU Normative Power
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## What Is the Democratic Peace Theory? Definition and Examples
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The Democratic Peace Theory states that countries with liberal democratic forms of government are less likely to go to war with one another than those with other forms of government. Proponents of the theory draw on the writings of German philosopher Immanuel Kant and, more recently, U.S. President Woodrow Wilson , who in his 1917 World War I message to Congress stated that “The world must be made safe for democracy.” Critics argue that the simple quality of being democratic in nature may not be the main reason for the historic tendency of peace between democracies.
## Key Takeaways
• The Democratic Peace Theory holds that democratic countries are less likely to go to war with one another than non-democratic countries.
• The theory evolved from the writings of German philosopher Immanuel Kant and the adoption of the 1832 Monroe Doctrine by the United States.
• The theory is based on the fact that declaring war in democratic countries requires citizen support and legislative approval.
• Critics of the theory argue that merely being democratic may not be the primary reason for peace between democracies.
## Democratic Peace Theory Definition
Dependent on the ideologies of liberalism , such as civil liberties and political freedom, the Democratic Peace Theory holds that democracies are hesitant to go to war with other democratic countries. Proponents cite several reasons for the tendency of democratic states to maintain peace, including:
• The citizens of democracies usually have some say over legislative decisions to declare war.
• In democracies, the voting public holds their elected leaders responsible for human and financial war losses.
• When held publicly accountable, government leaders are likely to create diplomatic institutions for resolving international tensions.
• Democracies rarely view countries with similar policies and form of government as hostile.
• Usually possessing more wealth that other states, democracies avoid war to preserve their resources.
The Democratic Peace Theory was first articulated by German philosopher Immanuel Kant in his 1795 essay entitled “ Perpetual Peace .” In this work, Kant argues that nations with constitutional republic governments are less likely to go to war because doing so requires the consent of the people—who would actually be fighting the war. While the kings and queens of monarchies can unilaterally declare war with little regard for their subjects’ safety, governments chosen by the people take the decision more seriously.
The United States first promoted the concepts of the Democratic Peace Theory in 1832 by adopting the Monroe Doctrine . In this historic piece of international policy, the U.S. affirmed that it would not tolerate any attempt by European monarchies to colonize any democratic nation in North or South America.
The democratic peace theory does not claim that democratic countries are generally more peaceful than nondemocratic countries. However, the theory’s claim that democratic countries rarely fight each other is widely regarded as true by international relations experts and further supported by history.
Kant’s “Perpetual Peace” essay remained largely unnoticed until the mid-1980s when the American international-relations scholar Michael Doyle cited it in arguing that the “zone of peace” envisioned by Kant had gradually become reality. After the Cold War, which pitted democratic states against communist states, the democratic peace theory became one of the most studied topics of research in international relations. This research has shown that while wars between non-democracies, or between democracies and non-democracies have been common, wars between democracies have been extremely rare.
Interest in the democratic peace theory has not been limited to the halls of academia. During the 1990s, U.S. President Bill Clinton featured it in many aspects of his administration’s foreign policy of spreading democracy throughout the world. Clinton’s foreign policy asserted that if the formerly autocratic nations of Eastern Europe and the collapsed Soviet Union converted to democracy, the United States and its allies in Europe would no longer need to restrain those countries militarily because democracies do not attack each other.
The democratic peace theory similarly influenced U.S. foreign policy in the Middle East in the aftermath of the September 11, 2001, terrorist attacks. U.S. policymakers believed that a zone of democracy equaled a zone of peace and security that supported President George W. Bush’s strategy of using military force to overthrow Saddam Hussein’s ruthless dictatorship in Iraq. Bush’s administration hoped that the democratization of Iraq would eventually result in the spread of democracy throughout the Middle East.
## Democracies and War in the 1900s
Perhaps the strongest evidence supporting the Democratic Peace Theory is the fact that there were no wars between democracies during the 20th century.
As the century began, the recently ended Spanish-American War had seen the United States defeat the monarchy of Spain in a struggle for control of the Spanish colony of Cuba.
In World War I , the U.S. allied with the democratic European empires to defeat the authoritarian and fascist empires of Germany, Austro-Hungary, Turkey, and their allies. This led to World War II and eventually the Cold War of the 1970s, during which the U.S. led a coalition of democratic nations in resisting the spread of authoritarian Soviet communism .
Most recently, in the Gulf War (1990-91), the Iraq War (2003-2011), and the ongoing war in Afghanistan , the United States, along with various democratic nations fought to counter international terrorism by radical jihadist factions of authoritarian Islamist governments. Indeed, after the September 11, 2001, terror attacks , the George W. Bush administration based its use military force to topple Saddam Hussein’s dictatorship in Iraq on the belief that it would bring democracy—thus peace—to the Middle East.
While the claim that democracies rarely fight each other has been widely accepted, there is less agreement on why this so-called democratic peace exists.
Some critics have argued that it was actually the Industrial Revolution that led to peace during the nineteenth and twentieth centuries. The resulting prosperity and economic stability made all of the newly modernized countries—democratic and nondemocratic—much less belligerent toward each other than in preindustrial times. Several factors arising from modernization may have generated a greater aversion to war among industrialized nations than democracy alone. Such factors included higher standards of living, less poverty, full employment, more leisure time, and the spread of consumerism. Modernized countries simply no longer felt the need to dominate each other in order to survive.
Democratic Peace Theory has also been criticized for failing to prove a cause-and-effect relationship between wars and types of government and the ease with which definitions of “democracy” and “war” can be manipulated to prove a non-existent trend. While its authors included very small, even bloodless wars between new and questionable democracies, one 2002 study contends that as many wars have been fought between democracies as might be statistically expected between non-democracies.
Other critics argue that throughout history, it has been the evolution of power, more than democracy or its absence that has determined peace or war. Specifically, they suggest that the effect called “liberal democratic peace” is really due to “realist” factors including military and economic alliances between democratic governments.
## Sources and Further Reference
• Owen, J. M. “ How Liberalism Produces Democratic Peace .” International Security (1994).
• Schwartz, Thomas and Skinner, Kiron K. (2002) “ The Myth of the Democratic Peace .” Foreign Policy Research Institute.
• Gat, Azar (2006). “ The Democratic Peace Theory Reframed: The Impact of Modernity .” Cambridge University Press.
• Pollard, Sidney (1981). “ Peaceful Conquest: The Industrialization of Europe, 1760–1970 .” Oxford University Press.
• Democracy Promotion as Foreign Policy
• U.S. Policy in the Middle East: 1945 to 2008
• American Manifest Destiny and Modern Foreign Policy
• Understanding the Bush Doctrine
• Impacts of the Iraq War on the Middle East
• Is Iraq a Democracy?
• Why Did the United States Go to War with Iraq?
• The Cold War in Europe
• What Is a Coalition Government?
• The Evolution of American Isolationism
• Origins of the Cold War in Europe
• Cold War Glossary
• U.S. Foreign Policy 101
• Did Oil Drive the US Invasion of Iraq?
• Ostpolitik: West Germany Talks to the East
• Iraq | Facts and History
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## DEMOCRATIC PEACE THEORY
from USQ Law Society Law Review Winter Edition 2021
by USQLS Law Review
## CHELSEA KEIRSNOWSKI
The democratic peace theory stems from basic principles devised by philosopher Immanuel Kant in 1795 that were later researched substantially in the 1960s.1 The theory proposes that democracies, especially those older than 3 years, are unlikely to be involved in wars against each other.2 It has been recognised that longevity of governance from a political party within a state’s regime further decreases the likelihood of interstate conflict.3 This essay will analyse the democratic peace theory and examine how a current sway toward authoritarianism in world politics is likely to bring increased global conflict. The heightened risk of war is linked to the erosion of stabilising democratic frameworks which inhibit belligerence and the disruption caused by changes in regimes and policy. This is intensified by instability caused by COVID19 and the rejection of mainstream politics and interstate agreements that previously ensured peace through interdependence.
A failure to ensure legitimacy and support through sufficient performance in authoritarian states, led to approximately 30 countries transitioning to democracy between 1974 and 1990, including the collapse of the Soviet Union and its occupied territories.4 Political scientist, Samuel P. Huntington, predicted this proliferation of democracy would not lead to world peace as each civilisation has their own culture that the population will choose to hold on to, rather than abandon for a different culture, such as by adopting a new way of living by becoming a democracy.5 He predicted these differences, such as the pressure initiated by the West on democratisation, and the retention by other civilisations of their culture, systems of rule and religion, will lead to conflict. Following Huntington’s hypothesis, circa 2007 began the rise of authoritarianism in South America and Africa, and growing support for populism in Eastern Europe. Reasons for this vary for each country but include a lack of repercussion without global governance, isolationism, and political and social instability.6 Recently, COVID-19 has accelerated the rise of authoritarianism. In some instances, there has been a genuine attempt to protect public health through decisive action and heightened regulations, however, some governments have profited from an opportunistic acquisition of power while global attention was preoccupied. The Democracy Index for 2020 showed democracy was in its worst state since the Index was developed by The Economist Intelligence Unit in 2006.7 A developing
1 Immanuel Kant, Perpetual Peace: A Philosophical Sketch (S. Sonnenschein, 1795); Dean Babst, ‘Elective Governments—A Force For Peace’ (1964) 3(1) The Wisconsin Sociologist 9, 9. 2 Spencer Weart, Never at War (Yale University Press, 1998). 3 Anais Marin, ‘Dictatorial peace? Comparing the conflict-proneness of authoritarian regimes in post-Soviet Eurasia: a research agenda’ (2015) 59 Research Gate 1, 18; Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297. 4 Samuel Huntington, ‘Democracy’s Third Wave’ (1991) 2(2) Journal of Democracy 12, 12. 5 Samuel Huntington, ‘The Clash of Civilizations?’ (1993) 72(3) Foreign Affairs 22, 40-1. 6 Ivan Krastev, ‘Eastern Europe’s Illiberal Revolution’, Foreign Affairs (Article, May/June 2018) <https://www.foreignaffairs.com/articles/hungary/2018-04-16/eastern-europes-illiberal-revolution>. 7 The Economist Intelligence Unit, ‘Global democracy has another bad year’, The Economist (Web Page, 22 January 2020) <https://www.economist.com/graphic-detail/2020/01/22/global-democracy-has-another-badyear>.
factor in the rise of authoritarianism is marked by the increase in power of China and Russia and the consequent rise in tensions with the US.
Whilst authoritarianism is not inherently belligerent, the frameworks of democratic institutions promote perpetual peace, while elements of authoritarianism often lead to the contrary, as indicated by the democratic peace theory. This essay will not use evidence based on whether democracies or authoritarian states initiate conflict because the initiating causes of conflict are often multifaceted, and each side tends to blame the other.8 Rather, this essay will engage the debate on the faults of democracies or authoritarian states which initiate conflict. Research concerning the correlation between democracy and lack of war uses different definitions of what constitutes as a democracy, war or armed conflict.9 Some researchers argue that many studies use restricting definitions, resulting in small sample sizes that ignore outliers that may refute the democratic peace theory.10 Despite this, most research shows that many of these outliers are states transitioning to democracy and that there are no wars between mature liberal democracies.11 A further study also supports that democratic states are also less likely to be involved in smaller militarised interstate disputes.12 Democratic peace is attributed to: requiring broad political support in order to mobilise for war; the ensured accountability of governments for their public decisions leading to increased hesitation to initiate war; the use of diplomatic efforts to resolve tensions peacefully; typically possessing wealth and resources that states do not wish to endanger; as well as the tendency to form alliances, collaborate, negotiate and their commercial interdependence.13 Further, democratic states see the same impediments in other democracies and thus can expect a peaceful relationship. Aversion to force is not so apparent in authoritarian foreign policies and this prompts democracies to act with more aggressive policies to protect their liberal democratic policies from being exploited.14 The repressive policies of authoritarian states can create instability through producing violent extremism and refugees attempting to flee.15 Authoritarian states have higher frequency internal systematic
8 Nils Gleditsch, Lene Christiansen and Håvard Hegre 2004, ‘Democratic Jihad? Military Intervention and Democracy’, PRIO (Conference Paper, 20 March 2004) <https://www.prio.org/Publications/Publication/?x=525>. 9 Rudolph Rummel, Power Kills: Democracy as a Method of Nonviolence (Transaction Publishers, 1997); James Ray, ‘A Lakatosian View of the Democratic Peace Research Program’ in Colin Elam and Miriam Elman (eds), Progress in International Relations Theory (MIT Press, 2003) 1; Spencer Weart, Never at War (Yale University Press, 1998). 10 James Ray, ‘Does Democracy Cause Peace?’ (1998) 1 Annual Review of Political Science 27, 89. 11 Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297, 297; Rudolph Rummel, Power Kills: Democracy as a Method of Nonviolence (Transaction Publishers, 1997). 12 James Ray, ‘A Lakatosian View of the Democratic Peace Research Program’ in Colin Elam and Miriam Elman (eds), Progress in International Relations Theory (MIT Press, 2003) 1. 13 Christopher Gelpi and Michael Griesdorf, ‘Winners or Losers? Democracies in International Crisis, 1918–94’ (2001) 95(3) American Political Science Review 633; James Ray, ‘A Lakatosian View of the Democratic Peace Research Program’ in Colin Elam and Miriam Elman (eds), Progress in International Relations Theory (MIT Press, 2003) 1; David Leblang and Steve Chan, ‘Explaining Wars Fought by Established Democracies: Do Institutional Constraints Matter?’ (2003) 56(4) Political Research Quarterly 385. 14 Christopher Gelpi and Michael Griesdorf, ‘Winners or Losers? Democracies in International Crisis, 1918–94’ (2001) 95(3) American Political Science Review 633. 15 Jacob Carozza 2017, ‘Democracy is Retreating, Authoritarianism is Rising’, Belfer Center for Science and International Affairs (Article, Fall/Winter 2017/8) <https://www.belfercenter.org/publication/democracyretreating-authoritarianism-rising>.
violence such as terrorism, genocide, politicide and democide.16 Whilst the amount of internal violence has not been proven to correlate with having a predisposition to intensify militarised interstate disputes, authoritarian states do not face the same institutionalised constraints that impede involvement in conflict.17 Therefore, the democratic peace theory can be understood through an analysis of the mechanics of democratic systems that promote peace.
Causes of war can be attributed to a multitude of factors, the lack of constitutional constraints in authoritarian regimes being one of them, however instability and regime changes also play a significant role. Whilst evidence shows democracies do not fight each other, there is not widespread support for monadic democratic peace, which claims democracies are less belligerent in general.18 No pair of personalist dictators or military regimes have been at war with each other since WWII either, alluding to the possibility of an illiberal peace.19 Studies of both the democratic peace and illiberal peace phenomenon have found evidence that both mature democracies and mature authoritarian states are non-belligerent, whilst states in transition are more likely to be involved in military conflict.20 This indicates that stability and length of a regime may be as important as democracy for contributing to regional and world peace. Political change, towards both democracy or authoritarianism, increases the likelihood of civil war. Thus, the current rise of authoritarianism may lead to conflict due to the political instability caused by the changing of regimes. Interestingly, transitions to democracy have been shown to cause more conflict than democratic backsliding.21 Government longevity is an indicator of stability as well as numerous other factors such as; lack of violence, the progress of the state’s human development, political legitimacy, and regime responsiveness in overcoming problems.22 Despite government longevity being one of many determining factors for peace, the quintessence of this being Italy changing its government sixty times in sixty years, there is evidence that transitions substantially contribute to instability.23 A study of
16 Alberto Abadie, ‘Poverty, Political Freedom, and the Roots of Terrorism’ (2004) NBER Working Paper Series 1; Barbara Harff, ‘No Lessons Learned from the Holocaust? Assessing Risks of Genocide and Political Mass Murder since 1955’ (2003) 97(1) American Political Science Review 57; Rudolph Rummel, Power Kills: Democracy as a Method of Nonviolence (Transaction Publishers, 1997). 17 Anais Marin, ‘Dictatorial peace? Comparing the conflict-proneness of authoritarian regimes in post-Soviet Eurasia: a research agenda’ (2015) 59 Research Gate 1, 21; Bruce Russett and John Oneal, Triangulating Peace: Democracy, Interdependence, and International Organizations (W.W Norton & Company, 2001). 18 Harald Müller and Jonas Wolff, ‘Dyadic Democratic Peace Strikes Back’, Academia (Conference Paper, 9 August 2004) <https://www.academia.edu/2486355/Dyadic_Democratic_Peace_Strikes_Back_Reconstructing_the_Social_Co nstructivist_Approach_After_the_Monadic_Renaissance>. 19 Mark Peceny, Caroline Beer and Shannon Sanchez-Terry, ‘Dictatorial Peace?’ (2002) 96(1) American Political Science Review 15. 20 Anais Marin, ‘Dictorial peace? Comparing the conflict-proneness of authoritarian regimes in post-Soviet Eurasia: a research agenda’ (2015) 59 Research Gate 1, 18; Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297. 21 Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297, 297. 22 Cecilia Sottilotta, ‘Political Stability in Authoritarian Regimes: Lessons from the Arab Uprisings’ (2013) (1301) Instituto Affari Internazionali 1, 3-4. 23 Cecilia Sottilotta, ‘Political Stability in Authoritarian Regimes: Lessons from the Arab Uprisings’ (2013) (1301) Instituto Affari Internazionali 1, 3; Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297, 297; Anais Marin, ‘Dictorial peace? Comparing the conflict-proneness of authoritarian regimes in post-Soviet Eurasia: a research agenda’ (2015) 59 Research Gate 1, 18.
authoritarian regimes in post-Soviet Eurasia showed that consolidated authoritarian regimes were the most pacific, as seen with Belarus, Turkmenistan, Azerbaijan and Tajikstan.24 Uzbekistan was the only exception as it more commonly initiated interstate conflict and fortified its borders during disputes with the Taleban.25 The states that attempted liberal reforms over the past 20 years; being Russia, Ukraine, Georgia, Moldova and Kyrgyzstan, were found to be more hawkish.26 Therefore, once the states transitioning to authoritarianism are consolidated, there may be peace, but until then, conflict is likely. This has been supported by numerous other studies.27 A study by Azar Gat, Professor of National Security at Tel Aviv University, which investigated transitions to democracy, found that the more prompt and untroubled the conversion was, the more peaceable the resulting nation would be.28 This supports the hypothesis that transitions and instability breed war. An example of this is the Thucydides Trap theory which proposes that when one state is at risk of being overtaken as global hegemon, is a likely result.29 Both the Thucydides Trap and the democratic peace theory warn of likely conflict due to the rising power of China threatening the position of the US as the current hegemon. This would be a significant sway in the weight of authoritarian global power which would likely be a destabilising transition.
The increasing significance of the authoritarian model will likely beget more conflict as this is already becoming apparent in global affairs. The decline in the Democracy Index since 2006 has been largely driven by the rise of authoritarianism in Latin America and Sub-Saharan Africa coupled with deterioration of democracy in Eastern Europe.30 Disenchantment with mainstream politics, Euroscepticism, xenophobia, disapproval of immigrants and a general distrust of a corrupt ‘elite’ has led to the rise of populism in Eastern Europe.31 Populism as a political approach appeals to citizens by proposing to give a voice to the people who feel they have been silenced by an upper class.32 Populism may initially appear democratic, and it may not lead to authoritarianism in all circumstances. However, the recent occurrences in Venezuela illuminate the danger populism poses. Venezuelan populist leader, Hugo Chavez, won the 1998 elections and subsequently manipulated the country’s ‘democracy’ to place the popular will of ‘ordinary’ citizens against whoever opposed with the changes, who are consequently branded as an evil ‘elite’. Mr Chavez used corrupt judges as a reason to grant himself the power to control the judiciary, thus overstepping the separation of powers that ensured he did not possess
24 Anais Marin, ‘Dictorial peace? Comparing the conflict-proneness of authoritarian regimes in post-Soviet Eurasia: a research agenda’ (2015) 59 Research Gate 1, 21. 25 Ibid 14. 26 Ibid 21. 27 Ursula Daxecker, ‘Perilous Polities? An Assessment of the Democratization-Conflict Linkage’ (2007) 13(4) European Journal of International Relations 527; Edward Mansfield and Jack Snyder, ‘Democratic Transitions, Institutional Strength, and War’ (2002) 56(2) International Organization 297. 28 Azar Gat, ‘The Democratic Peace Theory Reframed: The Impact of Modernity’ (2005) 58(1) World Politics 73, 79-80. 29 Graham Allison, ‘The Thucydides Trap’, FP (Article, 9 June 2017) <https://foreignpolicy.com/2017/06/09/the-thucydides-trap/>. 30 The Economist Intelligence Unit, ‘Global democracy has another bad year’, The Economist (Web Page, 22 January 2020) <https://www.economist.com/graphic-detail/2020/01/22/global-democracy-has-another-badyear>. 31 Jacques Rupnik, ‘Is East-Central Europe Backsliding? From Democratic Fatigue to Populist Backlash’ (2007) 18(4) Journal of Democracy 17, 18-22. 32 Rogers Brubaker, ‘Why Populism?’ (2017) 46(5) Theory & Society 357, 359.
overarching supremacy.33 Populism can guise consolidating power as giving power to the people, which current Venezuelan President, Nicolas Maduro, continued to do until democracy was eliminated.34 Similarly, liberal checks and balances have been undermined in Hungary and Poland by limiting the power of the judiciary; measures which are inconsistent with EU standards on judicial independence.35 Polish fears of Russian invasion led to the election of the populist Law and Justice party whose authoritarian policies appeared necessary to make firm and decisive measures.36 In Hungary, the government, led by Viktor Orban, controls the media so extensively that opposition parties are unable to campaign sufficiently, which is exacerbated by pro-democracy organisations being restricted by harsh regulations quelling their operation.37 The media is utilised by the government to broadcast propaganda that presents Muslim refugees as a severe threat to society and culture in order to make Orban’s authoritarian measures appear necessary to safeguard the community. In the same vein, Orban has used COVID-19 as an excuse to tighten his grip on the media and in March granted himself the power to rule by decree indefinitely. This power was subsequently reduced in June to limit Orban’s ability to alter laws on fundamental rights. However, this still leaves Hungary as more authoritarian than prior to the pandemic.38 In 2018 the European Parliament admonished Orban’s government as a “systemic threat” to the rule of law and democracy and gave a formal warning under Article 7 of the Treaty of the European Union.39 Whilst democracy is still more intact in the Czech Republic and Romania, there are increasing concerns these states are following a similar path to authoritarianism, as they also have put in place measures which impinge on judicial independence.40 Tear gas and water cannons were used on peaceful protesters against government corruption in Romania in 2018, highlighting the deteriorating state of their democracy.41 As democracy fails in Eastern Europe, so do the institutionalised frameworks that operate to maintain peace and security. By consolidating power, leaders grant themselves the ability to make decisions, such as potentially going to war, without the decision being reviewed. Whilst COVID-19 may have created extraordinary circumstances that require
33 Allan Brewer-Carias, Dismantling Democracy in Venezuela: The Chavez Authoritarian Experiment (Cambridge University Press, 2010) 24. 34 Adriana Boersner, ‘The Path Toward Authoritarianism in Venezuela’ (2019) Research Gate 1. 35 Zoll Fryderyk and Leah Wortham, ‘Judicial Independence and Accountability: Withstanding Political Stress in Poland’ (2019) 42(3) Fordham International Law Journal 875, 904. 36 Joanna Fomina and Jacek Kucharczyk, ‘The Specter Haunting Europe: Populism and Protest in Poland’ (2016) 27(4) Journal of Democracy 58, 60. 37 Peter Bajomi-Lazar, ‘The Party Colonisation of the Media: The Case of Hungary’ (2012) 27(1) East European Politics and Societies 69, 70. 38 Valerie Hopkins and Ben Hall, ‘Chill Descends Upon Hungary After Viktor Orban’s Power-Grab’, The Financial Times (Article, 19 March 2020) <https://www.ft.com/content/27243d36-bf9d-411f-89ed1d118ae639f8>; Lili Bayer, ‘Hungary Replaces Rule by Decree with State of Medical Crisis’, Politico, (Article 18 June 2020) <https://www.politico.eu/article/hungary-replaces-rule-by-decree-controversial-state-of-medicalcrisis/>. 39 Valentina Pop and Drew Hinshaw, ‘European Parliament Votes to Censor Hungary’, The Wall Street Journal (Article, 12 September 2018) <https://www.wsj.com/articles/eu-triggers-sanctions-procedure-against-hungaryover-rule-of-law-1536753206>. 40 Daniel Beers, ‘A Tale of Two Transitions: Exploring the Origins of Post-Communist Judicial Culture in Romania and the Czech Republic’ (2010) 18(1) Demokratizatsiya 28, 32. 41 Marius Stan and Vladmir Tismaneanu, ‘Democracy under siege in Romania’, Politic, (Article, 13 August 2018), <https://www.politico.eu/article/protest-piata-victoriei-bucharest-democracy-under-siege-in-romania/>.
strict regulations to protect public health, without standardised checks and balances in place, governments may not surrender this power resulting in permanent regression of democracy.
Membership in international organisations and unions may greatly reduce the likelihood of conflict because the operation of organisations such as the EU requires interdependence, collaboration, and diplomacy.42 Populist belief across Eastern Europe includes widely held Euroscepticism, mainly due to opposition of the EU requirements of liberal checks and balances, such as constitutional restraints on the government, as well as opposition to the expectations on immigration, which may now have become outdated due to COVID-19.43 There is a risk of the EU fading in relevance or disintegrating, which would leave Europe divided, akin to the divisions in Europe prior to WWI. Indicative of the wider risks posed by rising authoritarianism to international organisations and alliances, French President of the populist National Rally party, Marine Le Pen, has labelled NATO as ‘obsolete’, and Dutch populist party leader of the Party for Freedom, Geert Wilders, advocated for the Netherlands to leave the EU and NATO.44 A survey collaborated by the Pew Research Centre in 2016 asked Europeans if they felt their country should “deal with its own problems and let other countries deal with their own problems as best they can” or if they should “help other countries deal with their problems.” The highest percentages of respondents who believed in self-sufficiency were 83% in Greece, 77% in Hungary, 67% in Italy and 65% in Poland.45 This hostility to alliances correlates with the fact that Greece, Hungary and Poland were populist at the time the poll was conducted and Italy was classified as a “flawed democracy” by the Economist Intelligence Unit and is currently experiencing rising support for their populist party The Five Star Movement.46 Isolationist attitudes pose a unique risk amidst a pandemic where communication between nations is vital for viral containment and advancing medical knowledge. The circumstances of reduced interdependence between states, aggressive policies of populist governments and instability caused by this transition have security implications that heighten the possibility of war, as predicted by the democratic peace theory.
Despite the democratic peace theory positing that the proliferation of democracy will contribute to global peace, it is likely that war will always be inevitable. Nonetheless, an investigation
42 Azar Gat, ‘The Democratic Peace Theory Reframed: The Impact of Modernity’ (2005) 58(1) World Politics 73, 77. 43 Matthijs Roodujn and Stijn van Kessel 2019, Populism and Euroscepticism in the European Union, Oxford Research Encyclopedias, (Article, August 2019) <https://oxfordre.com/politics/view/10.1093/acrefore/9780190228637.001.0001/acrefore-9780190228637-e1045>. 44 Geert Wilders, ‘Wilders’s Plan: Time for Liberation’, (Webpage, 9 November 2016) <https://www.gatestoneinstitute.org/9291/geertwilders-liberation>; Will Kirby, ‘Le Pen backs Trump’s claim that NATO is ‘obsolete’ & vows to pull France out of alliance’, Express Online, (Article, 29 March 2017) <http://www.express.co.uk/news/uk/785165/marine-le-pen-france-nato-obsolete-president-trump-ussr-vladimirputin-newsnight>. 45 Bruce Stokes, Richard Wike and Jacob Poushter, ‘Europeans Face the World Divided’, Pew Research Center, (Article, 13 June 2016) <https://www.pewresearch.org/global/2016/06/13/europeans-face-the-world-divided/>. 46 The Economist Intelligence Unit, ‘Global democracy has another bad year’, The Economist (Web Page, 22 January 2020) <https://www.economist.com/graphic-detail/2020/01/22/global-democracy-has-another-badyear>; Maria Lanzone, ‘The "post-modern" populism in Italy: The case of the Five Star Movement’, in Dwayne Woods and Barbara Wejnert (eds Many Faces of Populism: Current Perspectives (Emerald Group Publishing, 2014).
into the cause of war can help minimise the frequency of wars and the extent of their harm. The lack of constitutional constraints, rejection of measures to engrain interdependence, and the instability caused by the current change to authoritarian regimes has been shown to contribute to the chances of future conflict. However, this is not an exhaustive list of contributing factors and international relations scholars have investigated further into the political effects of a leader’s personality, geopolitical factors, such as sharing borders, and the possibility of benevolent authoritarianism. The world may currently be experiencing a wave of authoritarianism, however American political scientist, Francis Fukuyama, offering a differing perspective to that of Samuel P. Huntington’s ‘Clash of Civilisations’, predicts that despite experiencing a temporary backlash, which may last for long periods of time, the globe will eventually adopt liberal democracy as its permanent system.47 Whilst this may eventually result in increased peace between more numerous mature democracies, both Fukuyama’s and Huntington’s hypotheses show there are many changes and conflicts to come before world politics ever stabilises.
47 Francis Fukuyama, The End of History and the Last Man (Free Press, 1992).
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Introduction, general overviews.
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## Democratic Peace Theory by Dan Reiter LAST REVIEWED: 02 May 2019 LAST MODIFIED: 25 October 2012 DOI: 10.1093/obo/9780199756223-0014
Democratic peace is the proposition that democracies are more peaceful in their foreign relations. This idea dates back centuries, at least to Immanuel Kant and other 18th-century Enlightenment thinkers. In recent decades it has constituted a major research agenda, competing with and arguably supplanting other research agendas such as neo-realism. The democratic peace proposition has many possible empirical and theoretical forms. On the empirical side, some propose that democracies are more peaceful in their relations with all other states in the system (“monadic” democratic peace); some propose that democracies are more peaceful only in their relations with other democracies (“dyadic” democratic peace); others argue that the more democracies there are in a region or the international system, the more peaceful the region or international system will be (“systemic” democratic peace); and still others doubt the existence of any significant relationship between democracy and peace. Notably, most although not all empirical research on the democratic peace has employed quantitative methods of analysis. On the theoretical side, there are many different accounts of the relationship between democracy and peace, with most focusing on domestic political institutions, domestic political norms, and constructed identities. The democratic peace proposition is connected to many other propositions linking domestic politics and international relations, including that democracies are more likely to cooperate with each other, that democracies are more likely to win the wars they fight, that escalating military casualties degrade public support for war, that leaders initiate conflict to secure their domestic hold on power (the diversionary hypothesis), that democracies fight shorter wars, that different kinds of democracies experience different kinds of conflict behavior, that different kinds of authoritarian systems experience different kinds of conflict behavior, and others. The democratic peace also overlaps with related ideas such as the liberal peace and the commercial peace.
The democratic peace proposition has been lurking in Western thought for millennia, as Weart 1998 shows, but Kant 1991 provides its first modern formulation. The idea that global democracy would provide a solid foundation for global peace was restated in 1917 by Woodrow Wilson as a justification for American entry into World War I and then as part of his vision for a new world order. Modern political science first observed the dyadic democratic peace—that democracies tend not to fight each other—in the 1970s. The observation enjoyed greater attention in the 1980s in particular in two pathbreaking 1983 essays by Michael Doyle, reprinted in Doyle 2011 . It received fuller theoretical and empirical attention in the 1990s. Fukuyama 1992 , a famous argument that humanity had reached “the end of history,” incorporates the democratic peace proposition. Other scholars sought to develop the theory and push forward more advanced research designs in works such as Russett 1993 ; Ray 1995 ; and Rousseau, et al. 1996 . In the 2000s, proponents of the democratic peace responded to their critics and embedded the democratic peace in a broader Kantian peace ( Russett and Oneal 2001 ).
Doyle, Michael W. Liberal Peace: Selected Essays . New York: Routledge, 2011.
Contains a number of Doyle’s important essays, especially from the 1980s, that lay out the philosophical and theoretical basis of the democratic peace.
Fukuyama, Francis. The End of History and the Last Man . New York: Free Press, 1992.
Presents a Hegelian argument that humanity has at last achieved its penultimate form of political and economic organization, liberal democracy. The definitive intellectual statement that Western values triumphed in the Cold War.
Huth, Paul K., and Todd L. Allee. The Democratic Peace and Territorial Conflict in the Twentieth Century . Cambridge, UK: Cambridge University Press, 2002.
Application of the democratic peace to territorial conflict in the 20th century. Presents a massive new data set on territorial conflicts.
Kant, Immanuel. Kant’s Political Writings . 2d ed. Edited by Hans S. Reiss. Cambridge, UK: Cambridge University Press, 1991.
Central essay is on the “perpetual peace,” which presents Kant’s vision as to how republics can maintain world peace. Originally published in 1796.
Ray, James Lee. Democracy and International Conflict: An Evaluation of the Democratic Peace Proposition . Columbia: University of South Carolina Press, 1995.
Provides an extensive literature review on democratic peace literature up to the early 1990s as well as case studies of the Fashoda Crisis and Spanish-American War.
Rousseau, David L., Christopher Gelpi, Dan Reiter, and Paul K. Huth. “Assessing the Dyadic Nature of the Democratic Peace, 1918–1988.” American Political Science Review 90.3 (1996): 512–533.
DOI: 10.2307/2082606
Important, early empirical test of the democratic peace, presenting important research design advances.Available online by subscription.
Russett, Bruce. Grasping the Democratic Peace: Principles for a Post–Cold War World . Princeton, NJ: Princeton University Press, 1993.
The first book-length treatment of the democratic peace. Lays out the normative and institutional explanations of the democratic peace and presents a variety of different forms of rigorous evidence demonstrating the dyadic democratic peace, including sophisticated analysis of post-1945 conflict behavior.
Russett, Bruce, and John R. Oneal. Triangulating Peace: Democracy, Interdependence, and International Organizations . New York: Norton, 2001.
Embedded the democratic peace in a larger theoretical framework, the Kantian Peace, in which democracy, trade, international organization, and peace all mutually reinforce each other. Presented more sophisticated empirical tests, addressing many 1990s theoretical and empirical critiques. Also see Democratization .
Weart, Spencer R. Never at War: Why Democracies Will Never Fight One Another . New Haven, CT: Yale University Press, 1998.
Summarizes several years of work on democratic peace theory. Presents a narrative rather than statistical empirical tests. One main contribution is the analysis of democratic peace in pre-Napoleonic times, including ancient Greece and medieval Italy. Discusses the phenomena of democratic aggression and imperialism.
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#### IMAGES
1. Democratic Peace Theory by Poppy Nevin
2. PPT
3. Democratic peace theory
4. The Democratic Peace Theory by JoAnna Adkisson
5. Democratic Peace Theory by Alexia Lafosse
6. Democratic peace theory_&_the_unintended_consequences_of[1]
#### VIDEO
1. Contemporary political theory & DECLINE OF POLITICAL THEORY 9
2. The Limits of Democratization: What is the Democratic Peace?
3. The Best Argument for Democratic Peace Theory
4. Political Theory
5. What Do We Need To Do To Achieve a Just Peace? A “Conversation” with John Rawls
6. The Lie of Democratic Peace Theory
1. What Is the Traditional Democratic Theory?
The traditional democratic theory emphasizes the values of liberty, equality and justice in any system of governance. It promotes the rule of majority, while protecting minority rights and maintaining the readiness to compromise.
2. Fractals
Advertisement If you examine a bifurcation diagram closely, you begin to see interesting patterns. For example, start with a completed diagram, such as the one in the first picture. Next, zoom in on the first doubling point. It looks like a...
3. Despite tooling limitations, DAO optimists see new use cases for a democratic, token-based future
The adoption of decentralized autonomous organizations, or DAOs, has skyrocketed in the past year, and participants believe this is just the beginning, claiming more use cases will form in the subsector. DAOs are community-led groups that, ...
4. Democratic peace theory
Proponents of "democratic peace theory" argue that both liberal and republican forms of democracy are hesitant to engage in armed conflict with other
5. The Democratic Peace Theory
Furthermore, in line with Immanuel Kant's theory of perpetual peace, I argue that the global spread of democracy will result in greater
6. What Is the Democratic Peace Theory? Definition and Examples
Key Takeaways · The Democratic Peace Theory holds that democratic countries are less likely to go to war with one another than non-democratic
7. Democratic peace
Examples include American Samoa (U.S.) and Greenland (Denmark). The dominant state may control some of the weak state's affairs, such as defense, foreign
8. The Democratic Peace
The challenges posed in this journal to the theory of democratic peace
9. Democratic Peace Theory
Dyadic democratic peace is the most common formulation of the theory and proposes that democratic states are less likely to go to war with one another. Monadic
10. DEMOCRATIC PEACE THEORY
1 The theory proposes that democracies, especially those older than 3 years, are unlikely to be involved in wars against each other.2 It has been recognised
11. DEMOCRATIC PEACE THEORY
It explores an alternative, and in some cases complementary, explanation for war and peace that derives from a domestic coalitional approach to politics. The
12. Democratic Peace Theory as Practice
7 According to Doyle and other adherents of the democratic peace, liberal democratic states have been able to maintain peaceful relations amongst themselves
13. Democratic Peace Theory
Modern political science first observed the dyadic democratic peace—that democracies tend not to fight each other—in the 1970s. The observation
14. Hawks and Doves
occur, however, is by definition ex-ante: explanations for the democratic peace aim to. Page 22. Individuals under Threat. 10 find out what particularities
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# Percentage Worksheets
Home Basic Pre-Algebra Algebra Geometry Statistics Math Test Membership Login
Math Worksheets > Pre-Algebra Worksheets > Percent
## Percent Worksheets
Free percent worksheets help kids to understand the usage of percent in day to day life. Teachers, parents and home schoolers can use these printable percent worksheets to explore the knowledge of students in percentage.Percent activities contain converting decimals and fractions into percents, percents into fractions and decimals, calculating percent increase or decrease, finding percent of numbers and many percent math problems. Download All Percent Worksheets (770 KB)
### Conversions Worksheets:
Percent worksheets include converting percents into fractions and decimals and vice versa. Practice with more percent math problems to clearly understand the relationship between percents, fractions, decimals and ratio.
Decimals into percent Proper fractions into percent Improper fractions into percent Fractions into percent Percents into decimals Percents into fractions More Conversion Worksheets
### Percent Worksheets:
Percent activities help you to understand and calculate percent of numbers, practical application of percents, finding percentage for given numbers and more. Worksheets are suitable for grade 4 students and up.
Percent of numbers Percent of practical units A number is what % of given number? % of what number is given number?
### Methods of Finding Percent:
We can find percent in so many ways. These printable percent worksheets help to find percent of certain numbers using decimal, fraction and ratio method. For example, 20% of 50 can be found by converting 20% into decimal or fraction or ratio.
Decimal method Fraction method Ratio method
### Percent of Increase and Decrease:
If a certain number is increased or decreased in value, then finding the percent of increase or decrease will help to compare the growth. Kids can practice these free percent worksheets to understand the application of percents.
Calculate percent increase Calculate percent decrease % of increase or decrease
### Compare Percent and Decimal:
You need to be familiar with conversion of percents into decimals and vice versa before start taking these worksheets for practice. These free percent worksheets help to compare percent and decimals.
Comparing worksheet 1 Comparing worksheet 2
### Compare the Percents:
These printable percent worksheets help you to compare the percents of certain numbers and by how much. Percent activities provide here are suitable for grade 5 students and up.
Percent worksheet 1 Percent worksheet 2
Free Math Test Practice
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Home Interview Questions and AnswersPlacement Papers with Answers Tata Consultancy Services Placement Paper For Freshers Part-3
21) A power unit is there by the bank of a river of 900 meters. A cable is made from power unit to power a plant opposite to that of the river of 2000mts. The cost of the cable below water is Rs. 5/- per meter and cost of cable on the bank is Rs. 4/- per meter. Then find out the amount to be invested to connect those two stations?
Explanation:
Required length of wire = 2000 mts cost of cable below water = 900 * 5 = 4500 cost of cable on the bank of river= (2000 – 900) * 4 = 4400 Total cost = 4500 + 4400 = 8900.
22) A farmer had 20 hens. All but 2 died. How many hens are still alive?
Explanation:
“All but 2” means 2. So, all but 2 died means that there are only two hens are alive and others were dead. So, the answer is 2.
23) At this series: 7, 10, 8, 11, 9, 12… What number should come next?
Explanation:
This is a simple alternating addition and subtraction series. In the first pattern, 3 are added; in the second, 2 are subtracted.
24) A college has 10 basketball players. A 5-member team and a captain will be selected out of these 10 players. How many different selections can be made?
Explanation:
We can select the 5 member team out of the 10 in 10C5 ways = 252 ways. The captain can be selected from amongst the remaining 5 players in 5 ways. Therefore, total ways the selection of 5 players and a captain can be made = 252*5 = 1260.
25) There are 3 gentlemen in a meeting: Mr. Yellow, Mr. Green and Mr. Brown. They are wearing yellow, green and brown ties. Mr. Yellow says: “Did you notice that the colors of our ties are different from our names?” The person who is wearing the green tie says, “Yes, you are right!” Do you know Brown is wearing what colour of tie?
Explanation:
We know that Mr. Yellow was not wearing a yellow tie because of his statement. He also was not wearing the green tie because the one wearing the green tie agreed to his statement. Therefore, Mr. Yellow was wearing a brown tie. Mr. Green was wearing a yellow tie. And Mr. Brown was wearing the green tie.
26) g[0]=1,g[1]=-1,g[n]=2*g[n-1]-3*g[n-2] then calculate g[4]= ?
Explanation:
From the given function g[n] =2*g [n-1]-3*g [n-2] put values of n as first 2 then 3, then 4 and you will get the answer. G[2] = 2*g[1]-3*g[0]=-2-3 =-5 , G[3]=2*g[2]-3*g[1] =2*(-5)-3*(-1)=-7, G [4] =2*g[3]-3*g[2]=2*(-7)-3*(-5) =1
27) Ten years ago X was half of Y. If the ratio of their present ages is 3:4, what will be the total of their present ages?
Explanation:
Let X’s age 10 years ago = x years. Then, Y’s age 10 years ago = 2x years. (x + 10)/ (2x + 10) = 3/4 => 4 (x + 10) = 3 (2x + 10) x = 5 Total of their present ages = (x + 10 + 2x + 10) = 3x + 20 = 15 + 20 = 35 years.
28) A train can travel 50% faster than a car. Both start from point A at the same time and reach point B 75 kms away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. The speed of the car is:
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Question:
By what least number should 1536 be divided to get a perfect cube?
(a) 3
(b) 4
(c) 6
(d) 8
Solution:
(a) 3
$1536=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3=(2)^{3} \times(2)^{3} \times(2)^{3} \times 3$
Therefore, to get a perfect cube, we need to divide 1536 by 3.
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# Fourier Integral and the Dirac Delta
• Nov 30th 2006, 08:45 AM
topsquark
Fourier Integral and the Dirac Delta
This is going to end up being a simple question with a simple answer, but I just can't tweak the answer out of my Math Methods text.
I have the following statement:
$\displaystyle \dot{\Delta}(\vec{x},0) = i \int \frac{d^3k}{2(2 \pi )^3} \left [ e^{i \vec{k} \cdot \vec{x}} + e^{-i \vec{k} \cdot \vec{x}} \right ]$ ($\displaystyle \vec{k}$ and $\displaystyle \vec{x}$ are real 3-vectors and $\displaystyle \vec{k} \cdot \vec{x}$ is the usual dot product.)
I'm supposed to get that
$\displaystyle \dot{\Delta}(\vec{x},0) = -i \delta ^3 (\vec{x})$
But looking at the integral I'm thinking it ought to be:
$\displaystyle \dot{\Delta}(\vec{x},0) = \frac{i}{2} \left ( \delta ^3 (\vec{x}) + \delta ^3 (-\vec{x}) \right )$
which would give me 0, which obviously isn't true. So (sigh) what am I doing wrong?
Thanks!
-Dan
• Nov 30th 2006, 12:12 PM
CaptainBlack
Quote:
Originally Posted by topsquark
This is going to end up being a simple question with a simple answer, but I just can't tweak the answer out of my Math Methods text.
I have the following statement:
$\displaystyle \dot{\Delta}(\vec{x},0) = i \int \frac{d^3k}{2(2 \pi )^3} \left [ e^{i \vec{k} \cdot \vec{x}} + e^{-i \vec{k} \cdot \vec{x}} \right ]$ ($\displaystyle \vec{k}$ and $\displaystyle \vec{x}$ are real 3-vectors and $\displaystyle \vec{k} \cdot \vec{x}$ is the usual dot product.)
I'm supposed to get that
$\displaystyle \dot{\Delta}(\vec{x},0) = -i \delta ^3 (\vec{x})$
But looking at the integral I'm thinking it ought to be:
$\displaystyle \dot{\Delta}(\vec{x},0) = \frac{i}{2} \left ( \delta ^3 (\vec{x}) + \delta ^3 (-\vec{x}) \right )$
which would give me 0, which obviously isn't true. So (sigh) what am I doing wrong?
Thanks!
-Dan
Do you think this might be related to:
$\displaystyle \delta(x)=\delta(-x)\ ?$
and that you are using the other sense convention for what is the forward FT
RonL
• Nov 30th 2006, 12:50 PM
topsquark
Quote:
Originally Posted by CaptainBlack
Do you think this might be related to:
$\displaystyle \delta(x)=\delta(-x)\ ?$
and that you are using the other sense convention for what is the forward FT
RonL
Okay, I see what you are saying about the convention. And (duh!) I forgot that $\displaystyle \delta$ is even. I knew it was something simple! Thanks.
-Dan
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# Statistics Essay
Type of paper: Essay
Pages: 7
Words: 1925
Published: 2020/10/10
## Introduction
The company is interested in making one-sized shoes only and the data has been provided I will need to establish if the data is from a random variable before I proceed. It is important to observe the data for possible outliers which are clearly not from the sample for instance in a study to observe the number of youth between the ages of 15-25 engaged in smoking cigarettes a value of age maybe 81 will stand out as a data handling error. Someone may have been intending to enter 18 and in the rush they fail to realize that they entered 81. Alternatively other statistics could be used in place of the mean and variance which are in most cases the ones affected by outliers. In the place of the mean the median could be used and the interquartile range could be used to replace the standard deviation. However the median in many cases does not present the sample as required especially in cases where the sample values are skewed either to the right or the left. The semi-interquartile range could be used also as an alternative or the interquartile range and is usually preferred because it represents a middle value.
## Discussion
After I have established the nature of our data I will first determine the descriptive that is the mean, mode and median before I carry out tests to that will help us in understanding whether the sample distribution is identical to the distribution of a normally distributed random variable (Kanji 26). Key to our analysis is the relationship that the variables in the sample data have with each other and this will help me in making a decision whether the company should or should not make a one-sized shoe and from there I can present my conclusions and recommendations. The relationship between these variables can strongly be relied on if the distribution of the sample data is identical to the distribution of a normally distributed random variable. I also need to estimate the proportion of males in the population and establish parametrically if they differ with the proportions of females.
Since the data for both height and shoe size are both continuous and quantitative, I can establish if they come from a random variable. Although I are utilizing a non-parametric test to establish if the sample does indeed come from a random variable it does not jeopardize our estimates interpretation and inference. I carry out a runs test to establish if the sample data is from a random sample. I am using the null hypothesis that the data does not come from a random variable. The results indicate that the tests is insignificant and as such I fail to reject the null hypothesis that the data does not come from a random variable. The distribution of gender is most probably binomial and I need to carry out a binomial test to determine if the proportions of females is significantly similar to that of males. The observed proportions of Females-1 is 0.51 and that of males is 0.49. The test is insignificant thus I cannot reject the hypothesis that the proportion of females equals to 0.5. The test is insignificant even at 90% confidence level which is considered too loose a test (Kanji 208).
I would like to determine the basic statistics which I will use to establish the distribution of the sample. A normal distribution has the same mean, mode and median, thus we will be more interested in the measures of central tendency than in the measures of dispersion. I am particularly interested in the normal distribution in my analysis because when a sample is normally distributed I can make statistically inference about the population freely. Furthermore important tests like the t-test and the f-test can be carried out only on condition that the sample is normally populated. The z test also requires the normality assumption and working on a normalized data is not only easier but also presents reliable inferences and interpretation (Kanji 41). In our case I will utilize the coefficient of correlation to establish the degree of the relationship between gender and shoe size and also the magnitude of the relationship between height and shoe size. I will then test the hypothesis that the distribution of the sample is identical to that of the normal distribution. If there is no significant difference between the sample distribution and the normal distribution. If the case is so I will use the assumption of normality to carry out tests that will help us establish if the company can go ahead with the decision to make one size shoe.
The mean shoe size is 9.1429 while the median shoe size is 9, the modal shoe size is 7 and the skewness is 0.367, I obtained a kurtosis of -1.083 for the shoe size data. The mean height of the sample is 68.94 and the median height is 70 which is equivalent to the modal height. There are multiple modes in this sample and the smallest is selected. I obtained a variance of 16.323, the skewness was -0.233 and the kurtosis obtained was 0.336 for the height data. The skewness of a distribution measures the degree of symmetry of that distribution it is therefore imperative that a symmetrical distribution has a skewness of zero. If the skewness obtained is greater than zero then, the distribution is skewed to the right and if it is less than zero that is negative then the distribution is skewed to the left. Kurtosis is a measure of the extent to which the frequencies are distributed close to the mean. A bell shaped distribution will most likely have a kurtosis of three. A flat shaped distribution on the other hand will have a kurtosis which is more than three. Kurtosis will assist us in understanding better how the decision by the company to manufacture the same shoe size may impact on the population.
In the event that the sample is not normally distributed and we fail to establish a transformation of the variable that will be normally distributed, we will have to use the non-parametric alternatives which although they are not as strong as the parametric tests, they will also allow us to make an inference on the population using the sample we have. The fact that the mean, mode and median slightly differ in both the shoe size and the height data does not rule out our sample from being normally distributed. I have observed from the HISTOGRAMS and Q-Q plots and I have established that the distribution is somewhat close to that of a normal distribution. Since observation alone is non- conclusive I will need to carry out parametric test to determine if the distribution of the sample is identical to the distribution of a normally distributed random variable. I have used the one sample KOLMOGOROV- Smirnov Test to establish the normality of the sample distribution and I have established that the Test distribution is normal based on the calculations obtained from the data (Kanji 109).
Firstly, it is of interest to determine if the shoe size differs across gender. Graphs are used to generate boxplots that will help indicate if there is a significant difference between the shoe sizes based on gender. It is observed that the upper limit of mean of the female shoe size is below the lower limit of the mean shoe size of the male gender. The outliers for the mean shoe sizes do not differ much from the sample and as such they will not be ruled out from the sample. In some cases due to errors in data entry and handling outliers which are too far from the sample are observed. It is crucial to be able to establish if they are really from the sample or they are errors that may have been introduced during data handling. Although some outliers do fall in some of these categories, it can be safely concluded that the mean shoe size of the male gender is higher than the mean shoe size of the female gender which is actually the case in normality. The company has considered making one size of shoe regardless of height and gender.
Since now I have established that the distribution of the sample is normal I can establish using parametric tests whether or not there is a correlation between gender and shoes size. I therefore proceed to carry out parametric tests on our sample data. I have to carry out a test to ascertain whether there is a correlation between height and shoe size. Since both variables that is height and shoe size are quantitative, I will use the Pearson method of computing correlation coefficient between variables. The CORRELATION coefficient between height and shoe size is 0.853 indicating a strong positive correlation. The test is also significant at 99% confidence interval which increases the reliability of our estimated value and its interpretation. The fact that the correlation coefficient is positive indicates that height and shoe size change in the same direction that is an increase in height will lead to an increase in shoe size. The value 0.853 indicates that the size of the relationship is very large. The correlation coefficient also between gender and shoe size is also very large 0.804.
I used the dummy variables male-0 and female-1 and as expected the correlation coefficient is negative but in this case we are interested in the value since numbering for gender was just to enable us to compute. The magnitude of the CORRELATION coefficient indicates that the relationship between gender and shoe size is very large. The fact that the test is significant at 99% confidence level indicates that our estimate of the correlation coefficient and the interpretation of the result is highly reliable. In normal situations a 95% confidence level is considered appropriate so as to enable us to strike a balance between not having too strict (99%) a test and not using a too loose one (90%) either (Kanji 19). At the moment establishing the degree of relationship between gender and height may not be instrumental to the task at hand since we are interested in the outcomes of the company’s decision to make a one sized shoe for all which is not targeted at a specified gender.
The last two analyses on the correlation coefficient the first between height and shoe size and the second between gender and shoe size are both significant at 99% level of confidence. This coupled with the fact that our data fitted that of the normal distribution model greatly increases the reliability of our analysis and the subsequent interpretation. Shoe size is highly correlated to height and gender. I was also able to establish that the means differ across the different genders so the company will have a difficult time deciding on the standard shoe size without being discriminative on gender lines.
## Conclusion and Recommendation
The company should not go ahead with its decision to make a one sized shoe since it will have a negative impact on its clients and possibly its market share. As it is facing financial hardships, I would recommend that a further study is carried out to establish the shoe size and height of most of its clients and maybe come up with a range of shoe sizes that do not differ much and are targeted at retaining its loyal clients and its market share. Alternatively other cost cutting precautions could be adopted so as to achieve the current budgetary expectations of the company.
## Work Cited
Kanji, Gopal. 100 Statistical Tests. New York: SAGE, 2006. Print.
Appendix
Binomial Test
a Based on Z Approximation.
Statistics
a Multiple modes exist. The smallest value is shown
One-Sample Kolmogorov-Smirnov Test
a Test distribution is Normal.
b Calculated from data.
Binomial Test
a Based on Z Approximation.
Correlations
** Correlation is significant at the 0.01 level (2-tailed).
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When you learn math, you have to understand negative numbers. Unlike when you were learning only positive integers, learning the concept of zero and addition and subtraction will allow you to perform complex calculations.
It may seem strange to you that there are numbers less than zero. However, we use negative numbers in many aspects of our daily lives. Therefore, it is very important to use negative integers as well as positive integers for calculations.
So what is the concept of 0 and negative numbers? How can we think and calculate positive and negative numbers?
When learning mathematics, we all need to understand positive and negative numbers. We will explain how to calculate positive and negative integers, with some exercises.
## 0 (Zero) Is Used to Indicate a Criteria
In general, 0 (zero) indicates that there is nothing. For example, if you have \$1, and if you spend it, the money you have is \$0.
If non-existence means zero, the concept of a negative doesn’t make sense. When learning about negative integers, many people get confused because they may think that there are numbers that are smaller than non-existent ones.
However, 0 does not just indicate that it does not exist. It has other meanings as well. It is a criterion. By using zero as a standard, we use positive and negative numbers, depending on whether the number is greater or less than zero.
For example, we use temperature in our daily lives. Temperature has a standard of 0°C (32°F). If the temperature is lower than 0°C, the number is negative. In a city where it snows, many people will experience negative temperatures.
Given this fact, we can realize that it’s not strange to use negative numbers as well as positive numbers; since zero has a meaning of indicating a standard, many people use negative numbers in their daily lives.
### Positive Integers (Natural Numbers) and Negative Integers Indicate Increase or Decrease from a Reference
Once you learn that zero has the concept of a standard, you will understand what a positive and negative integer means. Positive integers (natural numbers) and negative integers indicate how far away they are from the reference point (0).
In the following figure, with zero as the reference point, the numbers to the right are positive integers (natural numbers).
On the other hand, numbers to the left of zero are negative integers. By using plus and minus, we can express an increase or decrease from a standard.
For example, let’s consider losing weight by going on a diet. In that case, the majority of people use their current weight as a reference point (zero). If you have lost 3kg of weight, then you can say that you have successfully gained -3kg of weight.
-You Can Replace Plus and Minus
Therefore, the number of pluses and minuses can be replaced. For example, the following have the same meaning.
• I’ve lost 3kg of weight.
• I’ve gained -3kg of weight.
Also, the following have the same meaning
• The temperature drops by 5°C.
• The temperature goes up by -5°C.
It is important to understand that these rephrases are possible when learning positive and negative numbers.
### How to Distinguish Between Large and Small Numbers, and Absolute Value
With positive and negative numbers, how can we distinguish between large and small numbers? In order to understand this, we need to understand the concept of absolute value.
We have understood that we can check positive and negative numbers based on zero. For large and small numbers, check how far they are from zero. Regardless of whether it is positive or negative, the absolute value is how far it is from zero.
For example, the absolute value of 4 is 4. On the other hand, the absolute value of -3 is 3. If we go three to the left from 0, we get -3. Since the distance from 0 is 3, the absolute value of -3 is 3.
Once we learn this concept, we will be able to distinguish between large and small numbers. For example, with 5 and 3, we can see that 5 is a larger number than 3. Therefore, we can express 5>3. On the other hand, which number is larger, -3 or -5?
The absolute value of -3 is 3. Also, the absolute value of -5 is 5. The important fact is that the larger the number (absolute value) of a negative number, the smaller the number becomes. The absolute value of -5 is greater than -3. Therefore, -3 is a larger number and can be expressed as $-3>-5$.
It may seem strange that even though the number is large, the minus sign makes it a small number. However, we also use this in our daily lives.
For example, between -3°C (26.6°F) and -20°C (-4°F), -20°C is colder. This is because -20°C is a larger negative number. Understand that even though the number (absolute value) is large, it will be a small number in a negative number.
## Addition and Subtraction of Positive Numbers
Understanding the concepts of zero and negative numbers will help you understand addition and subtraction for positive and negative numbers.
It is easy to understand positive and negative numbers if you learn about them from addition and subtraction of positive numbers. When dealing with positive numbers, there are two types.
• Subtraction of positive numbers
We will explain each of these.
### Using Parentheses in Addition of Positive Numbers
You have already learned about the addition of positive numbers in elementary school math. Therefore, you will be able to understand the addition of positive numbers to positive numbers without any difficulty. For example, the following calculation is easy.
• $3+1=4$
In this case, $3+1$ can also be expressed as follows.
• $3+(+1)$
• $(+3)+(+1)$
They all have the same meaning. In addition, you can use two +’s in a row. However, instead of writing ++, we use parentheses.
-Add a Positive Number to a Negative Number
How can we add a positive number to a negative number? For example, the following calculation.
• $-5+3$
It can also be expressed as $-5+(+3)$ or $(-5)+(+3)$. In any case, this calculation requires adding 3 to -5. In other words, you have to answer a number that is 3 greater than -5.
The absolute value of -5 is 5. By doing +3 to -5, we can go three steps to the right and get -2. Therefore, the answer is -2.
### Rules for Subtracting Positive Numbers
In contrast, how can we think about subtraction of positive numbers? For example, suppose we have the following calculation
• $3-2=1$
We have already learned about this calculation in elementary school. The important thing is that it can be replaced by the following equations.
• $3-(+2)=1$
• $(+3)-(+2)=1$
All positive numbers have a + hidden in them. For example, when we use the number 2 or 3, we can represent it as +2 or +3. For positive integers (natural numbers), the + can be omitted. Just to describe it in detail, $3-2$ can be represented as $3-(+2)$.
-Subtract a Positive number from a Negative number
Of course, it is common to subtract positive numbers (natural numbers) from negative numbers. For example, what is the answer to the following calculation?
• $-1-3$
This equation can also be replaced with $-1-(+3)$ or $(-1)-(+3)$. In order to subtract +3 from -1, the answer must be a number that is 3 less than -1.
• $-1-3=-4$
• $-1-(+3)=-4$
• $(-1)-(+3)=-4$
It is calculated like this.
## Addition and Subtraction of Negative Numbers
We’ve discussed positive numbers, and it’s not difficult to understand addition and subtraction for positive numbers. It is the addition and subtraction of negative numbers that are difficult.
• subtract the negative numbers
What do these mean? For example, when we calculate $1 – (-3)$, we get the following.
• $1-(-3)=1+3=4$
Many people try to learn how to calculate without understanding why. However, you have to understand the reason. So, we will explain including the reasons.
### Learn How to Add Negative Numbers by Replacing
Calculations involving the addition of negative numbers are frequently given. For example, how do we think about the following calculation?
• $2+(-4)$
As mentioned above, the numbers can be rephrased. For example, if you lose 2kg of weight, you can rephrase it as “I’ve gained -2kg of weight”. Even though it is expressed as an increase in weight, it is a -2 kg plus, so you have lost weight.
In this way, you can replace the positive with the negative. In other words, we can rephrase it as follows.
• $2-4$:4 less.
• $2+(-4)$: -4 higher
Subtract 4 from 2 is -2. A number 4 less than 2 is -2. Understand that adding a negative number is, in essence, the same as subtracting a positive number.
### Why Negative Subtraction Is Positive?
When it comes to understanding negative numbers, it is the subtraction of negative numbers that confuses many people. What we have explained so far are the following.
• Addition of positive numbers (positive integers)
• Subtraction of positive numbers (positive integers)
• Addition of negative numbers (negative integers)
Addition and subtraction of positive numbers are already taught in elementary school. Also, the addition of negative numbers can be rephrased as the subtraction of positive numbers (natural numbers). On the other hand, how can we calculate the following, for example?
• $2-(-3)$
We can also rephrase this calculation. The minus sign has the opposite meaning. As mentioned above, the expression “I’ve lost 2kg” can be reworded as “I’ve gained -2kg”.
So, how should we think about the expression “I lost -3kg”? Since minus has the opposite property, -3kg decreased means 3kg increased. The minus of a minus is a plus.
That’s why when you subtract a negative number, you can change it to a positive number.
• $2-(-3)=2+3=5$
It is important to note that minus has the opposite meaning. A lot of people don’t understand in mathematics why subtracting a negative value means a positive value. Once you understand that a negative has the opposite meaning and can be rephrased, you will understand this reason.
## Exercises: Addition and Subtraction of Positive and Negative Numbers
Q1: Express the size of the following numbers using the inequality sign.
1. $-2、4、-3$
2. $-0.3、-3、0$
For negative numbers, the higher the number (absolute value), the smaller the number. Since the concept is the opposite of positive numbers, the order of inequality is as follows.
1. $4>-2>-3$
2. $0>-0.3>-3$
For example, in (b), when comparing 0, 0.3 and 3, the order of the higher numbers is $3>0.3>0$. For a negative number, the opposite is true, the higher the number (absolute value), the smaller it is, so $0>-0.3>-3$.
Q2: Do the following calculation.
1. $7-(+3)$
2. $-0.5+(-0.2)$
3. $-3-(-6)$
4. $\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{1}{3}\right)$
5. $-3 +(-2)-(-6)-5$
In calculating positive and negative numbers, let’s rephrase them. In both junior high school math and high school math, everyone calculates after replacing signs. The point of substitution is as follows.
• $+$ and $+$ becomes $+$
• $+$ and $-$ becomes $-$
• $-$ and $+$ becomes $-$
• $-$ and $-$ becomes $+$
All addition and subtraction can be rephrased in this way, including decimals and fractions. Therefore, we can calculate as follows.
(a)
$7-(+3)=7-3=4$
(b)
$-0.5+(-0.2)=-0.5-0.2=-0.7$
(c)
$-3-(-6)=-3+6=3$
(d)
$\displaystyle\frac{1}{2}-\left(-\displaystyle\frac{1}{3}\right)$
$=\displaystyle\frac{1}{2}+\displaystyle\frac{1}{3}$
$=\displaystyle\frac{3}{6}+\displaystyle\frac{2}{6}=\displaystyle\frac{5}{6}$
(e)
$-3+(-2)-(-6)-5$
$=-3-2+6-5=-4$
You cannot solve a problem without rephrasing the pluses and minuses. Make sure you know whether you are adding or subtracting. If you change the signs, it is elementary school arithmetic content, so all you have to do is add and subtract.
Fractions are a little more complicated because you need to reword the signs and then adjust the denominator. However, calculating fractions should be taught in elementary school, so it is not difficult.
Q3: Do the following calculations.
The table below shows how many centimeters taller the four people in A to D are from 160 cm.
A B C D Difference from 160 cm (cm) 12 -4 7 -9
1. How many centimeters taller would the tallest person be than the shortest person?
2. Find the average height of the four people.
a. How many centimeters taller would the tallest person be than the shortest person?
The tallest person is A. And the shortest person is D. A is +12 cm taller and D is 9 cm shorter.
When calculating the difference between the numbers, you have to subtract. For example, according to the diagram above, A is +12 cm tall and C is +7 cm tall; the difference in height between A and C is $12cm-(+7cm)=12cm-7cm=5cm$. In the same way, we use subtraction in comparing numbers.
The height difference between A and D is calculated as follows.
• $12-(-9)=12+9=21cm$
Therefore, the tallest person is 21 cm taller than the shortest person.
b. Find the average height of the four people.
When calculating the average height, one way to calculate the average height is to calculate the respective heights from A to D. If you calculate the height of each of them, you will get the following
A B C D Height (cm) 172 cm 156 cm 167 cm 151 cm
After adding the heights from A to D, you can divide the heights by four to get the average of the heights. But can’t we find the average in an easier way?
In this regard, let’s try to calculate how much it differs from the average value. As shown below, the heights of the four people are different.
• A: +12 cm
• B: -4 cm
• C: 7 cm
• D: -9 cm
What is the average height difference between the four people? If you add up the height difference between the four people, it is as follows.
$12+(-4)+7+(-9)$
$=12-4+7-9=6cm$
The total difference in height between the 4 people is 6 cm. If you divide the total by the number of people, you can calculate the average value. In other words, we can get the average value of how many cm taller (or shorter) each person is from 160 cm.
If you divide 6 cm (total value) by 4 people, you get $6÷4=1.5cm$. This means that the average height of the 4 people is +1.5 cm taller than 160 cm. Therefore, the average height of the 4 people is 161.5 cm.
• $160+1.5=161.5$
## Use Positive and Negative Numbers to Add and Subtract
When learning negative numbers, many people are confused; they don’t understand why there are values lower than 0 (zero). Also, most people, even adults, cannot explain why subtraction of a negative number becomes a positive number.
However, as we’ve explained, we find that we use negative numbers in many situations, including in daily lives. 0 means a standard value, and the absolute value is used to compare the size of a number.
In addition, when we actually use plus and minus numbers to add and subtract, we often use parentheses. In calculations that use parentheses, it is always possible to replace signs. Even when solving difficult problems in high school and college, everyone uses replacement, so be sure to do the calculation after the replacement.
Once you learn these rules, you will be able to solve addition and subtraction problems with positive integers (natural numbers) and negative integers. The method is the same for decimals and fractions. Many people have a hard time solving problems with negative subtraction in particular, so make sure you understand including the reasons why.
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Recognitions:
Gold Member
## Work of an Object moving down Inclined Plane
1. The problem statement, all variables and given/known data
Experiment set up.
A 208g weight is attached via a pulley system to a block on an inclined plane. What is the work done by the suspended mass as the car is lowering at a constant velocity? And work done by gravity?
Distance weight moves down - 24.7cm
Incline is 30 degrees
2. Relevant equations
$F=ma$
$w=Fdcos(\theta)$
3. The attempt at a solution
So, in this case, work would be negative, right? Because the direction of the force from the suspended mass is going UP the incline, and the direction of moment is DOWN the incline?
The block is 474g
I'm not sure where to start. If I use the basic work formula and do
$\(-208*980)*24.7$?
PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
Recognitions: Homework Help The thing losing energy is doing the work. The description of the experiment is incomplete - we are told about the suspended mass and then some car is introduced out of nowhere... but it looks like you are expected to use conservation of energy. Your descriptions of what you have tried are also incomplete so it is not clear what you have done. Try expressing your working symbolically - do all the algebra before you put numbers in.
Recognitions:
Gold Member
Quote by Simon Bridge The thing losing energy is doing the work. The description of the experiment is incomplete - we are told about the suspended mass and then some car is introduced out of nowhere... but it looks like you are expected to use conservation of energy. Your descriptions of what you have tried are also incomplete so it is not clear what you have done. Try expressing your working symbolically - do all the algebra before you put numbers in.
Sorry, I wrote it really quickly.
The setup was an inclined plane with a pulley system which was set off the leg of the triangle. The pulley system was attached to the car. We had to set a weight on the pulley so that the car would go down the plane with a constant velocity.
I'm really not sure what to do. The car is moving down the car, therefore losing potential energy. If work can be defined as the difference in U.
So, if then ΔU= (474*980*12.5)-(474-980*0)= 5,806,500 erg?
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Suppose X ~ N(0,1), i.e. X is normally distributed with mean 0 and variance 1.
Let Y = X^2
Find the MGF of Y DIRECTLY...
Can't seem to get a good answer out. Any help would be appreciated.
Also...
If f(z) = e^(-z/2)/sqrt(2*pi*z) for z > 0
and f(z) = 0 for z <= 0
Find the MGF of Z.
THANKS ALOT guys!
2. Originally Posted by chakim
Suppose X ~ N(0,1), i.e. X is normally distributed with mean 0 and variance 1.
Let Y = X^2
Find the MGF of Y DIRECTLY...
Can't seem to get a good answer out. Any help would be appreciated.
Also...
If f(z) = e^(-z/2)/sqrt(2*pi*z) for z > 0
and f(z) = 0 for z <= 0
Find the MGF of Z.
THANKS ALOT guys!
$\displaystyle m(t) = \frac{1}{\sqrt{2 \pi}} \int e^{tx^2} e^{-x^2/2} \, dx = \frac{1}{\sqrt{2 \pi}} \int e^{-\left( \frac{1 - 2t}{2}\right) x^2} \, dx$.
Now make the substitution $\displaystyle u = x \sqrt{\frac{1 - 2t}{2}}$ and use a well known result.
It's simple to confirm the answer, see here: Chi-Squared Distribution -- from Wolfram MathWorld
I ended up getting to this stage
But wasn't sure how to transform that into a recognizable chi-squared distn.
Any help would be appreciated cheers.
4. Also, are the limits negative infinity to infinity, or from 0 to infinity considering that Y = X^2 accounts for only positive numbers?
Cheers
5. Originally Posted by chakim
Also, are the limits negative infinity to infinity, or from 0 to infinity considering that Y = X^2 accounts for only positive numbers?
Cheers
Since $\displaystyle m(t) = E\left(e^{X^2t}\right)$, what do you think ....?
Originally Posted by chakim
I ended up getting to this stage
But wasn't sure how to transform that into a recognizable chi-squared distn.
Any help would be appreciated cheers.
The integral limits are not correct.
The function of t can be taken out of the integration. You will then be left with a well known integral - I suggest you research it ....
6. Well firstly, considering
$\displaystyle m(t) = E\left(e^{X^2t}\right)$
Can't x take on any value between negative infinity and infinity? Since X ~ N(0,1)?
So the new limits from the substitution would be negative infinity to positive infinity?
And also, when I take the function of t out, i get...
I've researched a bit, and i've tried to integrate the function directly (cause its only an exponential) but the limits give me weird answers...
I'm really stuck mr fantastic...
7. Originally Posted by chakim
Well firstly, considering
$\displaystyle m(t) = E\left(e^{X^2t}\right)$
Can't x take on any value between negative infinity and infinity? Since X ~ N(0,1)?
So the new limits from the substitution would be negative infinity to positive infinity?
And also, when I take the function of t out, i get...
I've researched a bit, and i've tried to integrate the function directly (cause its only an exponential) but the limits give me weird answers...
I'm really stuck mr fantastic...
Your integral limits are correct. As far as the final integral, read this for example: Gaussian Integral -- from Wolfram MathWorld
(You really should know this at this level).
8. I'm in 1st year university, studying an actuarial science degree.
I haven't seen this "Gaussian Integral" before mr fantastic.
So I couldn't really understand where you were coming from...
I thought I could just integrate the integral directly considering it wasn't a tricky integral, but the limits don't give a suitable answer.
Is there any other method to integrate, say by recognition of a statistical distribution?
P.S you've been really helpful, and I appreciate it very much.
Thank you
9. Originally Posted by chakim
I'm in 1st year university, studying an actuarial science degree.
I haven't seen this "Gaussian Integral" before mr fantastic.
So I couldn't really understand where you were coming from...
I thought I could just integrate the integral directly considering it wasn't a tricky integral, but the limits don't give a suitable answer.
Is there any other method to integrate, say by recognition of a statistical distribution?
P.S you've been really helpful, and I appreciate it very much.
Thank you
Try comparing the integral with the integral of the pdf for a standard normal distribution and you will get the same well known result.
10. ahhh, cheers!
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Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
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Agnishom
Today 14:56:09
#### bobbym wrote:
Yes, I see that.
with a(0)=1,a(1)=3
This is for what?
anonimnystefy
Today 14:36:25
That looks like something that would please the OP.
By the way, I just though of a twist to the puzzle to make it just a bit harder.
bobbym
Today 09:17:38
Hi;
Welcome to the forum, 308 is the correct answer.
Batman
Today 08:47:13
What you could do is say the kids got a, b, c, d, d (the twins each got the same) and then seek quadruples satisfiing a+b+c+2d=13 than try each value of d.
d=0: 13 candies 3 kids place dividers in ccccccccccccc (c for CANDY!) and see the number of ways we can put the 3 dividers (3 kids left) so 15 choose 2 = 105
d=1: same, 13 choose 2 = 78
d=2: same logic, 11 choose 2 = 55
d=3: ditto I stop saying same 9 choose 2 = 36
d=4: 7 choose 2 = 21
d=5: 5 choose 2 = 10
d=6: 3 choose 2 = 3
105+78+55+36+21+10+3=308
I got 308
bobbym
2013-01-09 06:00:14
That is not what I meant. GF's are rather obscure and not taught. He/she will probably expect a solution in terms of ncr's.
anonimnystefy
2013-01-09 05:58:00
bobbym
2013-01-09 05:53:39
Hi;
Okay, somehow though I do not think these answers are what the OP will require.
anonimnystefy
2013-01-09 05:42:42
Partial fractions in Maxima and expanding by hand.
bobbym
2013-01-09 04:55:59
Hi;
Did them by maxima?
anonimnystefy
2013-01-09 04:45:54
Partial fractions, then expanding.
bobbym
2013-01-09 04:43:48
Hi;
I know how I got mine, may I ask how you got yours?
anonimnystefy
2013-01-09 04:42:17
I think the two are same... I just put mine in the form above because of the common expression 1+(-1)^n. It is noce to know that that part is 0 when n is odd.
bobbym
2013-01-09 04:21:51
Hi;
Yes, I believe it is. I got,
anonimnystefy
2013-01-09 04:14:29
So my formula in post #9 is correct?
bobbym
2013-01-09 04:11:56
Yes, that is correct.
I meant coming up with an analytical form for the coeffs of the expansion of the polynomial in post #3. But I guess that is not really that important.
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# Lesson 6
Squares and Square Roots
## 6.1: Math Talk: Four Squares (5 minutes)
### Warm-up
The purpose of this Math Talk is to elicit strategies and understandings students have for solving equations in one variable and thinking about the graph of an associated function. The connection between the graph of $$y = x^2$$ and solutions to quadratic equations should be familiar from earlier courses. These understandings help students develop fluency and will be helpful later in this lesson when students solve equations that involve squares and square roots.
### Launch
Display one problem at a time alongside the image showing the graph of $$y =x^2$$ and the 4 lines. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
### Student Facing
Find the solutions of each equation mentally.
$$x^2 = 4$$
$$x^2 = 2$$
$$x^2 = 0$$
$$x^2 = \text{-}1$$
### Student Response
For access, consult one of our IM Certified Partners.
### Activity Synthesis
Discuss the connections between the solutions of the equations and the graph of $$y=x^2$$.
Although $$x^2=2$$ has two solutions ($$\sqrt{2}$$ and $$\text{-}\sqrt{2}$$), we don’t “take the square root” of each side of the equation, because by convention, $$\sqrt{2}$$ means the positive square root of 2. If we want the negative square root, we need to write $$\text{-}\sqrt{2}$$. Students will study this convention more closely in the following activity, so it does not need to be discussed in depth at this time.
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)
## 6.2: Finding Square Roots (15 minutes)
### Activity
The purpose of this activity is for students to consider some reasons for the convention that $$\sqrt{x}$$ means the positive square root of $$x$$. Students think about what it would mean if the square root function were not a function. In the synthesis, the convention that makes $$\sqrt{x}$$ a function is introduced. It should be emphasized that there are still two square roots of every positive number, but the square root function only gives us one of those. Students will explore the consequences of this convention in future activities.
### Launch
Arrange students in groups of 2. Display the following for all to see:
$$\sqrt{4} + \sqrt{9} =$$ ?
Tell students that in this activity, they will think about what the answer to this question is. Give students 2 minutes to read the statement, think, and write down their answers individually, and another 2 minutes for pairs to share their thoughts. Follow with a whole-class discussion.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses for Clare’s question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “Why do you think only one of these could be the same as $$\sqrt{4}+\sqrt{9}$$?” or “Why do you think all of them could be the same as $$\sqrt{4}+\sqrt{9}$$?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify their reasoning for their responses to Clare’s question.
Design Principle(s): Optimize output (for justification); Cultivate conversation
### Student Facing
Clare was adding $$\sqrt{4}$$ and $$\sqrt{9}$$, and at first she wrote $$\sqrt{4} + \sqrt{9} = 2+3$$. But then she remembered that 2 and -2 both square to make 4, and that 3 and -3 both square to make 9. She wrote down all the possible combinations:
2 + 3 = 5
2 + (-3) = -1
(-2) + 3 = 1
(-2) + (-3) = -5
Then she wondered, “Which of these are the same as $$\sqrt{4} + \sqrt{9}$$? All of them? Or only some? Or just one?”
### Student Response
For access, consult one of our IM Certified Partners.
### Student Facing
#### Are you ready for more?
1. How many solutions are there to each equation?
1. $$x^3=8$$
2. $$y^3=\text -1$$
3. $$z^4 = 16$$
4. $$w^4 = \text -81$$
2. Write a rule to determine how many solutions there are to the equation $$x^n=m$$ where $$n$$ and $$m$$ are non-zero integers.
### Student Response
For access, consult one of our IM Certified Partners.
### Activity Synthesis
Invite students to share their answers and reasons. Here are some questions for discussion if needed:
• If only one of these numbers is the sum of $$\sqrt{4}$$ and $$\sqrt{9}$$, does that mean that 4 and 9 each have only one square root?
• If all four of these are the sum of $$\sqrt{4}$$ and $$\sqrt{9}$$, what would happen if we added $$\sqrt{16}$$ to them? How many answers would we get?
• If more than one of these numbers is the same as $$\sqrt{4} + \sqrt{9}$$, then are they the same as each other?
The goal of the discussion is for students to consider some reasons why we might want the operation of taking the square root to give us only one number. These questions do not have to be explored in depth at this time.
To conclude the discussion and preview the work ahead, display the following graphs and the functions they represent for all to see:
Tell students that these graphs represent the two possibilities that they have been thinking about. $$b=\sqrt{a}$$ is a function, because each value of $$a$$ has only one corresponding value of $$b$$. But $$d^2=c$$ is not a function, because many values of $$c$$ have two corresponding $$d$$ values. If we used a graph like $$d^2=c$$ to find square roots, we would not get one unique value for each square root. This is why there is a convention in mathematics which says that the symbol $$\sqrt{x}$$ means the positive square root of $$x$$. All positive numbers do have two square roots, but when we write $$\sqrt{x}$$, or $$x^{\frac12}$$, that means only the positive square root. We indicate the negative square root by writing $$\text- \sqrt{x}$$ or $$\text- x^{\frac12}$$.
## 6.3: One Solution or Two? (15 minutes)
### Activity
In this activity, students analyze the graphs of $$b=\sqrt{a}$$ and $$t=s^2$$ in order to think about solutions to equations like $$\sqrt{a}=4$$ and $$s^2=5$$. They build on the discussion from the previous activity to find that equations like $$\sqrt{a}=4$$ have only one solution, while equations like $$s^2=5$$ have exactly two solutions that are opposites of each other.
### Student Facing
1. The graph of $$b=\sqrt{a}$$ is shown.
1. Complete the table with the exact values and label the corresponding points on the graph with the exact values.
$$a$$ $$\sqrt{a}$$ 1 4 9 12 16 20
2. Label the point on the graph that shows the solution to $$\sqrt{a} = 4$$.
3. Label the point on the graph that shows the solution to $$\sqrt{a} = 5$$.
4. Label the point on the graph that shows the solution to $$\sqrt{a} = \sqrt{5}$$.
2. The graph of $$t = s^2$$ is shown.
1. Label the point(s) on the graph that show(s) the solution(s) to $$s^2 = 25$$.
2. Label the point(s) on the graph that show(s) the solution(s) to $$\sqrt{t} = 5$$.
3. Label the point(s) on the graph that show(s) the solution(s) to $$s^2 = 5$$.
### Student Response
For access, consult one of our IM Certified Partners.
### Anticipated Misconceptions
Since students usually see $$x$$-values on the horizontal axis and $$y$$-values on the vertical, they may look for $$a$$ or $$s$$ values on the wrong axis. Encourage students to annotate the graph by drawing horizontal or vertical lines that will intersect the curves at the point that represents the solution, or using some other method that is helpful for them. For example, when solving $$s^2=25$$, students can draw a line representing $$t=25$$ and see where it hits the graph of $$t=s^2$$, since these points represent the solutions.
### Activity Synthesis
The purpose of this discussion is to emphasize that equations like $$\sqrt{a}=5$$ have only one solution while equations like $$s^2=11$$ have two solutions. Students may find this confusing since we can talk about “the square roots” (plural) of a positive number, so it is important to remind students of the discussion in the previous activity and the convention that we use the symbol $$\sqrt{a}$$ for the positive square root of $$a$$ while $$\text-\sqrt{a}$$ is used for the negative square root of $$a$$. Display the two graphs and tables from the activity for all to see throughout the discussion. Here are some questions for discussion:
• “How can you use the graph of $$b=\sqrt{a}$$ to see that the equation $$\sqrt{a}=5$$ has only one solution?” (The horizontal line $$b=5$$ intersects the graph of $$b=\sqrt{a}$$ at one point, $$(25,5)$$.)
• “How can you use the graph of $$t=s^2$$ to see that the equation $$s^2=12$$ has two solutions? What are the exact values of the two solutions?” (The horizontal line $$t=12$$ intersects the graph of $$t=s^2$$ at two points, $$(\text- \sqrt{12},12)$$ and $$(\sqrt{12},12)$$. The two solutions are $$\text- \sqrt{12}$$ and $$\sqrt{12}$$.)
Speaking: MLR8 Discussion Supports. As students share their responses to the first question, press for details by asking how they know that the intersection of the horizontal line $$b=5$$ and the graph of $$b=\sqrt{a}$$ represents the solution of $$\sqrt{a}=5$$. Show concepts multi-modally by displaying the graph of $$b=\sqrt{a}$$ and drawing the horizontal line $$b=5$$. This will help students justify why equations such as $$\sqrt{a}=5$$ have only one solution.
Design Principle(s): Support sense-making; Optimize output (for justification)
Representation: Internalize Comprehension. Use color and annotations to illustrate connections between representations in a problem. For example, circle $$s^2=25$$ in a certain color and draw a horizontal line at 25 in the same color to show where it intersects the graph of $$t=s^2$$.
Supports accessibility for: Visual-spatial processing; Conceptual processing
## Lesson Synthesis
### Lesson Synthesis
In this lesson, students encountered equations of the form $$x^2 = a$$ and $$\sqrt{x} = a$$ where $$a$$ is positive. Here are some questions for discussion:
• “What do you know about the number $$\sqrt{17}$$?” (It is a number that squares to make 17. It is positive. It is a little greater than 4.)
• “What are all the numbers that square to make 17? In other words, what are the square roots of 17?” ($$\sqrt{17}$$ and $$\text- \sqrt{17}$$)
• “Let’s say $$a$$ is a positive number. What do you know about the number $$\sqrt{a}$$?” (It is a number that squares to make $$a$$. It is positive.)
• “What are all the numbers that square to make $$a$$? That is, what are the square roots of $$a$$?” ($$\sqrt{a}$$ and $$\text- \sqrt{a}$$)
• “Is it possible for $$\sqrt{a}$$ to be equal to -10? Explain your reasoning.” (No, it’s not possible because $$\sqrt{a}$$ is a positive number and -10 is not.)
## 6.4: Cool-down - Squares and Roots (5 minutes)
### Cool-Down
For access, consult one of our IM Certified Partners.
## Student Lesson Summary
### Student Facing
The symbol $$\sqrt{11}$$ represents the positive square root of 11. If we want to represent the negative square root, we write $$\text{-} \sqrt{11}$$.
The equation $$x^2 = 11$$ has two solutions, because $$\sqrt{11}^2=11$$, and also$$\left(\text{-}\sqrt{11}\right)^2=11$$.
The equation $$\sqrt{x} = 11$$ only has one solution, namely 121.
The equation $$\sqrt{x} = \sqrt{11}$$ only has one solution, namely 11.
The equation $$\sqrt{x} = \text{-}11$$ doesn’t have any solutions, because the left side is positive and the right side is negative, which is impossible, because a positive number cannot equal a negative number.
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# Thread: power series of arctan'x
1. ## power series of arctan'x
how could i expand something such as arctan'x (derivate of arctanx ... i.e. d/dx arctanx) into a power series. also how would you be able to find the radius of convergence for it?
so far i have managed to work out that:
arctan'x = $\displaystyle \frac{1}{1 + x^2}$
$\displaystyle \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n}$
how do you work out the "radius of convergence" though: i know it is : |x|< 1.. but how do you work it out please?
i tried it on $\displaystyle (-1)^n x^{2n}$
i ended up with
$\displaystyle a_{n+1} / a_{n} = \frac{|x|^{2n + 2}}{|x|^{2n}} = |x|^2/1$ as n tends to infinity... ...
so radius of convergence is |x|< 1...
is this working out correct?
2. If $\displaystyle |x|<1$ then,
$\displaystyle 1-x^2+x^4-x^6 + ... = \frac{1}{1+x^2}$
Integrate both side from $\displaystyle 0\mbox{ to }t$ where $\displaystyle |t|<1$ to get,
$\displaystyle \tan^{-1}t = t - \frac{t^3}{3}+...$
Now this hold if $\displaystyle -1 < t < 1$.
But it also hold when $\displaystyle t=1$.
That is a little more difficult to show.
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# RBSE Solutions for Class 8 Maths Chapter 7 Construction of Quadrilaterals Ex 7.3
RBSE Solutions for Class 8 Maths Chapter 7 Construction of Quadrilaterals Ex 7.3 is part of RBSE Solutions for Class 8 Maths. Here we have given Rajasthan Board RBSE Class 8 Maths Chapter 7 Construction of Quadrilaterals Exercise 7.3.
Board RBSE Textbook SIERT, Rajasthan Class Class 8 Subject Maths Chapter Chapter 7 Chapter Name Construction of Quadrilaterals Exercise Exercise 7.3 Number of Questions 5 Category RBSE Solutions
## Rajasthan Board RBSE Class 8 Maths Chapter 7 Construction of Quadrilaterals Ex 7.3
Question 1.
Construct a quadrilateral ABCD in which AB = BC = 3.0 cm., AD = CD = 5.0 cm. and ∠ABC = 120°.
Solution
First of all we draw a rough sketch of the given measurements :
Steps of Construction
1. First of all, we draw a line segment AB = 3.0 cm.
2. At point B(RBSESolutions.com)with the help of pencil and compass, construct ∠ABX = 120° with AB from BX cut BC = 3 cm.
3. Draw an arc of radius 5 cm. from point B and an arc of radius 5 cm. from point C intersecting each other at D. Join AD and CD. Thus, we obtained a required quadrilateral ABCD.
Question 2.
Construct a quadrilateral PQRS in which PQ = 2.8 cm., QR = 3.1 cm.. RS = 2.6 cm.. SP = 3.3 cm. and ∠P = 60°.
Solution
First of(RBSESolutions.com)all, we draw a rough sketch of the given measurements :
Steps of Construction
1. First of all, draw a line segment PQ = 2.8 cm.
2. At point P, we construct an angle of QPX = 60° with PQ with the help of pencil and compass. Take a arc of radius 3.3 cm. and cut PS from PX.
3. From(RBSESolutions.com)point S draw an arc of radius 2.6 cm. and from point Q an arc of radius 3.1 cm. draw intersecting each other at point R. Join SR and QR.
Thus, we obtained a required quadrilateral PQRS.
Question 3.
Construct a rectangle whose sides are 4.2 cm. and 2.5 cm. Measure its diagonal lengths.
Solution
Draw a rough sketch of given measurements :
Steps of Construction
1. First of all, we draw a line segment AB = 4.2 cm.
2. Construct ∠BAX = 90° at point A with AB with the help of a pencil and compass. An arc of 2.5 cm. cut from AX. Put D at the cut point.
3. Similarly, we construct ∠ABX = 90° with AB at point B and arc of 2.5 cm cut a point C from BY.
4. Join CD.
Thus, we get required ABCD
Measures(RBSESolutions.com)of AC and BD = 4.9 cm.
Question 4.
Construct a rhombus in which one angle? is 75° and one side is 5.2 cm.
Solution
First of all we draw a(RBSESolutions.com)rough sketch of the given measurements :
b
Steps of Construction
1. First, we draw a line segment AB = 5.2 cm.
2. At point B, we(RBSESolutions.com)draw ∠ABX = 75° with AB and cut an arc of 5.2 cm on BX. Mark the cut point as C.
3. Draw an arc of radius 5.2 cm. from C and cut an arc of radius 5.2 cm. from A. The point of intersection is marked as point D.
Thus, ABCD is,a required rhombus.
Question 5.
Draw a square with one side of 5 cm.
Solution
Draw a(RBSESolutions.com)rough sketch of the given measurements :
Steps of Construction
1. First of all, we draw a line segment AB = 5.1 cm.
2. Construct an angle ABX of 90° at B with the help of pencil and Compass.
3. Taking B(RBSESolutions.com)as a centre and radius 5.1 cm. is drawn an arc intersection BX at C.
4. Now taking A as centre, draw an arc of radius 5.1 cm. and taking C as centre draw an arc of 5.1 cm. intersection each other at point D.
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Always Right Answers To Community
# How to Find Price Demand Equation
It seems like a daunting task, but finding the price demand equation is actually not that difficult. There are a few steps that you need to follow in order to find this equation. First, you need to identify the two variables that are involved in the equation.
These variables are price and quantity demanded. Once you have identified these two variables, you need to determine how they are related to each other. This relationship is known as the demand curve.
The demand curve will show you how price and quantity demanded are related to each other. Once you have determined the demand curve, you can then find the price demand equation.
• Look at a demand curve and identify the price and quantity demanded
• Use the information from the demand curve to calculate the elasticity of demand
• Determine the point of intersection between the two axes on the graph
• This is known as the y-intercept
• Use the y-intercept and slope to write the equation in slope-intercept form, which is y = mx + b
Contents
## What is the Price Demand Equation
In economics, the price demand equation is an equation that describes the relationship between the price of a good or service and the quantity demanded by consumers. The equation is represented by the following formula: P = D(Q), where P is the price, D is the demand function, and Q is quantity demanded. The demand function shows how much consumers are willing and able to purchase at different prices.
The quantity demanded is the amount of a good or service that consumers are willing and able to purchase at a given price. The price demand equation can be used to predict how changes in price will affect consumer behavior. For example, if the price of a good increases, then we would expect the quantity demanded to decrease (assuming everything else remains constant).
This relationship is known as “the law of demand.” The law of demand states that there is an inverse relationship between prices and quantities demanded – as prices increase, quantities demanded decrease, and vice versa. There are many factors that can affect the shape of the demand curve, including income levels, tastes and preferences, substitute goods, etc.
A change in any of these factors can cause a shift in the entire curve. For example, if incomes rise then people will have more money to spend on goods and services overall and this will lead to an increase indemand (i.e., a rightward shift in the curve).
## Price Demand Equation Calculator
If you’re looking to calculate the price demand equation, there are a few things you need to know. First, what is the price demand equation? The price demand equation is a mathematical formula that helps economists predict how changes in prices will affect consumer demand.
In other words, it allows us to see how much people are willing to buy of a good or service at different prices. This information is important for businesses because it can help them set prices that will maximize their profits. Now that we know what the price demand equation is, let’s look at how to calculate it.
To do this, we’ll need two things: data on prices and data on quantity demanded. Once we have this data, we can plug it into the following formula: Price Demand Equation = (% Change in Quantity Demanded / % Change in Price) x Price
For example, let’s say that we want to calculate the price elasticity of demand for apples. We know that when the price of apples increases by 10%, the quantity demanded decreases by 5%. Plugging these values into our formula gives us:
Price Elasticity of Demand = (-5% / 10%) x \$1 = -0.5 Thus, we can conclude that the demand for apples is relatively inelastic – meaning that people aren’t very sensitive to changes in price when it comes to buying apples.
## Price Demand Equation to Revenue Function
In microeconomics, the price demand equation expresses the relationship between price and quantity demanded. It is usually represented as a straight line on a graph, with quantity demanded increasing as price decreases. The slope of the line represents the amount by which quantity demanded changes for each unit change in price.
The revenue function is a mathematical expression of a firm’s total revenue. It shows how much revenue a firm generates at different levels of output. The formula for the revenue function is: R(x) = p(x)*q(x), where p(x) is the price per unit of output and q(x) is the quantity of output sold.
The two equations are related because the revenue function is simply the product of price and quantity (the demand equation). This means that, all else being equal, an increase in price will lead to an increase in revenue, while a decrease in price will lead to a decrease in revenue. Of course, things are rarely equal in the real world, so it’s important to understand both equations in order to make accurate predictions about how changes in prices will affect a firm’s bottom line.
## Price Demand Function Example
In microeconomics, the price demand function is a mathematical function that indicates the quantity of a good or service that consumers are willing and able to purchase at various prices. The function is represented by a graph with price on the y-axis and quantity demanded on the x-axis. The price demand function can be used to predict how changes in price will affect consumer behavior.
For example, if the price of a good increases, we would expect the quantity demanded to decrease (assuming all other things remain constant). This relationship is known as the law of demand. Let’s take a look at an example.
Suppose we’re interested in predicting how changes in the price of coffee will impact sales at our local coffee shop. We could begin by creating a table that lists different prices for coffee and the corresponding quantities demanded: Price (\$ per cup) Quantity Demanded (cups per day)
\$2 10 \$3 9
## Demand Equation Example
A demand equation is a mathematical expression of the relationship between the quantity of a good or service that consumers are willing and able to purchase, and the price of that good or service. The demand equation can be used to predict how changes in price will affect the quantity demanded by consumers. In its most basic form, the demand equation is expressed as Q = f(P), where Q is the quantity of a good or service demanded, and P is the price of that good or service.
The function f represents the law of demand, which states that, all else being equal, as the price of a good or service increases, the quantity demanded by consumers decreases. The demand equation can be used to calculate the elasticity of demand for a good or service. Elasticity measures how much one variable changes in response to changes in another variable.
In this case, it measures how much changes in price affect changes in quantity demanded. If a small change in prices leads to a large change in quantity demanded (i.e., ifdemand is elastic), then demand is said to be elastic; if a large change in prices leads to only a small change in quantity demanded (i.e., ifdemand is inelastic), then demand is said to be inelastic.
## Conclusion
In order to find the price demand equation, one must first determine the quantity demanded at different prices. This can be done by creating a demand schedule or a demand curve. The demand schedule lists the quantity of a good or service that consumers are willing and able to purchase at various prices.
The demand curve is a graphical representation of the data from the demand schedule. Once thedemand schedule or curve has been created, one can then begin to find the price elasticity of demand. This measures how much the quantity demanded changes in response to a change in price.
There are three types of elasticity: perfectly inelastic, inelastic, and elastic. If the quantity demanded does not change at all when the price changes, then the Demand is said to be perfectly inelastic and has an Elasticity coefficient of zero (0). If a small change in price causes a large change in quantity demanded (i.e. people are very sensitive to changes in price), thenDemand is said to be highly elastic and has an Elasticity coefficient greater than one (+1).
Lastly, if a small changein price leads to only a small change in quantity demanded (i.e. people are not very sensitiveto changesinprice),thenDemandis saidto beinelasticandhasanElasticitycoefficientbetweenzeroandone(0
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Formatted question description: https://leetcode.ca/all/1975.html
# 1975. Maximum Matrix Sum
Medium
## Description
You are given an n x n integer matrix. You can do the following operation any number of times:
• Choose any two adjacent elements of matrix and multiply each of them by -1.
Two elements are considered adjacent if and only if they share a border.
Your goal is to maximize the summation of the matrix’s elements. Return the maximum sum of the matrix’s elements using the operation mentioned above.
Example 1:
Input: matrix = [[1,-1],[-1,1]]
Output: 4
Explanation: We can follow the following steps to reach sum equals 4:
• Multiply the 2 elements in the first row by -1.
• Multiply the 2 elements in the first column by -1.
Example 2:
Input: matrix = [[1,2,3],[-1,-2,-3],[1,2,3]]
Output: 16
Explanation: We can follow the following step to reach sum equals 16:
• Multiply the 2 last elements in the second row by -1.
Constraints:
• n == matrix.length == matrix[i].length
• 2 <= n <= 250
• -10^5 <= matrix[i][j] <= 10^5
## Solution
To get the maximum matrix sum, at most one element can be negative after performing the operation any number of times. If there is no element 0 and the number of negative elements is odd, then there has to be one negative element. Otherwise, all elements can be non-negative. If there has to be one negative element, it should have the minimum absolute value. Therefore, find the element with the minimum absolute value and make it negative if there has to be one negative element (the element can be positive in the original matrix), and calculate the sum of elements in the matrix.
class Solution {
public long maxMatrixSum(int[][] matrix) {
long sum = 0;
int minPositive = Integer.MAX_VALUE;
int minNegative = Integer.MIN_VALUE;
int positiveCount = 0, negativeCount = 0, zeroCount = 0;
int side = matrix.length;
for (int i = 0; i < side; i++) {
for (int j = 0; j < side; j++) {
int num = matrix[i][j];
sum += Math.abs(num);
if (num > 0) {
minPositive = Math.min(minPositive, num);
positiveCount++;
} else if (num < 0) {
minNegative = Math.max(minNegative, num);
negativeCount++;
} else
zeroCount++;
}
}
if (zeroCount > 0 || negativeCount % 2 == 0)
return sum;
if (positiveCount == 0)
return sum += minNegative * 2;
int minPositiveAbs = minPositive, minNegativeAbs = -minNegative;
sum -= Math.min(minPositiveAbs, minNegativeAbs) * 2;
return sum;
}
}
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# 47.98 kg to lbs - 47.98 kilograms to pounds
Do you want to know how much is 47.98 kg equal to lbs and how to convert 47.98 kg to lbs? You are in the right place. This whole article is dedicated to kilogram to pound conversion - theoretical and also practical. It is also needed/We also want to highlight that whole this article is devoted to only one amount of kilograms - that is one kilogram. So if you want to learn more about 47.98 kg to pound conversion - read on.
Before we get to the more practical part - it means 47.98 kg how much lbs calculation - we want to tell you few theoretical information about these two units - kilograms and pounds. So we are starting.
How to convert 47.98 kg to lbs? 47.98 kilograms it is equal 105.7777933076 pounds, so 47.98 kg is equal 105.7777933076 lbs.
## 47.98 kgs in pounds
We will start with the kilogram. The kilogram is a unit of mass. It is a base unit in a metric system, known also as International System of Units (in abbreviated form SI).
From time to time the kilogram is written as kilogramme. The symbol of this unit is kg.
The kilogram was defined first time in 1795. The kilogram was described as the mass of one liter of water. This definition was simply but impractical to use.
Then, in 1889 the kilogram was defined using the International Prototype of the Kilogram (in short form IPK). The International Prototype of the Kilogram was prepared of 90% platinum and 10 % iridium. The IPK was in use until 2019, when it was switched by a new definition.
Nowadays the definition of the kilogram is based on physical constants, especially Planck constant. Here is the official definition: “The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of c and ΔνCs.”
One kilogram is 0.001 tonne. It can be also divided into 100 decagrams and 1000 grams.
## 47.98 kilogram to pounds
You learned a little about kilogram, so now we can go to the pound. The pound is also a unit of mass. It is needed to underline that there are not only one kind of pound. What does it mean? For instance, there are also pound-force. In this article we want to concentrate only on pound-mass.
The pound is used in the British and United States customary systems of measurements. Naturally, this unit is used also in other systems. The symbol of the pound is lb or “.
The international avoirdupois pound has no descriptive definition. It is defined as exactly 0.45359237 kilograms. One avoirdupois pound could be divided into 16 avoirdupois ounces and 7000 grains.
The avoirdupois pound was implemented in the Weights and Measures Act 1963. The definition of the pound was placed in first section of this act: “The yard or the metre shall be the unit of measurement of length and the pound or the kilogram shall be the unit of measurement of mass by reference to which any measurement involving a measurement of length or mass shall be made in the United Kingdom; and- (a) the yard shall be 0.9144 metre exactly; (b) the pound shall be 0.45359237 kilogram exactly.”
### How many lbs is 47.98 kg?
47.98 kilogram is equal to 105.7777933076 pounds. If You want convert kilograms to pounds, multiply the kilogram value by 2.2046226218.
### 47.98 kg in lbs
Theoretical section is already behind us. In this section we want to tell you how much is 47.98 kg to lbs. Now you know that 47.98 kg = x lbs. So it is high time to know the answer. Have a look:
47.98 kilogram = 105.7777933076 pounds.
It is a correct outcome of how much 47.98 kg to pound. You may also round off the result. After rounding off your result will be exactly: 47.98 kg = 105.556 lbs.
You know 47.98 kg is how many lbs, so have a look how many kg 47.98 lbs: 47.98 pound = 0.45359237 kilograms.
Obviously, this time you can also round it off. After rounding off your result will be as following: 47.98 lb = 0.45 kgs.
We are also going to show you 47.98 kg to how many pounds and 47.98 pound how many kg outcomes in tables. Let’s see:
We are going to start with a chart for how much is 47.98 kg equal to pound.
### 47.98 Kilograms to Pounds conversion table
Kilograms (kg) Pounds (lb) Pounds (lbs) (rounded off to two decimal places)
47.98 105.7777933076 105.5560
Now look at a table for how many kilograms 47.98 pounds.
Pounds Kilograms Kilograms (rounded off to two decimal places
47.98 0.45359237 0.45
Now you learned how many 47.98 kg to lbs and how many kilograms 47.98 pound, so it is time to move on to the 47.98 kg to lbs formula.
### 47.98 kg to pounds
To convert 47.98 kg to us lbs a formula is needed. We will show you two formulas. Let’s begin with the first one:
Amount of kilograms * 2.20462262 = the 105.7777933076 outcome in pounds
The first formula give you the most exact result. In some situations even the smallest difference can be considerable. So if you want to get an exact result - this version of a formula will be the best for you/option to convert how many pounds are equivalent to 47.98 kilogram.
So go to the another version of a formula, which also enables calculations to know how much 47.98 kilogram in pounds.
The second version of a formula is down below, let’s see:
Amount of kilograms * 2.2 = the result in pounds
As you can see, this formula is simpler. It could be the best solution if you want to make a conversion of 47.98 kilogram to pounds in fast way, for instance, during shopping. You only have to remember that final outcome will be not so correct.
Now we want to show you these two formulas in practice. But before we are going to make a conversion of 47.98 kg to lbs we want to show you another way to know 47.98 kg to how many lbs totally effortless.
### 47.98 kg to lbs converter
An easier way to know what is 47.98 kilogram equal to in pounds is to use 47.98 kg lbs calculator. What is a kg to lb converter?
Calculator is an application. Calculator is based on longer version of a formula which we gave you above. Due to 47.98 kg pound calculator you can effortless convert 47.98 kg to lbs. You only need to enter amount of kilograms which you need to calculate and click ‘calculate’ button. The result will be shown in a flash.
So try to convert 47.98 kg into lbs using 47.98 kg vs pound calculator. We entered 47.98 as an amount of kilograms. Here is the outcome: 47.98 kilogram = 105.7777933076 pounds.
As you see, our 47.98 kg vs lbs converter is intuitive.
Now we can go to our primary issue - how to convert 47.98 kilograms to pounds on your own.
#### 47.98 kg to lbs conversion
We will start 47.98 kilogram equals to how many pounds conversion with the first version of a formula to get the most accurate result. A quick reminder of a formula:
Number of kilograms * 2.20462262 = 105.7777933076 the result in pounds
So what have you do to know how many pounds equal to 47.98 kilogram? Just multiply number of kilograms, this time 47.98, by 2.20462262. It is exactly 105.7777933076. So 47.98 kilogram is equal 105.7777933076.
It is also possible to round it off, for instance, to two decimal places. It gives 2.20. So 47.98 kilogram = 105.5560 pounds.
It is time for an example from everyday life. Let’s convert 47.98 kg gold in pounds. So 47.98 kg equal to how many lbs? And again - multiply 47.98 by 2.20462262. It is exactly 105.7777933076. So equivalent of 47.98 kilograms to pounds, if it comes to gold, is 105.7777933076.
In this example it is also possible to round off the result. It is the result after rounding off, this time to one decimal place - 47.98 kilogram 105.556 pounds.
Now we can move on to examples calculated with a short version of a formula.
#### How many 47.98 kg to lbs
Before we show you an example - a quick reminder of shorter formula:
Amount of kilograms * 2.2 = 105.556 the outcome in pounds
So 47.98 kg equal to how much lbs? As in the previous example you have to multiply amount of kilogram, this time 47.98, by 2.2. Have a look: 47.98 * 2.2 = 105.556. So 47.98 kilogram is 2.2 pounds.
Make another calculation with use of this version of a formula. Now convert something from everyday life, for instance, 47.98 kg to lbs weight of strawberries.
So convert - 47.98 kilogram of strawberries * 2.2 = 105.556 pounds of strawberries. So 47.98 kg to pound mass is exactly 105.556.
If you learned how much is 47.98 kilogram weight in pounds and can calculate it using two different versions of a formula, we can move on. Now we are going to show you all outcomes in tables.
#### Convert 47.98 kilogram to pounds
We are aware that results presented in tables are so much clearer for most of you. We understand it, so we gathered all these outcomes in tables for your convenience. Due to this you can easily compare 47.98 kg equivalent to lbs outcomes.
Start with a 47.98 kg equals lbs chart for the first formula:
Kilograms Pounds Pounds (after rounding off to two decimal places)
47.98 105.7777933076 105.5560
And now have a look at 47.98 kg equal pound table for the second formula:
Kilograms Pounds
47.98 105.556
As you can see, after rounding off, when it comes to how much 47.98 kilogram equals pounds, the results are the same. The bigger amount the more significant difference. Keep it in mind when you need to make bigger number than 47.98 kilograms pounds conversion.
#### How many kilograms 47.98 pound
Now you learned how to calculate 47.98 kilograms how much pounds but we want to show you something more. Are you curious what it is? What about 47.98 kilogram to pounds and ounces conversion?
We will show you how you can convert it little by little. Begin. How much is 47.98 kg in lbs and oz?
First thing you need to do is multiply amount of kilograms, this time 47.98, by 2.20462262. So 47.98 * 2.20462262 = 105.7777933076. One kilogram is exactly 2.20462262 pounds.
The integer part is number of pounds. So in this example there are 2 pounds.
To convert how much 47.98 kilogram is equal to pounds and ounces you need to multiply fraction part by 16. So multiply 20462262 by 16. It is exactly 327396192 ounces.
So your result is equal 2 pounds and 327396192 ounces. You can also round off ounces, for instance, to two places. Then your result is exactly 2 pounds and 33 ounces.
As you can see, conversion 47.98 kilogram in pounds and ounces easy.
The last conversion which we are going to show you is conversion of 47.98 foot pounds to kilograms meters. Both foot pounds and kilograms meters are units of work.
To convert it you need another formula. Before we show you it, look:
• 47.98 kilograms meters = 7.23301385 foot pounds,
• 47.98 foot pounds = 0.13825495 kilograms meters.
Now see a formula:
Number.RandomElement()) of foot pounds * 0.13825495 = the result in kilograms meters
So to convert 47.98 foot pounds to kilograms meters you need to multiply 47.98 by 0.13825495. It is exactly 0.13825495. So 47.98 foot pounds is equal 0.13825495 kilogram meters.
You can also round off this result, for example, to two decimal places. Then 47.98 foot pounds is 0.14 kilogram meters.
We hope that this calculation was as easy as 47.98 kilogram into pounds calculations.
This article was a big compendium about kilogram, pound and 47.98 kg to lbs in conversion. Due to this calculation you learned 47.98 kilogram is equivalent to how many pounds.
We showed you not only how to do a conversion 47.98 kilogram to metric pounds but also two other calculations - to check how many 47.98 kg in pounds and ounces and how many 47.98 foot pounds to kilograms meters.
We showed you also another solution to do 47.98 kilogram how many pounds calculations, this is with use of 47.98 kg en pound converter. It will be the best choice for those of you who do not like converting on your own at all or this time do not want to make @baseAmountStr kg how lbs conversions on your own.
We hope that now all of you are able to make 47.98 kilogram equal to how many pounds conversion - on your own or with use of our 47.98 kgs to pounds converter.
It is time to make your move! Let’s convert 47.98 kilogram mass to pounds in the best way for you.
Do you need to do other than 47.98 kilogram as pounds conversion? For example, for 10 kilograms? Check our other articles! We guarantee that conversions for other numbers of kilograms are so easy as for 47.98 kilogram equal many pounds.
### How much is 47.98 kg in pounds
We want to sum up this topic, that is how much is 47.98 kg in pounds , we prepared one more section. Here you can find the most important information about how much is 47.98 kg equal to lbs and how to convert 47.98 kg to lbs . Have a look.
What is the kilogram to pound conversion? It is a mathematical operation based on multiplying 2 numbers. Let’s see 47.98 kg to pound conversion formula . See it down below:
The number of kilograms * 2.20462262 = the result in pounds
So what is the result of the conversion of 47.98 kilogram to pounds? The accurate answer is 105.7777933076 pounds.
There is also another way to calculate how much 47.98 kilogram is equal to pounds with second, easier type of the formula. Check it down below.
The number of kilograms * 2.2 = the result in pounds
So this time, 47.98 kg equal to how much lbs ? The result is 105.7777933076 lbs.
How to convert 47.98 kg to lbs in just a moment? It is possible to use the 47.98 kg to lbs converter , which will do whole mathematical operation for you and give you an accurate answer .
#### Kilograms [kg]
The kilogram, or kilogramme, is the base unit of weight in the Metric system. It is the approximate weight of a cube of water 10 centimeters on a side.
#### Pounds [lbs]
A pound is a unit of weight commonly used in the United States and the British commonwealths. A pound is defined as exactly 0.45359237 kilograms.
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# Lesson 6Be the ChangePractice Understanding
### 1.
Graph the following linear equations on the grid. The equation has been graphed for you. For each new equation explain what the number does to the graph of . Pay attention to the -intercept, the -intercept, and the slope. Identify what changes in the graph and what stays the same.
### 2.
The graph of is given. For each equation, predict what you think the number will do to the graph. Then graph the equation.
Prediction:
Prediction:
## Set
For each relation given:
1. Identify whether or not the relation is a function. (If it’s not a function, skip b–d.)
2. Determine if the function is linear, exponential, quadratic, or neither.
3. Describe the type of growth.
4. Express the relation in the indicated form.
### 3.
The Driver Education course started with students. Students complete the course by passing the driving examination. After the first month, there were students in the course, after the next month , then
#### a.
Is it function? (If not, skip parts b–d.)
Yes
No
#### b.
Is the function linear, exponential, quadratic, or neither?
linear
exponential
neither
#### c.
How does it change?
#### d.
Make a graph. Label your axes and the scale.
### 4.
#### a.
Identify whether or not the relation is a function. (If it’s not a function, skip b–d.)
Yes
No
#### b.
Determine if the function is linear, exponential, quadratic, or neither.
linear
exponential
neither
#### c.
How does it change?
#### d.
Write the explicit equation.
### 5.
#### a.
Identify whether or not the relation is a function. (If it’s not a function, skip b–d.)
Yes
No
#### b.
Determine if the function is linear, exponential, quadratic, or neither.
linear
exponential
neither
#### c.
How does it change?
Create a table.
### 6.
Distance traveled in feet based on the speed in mph the baseball leaves the bat.
#### a.
Identify whether or not the relation is a function. (If it’s not a function, skip b–d.)
Yes
No
#### b.
Determine if the function is linear, exponential, quadratic, or neither.
linear
exponential
neither
#### c.
How does it change?
#### d.
Predict the distance the baseball flies, if it leaves the bat at a speed of .
## Go
### 7.
1. ___
2. ___
3. ___
I put in a savings account that pays interest compounded annually. I plan to leave it in the bank for years. What amount will I have then?
4. ___
The area of the triangles.
5. ___
6. ___
7. ___
$x$ $f\left(x\right)$ $-7.75$ $-\frac{1}{4}$ $\frac{1}{2}$ $11.6$ $7.75$ $\frac{1}{4}$ $-\frac{1}{2}$ $-11.6$
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# Fundamental confusions of calculus
Discussion in 'Physics & Math' started by arfa brane, Feb 11, 2012.
Not open for further replies.
1. ### arfa branecall me arfValued Senior Member
Messages:
7,743
But when you said $\frac {\partial} {\partial x}$ multiplied by a scalar is not a vector, you said it was "nonsense", you weren't misinterpreting what Penrose was saying? Because that's what he says in his book.
Actually what he says is the scalar should be a function of x (or whatever variable is being varied). But as others have noted that's a mere mathematical detail because a vector multiplied by any scalar is still a vector. That's something I learned in a linear algebra course.
Last edited: Feb 13, 2012
3. ### przyksquishyValued Senior Member
Messages:
3,203
No, I mean your attitude (talking down to others even when it was your fault you hadn't defined your example properly), and generally misrepresenting and revising history.
Also, asserting errors without being able to point them out (in an unrelated post at that), like you just did here.
I've already explained to you that in both those examples, the total derivative is taken through all the function's parameters. The analogue in your case would be to do something like
$\frac{\mathrm{d}f}{\mathrm{d}\theta} \,=\, \frac{\partial f}{\partial \theta} \,+\, \frac{\partial f}{\partial u} \, \frac{\mathrm{d}u}{\mathrm{d}\theta} \,+\, \frac{\partial f}{\partial v} \, \frac{\mathrm{d}v}{\mathrm{d}\theta} \,.$
But since you've already said you're not letting $v$ be a function of $\theta$, you're not doing the same thing as in those examples, and they therefore don't support your case.
I said nothing in my original post in this thread that I don't still stand by. If you think I'm getting something "wrong" now that I got "right" before, then either you didn't understand my original post, or you don't understand what I'm saying now, or both.
I wasn't ignoring anything. I'd answered your question. Then I asked you two similar questions. The first was just the partial derivative of an isolated expression. There was simply no $f$, $u$, or $v$, and it was irrelevant how you might have defined $f$ elsewhere. In the second - a problem I gave you remember - there was an $f$, but no $u$ or $v$. Just $\theta$ and $x$.
5. ### TrippyALEA IACTA ESTStaff Member
Messages:
10,890
Tach, a question, if I may.
How would you apply the chain rule to the following problem:
f(x) = a.sin[sup]2[/sup](bx+c)
I'm not concerned with seeing you work through the problem as a whole at this stage, I would simply like to know how you would define y, and define u.
7. ### AlphaNumericFully ionizedRegistered Senior Member
Messages:
6,702
So now you're claiming the definition of the partial derivative is wrong? Wow.
The impression I get is my opinion and I'm drawing an analogy between your behaviour and that of other members. Neither of them constitute personal attacks on you.
I question that because if you had working familiarity with Hamiltonian/Lagrangian mechanics you'd know that it contradicts your claims. I gave the pendulum example and it explicitly involves a sin term. You have yet to counter it. Much of the Hamiltonian mechanics can be formalised in terms of differential geometry and the whole tangent/cotangent spaces, which pertain to the $\partial_{x}$ vector space basis elements originally asked about. You didn't grasp that either. All of these things fit together into a nice coherent structure in line with what everyone is telling you.
But why don't you enlighten me as to your experience with Hamiltonian mechanics. Can you tell me the canonical momentum for $T = \frac{1}{2}m\dot{q}^{2}$? Do you need me to give you the definition or can you manage it yourself?
I hardly think saying "You're pulling a Reiku" counts as much of an attack. Besides, you are not exactly on the moral high ground when it comes to attitude and behaviour. I find it somewhat funny that you have no problem being extremely abrasive to others but when someone gets anywhere close to not complementing you you cry foul.
Considering how abrasive you have been in the past if you think my comments shouldn't be allowed then you are being a hypocrite. And no, this isn't an insult, it is a statement of fact in that you try to hold others to standards you do not hold yourself to. Not being chummy is not the same as being rude. A certain amount of opinion and commentary is allowed in discussions. People, including moderators, can say "Come on, you're being a bit thick there" or the like, provided it's infrequent and they go on to justify why they are saying it. If you cannot handle a bit of discussion which isn't all hugs and kisses perhaps the internet isn't for you.
I don't feel you're even reading half of what people say. I discussed this, the difference between explicit and implicit parametrisations and the way they are related to the different derivatives. Rather than discuss that you try to be patronising (which is all the more hypocritical given you complained I was supposedly talking down to you) by repeating (as if people haven't seen it already) the same expression again and again.
I see this thread is going the way of the mirrored wheel one. Regardless of whether you consider it 'unworthy' of me to say or not I'll point out that perhaps it's a bad sign when your significant participation in a thread leads to similar 'discussions' as when Motor Daddy starts talking about relativity or Farsight about electromagnetism.
You've been asked a few questions by myself and Trippy and others. If by morning you're still dancing around crying "The book, the book!" then this thread can be relocked and those of us who actually do this stuff for a living can get back to it.
8. ### TachBannedBanned
Messages:
5,265
It isn't , but what does this older error of yours have to do with the partial vs. total derivative we are discussing?
9. ### TachBannedBanned
Messages:
5,265
No, I am simply pointing out that your understanding of the issue is flawed. Twice I showed you the mathematical steps that outline your error.
10. ### TachBannedBanned
Messages:
5,265
So, why aren't you responding to the errors that I pointed out in this post? That post wasn't even calculus, it was basic algebra.
Which is precisely what I did for Pete's benefit. u is chosen as a function of $\theta$, v is chosen as afunction of $x$ resulting into exactly what I posted in post 18:
$\frac{\mathrm{d}f}{\mathrm{d}\theta} \,=\, \frac{\partial f}{\partial \theta} \,+\, \frac{\partial f}{\partial u} \, \frac{\mathrm{d}u}{\mathrm{d}\theta} \,$
Before you and the other jumped in with your ideas about partial vs. total derivatives, I was trying to explain to Pete the reason for the absence of the term: $\, \frac{\partial f}{\partial v} \, \frac{\mathrm{d}v}{\mathrm{d}\theta} \,.$
Last edited: Feb 14, 2012
11. ### arfa branecall me arfValued Senior Member
Messages:
7,743
If it isn't a vector you should be able to revisit Quarkhead's post and explain his mistake to him (and everyone else reading it).
What does a partial derivative like $\frac {\partial} {\partial x}$ of no function have to do with the discussion? I'll let you work it out.
12. ### TachBannedBanned
Messages:
5,265
Based on how you phrased the question, it seems that you need an explanation about what is being debated.
Say that you have two definitions of the function f:
1. $f_1(x)=3x+a sin^2(bx+c)$
and :
2. $f_2(x,u)=3x+u$
where $u=a sin^2(bx+c)$
then, the partial derivatives wrt x of the two functions are DIFFERENT (while the total derivatives are the SAME). Why is this?
$\frac{ \partial f_1}{\partial x}=3+ab sin(2(bx+c))$
$\frac{ \partial f_1}{\partial x}=3$
Finally:
$\frac{d f_1}{dx}=\frac{d f_2}{dx}$
There is video that Pete linked in that explains all this very well. There is a page from a very good calculus book that I linked in that explains this very well. Take your choice.
13. ### TrippyALEA IACTA ESTStaff Member
Messages:
10,890
None of that answered my question, Tach. I don't need an explanation of what is being debated. If I have questions, I will ask them.
I asked you a straightforward question.
Here it is again:
14. ### TachBannedBanned
Messages:
5,265
You mean that you did not recognise $\frac{ df}{d x}=ab sin(2(bx+c))$ ?
15. ### TrippyALEA IACTA ESTStaff Member
Messages:
10,890
Yes, I recognized it, but I didn't ask you to complete the derivation, did I?
I simply asked you to define the first step for me - nothing more.
Using the generalized equation I provided you, how would you define u and y to arrive at dy/dx using the chain rule?
That's all I'm interested in at this stage.
16. ### TachBannedBanned
Messages:
5,265
You see, you still fail to understand that $\lambda \frac {\partial} {\partial x}$ is not a vector because, contrary to your beliefs it doesn't have either sense, nor direction. Now, if , instead of your misgiuded attempt at multiplying $\frac {\partial} {\partial x}$ by a scalar, you tried (as most introductory books show) by a vector, then , you would get a vector, as in:
$\vec{e_x} \frac {\partial} {\partial x}$
The above (and not $\frac {\partial} {\partial x}$ as in your misconceptions) indeed points along the x-axis.
You can even have:
$\vec{e_z} \frac {\partial} {\partial x}$
as a vector. Comes up in the definition of the curl, for example.
Now, that I explained this to you, please let us return to the discussion of partial vs. total derivatives. Thank you.
17. ### TachBannedBanned
Messages:
5,265
Why are you so interested in this? It is obvious that there is a virtually infinite number of ways of doing that, here are two:
$y=au^2$
$u=sin(bx+c)$
$y=au$
$u=sin^2(bx+c)$
You can even go:
$y=au$
$u=sin^2(v)$
$v=bx+c$
Now that I have answered your questions, what are you attempting to accomplish with your post? Where are you going with it?
18. ### PeteIt's not rocket surgeryRegistered Senior Member
Messages:
10,167
przyk's numbers don't seem to gel. Try this instead:
\begin{align} f(t) &= \sin^2t \\ g(t) &= \cos^2t \\ a &= b = c = 1 \\ t &= 0 \\ x &= f(t) \\ &= \sin^2t \\ &= 0 \\ y &= g(t) \\ &= \cos^2t \\ &= 1 \end{align}
What's the temperature of the bug at t=0?
Last edited: Feb 14, 2012
19. ### TachBannedBanned
Messages:
5,265
It isn't only the numbers that "don't gel", it is the whole approach that is pure BS. But he will not own to it.
Huh? Et tu, Brutus?
20. ### PeteIt's not rocket surgeryRegistered Senior Member
Messages:
10,167
Whoops!
That should be x = sin^2t.
It's possible that several posters in this thread are thinking exactly that about you.
21. ### arfa branecall me arfValued Senior Member
Messages:
7,743
--http://en.wikipedia.org/wiki/Partial_derivatives
22. ### TachBannedBanned
Messages:
5,265
What does this have to do with my suggestion for you to calculate the total derivative wrt t of the function $T=e^{-z}(....)$?
23. ### przyksquishyValued Senior Member
Messages:
3,203
*sigh*
You're only delaying the inevitable, you know. $t = 5$, $a = b = c = 1$ and $f(t) = g(t) = \sin(\pi t)^{2}$. What's the bug's temperature?
If I happen to have made any other oversights, due to not thinking this deserves more than about five seconds of my time, I trust you're intelligent enough to figure out the point I was making.
And that may well be the appropriate thing to do. But if you leave $v$ and/or $x$ independent of $\theta$ you're wrong to call that a total derivative.
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# Warm-Up: Determine the slope to be positive, negative, zero, or undefined for the two sets of relations and the graph. 1.(2,1),(4,2),(6,3)3. 2. (3,0),
## Presentation on theme: "Warm-Up: Determine the slope to be positive, negative, zero, or undefined for the two sets of relations and the graph. 1.(2,1),(4,2),(6,3)3. 2. (3,0),"— Presentation transcript:
Warm-Up: Determine the slope to be positive, negative, zero, or undefined for the two sets of relations and the graph. 1.(2,1),(4,2),(6,3)3. 2. (3,0), (5,0), (7,0)
Warm-Up: Determine the slope to be positive, negative, zero, or undefined for the two sets of relations and the graph. 1.(2,1),(4,2),(6,3)3. Positive 2. (3,0), (5,0), (7,0) No slope Negative
Gathering Data “The bigger the better?” Today you are going to look at your hand and one(1) centimeter cube. Now make a bucket of cubes. Take a moment to write that number on your data collection sheet in Part 1. Now use the tape measure to measure your hand from the tip of your 3 rd finger to the bottom of your hand(meets the wrist). Write that number using centimeters.
Use the picture to make sure you measure correctly. *Reminder-Record your answers in centimeters.
“Scatter Plots and Correlation” Standard(s): MAFS.8.SP.1.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
“Scatter Plots” Graphs of data pairs for a real-world situation rarely fall in straight line. The arrangement of data can suggest a relationship (correlation) than can be modeled to help us draw conclusions about the situation.
Scatter Plots A scatter plot is a type of graph that relates 2 data sets by plotting the data as ordered pairs. They are used to determine the strength of a relationship, or correlation, between 2 sets of data. There are 3 general types of correlation that can be determined using a scatter plot.
Positive Correlation In a data set with a positive correlation, you note that as the domain increases, so does the range. Basically, we see that as the x-values increase, so does the y-value.
www.medizinfo.de This scatter plot shows a positive correlation between players height and weight, because generally as the weight increases so does the height.
This data shows that as the number of hours worked increases, so does the amount of earnings. We can surmise that the time worked determines our pay.
Negative Correlation In a data set with a negative correlation, you note that as the domain increases, the range decreases.
www This scatter plots shows a negative correlation between the unemployment rate and the GDP. This suggests that as the GDP increases the unemployment rate decreases. www.economistsview.typepad.com
“Negative Correlation” This scatter plot shows the relationship between the amount of money left after a day of shopping and the number of hours spent shopping.
“Negative Correlations” This is the same data from the previous slide in the form of a 2-column input- output table. Once it’s organized, we can interpret what that data means.
“Correlations” At this time, talk with your neighbor about what we learned so far. Then use your whiteboards and markers to write down 1 real-world scenario that has a positive correlation and 1 real-world scenario with a negative correlation. Identify the independent variable and the dependent variable.
“No Correlation” A scatter plot with no correlation is one in which a change in one data set has no effect on the other. This type of graph has no recognizable pattern.
www.education.com This scatter plot appears to have no correlation at all. There is no pattern in how the x- and y- values behave. Therefore, we can conclude that there is no correlation.
“No Correlation” Some items with no correlation include: 1.Age and number of A’s earned 2.Earnings and ethnicity 3.Hair color and hair length 4.Height and nail length 5.Number of siblings and color of hair
Gathering “More” Data Task We will conduct an experiment to determine whether our hand lengths affect the number of centimeter cubes we can grab from the bucket of cubes on your table/desk.
Directions Take turns(one at a time) attempting to grab as many centimeter cubes from the bucket of cubes in your group. You may not use any other body part(s) to assist you. If a cube falls, it cannot be counted in that trial. Complete 10 trials, recording the number of cubes on your data collection sheet grabbed each time. After you complete a total of 10 trials, find the mean, mode, and range of your data. The mean will serve as your dependent variable(y).
Get Ready … When prompted, you are going to “report out” one by one. The x-value will be your palm size in cm The y-value will be your mean cube value Everyone’s data will be recorded on the board for you to copy on your paper. Make sure you have as many data points as there are people in the class.
Now, graph that data!!!
Analyzing the Data What similarities do you notice about the creation of the graphs? What are the common differences evident in the scatter plots in your group? Do those similarities or differences change the correlations in the data? Why? Or Why Not?
“ Lesson Summary ” In this lesson, we’ve looked at how useful scatter plots are in helping us to determine correlations in 2 sets(bivariate) data. We understand that a set of data may imply a positive correlation, negative correlation, or no correlation at all. We further understand how the correlations help us draw conclusions about the data to further deepen our understanding of related versus unrelated data sets. We can use this data to help us make predictions about values not readily evident in a table of values or a set of ordered pairs.
Ticket Out!!! Write positive, negative, or no correlation for the following scenarios. 1.Age and dance ability 2.Earnings and ethnicity 3.Education and earnings 4.Hours worked and exhaustion 5.Hours of TV watched and quiz grades 6.Size of meal and caloric intake
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# woburn
We consider a man of mass m = 77 kg as shown in the figure below using crutches. The crutches each make an angle of è = 26 with the vertical. Half of the person's weight is supported by the crutches, the other half is supported by the normal forces acting on the soles of the feet. Assuming that the person is at rest, find the magnitude of the force supported by each crutch.
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posted by PsyDAG
2. m=77kg
angle=26 degrees
Normal force holds half the weight.
Positive being up,
Y direction:
(1/2)w-(F1)cos(26)-(F2)cos(26)
=(1/2)(77kg)(9.8m/s^2)-2Tcos(26) [since T1=T2
=> T=377.3N/2cos(26)
T=210N
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posted by Gharib
3. m=77kg
angle=26 degrees
Normal force holds half the weight.
Positive being up,
Y direction:
(1/2)w-(F1)cos(26)-(F2)cos(26)
=(1/2)(77kg)(9.8m/s^2)-2Fcos(26) [since F1=F2
=> F=377.3N/2cos(26)
F=210N
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posted by Gharib
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A certain culture of bacteria quadruples every hour. If a container with these bacteria was half full at 10:00 a.m., at what time was it one-eighth full?
(A) 9:00 a.m.
(B) 7:00 a.m.
(C) 6:00 a.m.
(D) 4:00 a.m.
(E) 2:00 a.m
It seems to be very simple but I am stuck here
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17 Jan 2006, 03:03
Statement says
Time Population
T X
T+1 4X
In other word if we assume
T+1 X
T X/4
10 AM = 1/2 Full
9 AM = 1/4*1/2 Full = 1/8 Full
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17 Jan 2006, 07:53
9 :00 am
as (1/2)/4 = 1/8 whis is what is asked
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17 Jan 2006, 14:15
9:00 AM
Population in the beginning = x/8
Population now = x/2 i.e 4 times that of in the beginning. So one hour has passed.
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17 Jan 2006, 21:33
Pop at 10 am is x/2 and population at t is x/8
so how much has the population increased
(x/2)/(x/8)=4 times and 4 times increase happens in 1 hr so (10-1)=9
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17 Jan 2006, 21:41
Good Job guys OA is A
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23 May 2007, 12:59
andy_gr8 wrote:
Pop at 10 am is x/2 and population at t is x/8
so how much has the population increased
(x/2)/(x/8)=4 times and 4 times increase happens in 1 hr so (10-1)=9
very clear explanation. thanks
now / previous level
(x/2) / (x/8)
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iGCSE (2021 Edition)
# 4.01 Function notation
## Interactive practice questions
Suppose that $\left(7,-6\right)$(7,6) is an ordered pair that satisfies the function $g$g. Write this situation using function notation.
Easy
Approx a minute
Sign up to try all questions
Consider the function $f\left(x\right)=8x+6$f(x)=8x+6.
Consider the function $f\left(x\right)=-9x^2-8x-4$f(x)=9x28x4.
Use the graph of the function $f\left(x\right)$f(x) to find each of the following values.
### Outcomes
#### 0606C1.1
Understand the terms: function, domain, range (image set), one-one function, inverse function and composition of functions.
#### 0606C1.2A
Use function notation. e.g. f(x) = sin x, f: x ↦ sin x, f(x) = lg x or f: x ↦ lg x.
#### 0606C6.1
Solve simple simultaneous equations in two unknowns by elimination or substitution.
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# Calculus and Vectors How to get an A+
## 4.5 An Algorithm for Curve Sketching
A Algorithm for Curve Sketching
1. Domain
denominator 0 (rational functions)
logarithmic argument > 0 (logarithmic functions)
2. Intercepts
f ( x) = 0 (x-intercepts or zeros)
numerator = 0 (for rational functions)
y int = f (0) (if exists)
3. Symmetry
f ( x) = f ( x) (even functions are symmetric about
the y-axis)
f ( x) = f ( x ) (odd functions are symmetric about
the origin)
f ( x + T ) = f ( x) (periodic functions have cycles)
4. Asymptotes
compute lim f ( x) (horizontal asymptote)
x
## Ex 1. Sketch the graph for y = f ( x ) = 3 x 5 5 x 3 .
Domain: x R .
Intercepts: f ( x) = 0 x = 0 or x = 5 / 3 , f (0) = 0
Symmetry:
f ( x) = 3( x) 5 5( x) 3 = 3 x 5 + 5 x 3 = f ( x)
The function y = f (x) is odd.
Asymptotes: none
First Derivative:
f ' ( x) = 15 x 4 15 x 2 = 15 x 2 ( x 2 1)
f ' ( x) = 0 x = 0 or x = 1
f (0) = 0, f (1) = 3 + 5 = 2, f (1) = 3 5 = 2
x
0
1
1
f ( x)
0
`
_
`
2
2
f ' ( x)
0
0
0
+
(1,2) is a local maximum point.
(1,2) is a local minimum point.
Second Derivative:
f ' ' ( x ) = 60 x 3 30 x = 30 x (2 x 2 1)
f ' ' ( x ) = 0 x = 0 or x = 1 / 2
f (0) = 0,
## zero of the denominator but not of the numerator)
compute long division (to find the oblique
asymptotes for rational functions)
f (1 / 2 ) = 7 2 / 8 1.24
x
1/ 2
f ( x)
7 2 /8
x a
5. First Derivative
compute f ' ( x)
find critical points ( f ' ( x) = 0 or f ' ( x) DNE)
create the sign chart for f ' ( x)
find intervals of increase/decrease
find the local extrema (using first derivative test)
and global extrema (if function is defined on a closed
interval)
_
+
f ' ' ( x)
f (1 / 2 ) = (1 / 2 ) 3 [3(1 / 2) 5) = 7 2 / 8 1.24
1/ 2
7 2 /8
0
## (1 / 2 ,7 2 / 8) and (1 / 2 ,7 2 / 8) are points of inflection.
Curve Sketching:
6. Second Derivative
compute f ' ' ( x)
find points where f ' ' ( x) = 0 or f ' ' ( x) DNE
create the sign chart for f ' ' ( x )
find points of inflection
find intervals of concavity upward/downward
check the local extrema using the second
derivative test (if necessary)
7. Curve Sketching
use broken lines to draw the asymptotes
plot x- and y- intercepts, extrema, and inflection
points
draw the curve near the asymptotes
sketch the curve
4.5 An Algorithm for Curve Sketching
2010 Iulia & Teodoru Gugoiu - Page 1 of 4
## Ex 2. Sketch the graph for y = f ( x) = x 3 6 x 2 + 9 x + 1 .
Domain: x R .
Intercepts:
f (1) = 1 6 9 + 1 = 15
f (1) = 1 6 + 9 + 1 = 5
f (0) = 1
There are no rational zeros.
Symmetry:
f ( x) = ( x) 3 6( x) 2 + 9( x) + 1
= x3 6x 2 9x + 1
f ( x) f ( x), f ( x) f ( x)
The function y = f ( x) is neither odd nor even.
Asymptotes: none
First Derivative:
f ' ( x) = 3x 2 12 x + 9 = 3( x 2 4 x + 3) = 3( x 1)( x 3)
f ' ( x) = 0 x = 1 or x = 3
f (1) = 5, f (3) = 27 54 + 27 + 1 = 1
x
3
1
f ( x)
5
_
`
_
1
f ' ( x)
0
0
+
+
(1,5) is a local maximum point.
(3,1) is a local minimum point.
Second Derivative:
f ' ' ( x) = 6 x 12 = 6( x 2)
f ' ' ( x) = 0 x = 2, f (2) = 8 24 + 18 + 1 = 3
x
2
f ( x)
3
f ' ' ( x)
0
+
Curve Sketching:
## Ex 3. Sketch the graph for y = f ( x) =
4x
x2 +1
Domain: x R .
Intercepts: f ( x) = 0 x = 0, f (0) = 0
Symmetry:
4( x )
4x
f ( x) =
= 2
= f ( x)
2
( x) + 1
x +1
The function y = f (x) is odd.
Asymptotes: y = 0 is a horizontal asymptote.
First Derivative:
4( x 2 + 1) 4 x(2 x) 4 4 x 2
4(1 x 2 )
=
=
f ' ( x) =
( x 2 + 1) 2
( x 2 + 1) 2 ( x 2 + 1) 2
4(1)
4
=
= 2
f ' ( x) = 0 x = 1, f (1) =
2
(1) + 1 2
x
1
1
f (x)
_
`
`
2
2
f ' ( x)
0
0
+
(1,2) is a local maximum point.
( 1,2) is a local minimum point.
Second Derivative:
8 x( x 2 + 1) 2 4(1 x 2 )(2)( x 2 + 1)(2 x)
f ' ' ( x) =
( x 2 + 1) 4
=
8 x 3 8 x 16 x + 16 x 3
2
( x + 1)
8 x 3 24 x
2
( x + 1)
8 x( x 2 3)
( x 2 + 1) 3
f ' ' ( x) = 0 x = 0 or x = 3
f (0) = 0,
f ( 3 ) =
4 3
( 3 ) 2 + 1
= 3
0
f ( x)
f ' ' ( x)
3
0
3
0
Curve Sketching:
## 4.5 An Algorithm for Curve Sketching
2010 Iulia & Teodoru Gugoiu - Page 2 of 4
## Ex 4. Sketch the graph for y = f ( x) =
Domain: x R \ {1} .
x2
.
x 1
Intercepts: f ( x) = 0 x = 0, f (0) = 0
Symmetry:
( x) 2
x2
f ( x) =
=
x 1
x +1
f ( x) f ( x), f ( x) f ( x)
The function y = f (x) is neither odd nor even.
Asymptotes:
x2 1 +1
1
f ( x) =
= x +1+
x 1
x 1
y = x + 1 is the equation of the oblique asymptote.
First Derivative:
(2 x)( x 1) x 2 x 2 2 x x( x 2)
f ' ( x) =
=
=
( x 1) 2
( x 1) 2 ( x 1) 2
f ' ( x) = 0 x = 0 or x = 2, f ' ( x) DNE at x = 1
f (0) = 0, f (2) = 4, f (1) DNE
x
0
1
2
f ( x)
0
`
_
_
DNE `
4
f ' ( x)
0
0
DNE
+
+
(0,0) is a local maximum point.
(1,4) is a local minimum point.
Second Derivative:
2( x 1)( x 1) 2 x( x 2)(2)( x 1)
f ' ' ( x) =
( x 1) 4
2[( x 1)( x 1) x( x 2)]
2
=
=
( x 1) 3
( x 1) 3
f ' ' (1) DNE
x
1
f (x)
DNE
f ' ' ( x)
DNE
+
There are no inflection points.
Curve Sketching:
## Ex 5. Sketch the graph for y = f ( x) = x(5 x) 2 / 3 .
Domain: x R .
Intercepts: f ( x) = 0 x = 0 or x = 5, f (0) = 0
Symmetry:
f ( x) = x(5 + x) 2 / 3 , f ( x) f ( x), f ( x) f ( x)
The function y = f (x) is neither odd nor even.
Asymptotes: The function behaves at infinity as x 5 / 3 .
There is no asymptote.
First Derivative:
2
f ' ( x) = (5 x) 2 / 3 + x (5 x) 1/ 3 (1)
3
3
2
= (5 x) 1/ 3 (5 x)1/ 3 (5 x) 2 / 3 + x (5 x) 1/ 3 (1)
3
3
3(5 x) 2 x
15 5 x
5(3 x)
=
=
=
3(5 x)1/ 3
3(5 x)1/ 3 3(5 x)1/ 3
f ' ( x) = 0 at x = 3,
x
3
f (x)
3
_
3 4
f ' ( x)
5
0
DNE
## (3,3 4 ) is a local maximum point.
(5,0) is a local minimum point.
Second Derivative:
5
f ' ' ( x) = [(1)(5 x ) 1/ 3 + (3 x)(1 / 3)(5 x) 4 / 3 (1)]
3
5 3
1
= (5 x) 1/ 3 (5 x) 4 / 3 (5 x) 4 / 3 + (3 x)(5 x) 4 / 3
3 3
3
5 3(5 x) + (3 x ) 5 2 x 12
10( x 6)
=
=
=
3
3 3(5 x ) 4 / 3 9(5 x) 4 / 3
3(5 x) 4 / 3
## f ' ' ( x) = 0 at x = 6, f (6) = 6(5 6) 2 / 3 = 6
x
5
6
f (x)
0
6
f ' ' ( x)
0
DNE
(6,6) is a point of inflection.
Curve Sketching:
## 4.5 An Algorithm for Curve Sketching
2010 Iulia & Teodoru Gugoiu - Page 3 of 4
## B Link between a function and its derivative
Consider a double differentiable function
y = f (x) ( f ' ( x) and f ' ' ( x) exist). Then:
1. f ' ( x) is the slope of the tangent at P( x, f ( x)) .
2. If f ' ( x) = 0 , then P( x, f ( x)) is a local extrema and
tangent is horizontal.
3. If f ' ( x ) > 0 , then the function y = f (x) is
increasing.
4. If f ' ( x) < 0 , then the function y = f (x) is
decreasing.
5. If f ' ' ( x) = 0 , then f ' ( x) has a local extrema and
y = f (x) has an inflection point.
6. If f ' ' ( x) > 0 , then f ' ( x) is increasing and y = f (x)
is concave upward.
7. If f ' ' ( x) < 0 , then f ' ( x) is decreasing and y = f (x)
is concave downward.
## Ex 6. The graphs of a function and its first and second
derivatives are represented on the same grid. Identify
each of them.
## Ex 8. In the next figure is given the graph of the
derivative f ' ( x) of a function f (x) .
## Ex 7. In the next figure is given the graph of a
function y = f (x) .
y
4
5
4
3
1
1
x
x
4
1
2
3
4
## a) Find intervals where the function f (x) is increasing or
a) Find the intervals where f ' ( x) is positive and
decreasing.
negative.
The function f (x) is increasing where f ' ( x ) > 0 : (3, ) .
f ' ( x ) > 0 where the function is increasing: (1,1) .
f ' ( x ) < 0 where the function is decreasing: ( ,1) or The function f (x) is decreasing where f ' ( x) < 0 :
(,0) or (0,3) .
(1, ) .
b) Find intervals where the graph of f (x) is concave
b) Estimate intervals where f ' ' ( x ) is positive and
upward or downward.
negative.
f ' ' ( x) > 0 where the graph is concave
The graph of f (x) is concave upward where f ' ( x) is
increasing: (,0) or (2, ) .
upward: (2,0) or (2, ) (approximate).
f ' ' ( x) < 0 where the graph is concave
downward: ( ,2) or (0,2) (approximate).
## The graph of f (x) is concave downward where f ' ( x) is
decreasing: (0,2) .
## Reading: Nelson Textbook, Pages 207-212
Homework: Nelson Textbook: Page 213 #4begij, 6, 7b, 9
4.5 An Algorithm for Curve Sketching
2010 Iulia & Teodoru Gugoiu - Page 4 of 4
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Rounding 795 to the Nearest Ten
Rounding numbers is a useful skill to have in mathematics. It’s the process of simplifying a number by replacing it with an approximate value that’s easier to work with. When rounding numbers, you can round to the nearest ten, hundred, thousand, and so on. For example, when rounding the number 795 to the nearest ten, the answer would be 800.
The Process of Rounding 795 to the Nearest Ten
Rounding numbers is a fairly straightforward process. To round 795 to the nearest ten, you have to look at the digit in the ones place. In this case, the number is 5. When the number in the ones place is 5 or higher, you round up to the next ten. Since the number in the ones place is 5, 795 should be rounded up to 800. If the number in the ones place was 4 or lower, you would round down to the lower ten. For example, if the number was 784, you would round it down to 780.
Using the Rounding Method in Mathematics
Rounding is a useful tool when performing calculations. For example, if you’re trying to find the average of several numbers, it helps to round each number to the nearest ten first. This makes it easier to calculate the average, since the numbers are simpler. Rounding can also be used when comparing numbers. For instance, if you’re trying to find out which of two numbers is larger, it’s easier to compare them once they’ve been rounded to the nearest ten.
Using Rounding in Everyday Life
Rounding can be useful in everyday life too. When shopping, it helps to round prices to the nearest ten or hundred to make it easier to compare prices. For example, if you’re trying to decide between two items that cost \$7.95 and \$8.45, it’s easier to compare them if you round them to the nearest ten first. This way, you can see that both items cost approximately \$8.
Conclusion
Rounding numbers is a useful skill to have in both mathematics and everyday life. To round 795 to the nearest ten, you must look at the digit in the ones place. If it’s 5 or higher, you round up to the next ten. In this case, 795 should be rounded up to 800. If the number in the ones place was 4 or lower, you would round down to the lower ten.
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# How Much Would the Mishkan Cost Today?
23
By Rabbi Jonathan Gross
In last week’s parshah, Moshe starts raising money to build the mishkan. I was wondering, if we were to start building a mishkan today, how much would it cost? Here are some calculations I came up with. Please feel free to comment and correct me if either my facts or math are wrong.
The tent itself was made up of 48 gold plated beams that were connected. 20 on each side plus 8 across one wall.
The beams were 10 x 1 x 1.5 amos. That comes to a surface area of 50 x 48 = 2,400 amot squared.
Although there are many opinions, let’s say an ammah is 18 inches. So the perimeter of the Mishkan required 43,200 square inches of gold plating.
The beams were made of gold plated wood. With current technology gold can be hammered into extremely thin sheets called gold leaf. The price of gold leaf depends on the thickness. An ounce of gold hammered into a sheet of 100th the thickness of aluminum oil could cover 100 square inches.
So for 43,200 square inches of gold plate we would need 432 ounces of gold.
Today gold is about \$1,235 an ounce. So the gold plating for the perimeter of the Mishkan would cost \$533,520.
That doesn’t seem so bad. But that is just the gold plating.
The silver adanim were the sockets that kept the beams together. There were 100 of them. Each one was a solid kikar. A kikar is 3,000 shekels. So they needed 300,000 shekels. They collected 1/2 shekel for the census. There were just over 600,000 people counted, bringing in just over 300,000 shekels of silver. Exactly enough for the adanim. (and some left over for the curtain hooks.)
A shekel is about 1/2 and ounce. 3,000 shekel is 1,500 ounces.
An ounce of silver today costs about \$15.75.
300,000 shekel of silver for the adanim would cost \$2,362,500.
So we are at about \$3 million. The beams were made of lumber, there were some other hooks, poles, and extensions made of precious metals, and then there’s the labor involved.
The Aron was made of gold plated wood.
With dimensions of 2.5 x 1.5 x 1.5 the surface area needed comes out to 15.75 square amos, and it was plated inside and outside so double that is 31.5 square amos x 18 equals 567 square inches.
Gold plating at 1 ounces per 100 square inches comes out 5.67 ounces.
With gold at \$1,235 per ounce = \$7,002.45
That does not include the cover of the ark, the Kaporet.
It is not clear from the Torah if the kaporet was solid gold, or if it was gold plated wood like the rest of the Aron. We are also not told the height of the Kaporet, The Gemara assumes that it was 1 tefach (there are six tefachim in an amah, making a tefach 3 inches).
If it was a solid slab of gold 2 amot x 1 amah x 1 tefach.
That means that the kaporet was 36 inches x 18 inches x 3 inches = 1,944 inches cubed.
In gold mass 1 inch cubed = .7 pounds.
That means that the kapores weighed 1,360.8 pounds.
At 16 ounces in a pound and gold at \$1,235 an ounce that’s \$26,889,408 And that does not include the keruvim, the twin statues on the kapores that were made of solid gold and stood 30 inches high. Assuming that they were only 1 inch thick they would cost \$778,050 each. \$1,556,100 in materials alone, before the artist’s costs.
(I think we can assume that the kapores was either a thin sheet of gold or made of gold plated wood. It doesn’t seem practical otherwise. Can you imagine being called on to open the ark in shul and when you get up there the gabbai tells you that you have to lift something that weighed 1300 pounds? If I am right then the kapores would have been closer to about \$600 before the keruvim.)
The menorah was 1 kikar of gold.
At 3,000 shekel per kikar, and 1/5 an ounce per shekel, that’s 1,500 ounces of gold.
At \$1,235 an ounce that comes to \$1,852,500 in materials, before the artist’s fee.
Leaving out the kapores, we see that the more expensive vessels of the Mishkan cost upwards of \$2 million at today’s prices. The price of gold plating, thanks to current technology, is considerably cheaper. The rest is lumber, fabrics, labor, and artistry.
Without doing a final tally, the Mishkan would be expensive but not astronomical.
The Beis Hamikdash was much bigger and way more expensive.
Hashem should bless us and bring about the day where we have to raise funds for projects like these soon.
Rabbi Jonathan Gross
{Matzav.com Newscenter}
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1. What’s the problem with the kapores weighing so much no one has to lift it and why are the cherubim only 1 inch thick weren’t they solid gold not hollow
The Aron carried itself and those who carried it so it could have weighed a ton or two and it wouldn’t make a difference
2. your logic that the kapores was plated because who could lift it might be mistaken when did they ever open it?
you statement about the menorah before artist costs is also mistaken wasn’t it thrown into the fire and created by nes?
3. Of course the Kappores was solid gold. It says clearly that it was made of zahav Tahor with the keruvim chiseled (miksha) out from
That Same slab. All gold including the
Keruvim.
4. “Mikdash HaS konenu yadecha” it says in Az Yashir. Rashi says this means the 3rd Beis HaMikdash will descend from Shamayim built and furnished all by HaS. No need for us to worry about the material costs or the artistic efforts. “All” we have to do is to rebuild the destroyed Mikdash that is within each of our hearts (destroyed through loshon hara, sinas chinam, etc.) and then HaS will do the rest.
5. easier calculation:
just under 30 kikar of gold. kikar is 48,000 grams. at about \$41 a gram, that’s \$2,000,000 per kikar, or \$60,000,000. silver was just over 100 kikar. at about \$0.53 per gram, that’s about \$25,500 per kikar, or \$2,550,000. copper was about 70 kikar. at about \$4.50 per kilo, that’s \$216 per kikar, or about \$15,000.
quantity of wood, furs,precious stones etc. unknown, but likely less than \$1,000,000.
6. A shekel is .825 ounces of silver so closer to \$4 million for the adanim
The kapores was 2.5 Amos by 1.5 Amos by 1 tefach which is 2551 pounds of gold closer to about \$40 million
7. It clear and open in the Pesukim that the Kapores was made of solid gold. The Kapores and Keruvim together were a solid piece made of one Kikar of gold.
8. The Keruvim were solid gold and ten Tefachim tall, the Kapores was Tefach think and solid gold. Your calculation is way off, since the Aron was not wood plated with gold, rather it was a solid gold outer box and a solid gold inner box with a wooden box in between. You don’t know this?
9. Lma’aseh raising funds for the MISHKAN would not be a problem at all, even if the cost were \$1 billion.
However if the criterion were that only donations with “pure” l’shem shamayim intentions would be accepted then it might be a little harder…
10. Moshe Rabbenu laid out an expense sheet of how much of each thing was used. Why all the speculation?
Use those numbers
11. This is very confusing. Does he believe that it happened according to tradition, or he does the only believe what it says in the book the way he interpreted it with his personal understanding?
The Aron according to tradition was three boxes: the outer and inner boxes work pure gold and the middle one was wooden, not one wooden box plated with gold.
There were gold strands in the clothing and in the cover of the mishkan. They got the gold strands by beating a sheet of gold flat and then cutting off strings from it. This means that they obviously did have the technology of making thin sheets of gold as they do today.
The Aron was certainly not made to be opened, and based on my little knowledge it in fact was never actually opened.
The author also dismissed all the other expensive materials and designs and types of wood etc. as if their price was trivial and concluded his article with the “fact” that it was expensive but not astronomical as if he wanted that conclusion in the first place. While I personally am not concerned if that’s the case or not, it sounds like that was what the author wanted to come to conclude with in any case.
12. This is in fact a very ignorant article, both in the way the author is calculating (there’s a tally later in Torah) and in the comparison of (even gold) standards then and now.
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# How do you simplify (5-isqrt3)/(5+isqrt3)?
Oct 27, 2016
The simplification $= \frac{11}{14} - \frac{i 5 \sqrt{3}}{14}$
#### Explanation:
To simplify a complex numbers, we must multiply by the conjugate of the denominator
if $z = {z}_{1} / {z}_{2}$ then $z = \frac{{z}_{1} {\overline{z}}_{2}}{{z}_{2} {\overline{z}}_{2}}$
In our case ${\overline{z}}_{2} = 5 - i \sqrt{3}$
${i}^{2} = - 1$
so $\frac{\left(5 - i \sqrt{3}\right) \left(5 - i \sqrt{3}\right)}{\left(5 + i \sqrt{3}\right) \left(5 - i \sqrt{3}\right)} = \frac{25 - 10 i \sqrt{3} - 3}{25 + 3}$
$= \frac{22 - 10 i \sqrt{3}}{28} = \frac{11}{14} - \frac{i 5 \sqrt{3}}{14}$
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Convert {3} to 1/8 rev | equilateral triangle to octants
# angle units conversion
## Amount: 1 equilateral triangle ({3}) of angle Equals: 1.33 octants (1/8 rev) in angle
Converting equilateral triangle to octants value in the angle units scale.
TOGGLE : from octants into equilateral triangles in the other way around.
## angle from equilateral triangle to octant conversion results
### Enter a new equilateral triangle number to convert
* Whole numbers, decimals or fractions (ie: 6, 5.33, 17 3/8)
* Precision is how many digits after decimal point (1 - 9)
Enter Amount :
Decimal Precision :
CONVERT : between other angle measuring units - complete list.
How many octants are in 1 equilateral triangle? The answer is: 1 {3} equals 1.33 1/8 rev
## 1.33 1/8 rev is converted to 1 of what?
The octants unit number 1.33 1/8 rev converts to 1 {3}, one equilateral triangle. It is the EQUAL angle value of 1 equilateral triangle but in the octants angle unit alternative.
{3}/1/8 rev angle conversion result From Symbol Equals Result Symbol 1 {3} = 1.33 1/8 rev
## Conversion chart - equilateral triangles to octants
1 equilateral triangle to octants = 1.33 1/8 rev
2 equilateral triangles to octants = 2.67 1/8 rev
3 equilateral triangles to octants = 4.00 1/8 rev
4 equilateral triangles to octants = 5.33 1/8 rev
5 equilateral triangles to octants = 6.67 1/8 rev
6 equilateral triangles to octants = 8.00 1/8 rev
7 equilateral triangles to octants = 9.33 1/8 rev
8 equilateral triangles to octants = 10.67 1/8 rev
9 equilateral triangles to octants = 12.00 1/8 rev
10 equilateral triangles to octants = 13.33 1/8 rev
11 equilateral triangles to octants = 14.67 1/8 rev
12 equilateral triangles to octants = 16.00 1/8 rev
13 equilateral triangles to octants = 17.33 1/8 rev
14 equilateral triangles to octants = 18.67 1/8 rev
15 equilateral triangles to octants = 20.00 1/8 rev
Category: main menuangle menuEquilateral triangles
Convert angle of equilateral triangle ({3}) and octants (1/8 rev) units in reverse from octants into equilateral triangles.
## Angles
This calculator is based on conversion of two angle units. An angle consists of two rays (as in sides of an angle sharing a common vertex or else called the endpoint.) Some belong to rotation measurements - spherical angles measured by arcs' lengths, pointing from the center, plus the radius. For a whole set of multiple units of angle on one page, try that Multiunit converter tool which has built in all angle unit-variations. Page with individual angle units.
# Converter type: angle units
First unit: equilateral triangle ({3}) is used for measuring angle.
Second: octant (1/8 rev) is unit of angle.
QUESTION:
15 {3} = ? 1/8 rev
ANSWER:
15 {3} = 20.00 1/8 rev
Abbreviation, or prefix, for equilateral triangle is:
{3}
Abbreviation for octant is:
1/8 rev
## Other applications for this angle calculator ...
With the above mentioned two-units calculating service it provides, this angle converter proved to be useful also as a teaching tool:
1. in practicing equilateral triangles and octants ( {3} vs. 1/8 rev ) measures exchange.
2. for conversion factors between unit pairs.
3. work with angle's values and properties.
To link to this angle equilateral triangle to octants online converter simply cut and paste the following.
The link to this tool will appear as: angle from equilateral triangle ({3}) to octants (1/8 rev) conversion.
I've done my best to build this site for you- Please send feedback to let me know how you enjoyed visiting.
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# Higher moments of linear regression residuals?
I previously asked this on Math StackExchange, with no success, but this post will add to that with some simulations.
Background
In the following linear regression with i.i.d $$\epsilon_i$$ $$(i = 1, \cdots, n)$$ with mean 0 finite moments of all order (say, $$\mu_2 = \sigma^2$$ for the centralized second moment, $$\mu_3$$ for centralized third moment, etc), \begin{align*} Y_i = X_i^\intercal\beta + \epsilon_i \end{align*} we know the least-squares estimator for $$\beta$$ is \begin{align*} \hat{\beta}=(X^\intercal X)^{-1}X^\intercal Y \end{align*} where $$X = (X_1, \cdots, X_n)^\intercal \in \mathbb{R}^{n\times p}$$, $$Y = (Y_1,\cdots, Y_n)^\intercal \in \mathbb{R}^n$$. Now, we can easily derive the variance of $$\hat{\epsilon} = Y - X\hat{\beta}$$ with \begin{align*} \text{Var}(\hat{\epsilon}) &= \text{Var}((I-H)Y) \\ &\overset{\heartsuit}{=} (I-H)\text{Var}(Y)(I-H)^\intercal \\ &= \sigma^2(I-H)(I-H)^\intercal \\ &\overset{\spadesuit}{=} \sigma^2(I-H) \end{align*} where $$H = X(X^\intercal X)^{-1}X^\intercal$$ is a projection matrix of rank $$p$$, thus justifying equality $$(\spadesuit)$$. Hence, this allows us to find an unbiased estimator for $$\sigma^2$$, since \begin{align*} \text{Trace}(\text{Var}(\hat{\epsilon})) = \sigma^2\text{Trace}(I-H) = \sigma^2(n-p) \implies \widehat{\sigma^2}=(n-p)^{-1}\|Y - \hat{Y}\|_2^2 \end{align*} An attempt for higher moments
How would we generalize this computation in coming up with unbiased estimators for the third centralized moments \begin{align*} \mu_3 \overset{\text{def}}{=} \mathbb{E}\epsilon^3_i \end{align*} or beyond? Here's my attempt. Define \begin{align*} \mathcal{S}(\hat{\epsilon}) = \mathbb{E}(\hat{\epsilon}_i - \mathbb{E}\hat{\epsilon}_i)^{\otimes 3} \end{align*} Expanding the tensor product, we get \begin{align*} \mathcal{S}(\hat{\epsilon}) &= [(\mathcal{S}(Y)\times_1 M)\times_2 M]\times_3 M \\ &=\mu_3[(I_{n\times n\times n}\times_1 M)\times_2 M]\times_3 M \end{align*} where $$M = I - H$$, and $$\times_i$$ is defined by the $$n$$-mode product. This just means that \begin{align*} [\mathcal{S}(\hat{\epsilon})]_{ijk} = \mu_3 \sum_{v=1}^{n}M_{iv}M_{jv}M_{kv} \end{align*} So the trace is \begin{align*} \text{Trace}(\mathcal{S}(\hat{\epsilon})) = \sum_{i=1}^{n}[\mathcal{S}(\hat{\epsilon})]_{iii} = \sum_{i=1}^{n}\sum_{v=1}^{n}M_{iv}^3 \end{align*} This all looks reasonable, considering that \begin{align*} \text{Trace}(\text{Var}(\hat{\epsilon})) = \sum_{i=1}^{n}\sum_{v=1}^{n}M_{iv}^2 \end{align*} I was curious what the third moment trace might equal, so I conducted some simulations:
p = 5
n = 100
X = matrix(rnorm(p*n), nrow = n, ncol = p)
H = X%*%solve(t(X)%*%X)%*%t(X)
M = diag(n) - H
#Trace of variance (second-moment outer product)
sum(M^2) #Always = n - p
#Trace of third-moment outer product
sum(M^3) #Not only doesn't equal n - p, but changes for each X!
This is somewhat disappointing, since we no longer have a degree of freedom interpretation for higher moments.
Future
Some questions remain:
1. Are there generalized projections (for higher-order tensors) such that the nice theory we have for second-moments/variances flow over?
2. What is the distribution of $$\text{Trace}(\mathcal{S}(\hat{\epsilon}))$$? How does it depend on $$X$$?
• It seems to me that under the (implicitly) general assumptions you have made--namely, only that $\epsilon_i$ are iid of zero mean and variance $\sigma^2$--it is impossible to obtain an unbiased estimator of any higher moment. After all, those moments might not exist or they might be infinite. For you to have any chance at a "nice theory," I think you will need to specify a parametric family of distributions for the $\epsilon_i.$
– whuber
Feb 1 '19 at 20:02
• Sure, I've edited. Assume finite moments of all orders. I shouldn't try and specify a fully parametric distribution for $\epsilon_i$, because linear model theory all follows without a parametric assumption on $\epsilon_i$ for the second moments, and I'd like to see if something similar can follow for third. If this is too restrictive, I might consider assuming $\epsilon_i$ are normal. Feb 1 '19 at 20:10
• Permit me to repeat then: for a nonparametric model it's unlikely you can find any unbiased estimator. If you assume normality the theory becomes almost trivial because the first two moments determine all the others.
– whuber
Feb 1 '19 at 20:18
• Even assuming normality doesn't make the theory trivial. The difficulty does not lie with $\mu_3$ or whatever higher moments, but rather the $3$-mode products from $M = I - H$, which is a function of $X$. Even if $X$, along with $\epsilon_i$, were assumed normal, the result isn't obvious. Feb 1 '19 at 20:29
• The best unbiased estimator of the odd central moments in the Normal case is $0.$ That looks pretty trivial to me. I'll agree, before you post any rejoinder, that this does not imply it is trivial to find unbiased estimators for the non-central moments: but it clearly points the way.
– whuber
Feb 1 '19 at 20:32
## 1 Answer
Your method here involves an attempt to derive the higher-order moments of the residual vector $$\hat{\boldsymbol{\varepsilon}}$$ from its relationship to the error vector $$\hat{\boldsymbol{\varepsilon}}$$. If you want to do that then you will need to prescribe a distributional family for the error vector. So, for example, the distribution of the trace is determined by the underlying distribution of the error vector --- you cannot derive its distribution without this. If you assume that the errors are normally distributed then it is possible to derive the higher-order moments of the residual vector, but the results are going to be quite complicated (and are probably best computed by simulation).
On the other hand, if you wish to proceed without an assumption about the error distribution, then you would not attempt to determine the moments of interest from the underlying error vector at all. Instead, you would simply use standard methods to estimate the higher-order moments empirically from the residuals. For large $$n$$ (and under mild assumptions about the regressors) the residuals become asymptotically uncorrelated, which should allow you to use standard sample estimators and have good convergence properties.
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Geometric interpretation of singular values - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:19:28Z http://mathoverflow.net/feeds/question/17384 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/17384/geometric-interpretation-of-singular-values Geometric interpretation of singular values Wilson 2010-03-07T15:32:35Z 2010-03-07T16:58:22Z <p>The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and the singular values of A describe the length of its major axes. </p> <p>More generally, what do the singular values of a matrix say about the geometry of the image of other objects? How about the unit L1 ball? This will be some polytope: is there some natural way to describe this shape in terms of singular values, or other properties of matrix A?</p> http://mathoverflow.net/questions/17384/geometric-interpretation-of-singular-values/17393#17393 Answer by t0rb3n for Geometric interpretation of singular values t0rb3n 2010-03-07T16:58:22Z 2010-03-07T16:58:22Z <p>It's all about how the object of interest looks after you choose the orthogonal base corresponding to the singular decomposition of A. Then, is only a matter of stretching, just as with the euclidean ball. In this special case it's so simple, because the ball looks the same in all orthogonal bases. But since orthogonal transformation is only about rotation and reflection, the singular values then again describe the stretching of you object after the appropriate transformation.</p>
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# I Why only normal subgroup is used to obtain group quotient
#### SVN
Hello!
As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient.
Yes, the bundle of cosets in this case will be different for actions from the right and from the left (although, their number will be the same). But why is that so crucial? We have our division without intersections anyway, do we?
Is there any special name for such «one-sided (pseudo)quotients». Are there any uses for them?
Related Linear and Abstract Algebra News on Phys.org
#### protonsarecool
For a general group G and a general (non normal) subgroup H, the set of left-cosets G/H, respectively the set of right-cosets H\G won't be groups, you need a normal subgroup (i.e. G/H = H\G) for that.
#### SVN
Not quite sure I understand how cosets can be groups themselves. They lack identity element, since it is already included in the generating subgroup (normal or non-normal).
#### protonsarecool
You want the set of cosets to be a group (ie. the quotient group).
Say, you have a group $G$ and a subgroup $H$. So if we want to define a product on $G/H$ (where the elements are now left-cosets), we do it like $(aH)\cdot(bH) = (ab)H$. However, this will only make sense iff the left-cosets $aH$ are the same as the right-cosets $Ha$, or $aHa^{-1}=H$. Note however that for a non normal H we have $aHa^{-1} \neq H$. This means in particular that $(aH)\cdot(a^{-1}H)\neq (a a^{-1})H = H$. For a normal H this does work, and the rest of the group axioms are satisfied by $G/H$ with the defined product as well. Only then can we call $G/H$ a quotient group, otherwise its just a set of left-cosets.
#### SVN
Thanks a lot for such a detailed explanation. It answers my question fully.
#### micromass
In the case of nonnormal $H$, the quotient still has a nice structure of a homogeneous space. That is, there is the obvious group action $G$ on $G/H$ by putting $g\cdot kH = gkH$. This is an important action not only in group theory, but also in geometry since a lot of nice geometries arise as homogeneous spaces.
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# How do you calculate the number of m2 that you need?
### To calculate the amount of m2 you will need for your floor, determine the area of your floor: measure the length of the room in meteres and the width of the room in metres, and multiply these (so length x width) to obtain the area of your floor. Keep in mind you will lose about 5% of your floor boards by sawing them into the correct length (installer wastage). If the space contains many croocked or rounded walls, calculate a 10% installer wastage.
If the room your engineered wood floor is going to be installed in is not a perfect rectangle or square, divide the room into squares to calculate the square metres needed. Calculate the area of each reactangle as explaines above, and add these to each other to obtain the total amount of square metres. Make sure all your measurements are in the same unit: do not use centimetres in one part and metres in the other, this will lead to mistakes!
Do not underestimate the needed amount of parquet; when having to place a second order there will be differences in colour between the two orders.
## Area- Rectangle
Measure the width
Measure the length
Area = width x length
# Calculating the meters for your plinths
To determine the amount of metres needed for the plinths, measure the outline of the room, always keeping in mind a 10% loss due to installer wastage.
Outline- Rectangle
Measure the length
Measure the width
Outline = Length x 2 + Width x 2
Example: The outline of a floor 12m long and 5m wide = (2x12 + 2x5) = 34 m1
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# What are prime numbers in math? Prime Number List
## What are prime numbers in math?
Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and itself. The smallest prime number is 2. There is only one number that is both prime and even, which is 2.
Prime Numbers: 2,3,5,7,11,13,17…
Since Euclid, prime numbers are considered to be infinite. Many questions about prime numbers still remain unanswered today. Many theorems have been put forward on prime numbers for centuries, and various formulas have been tried to be produced to find prime numbers.
## How to find prime numbers?
Prime numbers from 1 to 100 can be found as follows;
Write all numbers from 1 to 100 in tabular form
Cross all the multiples of 2 in the table.
Cross all the multiples of the number 3 in the table.
As the numbers get larger, multiply the multiples of all the numbers.
Multiply by 1 since the smallest prime number is 2
The remaining numbers are prime numbers.
## Prime Number List
### Prime numbers from 1 to 100;
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97’
### Table of Primes to the Top 10.000
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157,
163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331,
337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,
521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709,
719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919,
929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 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,
2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179,
2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371,
2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557,
2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731,
2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927,
2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163,
3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343,
3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539,
3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719,
3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919,
3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127,
4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327,
4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523,
4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733,
4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957,
4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153,
5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393,
5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569,
5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783,
5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987,
6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203,
6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373,
6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619,
6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829,
6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019,
7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247,
7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507,
7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687,
7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919
### Why 13 is a prime number?
The prime factorisation of 11 is 1 × 13 since 13 has only two factors 1 and itself, hence it is a prime number.
### What are the prime numbers from 1 to 50?
The prime numbers from 1 to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
### What is the smallest even prime number?
2 is the only even prime number.
| 4,247 | 7,341 |
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A176498 Number of elements less than 1/2 in the Cross Set which is the subset of the set of distinct resistances that can be produced using n equal resistors in series and/or parallel. 3
0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 9, 24, 58, 124, 312, 759, 1768, 4421, 10811, 27191, 68591, 174627, 441633, 1124795, 2866004 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,10 COMMENTS This sequence arises in the decomposition of the sets A(n + 1) of equivalent resistances, when n equal resistors are combined in series/parallel, into series parallel and cross sets respectively. All the elements of the parallel set are strictly less than 1 and all those of the series set are strictly greater than 1. The cross set is expected to be dense around 1 with very few elements below 1/2. Hence it is relevant to count the elements below 1/2. LINKS Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010). S. A. Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - N. J. A. Sloane, Oct 23 2012 EXAMPLE The order of the cross set is given by A176497: 0, 0, 0, 1, 4, 9, 25, 75, 195, 475, 1265, 3135, ... The sets corresponding n = 4 to n = 8 do not have a single element below 1/2. For n = 9 onwards we have a few elements which are less than 1/2; they are 1, 6, 9, 24, .... CROSSREFS Cf. A048211, A176497. Sequence in context: A215528 A155577 A084431 * A142877 A260168 A093153 Adjacent sequences: A176495 A176496 A176497 * A176499 A176500 A176501 KEYWORD more,nonn AUTHOR Sameen Ahmed Khan, Apr 21 2010 EXTENSIONS a(16)-a(25) from Antoine Mathys, Mar 20 2017 STATUS approved
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MidTerm_Jamie_Boyer
# MidTerm_Jamie_Boyer - EC/315 Midterm Examination...
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EC/315 Midterm Examination Name:_Jamie Boyer______________________________________ Directions: This test is open-book and open notes and covers the content from weeks 1 through week 4 of EC/315. The test will be typed and submitted in the Dropbox marked Midterm Exam. The midterm is due the last day of Week 4. PROBLEM 1 (Weight 20 points). NBC TV news, in a segment on the price of gasoline, reported last evening that the mean price nationwide is \$1.50 per gallon for self-serve regular unleaded. A random sample of 35 stations in the Milwaukee, WI, area revealed that the mean price was \$1.52 per gallon and that the standard deviation was \$0.05 per gallon. At the .05 significance level, can we conclude that the price of gasoline is higher in the Milwaukee area? Calculate the p- value and interpret. Ho: u < 1.50 H1>1.50 T=\$1.52-1.50 =\$2.37 \$0.05/sr35 2(.5000-.4911)=.0178=p value Reject Ho and accept H1 PROBLEM 2: (Weight 20 points) . Suppose Babsie generated the following probability distribution: X p(x) 5 .25 7 .30 10 .25 12 .05 15 .15 a. Is this probability distribution discrete or continuous? Explain your reasoning. This probability distribution is continuous because it is a variable that can assume one of an infinitely large number of values within limitations. b. Calculate the expected value of X. Show your work!! u=.25(5)+.30(7)+.25(10)+.05(12)+.15(15)=1.25+2.1+2.5+.6+2.25=8.7
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## This note was uploaded on 04/23/2011 for the course EC 315 taught by Professor Barcus during the Spring '10 term at Park.
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# P1 - itself C A needle of length 1 is spinning through the...
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Math 241, Problem Set #1 (due in class Fri., 9/9/11) Stewart, section 10.1, problems 4, 6, 10, 17(a), 18, 20, 30, 36. For the last of these problems, you may use the formula proved in Exercise 7.2.27 (which you don’t need to prove!). For some of these problems, the technique of “completing the square” will be useful. Also: A. Two ±ies ±y past one another between time 0 and time 1. At time t (with 0 t 1), one ±y has position (0 , 0 , t ) and the other has position (1 , 1 - t, 0). At what instant are they closest, and how far apart are they at that instant? (Hint: Minimizing the distance between the ±ies is equivalent to minimizing the square of the distance.) B. Let L be the length of a line segment in space, and let L x,y be the length of its projection on the x, y plane. Show that L x,y L (that is, the shadow of a line segment cannot be longer than the line segment
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Unformatted text preview: itself). C. A needle of length 1 is spinning through the air. Let L x,y ( t ) be the length of its projection on the x, y plane at time t ; let L x,z ( t ) be the length of its projection on the x, z plane at time t ; and let L y,z ( t ) be the length of its projection on the y, z plane at time t . Show that regardless of the fashion in which the needle is spinning, [ L x,y ( t )] 2 + [ L x,z ( t )] 2 + [ L y,z ( t )] 2 stays constant as t varies. Please don’t forget to write down who you worked on the assignment with (if nobody, then write “I worked alone”), and record how much time you spent on each problem (this doesn’t need to be exact) on the time-sheets I gave out in class....
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## This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.
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# Math, Literacy and Basic Skills Package (1st Term) - Kindergarten - Printable
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Numbers Patterns (from 1 to 10)
Cut and paste activity with numbers (from 1 to 2)
Matching additions with the correct results (with numbers 3 and 4)
Circle the number of objects asked (with number 5)
Painting the number of objectes asked (with number 6)
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Making 10. Guess the correct number to make 10
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## Calculus (3rd Edition)
(a) $u=x^2+9$. (b) $u=x^3$. (c) $u=\cos x$.
(a) Here, we can use the substitution $u=x^2+9$. (b) Here, we can use the substitution $u=x^3$. (c) Here, we can use the substitution $u=\cos x$.
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# Basic Algebra
Section 5.1: Exercises 40, 68, and 72 Section 5.2: Exercises 16, 62, and 80 Section 5.3: Exercises 58, 64, and 102 Section 5.4: Exercises 18, 26, 38, and 88 Section 5.5: Exercises 36, 44, 66, 72, and 82 Section 5.6: Exercises 18, 32, 54, 58, 66, 98, 100, and 104 98.Avoiding a collusion 100. Winter wheat 104. Venture capi
### price per viewable image area
An advertisement for LED HDTVs lists the following prices for their different sizes (all have the same aspect ratio 9:16): 22-inch \$ 247.99 42-inch \$ 1,299.99 46-inch \$ 1,449.99 52-inch \$ 1,799.99 55-inch \$ 1,999.0 Rearrange this list in terms of the price per viewable image area ( \$/in2 ) The Green people of planet Xy
### Earned Value.
Use the data from problem 3 to randomly pick 10 days of sales. as you would in a simulation. What is the EV and standard deviation for Burger Sales? Daily Burger Sales Probability 500 0.15 700 0.25 800 0.4 1000 0.2
### Monomials, Binomial and Trinomial Multiplying
1. FOIL is an acronym designed to help you multiply binomials, not monomials or trinomials. Do you think that you could create an acronym (it will have more than 4 letters) to help you multiply a binomial and a trinomial? Illustrate your method by multiplying (x + 2) times (x2 â?" 3x + 4), showing all steps. 2. Take a numb
### Graphing and Algebra Word Problems
1. Customers of a phone company can choose between two service plans for long distance calls. The first plan has a \$30 one time activation fee and charges 8 cents a minute. The second plan has no activation fee and charges 13 cents a minute. After how many minutes of long distance calls will the cost of the two plans be equal i
### Determining Velocity and Distance using Algebra
1) A ball is hit into the air. Its height, s, at time t, is given by the equation s(t)=-16t^2+74t+4. Determine the maximum height of the ball. Round to the nearest foot. _________feet 2) Solve for x: 2^5x-4=3^7x+4 Round to the nearest 0.001. x=________ 3) The population of New York City was 8 million people in the y
### Algebra: Logarithmic and Exponential Equations
1) Given A=Pe^rt, with A = 455, P = 5, and r = 0.61, solve for t. Round to the nearest 0.01. t=__________ 2) Solve for r: 8(3^r+1)+4=20 Round to the nearest 0.001. r=__________ 3) Solve for x: 2^5x-4=3^7x+4 Round to the nearest 0.001. x=________ 4) Solve for x: log_7(4x-16)=3 Give an exact answer. _______
### Solving Basic Algebra Questions
1) The spread of a virus in an isolated community is modeled by N(t)=1100/1+49e^-0.3t, where N(t) is the number of people infected after t days. a) Approximately how many people will be infected in 16 days? ___________ b) How long until 900 people have been infected? Round to the nearest day. _________days 2) Solve
### Calculating Solutions: Algebra Example Questions
1) Given A=Pe^rt, with A = 455, P = 5, and r = 0.61, solve for t. Round to the nearest 0.01. t=__________ 2) Solve for r: 8(3^r+1)+4=20 Round to the nearest 0.001. r=__________ 3) Solve for z: 2log_2(5+2z)=16 Round to the nearest 0.01. z=__________ 4) An investment offers 4.5% interest compounded continuously.
### Odd Degree Polynomials
1) Find the remainder when x^3+2x^2+5 is divided by x+2. __________ 2) The spread of a virus in an isolated community is modeled by N(t)=1100/1+49e^-0.3t, where N(t) is the number of people infected after t days. a) Approximately how many people will be infected in 16 days? ___________ 3) A ball is hit into the ai
### Algebra and Functions with Domain and Range
1) Given the function f(x)=sqrt(x+8), find the DOMAIN of the inverse function, f^-1(x). a) [0,â??) b) (-â??, 0] 2) Suppose that a person x inches tall sprains an ankle. An approximation for the length of a crutch (in inches) that a person might need is given by the function f(x)=19/26x+2. Find and interpret
### Horizontal Asymptote Intercepts
1) Which of the following is/are TRUE about the graph of y=6e^x ? Select all that apply. a) It has a range of (6, -â??) b) It has a horizontal asymptote of y=0 c) It has both an x-intercept and a y-intercept 2) The temperature, in degrees Fahrenheit, of a cup of coffee can be expressed by T(t)=80+105e^-0.045t, where t is
### Solving Quadratic Equations and Completing the Square
What is the quadratic equation? Explain where the quadratic equation comes from. In particular, how is the quadratic equation related to the process of completing the square?
### Quotient, remainder and polynomial equation
1)Find the quotient,q(x) , and the remainder,r(x) , when 5x^3-20x^2+23x-5 is divided by x^2-3x+3. q(x)= r(x)= 2)Find the remainder when x^3-3x^2-6 is divided by x-3. 3)Solve the polynomial equation x^3-2x^2+3x-6=0 given that x=2 is one solution. The SUM of all the real solutions is_______
### real solutions of the equation
1)Determine if the following statements are true or false: (1) If (3,0) is an x-intercept of f(x), then (x+3) is a factor of f(x). (2) If f(x) is a polynomial and f(9)=0 , then -9 is a zero of f(x). a)1 is true, 2 is false b)They are both false c)1 is false, 2 is true d)They are both true 2) Given that 3x
### Algebra To Find a Quotient and Remainder
1) Find the quotient,q(x), and the remainder,r(x), when 3x^3+9x^2-16x-51 is divided by x^2+5x+7. q(x)= r(x)= 2) Solve: x^3-1/2x^2+4/3=11/3x^2-2/3x+4/3 The smallest zero is ________ The largest zero is _________ 3) Given that 3x-2 is a factor of 3x^3-2x^2-15x+10, select all of the following that are real solution
### Polynomial Function Factored Completely
1) Factor completely: f(x)=x^3-3x^2-4x+12 _______________ 2) Find the remainder when x^3+2x^2+5 is divided by x-2. _______________ 3) Find an equation for a polynomial function, f(x) , of degree 3 with zeros 0, 1, -2. a) f(x)=x^3+x^2-2x b)f(x)=2x^3-3x^2+5x c)f(x)=x^3-5x^2+x d)f(x)=-2x^3+5x^2-3x
### polynomial equations.
1). Find the quotient, q(x) , and remainder, r(x) , when x^6-2x^5+4x^3-x+1 is divided by x+1. a) q(x)=x^5-3x^4+3x^3+x^2-x, and r(x)=1 b) q(x)=x^5-x^4-x^3+3x^2+3x-2, and r(x)=-1 c) q(x)=x^5-3x^4+7x^3-8x^2+9, and r(x)=1 d) q(x)=x^5-x^4+3x^3+2x^2+3,and r(x)=-1 2)How do the zeros of g(x)=4x^4-2x^3+3x^2-4x-12 compare to the
### Ito's formula and Ito's isometry
Hi, I have a question about martingale with respect to Brownian motion process, and I am looking for a detailed explanation. Thank you.
### Expressions
Buying a Home For most people, buying a house is a great investment that can offer security in an uncertain world, but buying a house is also a commitment. Application Practice Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive b
### Algebra .
1) Which of the following is/are TRUE for the graph of EVERY polynomial function? It must have either a minimum or a maximum value It must be continuous It must have a y-intercept It must have at least one x-intercept 2) Which of the following are odd degree polynomials? x(x-4)^2-11 x^3+x^2-4 (x^2+2)^5+2x x^5+x^7
### The maximum dosage of a drug
1) A ball is hit into the air. Its height, s, at time t, is given by the equation s(t)=-16t^2+88t+3 . Determine the maximum height of the ball. Round to the nearest foot. _____feet 2)The maximum dosage of a drug that can be given to an individual is a function of the individual's weight. The following table gives the maximum
### Algebra: Word Problems on Fencing
1)A rancher is going to construct a new rectangular pen for his emu farm. If he has 2600 feet of fencing, what is the maximum area of the pen? ____ft^2 2)A farmer is fencing a rectangular pen for his sheep using the straight portion of a river as one side of the rectangle. If the farmer has 1200 feet of fencing, find the d
### Quadratic equation in standard form
Please find a quadratic equation in standard form to model the data in the table. x y 3 24 4 14 5 8 6 6 7 8 y= Another problem is A ball is hit into the air. Its height in feet, s, at time t is given by the equation s(t)=-16^2+25t+4. a) Determine the height of the ball after 1 second. The height is ___ feet.
### Quadratic Function for Completing the Square
Rewrite the quadratic function y=2x^2+8x-5 in standard form by completing the square. y=_(x+_)^2-_ A quadratic function y=g(x) has x-intercepts at (2,0) and (18,0), and a leading coefficient of -5. Find the vertex of the graph of y=g(x) (__,__)
### Total Distance of Trip
A driver makes a 126 mile trip from Phoenix to Tucson with the cruise control set at either 55 mph or 75 mph the whole way. If the trip took 2 hours, how long did the driver travel at 55 mph? ____Hours
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• NEW! FREE Beat The GMAT Quizzes
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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote What is the remainder if 7^10 is divided by 100? This topic has 6 expert replies and 1 member reply Top Member What is the remainder if 7^10 is divided by 100? Timer 00:00 Your Answer A B C D E Global Stats Difficult What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 OA E Source: Manhattan Prep GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2244 messages Followed by: 17 members Upvotes: 43 Top Reply BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 The remainder when 7^10 is divided by 100 is equal to the last two digits of the expansion of 7^10. Notice that 7^2 = 49, or 50 - 1. So 7^4 = (50 - 1)^2 = 2500 - 100 + 1 = 2401. The last two digits of 7^2 is 49 and those of 7^4 is 01 or simply 1. Since 7^10 = 7^2 x 7^4 x 7^4, the last two digits of 7^10 is 49 x 1 x 1 = 49 Answer: E _________________ Scott Woodbury-Stewart Founder and CEO GMAT/MBA Expert GMAT Instructor Joined 09 Oct 2010 Posted: 1440 messages Followed by: 32 members Upvotes: 59 Top Reply BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 Source: Manhattan Prep First note that: > 1 is the remainder of 101 (=1*100+1) divided by 100 > 32 is the remainder of 532 (=5*100+32) divided by 100 > 47 is the remainder of 7847 (=78*100+47) divided by 100 $${7^{10}} = K \cdot 100 + R{\mkern 1mu} {\mkern 1mu} \,\,{\mkern 1mu} {\mkern 1mu} \left( {K\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R \le 99\,\,{\mathop{\rm int}} } \right){\mkern 1mu}$$ $$? = R$$ $${7^{10}} = {\left( {{7^2}} \right)^5} = {49^5}$$ $${49^2} = {\left( {50 - 1} \right)^2} = {5^2} \cdot {10^2} - 100 + 1 = M \cdot 100 + 1\,\,\,,\,\,\,M\,\,{\mathop{\rm int}} \ge 1\,\,\,\,\,\,\,\,\,\,\left( {M = {5^2} - 1} \right)$$ $${49^4} = {\left( {M \cdot 100 + 1} \right)^2} = {M^2} \cdot {10^4} + M \cdot 200 + 1 = N \cdot 100 + 1\,\,\,,\,\,\,\,N\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\,\,\,\,\,\left( {N = {M^2} \cdot {{10}^2} + 2M} \right)$$ $${49^5} = \left( {N \cdot 100 + 1} \right) \cdot 49 = K \cdot 100 + 49\,\,\,,\,\,\,\,K\,\,{\mathop{\rm int}} \,\, \ge 1\,\,\,\left( {K = 49N} \right)$$ $$? = 49$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator ( Math for the GMAT) English-speakers :: https://www.gmath.net Portuguese-speakers :: https://www.gmath.com.br Last edited by fskilnik@GMATH on Sun Oct 14, 2018 12:03 pm; edited 1 time in total Junior | Next Rank: 30 Posts Joined 03 Oct 2018 Posted: 10 messages If we are asked to calculate the remainder when 6*5 is divided by 4, we observe that we can divide each of the numbers 6 and 5 and multiply their remainders to get the required answer. For example , $$\frac{6}{4}$$ yields a remainder of 2 whereas $$\frac{5}{4}$$ gives 1. Multiplying 2 and 1 is 2 - the same as the remainder when 30 is divided by 4. We use this same principle in the above problem. We break the given product into small numbers whose remainder we can easily find. In this case we can write it as $$7^9\cdot7$$ = $$343^3\cdot7$$ When 343 is divided by 100, the remainder is 43. Our required answer is (43*43)*(43*7) =1849 * 301 The remainders when 1849 and 301 are divided by 100 are 49 and 1. The product of 49 and 1 is 49 ... which is the required answer since 49 divided by 100 would continue to yield a remainder of 49. GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15203 messages Followed by: 1861 members Upvotes: 13060 GMAT Score: 790 BTGmoderatorDC wrote: What is the remainder if 7^10 is divided by 100? A] 1 B] 43 C] 19 D] 70 E] 49 When a positive integer is divided by 100, the remainder is yielded by the last two digits: 123/100 = 1 R23 548/100 = 5 R48 692/100 = 6 R92 Thus: The remainder when 7¹⁰ is divided by 100 is equal to the last two digits of 7¹⁰. Calculate the last two digits for consecutive powers of 7 and look for a pattern: 7¹ --> 07 7² --> 49 7³ --> 43 7⁴ --> 01 7⁵ --> 07 The last two digits appear in a CYCLE OF 4: 07, 49, 43, 01...07, 49, 43, 01... Implication: When 7 is raised to a power that is a MULTIPLE OF 4 -- constituting the end of a cycle -- the last two digits will be 01. From there, the cycle will repeat: 07, 49, 43, 01... Since 8 is a multiple of 4, the last two digits for 7⁸ are 01. The cycle then repeats: 7⁹ ---> 07 7¹⁰ --> 49 The correct answer is E. _________________ Mitch Hunt Private Tutor for the GMAT and GRE [email protected] If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at [email protected]. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. GMAT/MBA Expert GMAT Instructor Joined 04 Dec 2012 Posted: 2033 messages Followed by: 238 members Upvotes: 1443 _________________ Ceilidh Erickson Manhattan Prep GMAT & GRE instructor EdM in Mind, Brain, and Education Harvard Graduate School of Education Manhattan Prep instructors all have 99th+ percentile scores and expert teaching experience. Sign up for a FREE TRIAL, and learn why we have the highest ratings in the GMAT industry! Free Manhattan Prep online events - The first class of every online Manhattan Prep course is free. Classes start every week. GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12744 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 Sweeeeeeeeeeeeet solution, Scott!! Cheers, Brent _________________ Brent Hanneson – Creator of GMATPrepNow.com Use our video course along with Sign up for our free Question of the Day emails And check out all of our free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months! GMAT/MBA Expert GMAT Instructor Joined 25 Apr 2015 Posted: 2244 messages Followed by: 17 members Upvotes: 43 Brent@GMATPrepNow wrote: Sweeeeeeeeeeeeet solution, Scott!! Cheers, Brent Thanks Brent! _________________ Scott Woodbury-Stewart Founder and CEO • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • FREE GMAT Exam Know how you'd score today for$0
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# ABS Function – Absolute Value in Excel, VBA, Google Sheets
Written by
Editorial Team
Reviewed by
Steve Rynearson
Last updated on March 16, 2024
This tutorial demonstrates how to use the Excel ABS Function in Excel to calculate the absolute value.
ABS Function Overview
The ABS Function Calculates the absolute value of a number. The absolute value is the number’s distance from zero. Example: The absolute value of -9 is 9.
To use the ABS Excel Worksheet Function, select a cell and type:
(Notice how the formula inputs appear)
### ABS Function Syntax and Inputs:
``=ABS(number)``
number – A number.
### Creating a absolute value graph
To create a cosine curve in Excel, we need to first choose our start and end points and then list out a lot of numbers. Let’s go from -1 up to 1 in increments of 0.1.
``=ABS(C3)``
Next, we’re going to add the 0.1 onto the angle and then calculate the cosine of that angle. Use the formula
``=C3+\$G\$2``
(the \$ signs lock G2 so the formula will always reference that 0.1 even if we copy the formula!)
Now highlight both the new angle and cosine function that we’ve calculated, hold the handle, and drag it down until our angle reaches 1.
If you chose “Scatter with Smooth Lines”, you might notice the graph is a bit wobbly close to (0,0), and it does not create a sharp point as expected at x=0. This is because of our chosen increment of 0.1. To fix this, either use a smaller increment like 0.01 or choose “Scatter with Straight Lines”.
The ABS Function works exactly the same in Google Sheets as in Excel:
Highlight the entire range of both angles and cosines, click insert, find the graphs and select “Scatter with Straight Lines”
## ABS Examples in VBA
You can also use ABS in VBA. This example will loop through cells A2:A4 and output the absolute value in cells B2:B4.
``````Sub Abs_Example1()
Dim cell As Range
For Each cell In Range("A2:A4")
cell.Offset(0, 1) = Abs(cell.Value)
Next cell
End Sub``````
The result will be as following.(please see B2:B4)
The following 2 examples both will return 12.
``MsgBox Abs(-12)``
``MsgBox Abs(12)``
To find a number closest to 2 when a number array (1.5, 3.1, 2.1, 2.2, 1.8) is given, you can use the following code.
``````Sub Abs_Example2()
Dim Numbers
Dim item
Dim closestValue As Double
Dim diff As Double
Dim minDiff As Double
minDiff = 100
Numbers = Array(1.5, 3.1, 2.1, 2.2, 1.8)
For Each item In Numbers
diff = Abs(item - 2)
If diff < minDiff Then
minDiff = diff
closestValue = item
End If
Next item
MsgBox "The closest value: " & closestValue
End Sub``````
The result will be 2.1 as following.
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# Information Entropy
In the late 1940’s, the American mathematician Claude Shannon looked to answer the question “what is information?”, and in so doing he single-handedly developed the subject of “Information Theory”; which turned out to be one of the fundamental cornerstones of all of computer science. Shannon’s key idea was basically to try to figure out how much actual “information” is contained within a stream of “data”.
As it turns out most things that contain a lot of data, actually contain very little real information. (Think of how often a 300 page book is simply a 3,000 word essay fleshed out to 100,000 words.) According to Shannon most data in a data stream is “redundant” and can be “compressed” to its pure information content. Shannon called this pure compressed content “information entropy”; and so the more incompressible a system is, the higher will be its information entropy!..
Information Entropy (which incidentally is sometimes referred to as Shannon’s Entropy”) is in effect, “the limit of compression of redundant data”; it is “pure information undiluted by repetitive redundancy”…
Mathematically, Information Entropy is usually represented by the letter (H).
H = – Ʃ(P(x) × log2(P(x))) where P(x) is the probability of an event
So for example the Information Entropy of a fair coin (probability of head or tail = ½)
H = – (½ log2(½) + ½ log2(½))
H = – ( ½ (-1) + ½ (-1) )
H = – ( -½ + -½ )
H = – ( -1 ) = 1 bit
The Information Entropy of a biased coin (e.g. probability of head = ¼ and probability of tail = ¾)
H = – (¼ log2(¼) + ¾ log2(¾))
H = – (¼ (-2) + ¾ (-0.415) )
H = – ( -0.50 + -0.31 )
H = – ( -0.81 ) = 0.81 bits
The Information Entropy of a fair die (probability of each number = 1/6)
H = – (1/6 log2(1/6) + 1/6 log2(1/6) + 1/6 log2(1/6) + 1/6 log2(1/6) + 1/6 log2(1/6) + 1/6 log2(1/6))
H = – (6 × 1/6 log2(1/6))
H = – (log2(1/6))
H = – ( -2.585 ) = 2.59 bits
If all the letters of the English language were equally likely then the Information Entropy would be
H = – (27 × 1/27 log2(1/27))
H = – (log2(1/27))
H = – ( -4.755 ) = 4.76 bits
But since they are not equally likely (with “e” having the highest probability and “z” the lowest), and since there are quite a few constraints (such as, “q” is nearly always followed by “u”, and the famous “i” before “e” except after “c”, etc), it means there is extra redundancy in the language which reduces the Information Entropy from 4.76 bits to approximately 1 bit.
The redundancy in any message, or data stream, is therefore equal to {the number of bits used to encode it} minus {the number of bits of Information Entropy}. Consequently, the redundancy in a message is related to the extent to which it is possible to “compress” it.
So in the simplest possible terms, Information Entropy is the “Limit of Compression of Data Redundancy”, it is a measure of “Pure Information”. And since pure information is itself is a measure of uncertainty (i.e. pure information is something that we did not previously know or could not predict), it could therefore be said that Information Entropy is the limit of compression of uncertainty — or more precisely
Information Entropy is the “Limit of Prediction”, the “Limit of Certainty”…
[Note: One final thing to pay attention to in the overall context of this website is that Information Entropy is the data equivalent to the mathematical “Law of Large Numbers” which is effectively a “Limit of Compression of Mathematical Redundancy”…]
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# Interactive Real Analysis
Next | Previous | Glossary | Map
## 1.4. Natural Numbers, Integers, and Rational Numbers
### Example 1.4.5:
Why does one need another number system more complicated than the rational numbers Q?
There are several reasons, many of which are explored in detail in the next chapters. Here are a few of them (which may use terms we are not yet familiar with).
• A simple equation like x2 - 9 = 0 does have a solution in Q, but another, just as simple equation x2 - 2 = 0 does not have a solution in Q.
• If we construct a right triangle for which two sides have length 1, then we could not measure the length of the remaining side if all we knew were rational numbers.
• We could not measure the circumference of any circle if all we knew were rational numbers.
• If we set x0 = 2 and then for each integer n > 0 compute the number successively, then each resulting number is a rational number, the sequence of numbers is getting smaller and smaller, but they seem to get closer and closer to some limit. However, this sequence of numbers does not converge to a rational number. The sequence looks like this (do you know its limit ?):
• x0 = 2
• x1 = 3/2 = 1.5
• x2 = 17/12 = 1.416...,
• x3 = 577/408 = 1.414215686 , ...
• and so on ...
• There are sets consisting of rational numbers that are bounded, but do not have a least upper bound in Q.
• Equations such as sin(x) = 1/2 or cos(x) = 0 do not have solutions in Q.
Next | Previous | Glossary | Map
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To determine your maximum heart rate, subtract your age from 220. For example, a 35-year-old woman’s maximum heart rate is 220 minus 35 — or 185 beats per minute. To enter the fat-burning zone, she’d want her heart rate to be 70 percent of 185, which is about 130 beats per minute.
You can calculate your maximum heart rate by subtracting your age from 220. For example, if you’re 45 years old, subtract 45 from 220 to get a maximum heart rate of 175. This is the average maximum number of times your heart should beat per minute during exercise.
Subsequently, What is 70% of my max heart rate?
Target heart rate during moderate intensity activities is about 50-70% of maximum heart rate, while during vigorous physical activity it’s about 70-85% of maximum.
Also, How do I calculate my maximum heart rate?
You can estimate your maximum heart rate based on your age. To estimate your maximum age-related heart rate, subtract your age from 220. For example, for a 50-year-old person, the estimated maximum age-related heart rate would be calculated as 220 – 50 years = 170 beats per minute (bpm).
How do I work out my maximum heart rate?
You can estimate your maximum heart rate based on your age. To estimate your maximum age-related heart rate, subtract your age from 220. For example, for a 50-year-old person, the estimated maximum age-related heart rate would be calculated as 220 – 50 years = 170 beats per minute (bpm).
Last Review : 9 days ago.
Table of Contents
## Is it dangerous to go over max heart rate?
When heart rate is too high Going higher than your maximum heart rate for long periods of time could be dangerous for your health. That’s especially true if you’re new to exercise.
## Is it dangerous for your heart rate to go over 200?
Some people – mostly younger people – can easily push their heart rate to over 200 beats per minute, while others already reach their limit with a heart rate of 170. However, this does not reveal anything about whether the person with a maximum heart rate of 220 is fitter than the one with a maximum heart rate of 180.
## What happens if you go over max heart rate?
It is possible to exceed the upper limit of your zone without any ill effects, as long as you do not have coronary artery disease or are at risk for a heart attack. What it may do, though, is leave you with a musculoskeletal injury. Exercising above 85% of your target heart rate could bring you sore joints and muscles.
## How do you calculate 70 percent of your maximum heart rate?
Or, here’s a simple way to do the math yourself. If you’re aiming for a target heart rate in the vigorous range of 70% to 85%, you can use the heart rate reserve (HRR) method to calculate it like this: Subtract your age from 220 to get your maximum heart rate.
## What if my heart rate is higher than maximum?
If you notice your HR is above its normal training values, you may be overtraining. Adjust your training plan by running less and recovering more. An abnormal spike in resting heart rate during training indicates possible illness or fatigue, so stop. Rest a day or two before returning to training.
## How do you calculate 75 max heart rate?
To determine your maximum heart rate, subtract your age from 220. Your target heart rate zone is determined based upon your maximum heart rate. You want to stay within 50—75 percent of your maximum heart rate during exercise, depending upon your fitness level.
## Is a heart rate of 240 dangerous?
Some people experience rapid heartbeats (paroxysmal supraventricular tachycardia), with heart rates rising up to 240 beats per minute. Other symptoms include palpitations, shortness of breath, fainting and possibly angina.
## Can you be fit and have a high heart rate?
Normally, physically fit people have lower heart rates and those who don’t exercise much have higher heart rates.
## What is considered too high of a heart rate when exercising?
You can calculate your maximum heart rate by subtracting your age from 220. For example, if you’re 45 years old, subtract 45 from 220 to get a maximum heart rate of 175. This is the average maximum number of times your heart should beat per minute during exercise.
## Is it bad to go above max heart rate?
When heart rate is too high Going higher than your maximum heart rate for long periods of time could be dangerous for your health. That’s especially true if you’re new to exercise.
## What is a dangerously high heart rate during exercise?
If your heart rate exceeds 185 beats per minute during exercise, it is dangerous for you. Your target heart rate zone is the range of heart rate that you should aim for if you want to become physically fit. It is calculated as 60 to 80 percent of your maximum heart rate.
## What is 70 percent of max heart rate?
Your fat-burning heart rate is at about 70 percent of your maximum heart rate. Your maximum heart rate is the maximum number of times your heart should beat during activity. To determine your maximum heart rate, subtract your age from 220.
## Does a high heart rate mean you are unfit?
an unfit person has a higher resting heart rate than a fit person. an unfit person has a higher heart rate when they are exercising at the same intensity. a fit person takes less time for their heart rate to return to resting values after taking part in exercise.
## What is too high of a heart rate while exercising?
You can calculate your maximum heart rate by subtracting your age from 220. For example, if you’re 45 years old, subtract 45 from 220 to get a maximum heart rate of 175. This is the average maximum number of times your heart should beat per minute during exercise.
## Is it bad if my heart rate goes over 200?
More oxygen is also going to the muscles. This means the heart beats fewer times per minute than it would in a nonathlete. However, an athlete’s heart rate may go up to 180 bpm to 200 bpm during exercise. Resting heart rates vary for everyone, including athletes.
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If you desire to find out more, climate please store reading, and also you won"t it is in disappointed.
You are watching: 24 is what percent of 120
## Step by step technique for calculating what percent of 120 is 24
We currently have our first value 120 and also the 2nd value 24. Let"s i think the unknown worth is Y which answer us will uncover out.
As we have actually all the forced values us need, now we can put castle in a straightforward mathematical formula as below:
STEP 1Y = 24/120
By multiplying both numerator and denominator by 100 we will certainly get:
STEP 2Y = 24/120 × 100/100 = 20/100
STEP 3Y = 20
Finally, we have discovered the value of Y i m sorry is 20 and that is our answer.
You can use a calculator to uncover what percent that 120 is 24, just go into 24 ÷ 120 × 100 and also you will gain your answer i beg your pardon is 20
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Here is a calculator come solve portion calculations such together what percent of 120 is 24. You deserve to solve this kind of calculation with your values by start them right into the calculator"s fields, and click "Calculate" to obtain the result and explanation.
What percent of
is
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## Sample questions, answers, and how to
Question: her uncle had actually 120 share of his own firm a few years earlier, and also now he has actually 24 that them. What percent the the share of his agency he has now?
Answer: He has 20 percent of shares of his agency now.
How To: The key words in this problem are "What Percent" because they permit us recognize that it"s the Percent the is missing. For this reason the 2 numbers the it gives us need to be the "Total" and the "Part" us have.
Part/Total = Percent
In this case, it"s the total that our uncle owned. Therefore we placed 120 top top the bottom the the portion and 24 ~ above top. Now we"re all set to figure out the part we don"t know; the Percent.
See more: There Are 30 Squares In This 4X4 Grid, How Many Squares In A 4X4 Grid ?
24/120 = Percent
To uncover the percent, all we need to do is convert the portion into that percent kind by multiply both top and bottom component by 100 and also here is the way to figure out what the Percent is:
24/120 × 100/100 = 20/100
20 = Percent
And that means he has 20 percent of the share of his company now.
## Another action by step method
Let"s resolve the equation for Y by an initial rewriting the as: 100% / 120 = Y% / 24
Drop the percentage marks to leveling your calculations: 100 / 120 = Y / 24
Multiply both sides by 24 to isolate Y top top the appropriate side the the equation: 24 ( 100 / 120 ) = Y
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# Galileo's Rolling Ball experiment
Extracts from this document...
Introduction
## Galileo’s Rolling Ball experiment
Aim:
Galileo in his rolling ball experiment investigated the acceleration of a ball rolling down an inclined plane, using a similar setup I will investigate how the time taken to roll down the inclined plane varies with the vertical height change.
Theory:
When two similar objects are thrown vertically downwards, they are in a state of free-fall. Both objects will hit the ground simultaneously; the force which causes these objects to fall down is the pull of gravity which is also the acceleration of these objects.
As the object falls down, its speed increases hence its acceleration increases.
Using the equation of motion;
S= u t + ½ a t2
Since u = o, we can ignore initial velocity so:
S = ½ a t2
Straight line equation: y = m x + c
The variables in this experiment are: S and t2
When compared with the straight line equation:
S = ½ a t2
y = m x
a sin
Middle
t2 g = 2
Prediction:
I think that as the ball will run down the slope its acceleration will increase and the time for the ball to roll down will decrease. Also if the vertical height (h) is increased, the time for the ball to roll down will decrease, i.e. it will travel faster due to increase in the force of gravity.
Diagram:
Method:
• First setup the apparatus as shown in the diagram above by:
• Placing a 2m ramp on a horizontal surface.
• Having the ramp at an angle so it makes a slope for the ball to run down from, the ramp will be supported on a clamp-stand.
• Put a mark on the ramp for where the ball will be released and where it will stop.
• Then measure the vertical height of the inclined slope and record it.
• Place a cup at the end of the ramp where the mark is, so when the ball bearing reaches the end it will make a sound which will make it easy to stop the stopwatch.
• Place a ball bearing at the highest point of the slope where the mark is.
• Release the ball and simultaneously start the stopwatch.
• When the ball bearing reaches to the bottom of the slope where the mark is, then stop the stopwatch upon hearing the sound made by the ball on contact with the cup.
• Repeat the experiment several times and get an average for all repeats to get a more accurate result.
Safety:
• When carrying out the experiment make sure the ramp is securely held on the clamp.
• Handle the ramp carefully when carrying it around. Do not swing it around.
• When changing the height, first remove the ramp then adjust the height before returning the ramp to its place.
Conclusion
Max % error (with 3 readings) = = 6.7%
Max % error (with 6 readings) = = 3.3%
From the above calculation it can be clearly seen that increasing the number of readings significantly reduces the % error by 3.4% or by half.
Another way of increasing the accuracy of the timing would be to use a motion sensor to record the time.
The maximum percentage error in the height measurement is:
= 10%
This error could be reduced by measuring the height of the clamp accurately using a ruler with a mm scale.
The percentage error in the experiment was:
Error in height + (2x error in time) = 10 + 6.7 = 16.7%
From looking at the graph the points are scattered further away from the line of best fit as the height was increased. This is expected because the percentage error is greatest at these values.
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Albert Teen
YOU ARE LEARNING:
Half-Life Calculations and Curves
# Half-Life Calculations and Curves
### Half-Life Calculations and Curves
Half-life is the time it takes for the number of radioactive nuclei in a sample to halve.
This lesson will look at how to calculate the half-life of a sample of radioactive substance. But first, do you remember what the activity of a sample of radioactive substance means?
Can you remember what the two definitions of half-life are?
If the activity of a radioactive sample goes from $500 \space Bq$ to $250 \space Bq$ in 3 hours, then what is this sample's half-life?
The activity of a radioactive sample goes from $600 \space Bq$ to $150 \space Bq$ in 4 hours. What is this sample's half-life?
So the half-life of a sample if a radioactive substance has 2 definitions.
1
The time it takes for the number of radioactive nuclei in a sample to halve.
The radioactive nuclei are the unstable isotopes in the sample.
2
The time it takes for the activity of a sample to halve.
The activity is the overall rate of decay of the sample, measured in Becquerels (Bq).
1
We can show the process of radioactive decay with a graph.
We have plotted the number of unstable nuclei in a sample against time in minutes.
2
As time goes on, the number of unstable nuclei that decay declines. Why is it a curved and not a straight line?
A) Because the rate of decay is constant. B) Because the number of nuclei that decline per minute also declines. C) Because the rate of decline of the nuclei increases.
3
So the graph shows that the number of unstable nuclei in a radioactive sample decreases over time.
This means that the rate of decay (the activity) decreases as well. This is why the line is curved and not straight.
4
What is the initial number of unstable nuclei in this sample?
5
To find the half-life for this sample, you should go down to ________ on the y-axis.
6
So to find the half life, we figure out how long it takes for the number of nuclei to halve. How long does it take for the nuclei to go from 1000 to 500?
7
We can check if what we calculated is correct by doing it again. How many nuclei are left after another 20 minutes have gone by?
1
We can also find the half-life from plotting the activity of a radioactive sample against time.
Notice that the y-axis here is activity (Bq), not number of nuclei.
2
What is the initial activity of this sample?
3
What is the half life of this particular sample?
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• Subject: Math
### Exploring the Ratio of Circumference to Diameter
This activity provides an opportunity for students to explore the ratio of the circumference of its circle to the length of its diameter in order to generalize the ratio for pi.
• Resource ID: GM3L2A
• Grade Range: 7
• Subject: Math
### Creating Nets for Three-Dimensional Figures
Given nets for three-dimensional figures, the student will apply the formulas for the total and lateral surface area of three-dimensional figures to solve problems using appropriate units of measure.
• Resource ID: MATH_IMG_001
• Grade Range: K–8
• Subject: Math
### Interactive Math Glossary
The Interactive Math Glossary is provided by the Texas Education Agency to help teachers explore and understand mathematics vocabulary.
• Resource ID: M7M2L16
• Grade Range: 7
• Subject: Math
### Finding the Probabilities of Dependent and Independent Events
Given problem situations, the student will find the probability of the dependent and independent events.
• Resource ID: M8M5L5
• Grade Range: 5–7
• Subject: Math
### Selecting and Using Representations for Collected Data
Given a variety of data (including line plots, line graphs, stem and leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams), the student will select and use an appropriate representation for presenting and displaying relationships among the collected data with and without the use of technology
• Resource ID: M7M2L20
• Grade Range: 7
• Subject: Math
### Recognizing Misuses of Graphical or Numerical Information
Given a problem situation, the student will analyze data presented in graphical or tabular form by evaluating the predictions and conclusions based on the information given.
• Resource ID: M7M2L6
• Grade Range: 7
• Subject: Math
### Using Multiplication by a Constant Factor
Given problems involving proportional relationships, the student will use multiplication by a constant factor to solve the problems.
• Resource ID: M7M2L2
• Grade Range: 7
• Subject: Math
### Predicting, Finding, and Justifying Data from a Table
Given data in table form, the student will use the data table to interpret solutions to problems.
• Resource ID: M7M2L4b
• Grade Range: 7
• Subject: Math
### Predicting, Finding, and Justifying Data from Verbal Descriptions
Given data in a verbal description, the student will use equations and tables to solve and interpret solutions to problems.
• Resource ID: M8M2L6*
• Grade Range: 6–8
• Subject: Math
### Determining Slopes from Equations, Graphs, and Tables
Given algebraic, tabular, and graphical representations of linear functions, the student will determine the slope of the relationship from each of the representations.
• Resource ID: M7M1L6
• Grade Range: 7
• Subject: Math
### Solving Problems that Include Fractions and Decimals
Given problem situations, the student will use addition, subtraction, multiplication, and division to solve problems involving positive and negative fractions and decimals.
• Resource ID: M7M2L7
• Grade Range: 7
• Subject: Math
### Estimating and Finding Solutions: Percents
Given problem situations involving percents, the student will estimate and find solutions to the problems.
• Resource ID: M7M2L9
• Grade Range: 7
• Subject: Math
### Defining Similarity
Given similar figures, the student will use critical attributes to define similarity.
• Resource ID: M7M2L18
• Grade Range: 7
• Subject: Math
### Making Inferences and Convincing Arguments about Samples and Populations
Given problem situations that include given or collected data, the student will analyze the data and make inferences and convincing arguments based on the data.
• Resource ID: M7M3L4
• Grade Range: 7
• Subject: Math
### Using Models to Solve Equations
Given a problem, the student will use concrete and pictorial models to solve equations and use symbols to record their actions.
• Resource ID: M7M3L11
• Grade Range: 7
• Subject: Math
### Estimating Measurements: Area
Given problem situations, students will be able to estimate and solve for area of polygons and other shapes.
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# Fundamental Solution for 1d heat equation
So this question says to take $u(x,t) = v(x^2/t)$ to solve the 1d heat equation. That is, $$u_t = u_{xx}$$ and it gives the general solution in the form $$v(z) = c\int_{0}^z e^{-s/4}s^{-1/2} ds + d$$ (you can verify that this indeed is a solution to the heat equation). Anyway, the question involves choosing $c$ such that this solution is the fundamental solution, so that the initial condition $\Phi(x,0) = \delta(x)$ holds. I'm not sure how to go about doing this, can you perhaps point me in the right direction?
I think that $c$ solves $$c = \frac{1}{\int_{-\infty}^\infty e^{-s^2/4t}\left(\frac{s^2}{t}\right)^{-1/2} ds }$$ Can someone verify this/explain why this?
-
## 1 Answer
The heat equation (also known as diffusion equation) conserves total mass, which by definition is the integral $M (t) = \int_{-\infty}^\infty u(x,t)\,dx$. (This can be proved by taking time derivative of $M$ and using the PDE.) Since $\delta$ is a unit mass, in order to satisfy $u(x,0)=\delta(x)$ we need $M$ to be $1$.
This is why we normalize the fundamental solution $$u(x,t)=\frac{c}{\sqrt{t}}e^{-x^2/(4t)}$$ so that its mass is $1$. Which leads to $$c^{-1} = \int_{-\infty}^{\infty} e^{-x^2/4}\,dt =\sqrt{4\pi}$$ Above, the mass is computed at time $t=1$ and equated to $1$. It could be computed in any other moment $t_0>0$; the result is the same.
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bit flip of a number x is choosing a bit in the binary representation of x and flipping it from either 0 to 1 or 1 to 0.
• For example, for x = 7, the binary representation is 111 and we may choose any bit (including any leading zeros not shown) and flip it. We can flip the first bit from the right to get 110, flip the second bit from the right to get 101, flip the fifth bit from the right (a leading zero) to get 10111, etc.
Given two integers start and goal, return the minimum number of bit flips to convert start to goal.
Example 1:
Input: start = 10, goal = 7
Output: 3
Explanation: The binary representation of 10 and 7 are 1010 and 0111 respectively. We can convert 10 to 7 in 3 steps:
- Flip the first bit from the right: 1010 -> 1011.
- Flip the third bit from the right: 1011 -> 1111.
- Flip the fourth bit from the right: 1111 -> 0111.
It can be shown we cannot convert 10 to 7 in less than 3 steps. Hence, we return 3.
Example 2:
Input: start = 3, goal = 4
Output: 3
Explanation: The binary representation of 3 and 4 are 011 and 100 respectively. We can convert 3 to 4 in 3 steps:
- Flip the first bit from the right: 011 -> 010.
- Flip the second bit from the right: 010 -> 000.
- Flip the third bit from the right: 000 -> 100.
It can be shown we cannot convert 3 to 4 in less than 3 steps. Hence, we return 3.
Constraints:
• 0 <= start, goal <= 109
Solution: XOR
start ^ goal will give us the bitwise difference of start and goal in binary format.
ans = # of 1 ones in the xor-ed results.
For C++, we can use __builtin_popcount or bitset<32>::count() to get the number of bits set for a given integer.
Time complexity: O(1)
Space complexity: O(1)
## C++
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# Multiplying Negative Numbers Worksheet
The Bad Amounts Worksheet is a great way to start off instructing your kids the very idea of bad amounts. A poor variety is any number that is under zero. It could be additional or subtracted. The minus sign indicates the negative variety. Also you can publish negative figures in parentheses. Beneath is actually a worksheet to help you get started off. This worksheet has an array of adverse numbers from -10 to 10. Multiplying Negative Numbers Worksheet.
Negative phone numbers are quite a lot whose value is less than no
A poor quantity includes a worth below absolutely no. It can be indicated on a quantity series in two techniques: with the optimistic amount created because the very first digit, along with the adverse amount created as the very last digit. A positive quantity is written with a plus indication ( ) well before it, but it is optionally available to create it doing this. If the number is not written with a plus sign, it is assumed to be a positive number.
## They can be displayed by way of a minus indication
In historic Greece, unfavorable phone numbers were actually not utilized. These were ignored, as his or her mathematics was according to geometrical principles. When European scholars started out translating historical Arabic texts from Northern Africa, they came to identify negative numbers and appreciated them. These days, unfavorable amounts are depicted with a minus indication. To understand more about the history and origins of adverse figures, look at this report. Then, try these good examples to discover how negative numbers have progressed as time passes.
## They could be additional or subtracted
Positive numbers and negative numbers are easy to add and subtract because the sign of the numbers is the same, as you might already know. Negative numbers, on the other hand, have a larger absolute value, but they are closer to than positive numbers are. These numbers have some special rules for arithmetic, but they can still be added and subtracted just like positive ones. You can even subtract and add adverse numbers using a amount range and implement exactly the same regulations for subtraction and addition while you do for beneficial amounts.
## They can be depicted from a number in parentheses
A poor variety is symbolized by way of a number encased in parentheses. The adverse indication is transformed into its binary comparable, along with the two’s complement is held in the same devote memory. Sometimes a negative number is represented by a positive number, though the result is always negative. When this happens, the parentheses needs to be provided. If you have any questions about the meaning of negative numbers, you should consult a book on math.
## They are often divided by a positive amount
Negative numbers can be divided and multiplied like positive numbers. They can be divided by other negative phone numbers. However, they are not equal to one another. At the first try you flourish a negative quantity by way of a positive variety, you will definitely get absolutely nothing because of this. To create the solution, you must select which sign your answer ought to have. It is actually simpler to recall a negative variety when it is designed in mounting brackets.
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# How do you find the derivative of f(x)=ax+b?
Sep 25, 2017
${f}^{'} \left(x\right) = a$
#### Explanation:
$f \left(x\right) = a x + b$
Take derivative on both sides:
$\frac{d}{\mathrm{dx}} \left(f \left(x\right)\right) = \frac{d}{\mathrm{dx}} \left(a x + b\right)$
Apply the sum/difference rule for derivative which is stated as:
$\frac{d}{\mathrm{dx}} \left(f + g\right) = \frac{d}{\mathrm{dx}} \left(f\right) + \frac{d}{\mathrm{dx}} \left(g\right)$
So that we will have:
${f}^{'} \left(x\right) = \frac{d}{\mathrm{dx}} \left(a x\right) + \frac{d}{\mathrm{dx}} \left(b\right)$
Remember the derivative of a constant is zero, so that we will have:
${f}^{'} \left(x\right) = \frac{d}{\mathrm{dx}} \left(a x\right) + 0$
Take the constant out by applying $\frac{d}{\mathrm{dx}} \left(a \cdot f\right) = a \cdot \frac{d}{\mathrm{dx}} \left(f\right)$
${f}^{'} \left(x\right) = a \cdot \frac{d}{\mathrm{dx}} \left(x\right)$
Apply the common derivative rule $\frac{d}{\mathrm{dx}} \left(x\right) = 1$
${f}^{'} \left(x\right) = a \cdot 1$
${f}^{'} \left(x\right) = a$
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# Rounding to the Nearest 10th
When working on word problems, we are often asked to round to the nearest tenth. This can make a really long, and possibly challenging answer easier to work with.
In order to round to the nearest tenth, we need to know where the tenths place is. Here is a visual look at place value:
The tenths place is to the right of the decimal point. Our rounded answer will stop at the tenths place.
We use the hundredths place to help us determine the value that needs to be in the tenths place.
If the value in the hundredths place is 5 or above "we give it a shove."
If the value is four or below, "we let it go."
In this example, the number is 6, which is "5 or above." So we give it a shove. This means that we round it up from 5 to 6.
Is 0.56 closer to 0.50 or 0.60? We know that 56 is closer to 60, so 0.56 is closer to 0.60 or 0.6. Therefore, we round the 5 up to a 6.
Here are some more examples. Round each number to the nearest tenth.
#1.
Step 1: Locate the tenths place.
Step 2: Look to the right of the tenths place and use the number to determine if you will round up or stay the same.
Notice that the number to the right is less than 5. This means that the 1 will not round up to a 2. Instead, it will stay the same.
Step 3: Write the final answer that ends at the tenths place.
#2.
Step 1: Locate the tenths place.
Step 2: Look to the right of the tenths place and use the number to determine if you will round up or stay the same.
Here, we have a 9 to the right of the tenths place. This is above 5, so we will round up from 2 to 3.
Step 3: Write the final answer that ends at the tenths place.
#3.
Step 1: Locate the tenths place.
Step 2: Look to the right of the tenths place and use the number to determine if you will round up or stay the same.
This is an interesting example. The 7 tells us to round up. However, when we round up the nine becomes a ten. When this happens the nine becomes a zero and the place to the left is one bigger. Here is a look at the 9 becoming a 10. Notice that the 1 in the ten is under the next place value to the left.
We can think of this as the 29 rounding up to a 30.
Step 3: Write the final answer that ends at the tenths place.
Let's Review:
After you locate the tenths place, you will look to the right. If the number to the right is 5 or greater, you will round the tenths place up to the next digit.
If the number to the right is 4 or less, you will leave the tenths place alone.
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### Course: Algebra (all content)>Unit 7
Lesson 7: Recognizing functions
# Does a vertical line represent a function?
Explaining why a vertical line doesn't represent a function. Created by Sal Khan.
## Want to join the conversation?
• can a horizontal line represent a function?
• y = 5 is a horizontal line and is indeed a function.
• could he have just used the vertical line test?
• Yes, he could've. If he did that, then he would've noticed that the relation intersects the vertical line x=-2 at infinitely many points. This is because the relation is x=-2, so obviously it intersects it at infinitely many points.
However, I think Sal was trying to demonstrate a more rigorous way of testing a relation for being a function. Instead of just doing a vague, vertical line test, he used the definition of a function to test the relation for being a function.
I hope this helps!
• 3.14159, he actually started defining pi :)
• yes I saw that to. :)
• Can there be many domain but getting only one range? If the line will be horizontal will it be a function?
• One domain and one range although the domain can consist of the union of various regions on the x-axis. EG {-100 < x < -10} U {-1 < x < 1} U {10 < x < 100}. the U is the symbol of union.
A horizontal line is a function, but a pretty boring one since no matter what x value you input, the output will always be the same. EG f(x)=5. No matter what x is, the output is always 5. As you can see, the output value does not depend on the input value x.
• So is Sal saying that x -> f(x) -> infinity is not a function? If you just wrote the infinity sign could it be considered as only one output?
• Infinity cannot be a single output. This rhetorical question I'm about to give you came from another user: "Think of the biggest, biggest, biggest number you can then keep adding 1." There is no definite answer for infinity, so it can't be considered as a single output.
• is a function with multiple outputs a logarithm?
• No. A function, by definition, can not have multiple outs for a specific input value. Each input can create only one output to be a function. Thus, any equation that doesn't meet this definition would not be a function.
FYI.. there are logarithmic functions.
• Would a drastically curved line on a graph represent a function? Does the curve have to go through x and y to be a function?
(1 vote)
• For a relation to be a function, use the Vertical Line Test: Draw a vertical line anywhere on the graph, and if it never hits the graph more than once, it is a function. If your vertical line hits twice or more, it's not a function. For example, a circle is not a function because when you draw a vertical line on top of its graph, the vertical line will cut through the circle twice.
• What's the vertical and horizontal line test?
(1 vote)
• The vertical line test is used to determine if a graph of a relationship is a function or not. if you can draw any vertical line that intersects more than one point on the relationship, then it is not a function. This is based on the fact that a vertical line is a constant value of x, so if there is one input, x, with more than two outputs, y, then it breaks the function rule. A horizontal line test does not have as much meaning.
• Can a vertical line be represented by an equation?
• Yes. Points on a vertical line would all have the same X value, but could take on any value for Y. So there would be no Y value in the equation.
For instance, if the vertical line crossed the X-axis at x=2, then x=2 would be its equation!
• does anyone know how to solve this? :
"solve by graphing and check" y= -2x-2 and y=-4 ??
• Graph to solve: y = -2x -2 and y = -4
y = -2x -2
Is in Slope Intercept Form, which means we can look at the equation and see the slope of the line is -2, (because it's the coefficient of x), the second term is the y intercept and also -2, (where the line crosses the y-axis).
y-intercept of -2, means…
when x is zero, y is -2.
So we're able to graph that Point.
Point at: (0, -2) ←y intercept
★ Since the Slope is Negative, -2, we know the line is decreasing, meaning it goes ↘️ downward from left to right, and…
•A Slope of -2, means every time we move one unit Right, we move two Down.
★So let's do that from the y intercept!
If we're at (0, -2) and need to…
move one to the Right,
that's plus one on x axis…
x = 0 + 1 = 1
and for y we need two Down,
that's minus 2 on y axis…
y = -2 -2 = -4
Now we have another coordinate on the same line!
Point at: (1, -4)
★(0, -2) and (1, -4)
Draw a line through both coordinate Points, to finish graphing the first equation.
• Now to graph…
y = -4
Notice there is no written slope in this equation, just a y intercept, this is because it's a…
↔️ a Horizontal Line, slope = 0
It runs flat crossing y-axis at -4.
Find -4 on y-axis, draw a straight, side to side, ↔️ Horizontal Line through it.
Look to see where these two lines cross.
That (x, y) coordinate is the Point of Intersection between these two lines, the only x and y coordinate pair that make both of these equations True Statements.
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# algebra
posted by .
how would i find an equation of a line that goes through points(1,6) and (3,10)??
thanks
A straight line is
y=mx+b
Substitute the points to make two equations.
6=m(1)+b
10=m(3)+b
Two equations; two unknowns, m and b.
Solve for m and b, then plug back into y = mx + b. That will the the equation for the straight line.
i don't understand?
You know how to solve two equations simultaneously. There are two unknowns, m and b. Solve them for m and b, list the values here and I will show you how to construct the equation for the line. Here are the two equations from above.
eqn 1: 6=m(1)+b
eqn 2: 10=m(3)+b
To start you in the right directin, here is what you do to solve the equations.
1. Multiply eqn 1 by -1
2. Add the result + eqn 2.
3. That will eliminate b. solve for m.
4. Plug m back into either equation and solve for b.
5. Post values for m and b and I shall show you how to write the equation for the line that passes through those points.
P=H-F/2H TO SOLVE FOR H
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Question
Let f:[−1,1]→R be defined as f(x)=ax2+bx+c for all x∈[−1,1], where a,b,c∈R such that f(−1)=2,f′(−1)=1 and for x∈(−1,1) the maximum value of f′′(x) is 12. If f(x)≤α,x∈[−1,1], then the least value of α is equal to
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Solution
f(x)=ax2+bx+cf′(x)=2ax+bf′′(x)=2a We know f(−1)=2⇒a−b+c=2⋯(1)f′(−1)=1⇒b−2a=1⋯(2)f′′(x)≤12⇒a≤14⋯(3) From equations (1) and (2), we get b=1+2a, c=3+a⇒f(x)=ax2+(1+2a)x+(3+a)⇒f(x)=a(x+1)2+(x+3) To get the maximum value of f(x), a should be maximum, so a=14f(x)=(x+1)24+(x+3)⇒f(x)∈[2,5], x∈[−1,1] As f(x)≤α,x∈[−1,1] ∴α=5
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# If the product xy is negative, which of the following must be true?
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If the product xy is negative, which of the following must be true? [#permalink]
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06 Nov 2018, 00:08
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If the product xy is negative, which of the following must be true?
A. x < 0
B. y < 0
C. x/y > 0
D. x/y < 0
E. x + y < 0
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Re: If the product xy is negative, which of the following must be true? [#permalink]
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06 Nov 2018, 00:41
1
Bunuel wrote:
If the product xy is negative, which of the following must be true?
A. x < 0
B. y < 0
C. x/y > 0
D. x/y < 0
E. x + y < 0
xy<0.
It means either x or y is negative. we are not sure which one is negative.
Option C) x/y <0. All time.
One of them is negative . thus ultimate result is negative.
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Re: If the product xy is negative, which of the following must be true? [#permalink]
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22 Nov 2019, 23:22
XY<0 => Either one of X or Y is negative. In such a case X/Y is always negative.
Ans D
Re: If the product xy is negative, which of the following must be true? [#permalink] 22 Nov 2019, 23:22
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# Simplify the fraction (x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1].
We have to simplify (x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1]
(x^2-y^2+2x+1)/[(x+y)^2+2(x+y)+1]
=> (x^2-y^2+2x+1)/(x + y + 1)^2
=> (x^2 +2x+1 - y^2)/(x + y + 1)^2
=> [(x + 1)^2 - y^2]/(x + y + 1)^2
=> [(x + 1 - y)(x + 1 +y)/(x + y + 1)^2
=> [(x + 1 - y)/(x + y + 1)
The required result is (x - y + 1)/ (x + y + 1)
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# Bond Price Arithmetic
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1 1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously compounded rates. To learn how to handle cash flows that are unequally spaced, or where there are fractional periods of time to particular cash flows. To understand the market convention of quoting prices, computing accrued interest and communicating prices in a yield form. To set the stage for a deeper analysis of fixed income products. 1.1 FUTURE VALUE AND COMPOUNDING INTERVALS Let \$P be invested at a simple interest rate of y% per year for one year. The future value of the investment after one year is V 1 where: V 1 = P (1 + y) and after n years the value is V n where: V n = P (1 + y) n 1
2 2 CHAPTER 1: BOND PRICE ARITHMETIC If interest is compounded semi-annually then after n years: V n = P [1 + y 2 ]2n If interest is compounded m times per year then after n years: V n = P [1 + y m ]m n As the compounding interval gets smaller and smaller, i.e. as m, the accumulated value after n years increases, because interest is being earned on interest. If interest is compounded continuously at rate y, then after n years the accumulated value is: V n = lim m P [1 + y m ]m n Mathematicians have shown that this limit can be expressed in a simple way. In particular, lim m [1 + y m ]m n = e yn where e x is the exponential function that can be written as follows: 1 e x =1+x + x2 2 + x3 +...for all values of x. 6 Hence, with continuous compounding, the future value is: V n = Pe yn Example The future value of a \$100 investment compounded at 10% per year simple interest is \$110; compounded semiannually the future value is 100(1.05) 2 = \$110.25; and compounded continuously is 100e 0.10 = \$ Given one method of computing interest, it is possible to find another compounding rate that leads to the same terminal wealth. For example, assume the semi-annual compounding rate is y. Then after n years we have: V n = P [1 + y 2 ]2n 1 The exponential expansion shows that when x is very small, e x 1+x. In this case x is a simple return. For larger values of x, the higher order terms become important.
3 CHAPTER 1: ANNUALIZING HOLDING PERIOD RETURNS 3 The continuous compounding rate that leads to the same terminal wealth can be established by solving the equation for y : V n = P [1 + y 2 ]2n = Pe y n Taking logarithms on both sides leads to y n = ln[(1 + y 2 )2n ] = 2n ln[(1 + y 2 )] or y = 2 ln[(1 + y 2 )] Example A semiannual rate of 10% per year is given. The equivalent continuously compounded yield is y = 2 ln[(1 + y )] = 2 ln(1.05) = 9.758% ANNUALIZING HOLDING PERIOD RETURNS The price of a contract that promises to pay \$100 in 0.25 years is \$98.0. Let R represent the return obtained over the period. The holding period yield is R = = or 2.04%. The holding period yield does not adjust for the length of the period. To make comparisons between investments held for different time periods, it is common to annualize the yield. This is usually done in one of two ways, either as simple interest, or as compounded interest. Example (i) The annualized simple interest in the last example is given by multiplying the holding period yield by the number of periods in the year, namely 4. Specifically, the annualized yield is = 8.16% (ii) The compounded rate of return in the last example is given by(1 + R) n 1, where n = 4. This value is (1.0204) 4 1=8.42%. In the above example the compounding interval was taken to be quarterly. In many cases the investment period could be quite small, for example one
4 4 CHAPTER 1: BOND PRICE ARITHMETIC day. In this case the compounded annualized return is (1 + R) 365 1, where R is the one day return. If the holding period is small, then the calculation of annualized return can be approximated by continuous compounding. Specifically, for R close to zero, and n large, (1 + R) n e nr. Example An investment offers a daily rate of return of A one million dollar investment for one day grows to (1, 000, 000)( ) = \$1, 000, 250. The annual rate, approximated by continuous compounding, is e 365( ) 1= 9.554% Given the annualized continuously compounded return is y = , the simple return for a quarter of a year is e ( )(0.25) 1=2.417%. In all calculations care must be taken that the annual interest rate used is consistent in all calculations. For example, if a security returns 10% over a six month period, then the equivalent continuous compounded return is obtained by solving the equation e y(0.5) =1.10. Equivalently, y = log(1.10)/0.5 = 19.06% Compounding Over Fractional Periods The future value of \$P over 2 years when compounding is semi annual is P (1 + y 2 )4. Raising (1 + y ) to the power of 4 reflects four semiannual interest 2 payments. If the time horizon is not a multiple of six months, then establishing the future value is a problem. For example, if the time horizon is 2.25 years, the future value could be written as P (1 + y 2 )4 (1 + y 2 )0.5. The handling of the fractional period is not altogether satisfactory, and there is no real theory to justify this calculation. However, this calculation is one popular market convention. If compounding was done quarterly, then the answer to the above problem is P (1+ y 4 )9. Of course, if the time horizon was 2.26 years, then compounding quarterly would not solve the problem, and we would again encounter the problem of computing interest over a fraction of a period. If compounding is done continuously then the problem of handling fractional periods disappears. The future value of P dollars over T years is Pe yt.
5 CHAPTER 1: DISCOUNTING DISCOUNTING The present value of one dollar that is received after n years, assuming the discount rate is y% per year with annual compounding, is given by PV =1 1 (1 + y) n If compounding is done m times per year, the present value is: 1 PV =1 (1 + y m )n m If the one dollar is discounted continuously at the rate of 100y% per year, the present value is: PV =1 e y n 1.4 BOND PRICES AND YIELD -TO- MATURITY A coupon bond is a bond that pays fixed cash flows for a fixed number of periods, n say. Typically, the cash flows in all the periods are equal. At the last period a balloon payment, referred to as the face value of the bond, is also paid out. Typically, the coupon is expressed as a fraction of the face value of the bond. In what follows we will take c to be the coupon rate, and C = c F to be the dollar coupon. If the coupons are annual coupons, of size C, and the face value is F, then the yield-to-maturity of the bond is the discount rate, y, that makes the following equation true. B 0 = C 1+y + C (1 + y) C + F (1 + y) n where B 0 is the actual market price of the bond. The coupon of a bond refers to the dollar payout that is made in each year. If coupons are paid annually then each cash flow is of C dollars. Payments at frequencies of once a year are appropriate for typical bonds that are traded in the Eurobond market. For bonds issued in the US, however, the typical convention is for coupon payments to be made semiannually. Such a bond would therefore pay half its coupon payment every six months. In this case, the yield-to-maturity of a bond that matures in exactly n years, is the value for y that solves the following equation: B 0 = C/2 1+y/2 + C/2 C/2+F (1 + y/2) 2 (1 + y/2) 2 n (1.1)
6 6 CHAPTER 1: BOND PRICE ARITHMETIC Example Consider a bond with a 10% coupon rate and 10 years to maturity. Assume the face value is \$100 and its price is \$102. The bond will pay 20 coupons of \$5.0 each, plus the face value of \$100 at the end of 10 years. The value of y that solves the above equation is given by y =9.6834%. Clearly, the yield-to-maturity of a bond that pays coupons semiannually is not directly comparable to the yield-to-maturity of a bond that pays coupons annually, since the compounding intervals are different. 1.5 ANNUITIES AND PERPETUITIES An annuity pays the holder money periodically according to a given schedule. A perpetuity pays a fixed sum periodically forever. Suppose C dollars are paid every period, and suppose the per period interest rate is y. Then the value of the perpetuity is: P 0 = i=1 C (1 + y) i The terms in the sum represent a geometric series and there is a standard formula for this sum. In particular, it can be shown that 2 P 0 = i=1 C (1 + y) i = C y (1.2) As an example, if a perpetuity paid out \$100 each year and interest rates were 10% per year, then the perpetuity is worth 100/0.10 = \$ To see this let a = 1. Let Sn be the sum of the first n terms of the cash flows of the (1+y) perpetuity. That is S n = ac + a 2 C a n C Now, multiply both sides of the equation by a to yield as n = a 2 C a n C + a n+1 C. Subtracting the equations lead to (1 a)s n = ac a n+1 C Hence S n = ac an+1 C 1 a. Substituting for a and letting n leads to limit n S n = C y.
7 CHAPTER 1: ANNUITIES AND PERPETUITIES 7 The value of a deferred perpetuity that starts in n years time, with a first cash flow in year n + 1, is given by the present value of a perpetuity or ( ) 1 C P n = (1 + y) n (1.3) y By buying a perpetuity and simultaneously selling a deferred perpetuity that starts in n years time, permits the investor to receive n cash flows over the next n consecutive years. This pattern of cash flows is called an n-period fixed annuity. The value of this annuity, A 0 say, is clearly: A 0 = P 0 P n = C y [1 1 (1 + y) n ] (1.4) Rewriting the Bond Pricing Equation A coupon bond with n annual payments \$C and face value \$F can be viewed as an n period annuity together with a terminal balloon payment equal to F. The value of a bond can therefore be expressed as B 0 = C y [1 1 (1 + y) ]+ F (1.5) n (1 + y) n where y is the per period yield-to-maturity of the bond. When F =\$1.0, the coupon is given by C = c 1=c. Ify = c then from the above equation, it can be seen that B 0 = 1. Hence, when the coupon is set at the yield to maturity, the price of a bond will equal its face value. Such a bond is said to trade at par. If the coupon is above (below) the yield-tomaturity, then the bond price will be set above (below) the face value. Such bonds are referred to as premium (discounted) bonds. Unequal Intervals Between Cash Flows So far we have assummed that the time between consecutive cash flows is equal. For example, viewed from a coupon date, the yield to maturity of a bond with semi annual cash flows is linked to its market price by the bond pricing equation: B 0 = m j=1 C/2 (1 + y/2) j + F (1 + y/2) m where y is the annual yield to maturity, C is the annual coupon and m is the number of coupon payouts remaining to maturity. In this equation, the first coupon is paid out at date 1, in six months time. If the first of the m
8 8 CHAPTER 1: BOND PRICE ARITHMETIC cash flows occurred at date 0, then the price of the bond is: m C/2 B 0 = (1 + y/2) j 1 + F (1 + y/2) m 1 j=1 If the first coupon date is not immediate but occurs before 6 months, then the above equation must be modified. Specifically, the above equation can be used to price all the cash flows from the first cash flow date. This value, is then discounted to the present date. Specifically, the yield-to-maturity of a coupon bond is defined to be the value of y that solves the equation: B 0 = 1 (1 + y/2) p m j=1 C/2 (1 + y/2) j 1 + F (1.6) (1 + y/2) m 1 where p = t n /t b and t n is the number of days from the settlement date to the next coupon payment, and t b is the number of days between the last coupon date and the next coupon date. In this equation we have assumed that the total number of coupons to be paid is m. This way of handling fractional periods is the market convention used in the US Treasury bond market. 1.6 PRICE QUOTATIONS AND ACCRUED INTEREST If a coupon bond is sold midway between coupon dates, then the buyer has to compensate the seller for half of the next coupon payment. In general, for Treasury bonds, the accrued interest, AI, that must be paid to the previous owner of the bond is determined by a straight line interpolation based on the fraction of time between coupon dates that the bond has been held. Specifically, AI = t l t b where t l is the time in days since the last coupon date, and t b is the time between the last and next coupon date. The computation of accrued interest using this convention is termed actual/actual. The first actual refers to the fact that the actual days betwen coupons are used in the calculation. The second actual refers to the fact that the actual number of days in a year are used. The above convention is standard for Treasury bonds traded in the US. Other methods of computing accrued interest that apply in different markets will be considered later. Market convention requires that US Treasury bond price quotations be reported in a particular way. A face value of \$100 is assumed and the quotation ignores the accrued interest. The actual cost, or invoice price of a bond, corresponding to B 0 in the equation (1.6) given a quotation is: Invoice Price = Quoted Price + Accrued Interst
9 CHAPTER 1: PRICE QUOTATIONS AND ACCRUED INTEREST 9 The specific rule for computing accrued interest and translating quoted prices from a newspaper into market prices vary according to the particular fixed income product. Example A Eurobond is a bond issued by a non European firm in Europe. Typically, interest is paid annually, and yields are simple annual yields. The accrued interest in this market are not based on actual/actual, but rather on 30/360. In this convention each month is counted as having 30 days and each year has 360 days. Say a bond pays coupons on August 1st of each year and the settlement date for the transaction falls on April 10th. The seller has held the bond for 8 months and 10 days. Under this convention the accrued interest is based on = Specifically, the accrued interest is 25/36 th of the annual coupon. This accrued interest is added to the quoted price to obtain an invoice price. Given the invoice price, a yield for this product can be obtained using the appropriate bond pricing equation. Specific products and the market conventions related to compounding frequency, quotation format, and the handling of accrued interest will be discussed in more detail in future chapters. The important point here is that the conventions are market specific. The accrued interest convention makes the quoted price process smooth over time. Actual market prices of bonds fall at coupon dates. Just before a coupon, the price of a bond with n years to maturity is B 0 = C 2 + 2n i=1 C/2 (1 + y/2) i (1 + y/2) 2n Since the seller has held the bond over the entire period, (t l = t b ) the accrued interest is C 2 and the quoted price, Q 0 say, is the above market price less C 2,or Q 0 = 2n i=1 C/2 (1 + y/2) i (1 + y/2) 2n Immediately after the coupon has been paid, the bond price is given by B + 0 = 2n i=1 C/2 (1 + y/2) i (1 + y/2) 2n The drop in price, B 0 + B 0, equals the actual coupon paid out. Since the new accrued interest is now zero ( t l = 0), the new quoted price equals the new market price, which in turn equals the old quoted price. That is, quoted prices remain unchanged.
10 10 CHAPTER 1: BOND PRICE ARITHMETIC 1.7 COMMON INTEREST RATE CONVENTIONS Securities are issued with cash flows that occur at different time intervals. To compare rates it is often necessary to switch from one type of rate, based on a particular compounding interval, to another rate. Example A rate of 9% semi-annual is equivalent to a ( )2 1=9.2025% annual rate. A 9% semi-annual rate is also equivalent to a daily rate of ( ) = % per day. On annualizing this rate we obtain = A 9% semi-annual rate is equivalent to a daily rate of %. Over a 100 day period, the rate is ( ) 100 1= or %. Annualizing this rate we obtain =8.9097%. The effective annualized rate of this loan for 100 days is %. Table 1.1 shows the market convention of rates in particular markets. Table 1.1 Market Convention of Rates in Particular Markets UK Money Markets Annual Actual/365 US & Euromoney Markets Annual Actual/360 US Treasury Bonds Semi-annual Actual/365 Eurobonds Annual 30/360 US Federal Agencies, Municipals, Corporates Semi-annual 30/360 US Commercial Paper, Bankers Acceptances Discount Basis, Actual/360 Commercial Paper Discount Basis, Actual/365 Examples (i) Assume the semiannual coupon periods are divided into 181 and 184 days. Assume 10m dollars are borrowed at 10 % semiannual actual/365. Then, the coupon payments of \$1m over the year would be split up into payments of 10m = \$495, , and 10m = \$504, (ii) The same loan done on a 30/360 basis would have two cash flows of \$500,000 each. The annual total is the same, but the size and timing of the individual cash flows are different.
11 CHAPTER 1: YIELDS AS A METHOD OF COMMUNICATING PRICES 11 (iii) Table 1.2 shows the effective annual rates of a 10% quotation for several market conventions. Table 1.2 Examples of Market Conventions Convention Computation Effective Annual Rate Annual Actual/365 ( ) 1 =10.0% Annual Actual/360 ( ) 1 =10.14% Semi-annual Actual/365 ( )2 1 =10.25% Semi-annual Actual/360 ( )2 1 =10.40% Monthly Actual/365 ( )12 1 =10.47% Monthly Actual/360 ( )12 1 =10.62% 1.8 YIELDS AS A METHOD OF COMMUNICATING PRICES The invoice price of a bond is the amount of dollars one requires in order to purchase it. Once you know the price, you can compute its yield using an appropriate formula. Conversely, if the yield of a bond is given, then provided you understand the market convention associated with the fixed income product, the unique price of the bond can be established, the accrued interest computed, and a quoted price can be established. The mapping from yields to quoted prices requires understanding the compounding mechanism (eg. annual or semiannual), the handling of fractional periods and the computation mechanism for accrued interest. Given these rules, prices can be quoted in yield form. While yields associated with different fixed income products may be useful for communicating price information, one has to be careful in interpreting these numbers. Higher yields do not necessarily imply higher returns, or higher risks. As a simple example, comparing yields of a coupon bond that pays annually, with a coupon bond that pays semiannually may be misleading. While in some cases the yield of a fixed income product may have a simple economic interpretation, in others no simple interpretation exists. For example, consider a straight default free coupon bond. Its price is the present value of the bonds cash flows using the yield as a discount rate. On the other hand, consider a coupon bond that has a call feature. The yield that is given
12 12 CHAPTER 1: BOND PRICE ARITHMETIC to characterize its price cannot be interpreted as a discount rate for all the promised cash flows to the maturity date. 3 In general, then, while yields are often used to characterize prices of fixed income products, in general they may not have simple economic interpretations, and certainly do not provide a common ground by which their relative benefits can be accessed. Given a bond price, there is no theoretical reason why coupon bonds have to have their yields to maturity computed according to any market convention. For example, we could define the continuously compounded yield to maturity of a bond that has face value F, and pays C dollars at times t 1, t 2,...,t n and face value F, is given by the value y that solves the equation. B = Ce yt1 + Ce yt2 + Ce yt (F + C)e ytn (1.7) In this equation the times t 1, t 2..., t n are all expressed in years and need not be equidistant. This definition of a yield to maturity is as valid as any other definition, but is not adopted in any specific market as a normal market convention. 1.9 CONCLUSION The purpose of this chapter has been to review the basics of discounting at annual, semiannual and continuously compounded rates and to obtain some insight into how prices are connected to specific yields according to market conventions. In order to obtain the invoice price of a bond, its quoted price may have to be adjusted by accrued interest. The computation of accrued interest varies according to the particular product. We illustrated the adjustment for Treasury bonds, where the actual/actual rule holds and for eurobonds, where a 30/360 rule holds. Given the invoice price, the quoted yield for the particular fixed income product can also be obtained. The way in which the yield is computed also depends on the particular product. Treasury bond yields, for example, are reported in semiannual form while Eurobonds are reported using annual compounding. Given the market convention, price information can be conveyed using their appropriate yields. In general, however, the particular yield-to-maturity statistic that is computed for a product may not provide useful economic information relating to its potential return or risk. 3 We shall explore this in another chapter. The problem for callable bonds is that the exact number of future cash flows is not certain since the bond can be called at any time after the call date.
13 CHAPTER 1: EXERCISES EXERCISES 1. An investment requires an initial investment of \$100 and guarantees \$104 back in 0.25 years. (a) Compute the holding period return. (b) Compute the simple annualized yield. (c) Compute the compounded annualized yield for the investment, assuming quarterly compounding. (d) What is the annualized continuously compounded rate of return for this investment. 2. A discount bond with a maturity of 5 years and a face value of \$1000 is priced at \$ (a) Compute the continuously compounded yield -to-maturity. (b) Compute the semi annualized yield to maturity. 3. A discount bond with a face value of \$1000 is currently priced at \$ The maturity of the bond is 6 years. The bond, however, is callable in 3 years for a price of \$ (a) Compute the continuously compounded yield-to-maturity. (b) The yield to call is the yield to maturity obtained under the assumption that the call date is the maturity date. Compute the continuously compounded yield-to-call. (c) Interpret the above two numbers and comment on the potential problems with interpreting these two yield measures. 4. The quoted price of a Treasury bond with settlement date January 6 th 1999 is \$ The bond s coupon is 4 1/4. It matures on November 15 th The number of days in the current coupon period is 182, and the number of days from settlement to the next coupon date is 130 days. Compute the accrued interest, the invoice price of the bond, and the semiannual yield to maturity. 5. In this problem you will learn how to use excel to compute prices of coupon bonds when cash flows are equally spaced. In particular, you will compute bond prices four different ways. The main idea here is to show that the analytical solution for the bond price is helpful, and to introduce you to excel s PRICE function that produces a clean price and is fairly useful. The benchmark model we will solve is a 5 year maturity bond paying annual coupons rate of 5% seminanually. The face value is \$100. The yield-to-maturity is given as 6%. In excel set the inputs up as follows:
14 14 CHAPTER 1: BOND PRICE ARITHMETIC INPUTS Annual Coupon Rate (AC) 0.05 Yield to maturity (Y) 0.06 Number of payments per year (num) 2 Number of Periods (N) 10 Face Value (FV) 100 OUTPUTS Discount rate/period (Rate) 0.03 Coupon payment (c) 2.5 For each of these variables label them using Insert,Name,Define. Excel will now recognise these variables when you refer to them. Now we are ready to compute bond prices. (a) Set up 11 colums numbered 0 to 10. These refer to the time periods. There will be three rows under these columns. The first row is called time (in years). For this problem it will be the period number divided by 2. The second row will contain the cash flows. For this problem it will be a row of 2.5 dollars starting from period 1 and ending in period 9. In period 10 there will be a cash flow of The final row will then contain the present value of each of these cash flows. The bond price is then obtained by adding these numbers up. Confirm that you obtain a value of Note that if we change the number of periods, we will have to add more columns in our spreadsheet. So this method is not very useful. (b) Now repeat the exercise of pricing this bond, but this time use the analytical formula for bond pricing. So in one equation, using the variable names, you can obtain the price. This formula has an advantage over (a) in that the number of periods can be changed and the price will automatically update. (c)now compute the bond price using the PV function in excel. This function requires the Rate (Rate), Number of periods (N), coup (c), and face value, (FV), as inputs. (d) Finally, compute the bond price using the PRICE function in excel. This function requires the settlement date, the maturity date, annual coupon rate, yield-to-maturity, face value and the number of payments. To use it for an example make up a settle date (eg 01/01/2000) and then add 5 years to get the maturity date. To do this use the excel DATE command eg DATE(2000+5,1,1). This will give you a maturity date exactly 5 years later. In general the PRICE function gives you a quoted, flat, or clean price. The actual invoice, full, or dirty price is obtained by adding on the accrued interest. In this above problem there is no accrued interest so the clean and dirty prices are equal. We will use the PRICE function in the next chapter.
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# Find the Length of the Side and Perimeter of an Equilateral Triangle Whose Height is √ 3 Cm. - Geometry
ConceptSimilarity in Right Angled Triangles
#### Question
Find the length of the side and perimeter of an equilateral triangle whose height is $\sqrt{3}$ cm.
#### Solution
Since, ABC is an equilateral triangle, CD is the perpendicular bisector of AB.
Now, According to Pythagoras theorem,
In ∆ACD
${AC}^2 = {AD}^2 + {CD}^2$
$\Rightarrow \left( 2a \right)^2 = a^2 + \left( \sqrt{3} \right)^2$
$\Rightarrow 4 a^2 - a^2 = 3$
$\Rightarrow 3 a^2 = 3$
$\Rightarrow a^2 = 1$
$\Rightarrow a = 1 cm$
Hence, the length of the side of an equilateral triangle is 2 cm.
Now,
Perimeter of the triangle = (2 + 2 + 2) cm
= 6 cm
Hence, perimeter of an equilateral triangle is 6 cm.
Is there an error in this question or solution?
#### APPEARS IN
Balbharati Solution for Balbharati Class 10 Mathematics 2 Geometry (2018 to Current)
Chapter 2: Pythagoras Theorem
Problem Set 2 | Q: 5 | Page no. 44
#### Video TutorialsVIEW ALL [4]
Solution Find the Length of the Side and Perimeter of an Equilateral Triangle Whose Height is √ 3 Cm. Concept: Similarity in Right Angled Triangles.
S
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# insert 1 rational number between 4/3 and 7/5
properties of rational numbers
Rational number
A rational number is expressed as a fraction p/q when the denominator q is not equals to 0.
Given rationals are 4/3 and 7/5
Formula for rational number between x and y when x < y then 1/2(x+y)
x = 4/3 and y = 7/5
1/2(4/3+7/5)
= 1/2(4*5+7*3)/15
= 1/2(20+21)/15
= 41/30
Required rational 41/30.
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# Number of Vaccinations Activity
In the new course "Python in Data Science", there is an activity called "Number of Vaccinations" which is very easy but I can't get the answer right. Here is the link for the activity, and any help would be highly appreciated, Thank you. The link: https://www.sololearn.com/learning/1161/4777/12269/1
19th Mar 2021, 11:45 AM
BasharGh123
+ 5
you must carefully read the hint provided: Hint: Think about the data this way: it contains 20 values, each representing the number of vaccinations the corresponding person had. then, you could visualize better the data set by writtng it: ds = [ 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0 ] hope this help! :)
19th Mar 2021, 12:25 PM
visph
+ 5
vac_nums = [0,0,0,0,0, 1,1,1,1,1,1,1,1, 2,2,2,2, 3,3,3 ] #your code goes here print((0*5+8*1+4*2+3*3)/20)
25th Jan 2022, 9:24 PM
IbrahimCPS
+ 3
Note the variance is the squared difference between the mean and the values. So all you need to do is to subtract the mean from the value and square it. def variance(X): mean = sum(X)/len(X) tot = 0.0 for x in X: tot = tot + (x - mean)**2 return tot/len(X) sample=[0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3] print(variance(sample)) This code works perfectly 👌.
27th Aug 2021, 11:53 AM
Dramani Yinsongte Kizito
+ 2
Here's my solution using a dictionary to allow me to add the values easily, please consider reading the code carefully https://code.sololearn.com/c8y85eVu6Z8i/?ref=app
4th Apr 2021, 10:09 AM
+ 2
LIN, ZHI-JIA yeah I noticed it but it just didn't accept this way for a reason I don't know yet
10th Apr 2021, 6:35 PM
+ 1
This is my solution ! Number of Vaccinations: vac_nums = [0,0,0,0,0, 1,1,1,1,1,1,1,1, 2,2,2,2, 3,3,3 ] #your code goes here mean = sum(vac_nums)/len(vac_nums) diff = 0.0 for i in vac_nums: diff = diff + (mean - i)**2 variance = diff / len(vac_nums) print(variance)
1st Aug 2022, 5:48 AM
Abhishek Verma
0
this is your data [0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3] the result is print ((0*5+1*8+2*4+3*3)/20)
2nd Aug 2021, 7:25 PM
Bouzidi Ikram
0
nums={0:5,1:8,2:4,3:3} arr=[] for key,val in nums.items(): for i in range(key): arr.append(val) #arr will be [8,4,4,3,3,3] #mean print(sum(arr)/20)
11th Oct 2021, 7:48 PM
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# Explaining why (or whether) zero and one are prime, composite or neither to younger children
There are lots of discussions out there about whether $$1$$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $$\mathbb{Z}$$ by the standard abuse of terminology ever since Kummer).
However, I haven't seen a discussion of the related question on this site - namely:
How should one talk about the question of $$1$$ or $$0$$ being prime (or composite, or neither) with primary or middle school children?
I'm particularly interested in the question about zero.
The fundamental theorem of arithmetic uniqueness rationale is probably a little heady for them (though this book gives it a great try) or even college students at times. But I also don't really like the idea of saying "because we said so" when so much of school math feels like this to students.
Semi-sarcastic, but still possible final thoughts:
If you think you can adapt Conway's notion of factoring all integers via $$-1$$/$$1$$ as "prime powers", that would be great. Also on topic here would be discussion of non-uniqueness of factorization for this age group. But I think that those are probably asking a bit much.
• Yes, this really did come up at bedtime tonight. Commented Oct 27, 2019 at 1:15
• Re "because we said so": But that's exactly it... One could only try to explain why it makes sense to "say so", e.g. as in the accepted answer of the question you linked. Commented Oct 27, 2019 at 11:57
• @Namaste You have to omit 'distinct' though, otherwise squares of primes are also included.
– Paul
Commented Oct 27, 2019 at 21:47
• For what it's worth, some of your questions in your more recent comment could definitely be turned into good questions on this site; I know that many even much older students struggle realizing that one can think of integers as also rational via $5 = 5/1$, asking whether that is allowed, and again for different developmental stages there could well be different strategies. Commented Oct 28, 2019 at 20:36
• Some Greeks, Euclid for instance, did not even consider one a number. A number meant more than one. Indeed it is generally what is meant English by “I have a number things to do today.” Commented Oct 28, 2019 at 21:32
"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each other. Does that seem like it would be a good idea?" is both more inclusive and more correct.
I don't even think it's that hairy to talk through the FTA with anyone old enough to understand primes. Have everybody take out a sheet of paper and express, say, $$12$$ as a product of the smallest numbers possible (with repetitions being okay). You take a sheet of paper and write $$1\cdot 2\cdot 3\cdot 1\cdot 1\cdot 2$$. Then you can discuss what everyone's paper has in common and if there are any differences. Then point out that people a while ago realized that everyone's papers would have a lot more in common if they would just agree that $$1$$ isn't a prime number, and that's what lead to the convention being established.
• I like the sheet of paper idea for an activity that keeps things from giving the entire statement of FTA while maintaining the spirit. I may take exception to the first paragraph a bit, at least in terms of it feeling any better than "we said so", as it still doesn't really tell them why and feels like a conversation closer. Of course, that might just mean I'm raising too inquisitive of children :) Commented Oct 28, 2019 at 3:28
• Yeah, I'm a big fan of constructivism, especially in math education. Some things that are hard to teach can be relatively easy when we give students the environment in which they can "teach themselves". It's not a magic bullet for every lesson, but I like using it when it makes sense. Commented Oct 28, 2019 at 13:12
• And it is ultimately a fair question. You can tell a particularly precocious young person that mathematicians create new theories by bending "the rules" and seeing what changes. If the new theory turns out to be useful, then you're an applied mathematician. If the new theory is cool but nobody can find a use for it, then you're a pure mathematician and (as my college advisor would joke) you just need to wait 150 years for physicists and engineers to evolve to the point where they can think of an application. Commented Oct 28, 2019 at 13:33
• "Because 0 x 0 = 0 and 1 x 1 = 1 but 2 x 2 ≠ 2, 3 x 3 ≠ 3 ... If p x p ≠ p, and p can only be divided by itself or 1 and result in an integer, then p is prime. Because 0 x 0 = 0 and 1 x 1 = 1 neither 0 nor 1 can be prime. That makes 12 uniquely equal to 2 x 2 x 3. If we make the rule that you have to put the prime factors in increasing order, every integer larger than 1 either is a prime itself, or is a composite that can be written as the product of two or more primes, which will always be the same for that number." Commented Oct 28, 2019 at 22:00
There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ 3&\quad 6 &\quad 9 &\quad 12 &\quad\cdots\\ 4&\quad 8 &\quad 12 &\quad 16 &\quad\cdots\\ \vdots&\quad \vdots &\quad \vdots &\quad \vdots &\quad\ddots\\ \end{alignat} but going up to $$10$$.
What numbers appear in this table somewhere? All of them. (Advanced language: all postive integers) because they are all in the first row.
What numbers appear only once in this table? Just the number $$1$$. Any other number appears at least twice, once in the first row and once in the first column (and possibly elsewhere).
What numbers appear exactly twice in this table? The numbers $${}\ge 2$$ that do not appear except in the first row and first column.
Definition these are called "prime numbers".
What numbers appears three or more times in this table? All the numbers in the table when you omit the first row and the first column.
Definition these are called "composite numbers".
• This feels very procedural to me, and doesn't explain why we bother calling numbers prime. I would not think it would help a child understand. Commented Oct 27, 2019 at 17:47
• The OP said he didn't like the explanation "because we said so". But that's exactly what this answer does, with a slight twist "because the table says so".
– IMil
Commented Oct 28, 2019 at 0:38
• @IMil true but I think this one is a bit better than "I said so", because of the visualization. Commented Oct 28, 2019 at 3:23
• @SueVanHattum it might not help us say why we call them prime, but one advantage is it might lead to exploration of how many times a number shows up in the table etc. I agree it couldn't be an entire answer but may help visualizing. And note that extended to zero we would certainly have lots of zeros so zero isn't prime in this context. Commented Oct 28, 2019 at 3:25
• Another fun question: which numbers appear an odd number of times? Commented Oct 28, 2019 at 19:05
FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html
One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from prime-ness because many other higher theorems would be complicated by leaving it as prime. Essentially "prime" -> "prime*". The definition of primeness was tweaked to exclude one.
This definition change is glossed over with the "different from itself" formalism that doesn't sound as awkward as saying "except one". But clearly the reason for the change and the effect of it was the same as if we had adjusted the definition with a suspicious sounding "except one". I personally think just saying "except one" is a little more direct and revealing. (Within the definition of prime. I'm OK with the change to prime*. But let's be real...we did it to vote one off the island. If it didn't simplify a lot of higher math statements, we would not have made the change. Certainly wouldn't have made the change if it complicated them!)
P.s. I personally think a too dogmatic "one is not a prime...how dare you think that...you are just wrong" stance is too harsh to give to the child. Just being honest and saying they tweaked the definition because it makes later math simpler is more honest and less upsetting, even though it leaves an impression of capriciousness.
• "If it didn't simplify a lot of higher math statements" - even if beyond a grade schooler, I'd love to hear the simplest thing this quote references. Commented Oct 27, 2019 at 12:59
• @JoeTaxpayer: FTA: "There is a unique way to write any natural number above 1 as a product of primes." Not true if 1 is prime, e.g., $5 = 1 \times 5 = 1 \times 1 \times 5$, etc. Options to resolve: (1) modify/expand the statement with some exception about 1, or (2) say 1 isn't a prime number. Commented Oct 27, 2019 at 20:19
• @DanielR.Collins: you can easily recover uniqueness of prime factorization by demanding that factors are unique after simplifying (since $1^k = 1$ for all natural $k$). Commented Oct 28, 2019 at 21:23
• @nomen: That would fit into my 1st category of options, because that's only a statement you can make about the natural number $1$ (and seems like a burdensome way of expressing it, IMO). Commented Nov 1, 2019 at 4:02
• Demanding that factors are expressed in lowest terms is burdensome? Commented Nov 1, 2019 at 5:26
How should one talk about the question of 1 or 0 being prime ... with primary or middle school children?
Depending on what you did before you will have an easy or a hard task:
If the children were told:
A prime number is a natural number which cannot be divided by other numbers than by 1 and by itself.
... you will have problems explaining why 1 is not a prime number because 1 is a natural number that cannot be divided by any other number than by 1 and by itself.
However, if they were told:
A prime number is a natural number which can be divided by exactly two numbers: By 1 and by itself.
... it will be easy to explain why 1 is not a prime number: 1 is only divisible by 1, so it is not divisible by exactly two numbers, but by only one number.
This means that the key is that the children are told a more or less correct definition of the word "prime number"; otherwise you will later have problems explaining why 1 is not a prime number.
• Best answer so far. While other answers are also correct, they are so convoluted. This is the answer that my teacher gave when I was like 8 years old in class. Based on response, most if not all of the class understood why 1 is not a prime in an instant. Commented Oct 30, 2019 at 7:01
• @Namaste "A prime number is a natural number which can only be divided by two unique numbers: By 1 and by itself." Commented Oct 30, 2019 at 8:09
A good way to lead to the uniqueness of prime factorization and the convention that $$1$$ is not a prime is to build factor trees (that's common in elementary school these days in fourth grade, sometimes third grade).
24 24 24
8 3 6 4 2 12
2 4 3 2 2 2 3 4
2 2 2 2
The leaf labels always turn out the same, up to order, and $$1$$ never shows up.
I did this once in a math club. Later in the day, in the classroom, Alejandro, who's in the club, volunteered the definition "A number that only $$1$$ and itself go into, except that $$1$$ is not a prime."
The teacher asked "Why isn't $$1$$ a prime?"
"Because Dr. Bolker says so."
So appeal to authority often wins over thought.
• In fact, this is one of the things one of my kids brought up while we talked about it. I suppose as long as they don't use $-2\cdot -2=4$ then we are okay, because $0$ won't show up in these - except with $0$ itself! But then the question is whether zero should be composite. Commented Oct 28, 2019 at 3:12
• @kcrisman Yes, it is composite by the definition that 0 is the product of 1 and itself - but it's also the product of any other number and itself as well, while primes can only be factored in one way (discounting negatives). Commented Oct 28, 2019 at 17:37
• I think the point is not so much that people reading this don't know that $0$ is the product of various other numbers with itself - surely they all do! Rather, the point is to think from the point of view of a younger person who may not be convinced/interested in formal definitions, or who may wonder how many "categories" there are of numbers. See e.g. Dr. Math for one person's query about exactly this distinction. Commented Oct 28, 2019 at 19:29
This one is really very simple. First, tell them what a prime number is:
A prime number has exactly two different factors.
(If they don't know what factors are, and they ask about primes, the correct answer is "well, first you have to know about factors...")
With that definition, it is very easy to figure out 0 and 1. Is 1 a prime? No, because it only has one factor. Is 0 prime? No, because every number is a factor of 0.
Of course, the next question is likely to be "why is that the definition?" or "what's their purpose?" or some such. The answer to that is also simple - every number bigger than one is made up of prime factors. And, for every number, there is exactly one combination of primes that makes that number.
If you build each number n using n square blocks in rectangular configurations, there are multiple configurations for each composite number. (4 is 4 by 1 or 2 by 2.) The primes are the ones that can only be built as a 1 by n rectangle.
It seems clear that 0 would be neither prime nor composite, when looked at this way.
The easiest way to understand why we don't call 1 prime is that when we factor a number like 12 down to primes, we like having just one answer. (Which is often true in mathematics.)
I was asked "Why not teach that the factorization is 1𝑛×23×32?"
The prime factorization describes how to break the number down into factors, emphasis on 'down'. 1 doesn't break it down into smaller factors, so it's not useful.
The preceding question, "Why is having one answer a good thing?", is harder to answer. My answer to it, for now, is a bit of an exploration of thoughts.
I think it feels natural to want one answer, but I'd have to have more experience with young kids to know whether it seems natural to them. Perhaps for me it comes from thinking about functions, and wanting just one thing to come out, when you put something in.
I know that square root can give even a good math student trouble, because it normally has one answer, but when we square root both sides of an equation, we put that plus or minus in front to give two answers. My student assistant had trouble with that when he was tutoring. He wanted to put the plus or minus in front when he was checking an answer.
I don't know if that helps answer this question, but I hope that it shows that things that can have more than one answer get confusing for students. (In fact, one thing many people like about math is that there is one right answer.)
• If "the primes are the ones that can only be built as a 1 by n rectangle" then 1 is prime, right? Commented Oct 27, 2019 at 18:43
• I wish there were a way to think of this one for 0 and 1, as it is probably the best way to approach it. I think that is the trouble; how to find an argument that seems "reasonable" to concrete young thinkers and isn't too abstract? Commented Oct 28, 2019 at 3:14
I'm kind of restating what other answers have said, but I wanted to practice expressing it in the clearest, most concise way I could think of. (Coincidentally, this came up with my partner tonight, so I got a test-run with it, and got an entirely satisfying result).
Consider only natural numbers (i.e, positive integers). It seems like the number of divisors for different numbers is interesting and important. The following terms indicate how many distinct divisors a number has:
• Unit: A number that has exactly 1 divisor. Only the number 1 satisfies this criterion.
• Prime: A number that has exactly 2 divisors. Numbers such as 2, 3, 5, 7, etc. are in this category.
• Composite: A number that has 3 or more divisors. Numbers such as 4, 6, 8, 9, etc. are in this set.
In short, we don't call $$1$$ "prime" because it has a unique number of distinct divisors; just a single one.
• Clear and concise, indeed. I'm surprised no one had used the term "unit" yet! I was sort of expecting it. Commented Nov 1, 2019 at 11:44
We don't need the full FTA upfront if we limit our discussion for the moment to obvious examples of what's necessary for a factorization to be unique. The FTA provides analogous sufficiency conditions they'll probably guess on their own, even if they don't know how it's proven. You can say,
$$1$$ isn't considered prime because then there wouldn't be unique prime factorizations of anything, but because $$1$$ isn't prime, it's not composite either because it has no prime factors. $$0$$ isn't prime because every integer is a factor of it, but because $$0$$ isn't prime, it's not composite either because you can't write it as a product of prime numbers.
The worst reaction a child can have to that is, "Oh, so you're saying if only we do start prime numbers at $$2$$, everything from that point will either be divisible only by itself and $$1$$, or will have a unique factorization in terms of such things?" And you can say, "yes, that can be proven, but it's a bit heavy for now; and in fact you don't need to consider those two different cases, if you'll say the prime factorization of a prime number is just that prime number". The best reaction they can have is to figure out the reply on their own.
Now, if a child does want to know how any of this is proven, you can probably make that fairly accessible by exploiting their intuition, rather than yammering on about strong induction, but that's another issue.
I would start by showing them, on paper, what they already know - that in the context of multiplication the number 1 is useless. It is the identity function. It simply reflects the original number. It is a mirror. "And just like your reflection in a mirror is not a real person, neither is 1 a real number when multiplying (not to be confused with a Real number.) Since primes are only in the context of multiplication 1 isn't relevant and should not be considered in the family of primes." I would further add that if they pursue very advanced mathematics in college there are other, more formal reasons to not let it be in the family of prime numbers.
Similarly, zero changes every number to itself. "Just like a black hole absorbs matter and energy, zero, when multiplying, destroys every other number. You have lost any information in the equation when you multiply by zero. Prime numbers are useful in solving real world problems like making your text messages unreadable to everyone except your friend. Using zero as a prime would destroy the data and not allow your friend to read your text. It doesn't work as a prime number."
(If your context does not include negative numbers, turn all the negatives below positive. This almost won't change the discussion.)
Everything divides zero, so zero can't be prime. $$0 \cdot 7 = 0$$ means $$0$$ and $$7$$ divide $$0$$. $$0 \cdot -8 = 0$$ means minus eight also divides zero. Can we see that everything divides zero, so zero is very far from being prime.
Primes are numbers that are divisible by exactly two different positive numbers. (Note that this also holds true for negative integers that are prime.) Every number is divisible by one, so that must be one of the positive divisors of a prime. Every number is divisible by its magnitude ("itself" if only talking about positives), so that must be the other positive divisor of a prime. Non-primes must have more positive divisors. If we take all the positive numbers bigger than one, take them in pairs and multiply them together, we get all the non-primes.
\begin{align*} 2 \cdot 2 &= 4, 2 \cdot 3 = 6, 2 \cdot 4 = 8, \dots \\ 3 \cdot 2 &= 6, 3 \cdot 3 = 9, 3 \cdot 4 = 12, \dots \end{align*} (This could be a good time to remind/discuss multiples of a number and to remind/discuss commutativity of multiplication to reduce redundant calculations.) \begin{align*} 4 \cdot 4 &= 16, 4 \cdot 5 = 20, 4 \cdot 6 = 24, \dots \\ 5 \cdot 5 &= 25, 5 \cdot 6 = 30, 5 \cdot 7 = 35, \dots \end{align*} Here might be a good time to point out that the smallest number we get in each of these lists is the square of the number used in every product on that row. And the products get larger as we go to the right. So is it possible that there are any composites less than $$25$$ we have missed?
Let's list our composites up to $$10$$: \begin{align*} 4 = 2 \cdot 2 &\text{, so 2 also divides 4.} \\ 6 = 2 \cdot 3 &\text{, so 2 also divides 6.} \\ 8 = 2 \cdot 4 &\text{, so 2 and 4 also divide 8.} \\ 9 = 3 \cdot 3 &\text{, so 3 also divides 9.} \end{align*} This means the ones we did not produce in the table above, $$2$$, $$3$$, $$5$$, and $$7$$ must be prime -- they are only divisible by $$1$$ and themselves.
We can test this by checking each one for divisibility by smaller numbers. For two, there is nothing to check since there are no smaller positive numbers between one and two, so two is prime. For three, we see that two does not divide three, so three is prime. For five, we check two, three, and four, and discover five is prime. (This is a good time to notice that if four divides five, then two divides five, so we really only need to test for divisibility by primes.) We easily check that seven has no divisors among two, three, four, five, and six. (This could be a good time to discuss that we only need to test divisors whose square is smaller than seven, otherwise the cofactor is smaller and we have already checked the smaller potential divisors.)
When I was at school, after being taught about integer division, I was told:
"A natural number is prime if it has exactly 4 integer divisors"
Then 2 is prime, as it can be divided by -2, -1, 1, and 2.
Then one can be divided by -1 and 1, and those are only 2 divisors. Hence not prime.
Zero can be divided by anything but itself, yielding zero, hence not exactly 4 options.
• This is an interesting alternative to "A natural number is prime if it has exactly two (distinct) positive divisors." Commented Mar 9, 2022 at 14:30
\begin{align} & \begin{array}{cccccccccccccccccccc} & & & & & & & & & 840 \\[12pt] = {} & & & & & 28 & & & & \times & & & & & 30 \\[12pt] = {} & & & & 4 & \times & 7 & & & \times & & & & 5 & \times & 6 \\[12pt] = {} & & & 2 & \times & 2 & \times & 7 & & \times & & & 5 & \times & 2 & \times & 3 \end{array} \\[10pt] & = 1\times2\times1\times1\times2\times7\times1\times5\times2\times1\times1\times3\times 1\times 1 \\[10pt] & = 1\times1\times1\times2\times \cdots \end{align} Once you start taking out $$1\text{s,}$$ you can keep doing that without adding any more information about the factorization of the number you started with. So the number $$1$$ plays a different role from the role of a number that you factor and the role of a number that you end up with when you're done factoring.
• 84 should be 840. Commented Oct 29, 2019 at 8:09
The problem here is defining primes in their own right rather than defining them in terms of factorization.
Start with a number like 30. Writing 30 = 2 x 15 tells us something new; writing 30 = 1 × 30 doesn't. Writing 30 = 2 x 3 x 5 tells us something new again, whereas writing 30 = -2 x -15 doesn't.
Once we get to 30 = 2 x 3 x 5, we can't break down any of the components any further in a way that tells us anything new.
Now define primes as those numbers we can't break down any further in a way that tells us anything new.
• Hmm, of course the point of what we define first (factoring or primes or something else) is well taken. But how does that apply to 1 and 0 in your paradigm? They certainly can't be broken down any further in this sense to tell us something new. Commented Nov 1, 2019 at 1:44
• Well, do you want a formalization or do you want something that's easy to understand? Start with a large positive number and factor it until you can't factor it any more. The terms you're left with are primes. That's the motivation for defining primes in the first place. Investigate the phenomenon, then make names for its salient features. Commented Nov 1, 2019 at 2:03
• In doing this you'll discover why zero and one don't fit into the picture, and then it's obvious why you don't want to consider them primes. Commented Nov 1, 2019 at 2:08
• No, what I mean is that if you start with zero and one and use this same argument, then the break down into (from a kid's point of view) ... 0 and 1, respectively. Which might lead a kid to say they should also be prime. Do you follow the reasoning? Indeed, a similar issue with negative numbers would lead one to say -1 is prime as well (see my reference to John Horton Conway in the OP). Commented Nov 1, 2019 at 11:42
• Yeah, you have to get a full understanding of the situation before you try to say what a prime is. And you choose the definition of prime because it's exactly the thing you want to describe this behavior. Commented Nov 1, 2019 at 12:50
Primes are called primes because all other integers above 1 are (multiplicatively) "built out of them."
You can't build anything else (multiplicatively) out of 1s. No matter how many 1s you multiply together.
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# Difference between revisions of "Area"
In mathematics, area refers to the size of the region that a two-dimensional figure occupies.
It is often possible to find the area of a region bounded by parts of circles and line segments through elementary means. One can find the area of even more complex regions via the use of calculus.
Rectangles are the most basic figures whose area we can study. It makes sense that the area of a rectangle with length $l$ and width $w$ is simply $l\cdot w$.
Once we know the area of a rectangle, we can easily find the area of a triangle by just noting that if our triangle has base $b$ and height $h$, then the rectangle with length $b$ and width $h$ has exactly twice as much area as the original triangle. Thus, the area of a triangle is
$A=\frac 12 bh.$
We can now find the area of any polygon by breaking it up into triangles.
## Notation
The letters $A$ and $K$ are frequently used to stand for area. When there are multiple regions under consideration, subscripts are often employed: $A_1, K_2,\ldots$ might be used to denote the areas of particular regions, or $A_{ABC}, K_{BCD},\ldots$. For example, $K_{ABCDEF}$ would mean the area of hexagon $ABCDEF$.
An alternative notation is to use square brackets around the name of the region to denote its area, e.g. $[ABC]$ for the area of triangle $\triangle ABC$.
## Area of Regular Polygons
The area of any regular polygon can be found as follows:
Inscribe the figure, with $n$ sides of length $s$, in a circle and draw a line from two adjacent vertices to the circumcenter. This creates a triangle that is $\frac{1}{n},$ of the total area (consider the regular octagon below as an example).
Drawing the apothem creates two right triangles, each with an angle of $\frac{180}{n}^{\circ}$ at the top vertex. If the polygon has side length $s$, the height of the triangle can be found using trigonometry to be of length $\frac s2 \cot \frac{180}{n}^{\circ}$.
The area of each triangle is $\frac12$ the base times the height, which can also be expressed as $\frac{s^2}{4} \cot\frac{180}{n}^{\circ}$ and the area of the entire polygon is $\frac{n\cdot s^2}{4} \cot\frac{180}{n}^{\circ}$.
## Area of Triangle
There are many ways to find the area of a triangle. In all of these formulae, ${K}$ will be used to indicate area.
• $K=\frac{bh}{2}$ where $b$ is a base and $h$ is the altitude of the triangle to that base.
• Heron's formula: $K=\sqrt{s(s-a)(s-b)(s-c)}$, where $a, b$ and $c$ are the lengths of the sides and $s$ is the semi-perimeter $s=\frac{a+b+c}{2}$.
• $K=rs$, where $r$ is the radius of the incircle and s is the semi-perimeter.
• $K=\frac{ab\sin{\theta}}{2}$ where $a$ and $b$ are adjacent sides of the triangle and $\theta$ is the measure of the angle between them.
• $K=\frac{abc}{4R}$, where $a,b,c$ are the lengths of the sides of the triangle and $R$ is the circumradius.
• $K=4\sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}$, where $H=\frac{(h_a^{-1}+h_b^{-1}+h_c^{-1})}{2}$ and the triangle has altitudes $h_a$, $h_b$, $h_c$.
To find the area of most quadrilaterals, you must divide the quadrilateral up into smaller triangles and find the area of each triangle. However, some quadrilaterals have special formulas to find their areas. Again, $K$ is the area.
• Kite - $K=\frac{d_1\cdot d_2}{2}$ where the $d$s represent the lengths of the diagonals of the kite.
• Parallelogram - ${K=bh}$, where $b$ is the base and $h$ is the height to that base.
• Trapezoid - $K=\frac{b_1+b_2}{2}\cdot h$, where the $b$s are the parallel sides and $h$ is the distance between those bases.
• Rhombus - a special case of a kite and parallelogram, so either formula may be used here.
• Rectangle - ${K=lw}$, where $l$ is the length of the rectangle and $w$ is the width. (This is a special case of the formula for a parallelogram where the height and a side happen to coincide.)
• Square - $K=s^2$, where $s$ is the length of a side.
• Any quadrilateral - $K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^2\left(\dfrac{B+D}{2}\right)}$, where $s$ is the semiperimeter, $a$, $b$, $c$, and $d$ are the side lengths, and $B$ and $D$ are the measures of angles $B$ and $D$, respectively.
• Cyclic quadrilateral - $K=\sqrt{(s-a)(s-b)(s-c)(s-d)}$ where $s$ is the semiperimeter and $a$, $b$, $c$, and $d$ are the side lengths. (This is a special case of the formula for the area of any quadrilateral; $\cos^2\left(\dfrac{B+D}{2}\right)=0$.)
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# Satz von Menger
Menger's theorem In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs.
Inhalt 1 Edge connectivity 2 Vertex connectivity 3 Short proof 4 Other proofs 5 Infinite graphs 6 Siehe auch 7 Verweise 8 Weiterlesen 9 External links Edge connectivity The edge-connectivity version of Menger's theorem is as follows: Let G be a finite undirected graph and x and y two distinct vertices. Then the size of the minimum edge cut for x and y (the minimum number of edges whose removal disconnects x and y) is equal to the maximum number of pairwise edge-independent paths from x to y. Extended to all pairs: a graph is k-edge-connected (it remains connected after removing fewer than k edges) if and only if every pair of vertices has k edge-disjoint paths in between. Vertex connectivity The vertex-connectivity statement of Menger's theorem is as follows: Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally vertex-disjoint paths from x to y. Extended to all pairs: a graph is k-vertex-connected (it has more than k vertices and it remains connected after removing fewer than k vertices) if and only if every pair of vertices has at least k internally vertex-disjoint paths in between.
All these statements (in both edge and vertex versions) remain true in directed graphs (when considering directed paths).
Short proof Most direct proofs consider a more general statement to allow proving it by induction. It is also convenient to use definitions that include some degenerate cases. The following proof for undirected graphs works without change for directed graphs or multi-graphs, provided we take path to mean directed path.
For sets of vertices A,B ⊂ G (not necessarily disjoint), an AB-path is a path in G with a starting vertex in A, a final vertex in B, and no internal vertices in A or B. We allow a path with a single vertex in A ∩ B and zero edges. An AB-separator of size k is a set S of k vertices (which may intersect A and B) such that G−S contains no AB-path. An AB-connector of size k is a union of k vertex-disjoint AB-paths.
Satz: The minimum size of an AB-separator is equal to the maximum size of an AB-connector.
Mit anderen Worten, if no k−1 vertices disconnect A from B, then there exist k disjoint paths from A to B. This variant implies the above vertex-connectivity statement: for x,y ∈ G in the previous section, apply the current theorem to G−{x,j} with A = N(x), B = N(j), the neighboring vertices of x,j. Then a set of vertices disconnecting x and y is the same thing as an AB-separator, and removing the end vertices in a set of independent xy-paths gives an AB-connector.
Proof of the Theorem:[1] Induction on the number of edges in G. For G with no edges, the minimum AB-separator is A ∩ B, which is itself an AB-connector consisting of single-vertex paths.
For G having an edge e, we may assume by induction that the Theorem holds for G−e. If G−e has a minimal AB-separator of size k, then there is an AB-connector of size k in G−e, and hence in G.
An illustration for the proof.
Andernfalls, let S be a AB-separator of G−e of size less than k, so that every AB-path in G contains a vertex of S or the edge e. The size of S must be k-1, since if it was less, S together with either endpoint of e would be a better AB-separator of G. In G−S there is an AB-path through e, since S alone is too small to be an AB-separator of G. Let v1 be the earlier and v2 be the later vertex of e on such a path. Then v1 is reachable from A but not from B in G−S−e, while v2 is reachable from B but not from A.
Jetzt, let S1 = S ∪ {v1}, and consider a minimum AS1-separator T in G−e. Since v2 is not reachable from A in G−S1, T is also an AS1-separator in G. Then T is also an AB-separator in G (because every AB-path intersects S1). Hence it has size at least k. Durch Induktion, G−e contains an AS1-connector C1 of size k. Because of its size, the endpoints of the paths in it must be exactly S1.
Ähnlich, letting S2 = S ∪ {v2}, a minimum S2B-separator has size k, and there is an S2B-connector C2 of size k, with paths whose starting points are exactly S2.
Außerdem, since S1 disconnects G, every path in C1 is internally disjoint from every path in C2, and we can define an AB-connector of size k in G by concatenating paths (k−1 paths through S and one path going through e=v1v2). Q.E.D.
Other proofs The directed edge version of the theorem easily implies the other versions. To infer the directed graph vertex version, it suffices to split each vertex v into two vertices v1, v2, with all ingoing edges going to v1, all outgoing edges going from v2, and an additional edge from v1 to v2. The directed versions of the theorem immediately imply undirected versions: it suffices to replace each edge of an undirected graph with a pair of directed edges (a digon).
The directed edge version in turn follows from its weighted variant, the max-flow min-cut theorem. Its proofs are often correctness proofs for max flow algorithms. It is also a special case of the still more general (stark) duality theorem for linear programs.
A formulation that for finite digraphs is equivalent to the above formulation is: Let A and B be sets of vertices in a finite digraph G. Then there exists a family P of disjoint AB-paths and an AB-separating set that consists of exactly one vertex from each path in P.
In this version the theorem follows in fairly easily from Kőnig's theorem: in a bipartite graph, the minimal size of a cover is equal to the maximal size of a matching.
This is done as follows: replace every vertex v in the original digraph D by two vertices v' , v'', and every edge uv by the edge u'v''; additionally, include the edges v'v'' for every vertex v that is neither in A nor B. This results in a bipartite graph, whose one side consists of the vertices v' , and the other of the vertices v''.
Applying Kőnig's theorem we obtain a matching M and a cover C of the same size. Im Speziellen, exactly one endpoint of each edge of M is in C. Add to C all vertices a'', for a in A, and all vertices b' , for b in B. Let P be the set of all AB-paths composed of edges uv in D such that u'v'' belongs to M. Let Q in the original graph consist of all vertices v such that both v' and v'' belong to C. It is straightforward to check that Q is an AB-separating set, that every path in the family P contains precisely one vertex from Q, and every vertex in Q lies on a path from P, as desired.[2] Infinite graphs Menger's theorem holds for infinite graphs, and in that context it applies to the minimum cut between any two elements that are either vertices or ends of the graph (Halin 1974). The following result of Ron Aharoni and Eli Berger was originally a conjecture proposed by Paul Erdős, and before being proved was known as the Erdős–Menger conjecture. It is equivalent to Menger's theorem when the graph is finite.
Let A and B be sets of vertices in a (possibly infinite) digraph G. Then there exists a family P of disjoint A-B-paths and a separating set which consists of exactly one vertex from each path in P. See also Gammoid k-vertex-connected graph k-edge-connected graph Vertex separator References ^ Göring, Frank (2000). "Short proof of Menger's theorem". Diskrete Mathematik. 219 (1-3): 295–296. doi:10.1016/S0012-365X(00)00088-1. ^ Aharoni, Ron (1983). "Menger's theorem for graphs containing no infinite paths". Europäische Zeitschrift für Kombinatorik. 4 (3): 201–4. doi:10.1016/S0195-6698(83)80012-2. Further reading Menger, Karl (1927). "Zur allgemeinen Kurventheorie". Fund. Mathematik. 10: 96–115. Aharoni, Ron; Berger, Eli (2008). "Menger's theorem for infinite graphs". Inventiones mathematicae. 176: 1. arXiv:math/0509397. Bibcode:2009InMat.176....1A. doi:10.1007/s00222-008-0157-3. Halin, R. (1974). "A note on Menger's theorem for infinite locally finite graphs". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 40: 111. doi:10.1007/BF02993589. External links A Proof of Menger's Theorem Menger's Theorems and Max-Flow-Min-Cut Network flow[permanent dead link] Max-Flow-Min-Cut[permanent dead link] Kategorien: Graph connectivityNetwork theoryTheorems in graph theory
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Intuitionistic Logic Explorer < Previous Next > Nearby theorems Mirrors > Home > ILE Home > Th. List > indi GIF version
Theorem indi 3212
Description: Distributive law for intersection over union. Exercise 10 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
indi (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Proof of Theorem indi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 andi 765 . . . 4 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
2 elin 3156 . . . . 5 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3 elin 3156 . . . . 5 (𝑥 ∈ (𝐴𝐶) ↔ (𝑥𝐴𝑥𝐶))
42, 3orbi12i 714 . . . 4 ((𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)) ↔ ((𝑥𝐴𝑥𝐵) ∨ (𝑥𝐴𝑥𝐶)))
51, 4bitr4i 185 . . 3 ((𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
6 elun 3114 . . . 4 (𝑥 ∈ (𝐵𝐶) ↔ (𝑥𝐵𝑥𝐶))
76anbi2i 445 . . 3 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑥𝐵𝑥𝐶)))
8 elun 3114 . . 3 (𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)) ↔ (𝑥 ∈ (𝐴𝐵) ∨ 𝑥 ∈ (𝐴𝐶)))
95, 7, 83bitr4i 210 . 2 ((𝑥𝐴𝑥 ∈ (𝐵𝐶)) ↔ 𝑥 ∈ ((𝐴𝐵) ∪ (𝐴𝐶)))
109ineqri 3160 1 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class Syntax hints: ∧ wa 102 ∨ wo 662 = wceq 1285 ∈ wcel 1434 ∪ cun 2972 ∩ cin 2973 This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-v 2604 df-un 2978 df-in 2980 This theorem is referenced by: indir 3214 undisj2 3303 disjssun 3308 difdifdirss 3328 disjpr2 3458 diftpsn3 3529 resundi 4647
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Prove that the roots of cyclotomic polynomial $\Phi_{p-1}(x) \equiv 0 (mod~p)$ are exactly the primitive roots mod p
$$p$$ is a prime, and $$\Phi_{p-1}(x)$$ denote the cyclotomic polynomial of order $$p-1$$. And I want to show the following:
$$g$$ is a solution of the congruence $$\Phi_{p-1}(x) \equiv 0 (mod~p)$$ if and only if $$g$$ is a primitive root (mod p)
that is:
$$\Phi_{p-1}(g) \equiv 0 (mod~p) \iff g$$ is a primitive root (mod p)
Here is some properties about cyclotomic polynomial:
$${\textstyle \prod_{d|n}^{}}\Phi_{d}(x) = x^n-1\tag{1}$$
$$\Phi_{n}(x) = {\textstyle \prod_{d|n}^{}}(x^d-1)^{\mu(n/d)}\tag{2}$$
$$\mu(x)$$ is the Möbius inversion formula
• In general, when $p\nmid n$, the roots of $\Phi_n$ in $\mathbb F_p$ are going to be precisely the elements of order $n$ in $\mathbb F_p^\times$. This shouldn't be too hard for you to prove by induction. Commented Nov 20, 2022 at 8:58
• @Wojowu Could you give me some tips beacuse I have tried the induction but I failed. Commented Nov 20, 2022 at 9:06
• Firstly show that $x^n-1$ has no double roots. After you do that, note that if a root $a$ has order $d<n$, then $d\mid n$, and so $\Phi_d(a)=0$, and as there are no double roots, this implies $\Phi_n(a)\neq 0$. Commented Nov 20, 2022 at 9:37
• I didn't write it in the cleanest possible way, but this should help you. Commented Nov 20, 2022 at 9:42
• @Wojowu Thanks you so much and I wonder the roots are whether congruence roots or the normal roots? Commented Nov 20, 2022 at 9:51
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PRE-CAL MATH TEST SOLUTIONS
A/
1)Test the following to see if they are functions. If they are
i) state one-to-one or many-to-one ii) state the domain.
If they're not, say why.
a) x = y 2 not a functionsince b) y = x2 - 3x - 28a function, many-to-onedomain is R c) y =a 1-1 functiondomain R but x ! 10 d) f(x) =not a functionsame reason as a) e) y = x is a 1 to 1 functiondomain R.
.
2) Write the equation of the line:
a) through (-2, 3) parallel to y = -5x + 7
.
b) through (-2, 3) and ( 1, 5)
slope = 2 / 3 so
.
c) through the mid-point of A (-6, 3) B (4, 7) perpendicular to 3x - 5y + 10 = 0
slope of 3x - 5y + 10 = 0 is 3 / 5, so perpendicular slope = -5 / 3
midpoint is (-1, 5)
.
d) through (-6, 3) parallel to the Y-axis
x = - 6
.
e) that bisects A (1, -3) B (7, - 7) at right angles.
midpoint is (4, - 5); slope is - 4 / 6 = - 2/3 so perpendicular slope = 3 / 2
.
.
3) State the slope, y-intercept and x-intercept of 2x + 3y =24
slope = -A / B = -2 / 3
y-intercept : set x = 0 so b = 8
x-intercept : set y = 0 so a = 12
.
.
B/ 1) f(x) = 2x2 - 60x + 25
a) h = -b / 2a = 60 / 4 = 15, k = f(h) = - 425
f(x) = 2(x - 15) 2 - 425
b) the graph is a parabola
i) it opens upwards
ii) the vertex is ( 15, - 425 )
iii) the equation of the axis of symmetry is x = 15
iv) domain is R, range is [- 425, º ] , extreme value is - 425
c) draw the graph.
.
.
2) Find the x-intercepts or ZEROS (if any) for:
a) y = x2 - 5x x(x - 5) = 0 x = 0 or x = 5 b) y = x2 - 4x - 21 (x - 7)(x + 3) = 0x = 7 or x = - 3 c) y = x2 - 4x + 5it doesn't factor, use quadratic formula b 2 - 4ac = -4 < 0, no roots, min k = 1
.
3) A farmer has 60 meters of fencing to enclose a rectagular field along a riverbank.
The side along the river needs no fence since his cows can't swim.
.
.
.
.
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The other day by chance I happened to look at Richard Kenyon’s web page, and was struck by a very beautiful animated image there. The image is of a region tiled by colored squares, which are slowly rotating. As the squares rotate, they change size in such a way that the new (skewed, resized) squares still tile the same region. I thought it might be fun to try to guess how the image was constructed, and to produce my own version of his image.
I already know a little bit about square tilings. This is a subject with a history, going back at least to the work of Tutte and his colleagues. The basic problem is just to tile a rectangular region by squares. Easy enough, you say.
Well, yes; if the rectangle has sides which are rationally related, in can be filled up by squares with commensurable side lengths pretty easily. Here a 4 by 8 rectangle is filled with 7 squares of edge length 1, 1 square of edge length 3, and 1 square of edge length 4. It’s more amusing to look for a tiling in which all the squares have different lengths. One well-known tiling, found by Tutte’s colleague Stone, is as follows:
Obviously the problem becomes more interesting and challenging if one starts in advance with the combinatorics of a square tiling, and then tries to assign edge lengths to squares in such a way that they fit together nicely. One elegant method, developed by Brooks, Smith, Stone and Tutte, assigns a directed graph to the tiling, with one vertex for each vertical edge (say) and one directed edge for each square. Here’s the graph associated to Stone’s tiling:
The condition that the sum of square lengths on either side of a vertical edge sum to the length of that edge implies that the incoming edge weights and the outgoing edge weights at each vertex sum to the same value (except for at the leftmost and rightmost vertices). On the other hand, the fact that the squares are all square implies that each edge weight (as above) is equal to the length of its projection to a horizontal line; this means that the sum of edge weights around each loop in the graph (with sign changed when the orientation disagrees with the orientation on the loop) is equal to zero. These two conditions are precisely Kirchoff’s two laws for the current flowing through an electrical network where every edge has resistance 1, and the voltage difference between left and right vertices is the width of the rectangle. There is a unique solution; it might have some weights negative, in which case we can reverse the orientation of the edge so that the weights are all positive, and determine a square tiling with slightly different combinatorics. By the way, the uniqueness of the solution has an interesting (and well-known) consequence: since Kirchoff’s laws both impose linear conditions on the edge weights, the space of solutions is a rational affine space (in units for which the width is equal to 1). Since this space of solutions consists of a single point, this point has rational coordinates; this implies in particular that the height of the rectangle is a rational multiple of the width, and so are the widths of the squares.
In more homological language, the assignment of weights to edges is a (simplicial) 1-chain. The condition that the incoming and outgoing edge weights at each vertex have equal sum says that this 1-chain is actually a (relative) 1-cycle; i.e. that it is closed. The condition that the sum around every loop is zero says that if we think of this 1-chain as a 1-cochain it is actually a 1-cocycle; i.e. it is co-closed. A (co)-chain which is both closed and co-closed is said to be harmonic, and the uniqueness of a solution corresponds to the uniqueness of a harmonic representative of a (relative co-) homology class.
Incidentally, if we form the graph with one vertex for each vertical horizontal line and one edge for each square, this will be the (planar) dual to the graph above. Edges in one graph correspond to edges in the other, and the closed condition for one set of edge weights becomes the co-closed condition for the other, and vice versa.
Now instead of considering a square tiling of a rectangle, let’s consider a square tilings of a Euclidean torus. A combinatorial tiling gives us a graph, and a harmonic 1-cycle gives us a square tiling with the desired combinatorics. Changing the 1-cycle by rescaling it just rescales the torus and all the squares by the same factor, which is not very interesting. However, there is something interesting we can do. The homology of a torus is 2-dimensional, so we can consider a 1-parameter family of homology classes whose projective classes are changing, and a 1-parameter family of harmonic 1-cycles and of square tilings.
Let’s start with the simplest possible example. We fix a graph G embedded in the torus. Since we want G to be able to carry every homology class, we need at least two edges. So let’s take as G the graph with one vertex and two edges, one of which wraps horizontally once around the torus, and one of which wraps vertically around. Any assignment of weights to the edges will be both closed and co-closed, so a 1-parameter family is given by taking weights cos(t), sin(t) for t in the unit circle. The resulting square tilings of the torus have two squares, one of side length cos(t) and one of side length sin(t). The total area of the torus is thus normalized to be 1. The pattern of tilings “rotates” with t as follows:
(click on the image to see it rotate)
OK, how about a more complicated example? Let’s let G be some complicated embedded graph on the torus (so that it can carry any homology class). For the sake of concreteness, let’s let G be the following graph:
G has 10 edges (corresponding to 10 squares in the tiling), 5 vertices and 5 complementary faces. There are 5 vertex conditions and 5 face conditions; however, this system of 10 equations is redundant, and has a 2 dimensional space of solutions.
Weights on the edges of G form a vector space, and there is an inner product on this space which is just the ordinary Euclidean inner product with co-ordinates the weights on each each edge. We want to normalize our weights to have length (i.e. square root of their inner product with themselves) equal to 1, so that the resulting torus will have area 1. All we need to do is find two orthogonal weights M and L which are closed and co-closed, orthogonal to each other (i.e. the inner product of M and L is zero) and of length 1, and then we can form the family cos(t)M + sin(t)L of weights, and the associated square tilings.
The resulting rotating family of tilings is as follows:
(click on image to see it rotate)
Something else is needed to get the “spiraling” evident in Kenyon’s picture. For our square tilings of a torus above, the result of laying down a sequence of squares that winds once around a loop in the torus is to displace the tiling by a translation of the plane; this translation is called the holonomy around the loop, and only depends on its homotopy class (actually: on its homology class). Essentially, this is the result of integrating the (dual) 1-form associated to the weight. An educated guess is that in Kenyon’s picture, the holonomy is not a translation, but rather a dilation of the plane, centered at some point. At the level of homology, one can think of the dilation factor around a loop as a representation of the fundamental group, and we need to consider (harmonic) 1-cycles with coefficients twisted by this representation.
How to translate this into the language of square tilings and weights? Instead of thinking of a weight on the graph G, let’s let G~ denote the lift of G to the universal cover of the torus; i.e. G~ is a periodic graph in the plane. A twisted weight on G with coefficients in a representation is the same thing as a weight on G~ that transforms according to the given representation. For the sake of simplicity, let’s work with the graph G with one vertex and two edges as in the first example above, so that G~ has one vertex, one horizontal edge, and one vertical edge for each pair of integers. Pick a pair of edges H,V of G~, going to the right and up incoming to the vertex (0,0) respectively and let h,v be the weights on these edges.
If we let A denote the multiplication factor for horizontal translation, and B the multiplication factor for vertical translation, the vertex equation at (0,0) is
$h(A-1)+v(B-1)=0$
The vertex equations at every other vertex are obtained from this one by scaling by power of A and B, so they are satisfied if this one is. The face equation for the face with vertices (-1,-1), (0,-1), (0,0), (-1,0) is
$h(B^{-1}-1)+v(1-A^{-1})=0$
Eliminating h from this pair of equations and dividing out by v gives
$A+A^{-1}+B+B^{-1}=4$
In order to enforce spiraling, we would like moving “horizontally” some fixed number of steps to be the same as moving “vertically” some (other) fixed number of steps; this can be imposed by setting $A^pB^q=1$ for some coprime integers p,q. With these constraints, there is a unique solution h,v in complex numbers, up to scale. The real part of any such solution gives a “spiral” tiling, and the 1-parameter family obtained by multiplying by $e^{it}$ before taking the real part gives a rotating spiral.
Let’s try an example. Taking p=2,q=1 gives $A=(-3-\sqrt{5})/2=-2.618033$ and $B=A^{-2}=0.145898$. There is a totally real solution, giving rise to the following “degenerate” spiral:
Since this solution is totally real, it can’t be “rotated”. Hmm, I wasn’t expecting that. OK, taking p=3,q=1 gives $A=-0.742934-1.52909i$ and $B=0.198893-0.0432177i$.
(click on image to see it rotate)
Success!
Getting more squares in the picture is a matter of spiraling slower, which can be achieved by taking p and q bigger. Let’s try p=7,q=1.
(click on image to see it rotate)
If you want to have a play with this yourself, the source of the .eps file that generated these figures is below. To change the amount of spiraling, change the values of A and B, subject to the constraint that $A+A^{-1}+B+B^{-1}=4$. The resulting .eps file can be transformed to a layered .pdf (eg using Preview on a Mac) then to a .gif (eg in gimp). The case q=1 is pretty easy, since then A is the root of $x^{2p} + x^{p+1} - 4x^p + x^{p-1} + 1 = 0$ with smallest (nonzero) argument, and $B=A^{-p}$. Wolframalpha will cough up the values of A and B if you coax it long enough.
(Update January 16): Just for fun, here’s the tiling with p=101, q=1 (warning: the .gif file is quite large!)
(click on image to see it rotate)
%!PS-Adobe-2.0 EPSF-2.0 %%BoundingBox: 0 0 400 400 gsave 400 400 scale 1 20 div setlinewidth 1 setlinejoin 0.5 0.5 translate /square{4 dict begin /z exch def /y exch def /x exch def gsave newpath rand 10 mod 10 div rand 10 mod 10 div rand 10 mod 10 div setrgbcolor x y moveto x z add y lineto x z add y z add lineto x y z add lineto closepath fill stroke grestore end } def /simple_edge_squares{4 dict begin /v exch def /n v length def 0 1 n 1 sub{ /i exch def 0 0 v i get square 0 v i get translate } for end} def /rcmul{2 dict begin /t exch def /z exch def [ z 0 get t mul z 1 get t mul] end} def /ccmul{2 dict begin /w exch def /z exch def [ z 0 get w 0 get mul z 1 get w 1 get mul sub z 0 get w 1 get mul z 1 get w 0 get mul add ] end} def /cconj{1 dict begin /z exch def [ z 0 get 0 z 1 get sub ] end} def /cnorm{1 dict begin % |z|^2 /z exch def z 0 get dup mul z 1 get dup mul add end} def /ccdiv{2 dict begin % w/z = w*zbar/|z|^2 /z exch def /w exch def w z cconj ccmul 1 z cnorm div rcmul end} def /ccadd{2 dict begin /w exch def /z exch def [ z 0 get w 0 get add z 1 get w 1 get add ] end} def /creal{1 dict begin /z exch def z 0 get end} def /cimag{1 dict begin /z exch def z 1 get end} def 0 5 355 { /t exch def 0 srand gsave /A [0.71469 -0.870643] def % A is root of x^14+x^8-4x^7+x^6+1=0 /B [0.432505 -0.0429583] def % B = A^-7 % check: A^3B=1 /h [t cos t sin] def /v h A [-1 0] ccadd ccmul [1 0] B -1 rcmul ccadd ccdiv def % % h*(A-1)/(1-B) /Ainv [1 0] A ccdiv def /Aser [1 0] Ainv ccadd Ainv Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul ccadd Ainv Ainv ccmul Ainv ccmul Ainv ccmul Ainv ccmul ccadd def /Acom [1 0] Ainv -1 rcmul ccadd def /htran h Acom ccdiv def /vtran v Acom ccdiv def t rotate htran creal vtran creal -1 mul translate [h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares 1 1 50{ h -1 rcmul creal v creal translate /h h A ccdiv def /v v A ccdiv def [h A ccmul creal v B ccmul creal h [-1 0] ccmul creal v [-1 0] ccmul creal] simple_edge_squares } for showpage grestore } for grestore %eof
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# Write the first six terms of the sequence given by : a1=a2=1 an=2*an-1 + 3*an-2, n>2 n n.n-1,n-2 is an index.
crmhaske | College Teacher | (Level 3) Associate Educator
Posted on
As the first two terms are known, we only need to find the next four terms as follows:
a1 = 1
a2 = 1
a3 = 2(a2) + 3(a1) = 2(1) + 3(1) = 5
a4 = 2(a3) + 3(a2) = 2(5) + 3(1) = 10 + 3 = 13
a5 = 2(a4) + 3(a3) = 2(13) + 3(5) = 26 + 15 = 41
a6 = 2(a5) + 3(a4) = 2(41) + 3(13) = 82 + 39 = 121
Therefore the first six terms are: 1, 1, 5, 13, 41, 121
hala718 | High School Teacher | (Level 1) Educator Emeritus
Posted on
a1= a2 = 1
an = 2*an-1 + 3*an-2
a1= 1
a2 = 1
a3= 2*a2 + 3*a1= 2*1 + 3*1 = 5
a4 = 2*a3 + 3*a2 = 2*5 + 3*1 = 13
a5 = 2*a4 + 3*a3 = 2*13 + 3*5 = 26+15 = 41
a6= 2*a5 + 3*a4 = 2*41 + 3*13 = 82+ 39 = 121
Then the first 6 terms are:
1, 1, 5,13, 41, 121
giorgiana1976 | College Teacher | (Level 3) Valedictorian
Posted on
Because the first and the second term are known, we'll have to write the next 4 terms.
We'll use the formula of the general term, given by enunciation:
an=2*an-1 + 3*an-2
Now, we'll put n = 3:
a3 = 2*a2 + 3*a1
a3 = 2*1 + 3*1
a3 = 2+3
a3 = 5
a4 = 2*a3 + 3*a2
a4 = 2*5 + 3*1
a4 = 10 + 3
a4 = 13
a5 = 2*a4 + 3*a3
a5 = 2*13 + 3*5
a5 = 26 + 15
a5 = 41
a6 = 2*a5 + 3*a4
a6 = 2*41 + 3*13
a6 = 82 + 39
a6 = 121
The 6 terms of the sequence are: 1, 1, 5, 13, 41, 121.
neela | High School Teacher | (Level 3) Valedictorian
Posted on
a1 = a2 =1
an = 2an-1 +3an-2 n > 2.
Applying the relation, for n =3,4,5,6
a3 = 2a2+3a1 =2+3 = 5,
a4 =2a3+3a2 = 2*5 +3*1 = 13
a5 = 2a4+3a3 = 2*13+3*5 =41
a6 = 2a5+3a4 = 2*41+3*13 = 121.
So 1, 1, 4, 13, 41, and 121 are the first 6 terms.
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# IBPS RRB PO/Clerk Mains Quantitative Aptitude Quiz: 22nd September 2019
IBPS RRB PO/Clerk Quantitative Quiz
Do you follow a proper strategy or Study Plan for IBPS RRB Mains 2019? Are you aiming IBPS RRB 2019 this time? If yes, then this is the section which can help you to do wonders if practiced well. A good attempt with a mix of accuracy can help you fetch good marks. The reasoning is a game of wits and mind. It is all about logics that a question may have. Speed and accuracy are what that matters the most in this section. The only way to achieve an ambitious goal is by practicing only. So, attempt the quiz of Reasoning ability that inculcates the important questions from the important topics. Do not miss out to practice the Quantitative Aptitude Quiz that is being provided on Bankersadda.
Q6. Raj invested Rs. X in a scheme for 2 year which offered S.I. at the rate of 15% per annum and Adarsh invested Rs. (X + 5000) in another scheme for same period of time on Raj invested on C.I. at the rate of 20% per annum. If both got total interest of Rs. 29950, then find amount invested by Adarsh ?
(a) Rs. 45500
(b) Rs. 25000
(c) Rs. 27500
(d) Rs.37500
(e)Rs. 42500
Q7. In an election, which is between two candidates, 80% of the registered voters casted their votes, out of which 4% votes declared invalid. A candidate got 27648 votes which were 75% of the valid votes. The total number of voters enrolled in the elections was?
(a) 40000
(b) 45000
(c) 48000
(d) 49000
(e) 36000
Q8. A, B and C enter into a partnership business. A invested Rs. 24000 for whole the year, B invested Rs. 32000 first and after four months he increased its investment by Rs. 8000 and C invested Rs. 30000 for the first nine months and after that withdrawn Rs. 6000. If at the end of a year B gets total profit of Rs. 31200 then find the sum of profit of A and C?
(a) Rs. 36500
(b) Rs. 43875
(c) Rs. 44500
(d) Rs. 46500
(e) Rs. 48500
Q9. Two trains of lengths 240 m and 160 m are running in the same direction with the speed of 80 kmph and 100 kmph respectively. The time taken by them to cross each other is ?
(a) 56 sec
(b) 64 sec
(c) 72 sec
(d) 60 sec
(e) 48 sec
Q10.A bag contains 6 Red, 4 blue and 8 while ball, if three balls are drawn at random, find probability that one is Red and two are blue ?
Directions (11-15): Given below the table shows total employees working in five different companies, % of employees out of total employee, who prefer two newspapers (The Hindu and Economic times).
Q11. What is the ratio between employees preferred The Hindu from company E and D together to employees preferred economic times from company E, A and D together ?
(a)79:63
(b)77:67
(c)63:74
(d)74:63
(e) None of these
Q12. If 120 employee from an another company F prefer economic times, then what is average of employees preferred Economic times from all the six companies ?
(a) 123
(b)133
(c) 143
(d)153
(e)163
Q13. Employees who preferred The Hindu from company B and C together, what percent of more than employees who preferred economic times from company B and D together ?
Q14. What is sum of employee who do not preferred The Hindu and Economic times from company B and E together?
(a) 1106
(b) 1060
(c) 1080
(d) 1108
(e) 1118
Q15. If employee who do not preferred The Hindu and Economic times from company A, they all prefer only two newspapers i.e. The time of India and Indian express equally. Then find employee who prefer The times of India from company A are what percent of employee prefer The Hindu from company D and C together?
For 200+ most important arithmetic questions
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All the Best BA’ians for IBPS RRB PO/Clerk Main
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# algebra
How do I divide 3sqrt40 by 3sqrt5?
1. 👍 0
2. 👎 0
3. 👁 117
1. sqrt40=sqrt(5*2*4)=2sqrt5*sqrt2
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2. 👎 0
👨🏫
bobpursley
2. 3√40
-----------
3√5
√40
=-----------
√5
=√(40/5)
=√8
=√2^2*2
=2√2
1. 👍 0
2. 👎 0
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# Babylonian mathematics quadratic equations. Babylonian Pythagoras 2019-02-10
Babylonian mathematics quadratic equations Rating: 7,2/10 226 reviews
## Babylonian mathematics
The many uses of spreadsheets: There are many people that… 955 Words 4 Pages Case studies, surveys, and naturalistic observations are a few research methods used by psychologists to facilitate the understanding of behavior. A History of Mathematics 2nd rev. She is enjoying the fact the Mathnasium instructors teach the concepts in different ways. Robert Coolman, Live Science Contributor on. Even though we have a ways to go, I like how they assess and challenge him to work through problem solving techniques and strategies! Ancient Egyptian method of division Unit fractions could also be used for simple division sums. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.
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## Rectangular
The Historical Roots of Elementary Mathematics reprint ed. Firstly, the number 60 is a , having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 including those that are themselves composite , facilitating calculations with. Quadratic Equations and the n3 + n2 table One important table for Babylonian algebra was that of the values of n3 + n2 for integer values of n from 1 to 30. For example, if they needed to divide 3 loaves among 5 people, they would first divide two of the loaves into thirds and the third loaf into fifths, then they would divide the left over third from the second loaf into five pieces. The solution given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40.
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## Babylonian Pythagoras
The shorter side has length 4cm. Hundreds of these 'rectangular' problems are known. The article gives some background to how the civilisation came about and the mathematical background which they inherited. The technique of graphing as it is practiced today is based on the work of René Descartes. Felt they listened to my concerns of my childs challenges and they helped with focusing on my son's weaknesses and challenges as well as give him the confidence he needed.
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In the case when the coefficient of x is odd, we will need to use fractions. As a final comment on what these four tablets tell us of Babylonian mathematics we must be careful to realise that almost all of the mathematical achievements of the Babylonians, even if they were all recorded on clay tablets, will have been lost and even if these four may be seen as especially important among those surviving they may not represent the best of Babylonian mathematics. The Old Babylonians had no measurement of angle, which to us is such a basic part of geometry. Conversely we must be careful not to underestimate the significance of the mathematics just because it has been produced by mathematicians who thought very differently from today's mathematicians. She usually can't wait to get there and wishes she could attend everyday. There are some examples of virtuoso work, and it is not clear to us how much mathematical training the average scribe had.
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## METHOD OF BABYLONIANS
There is no algebraic solution in Euclid. The technique of completing the square that we have gone through in this chapter will be used to find the axis of symmetry of the parabola. Tables, such as multiplication tables have a very simple structure. Rectangular Old Babylonian 'Quadratic' Problems Old Babylonian mathematicians were much taken with problems involving two unknowns and square roots, what we would term 'quadratic' problems. Setting aside claims that the pyramids are first known structures to observe the golden ratio of 1 : 1.
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## Babylonian mathematics
A problem on a tablet from Babylonian times states that the area of a rectangle is 1, 0 and its length exceeds its width by 7. The first column is harder to understand, particularly since damage to the tablet means that part of it is missing. The Indians Throughout antiquity various rules were given for special cases and types of quadratics. In respect of content there is scarcely any difference between the two groups of texts. The errors are readily seen to be genuine errors, however, for example 8,1 has been copied by the scribe as 9,1. . Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths 29 or 30 days; 12 or 13 months respectively.
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## An Overview of Babylonian Mathematics
We preserve the modern notation x and y as each step for clarity but we do the calculations in sexagesimal notation as of course does the tablet. Why did older civilizations need to solve equations of this form in the first place? Ancient Babylonian origins To offer some insight into where the quadratic formula comes from and why it works, let us examine a procedure used on an ancient Babylonian clay tablet from around 1800 B. The problem for scholars is rather as if you were to try to reconstruct modern mathematics armed only with the exercises from the text-book and a few worked examples. Rutten, Textes mathématiques de Suse, Mémoires de la Mission archéologique en Iran vol. These values are called the solutions of the equation. The triples are too many and too large to have been obtained by brute force. See also Beckmann, Petr 1971 , , New York: St.
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## Rectangular
The essential idea for solving a linear equation is to isolate the unknown. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. We review some special second order ordinary differential equations. The Egyptians gave the solution as a sequence of unexplained steps which basically use ideas of proportion. There are many quadratics that have irrational solutions, or in some cases no real solutions at all.
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## The History Behind The Quadratic Formula
We seek to find the value s of which make the statement true, or to show that there are no such values. Suppose the quadratic equation you are looking at is with and both positive numbers. The Sumerians developed the earliest known writing system - a pictographic writing system known as cuneiform script, using wedge-shaped characters inscribed on baked clay tablets - and this has meant that we actually have more knowledge of ancient Sumerian and Babylonian mathematics than of early. This is certainly possible and the Babylonians' understanding of quadratics adds some weight to the claim. At first, the archaeological evidence was such that Old Babylonian mathematics seemed to appear fully-formed out of nowhere, flourish briefly and then disappear again for a thousand years. The Babylonians did not have an algorithm for.
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## Rectangular
A year later he continues to have success in math. A plane intersecting a cone makes a parabola. Its size is not known. In this case, the positive value is of greater physical significance, because a rectangle shouldn't have negative width. It's hard to remember reciprocals, so they made tables for them. Changes in the height of a parabola are proportional to changes in the square of that parabola's width. The quadratic formula Here's how students are instructed to solve this equation today.
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# Homeschool Physics Experiment You Can Do With Your Kids
The reason why things bounce, fly, zoom, and splat are described by the Laws of Physical Motion most kids learn in their high school physics class. But you don’t have to wait until your kid hits puberty to have fun with physics – you can start right now. Kids across the globe use the law of gravitation everyday to put the zing in their games, from basketball games to skateboarding. Let’s find out how they do it.
Let’s take a look at the first law of motion. When you place a ball on the floor, it stays put. A science textbook will tell you this: An object at rest tends to stay at rest unless acted upon by an external force. Your foot is the external force. Kick it!
What about when the ball whacks into something? Checking back in with the science textbook: An object in motion tends to stay in motion unless acted upon by an external force. After you kicked the ball (external force), it flies through the air until it smacks into something.
But there are two other forces acting on the ball that you can’t see. One force is air resistance. The ball is hitting the air molecules when it flies through the air, which slows it down. The other force is gravitational. Gravity is inherent in anything that has mass (including you!), but you need something the size of a planet before you can begin to see the effects it has on other objects. If you tossed your ball in space (away from any nearby gravitational pulls like black holes or galaxies), it would continue in a straight line forever. There aren’t any molecules for it to collide with, and no gravitational effects to pull it off-course.
There is one more idea that you need to understand – acceleration. A ball at rest has a position you can chart on a map (latitude, longitude, and altitude), but no velocity or acceleration. It’s not moving. When you decide to stir things up and kick the ball, that’s when it gets interesting. The second your toe touches the ball, things start to change. Velocity is the change in position. If you kick the ball ten feet, and it takes five seconds to go the distance, the average speed of the ball is 2 feet per second (about 1.4 MPH).
The trickier part of this scenario has to do with acceleration, which is the change of velocity. When you drive on the freeway at a constant 65 MPH, your acceleration is zero. Your speed does not change, so you have no acceleration. Your position is constantly changing, but you have constant speed. When you get on the freeway, your speed changes from zero to 65 MPH in ten seconds. Your acceleration is greatest when your foot first hits the gas – when your speed changes the most.
There’s an interesting effect that happens when you travel in a curve. You can feel the effect of a different type of acceleration when you suddenly turn your car to the right – you will feel a push to the left. If you are going fast enough and you take the turn hard enough, you can get slammed against the door. So – who pushed you?
Think back to the first law of motion. An object in motion tends to stay in motion unless acted upon by an external force. This is the amazing part – the car is the external force. Your body was the object in motion, wanting to stay in motion in a straight line. The car turns, and your body still tries to maintain its straight path, but the car itself gets in the way. When you slam into the car door, the car is turning itself into your path, forcing you to change direction.
This effect is true when you travel in a car or in a roller coaster. It’s the reason the water stays in the bucket when you swing it over your head. Physical motion is everywhere, challenging toddlers learning to walk as well as Olympic downhill skiers to go the distance.
Let’s try these ideas out.
Bucket Splash Fill a bucket half-full with water. Grasp the handle and swing it over your head in a circle in the vertical direction. Try spinning around while holding the handle out in front of your chest to swing it in the horizontal plane. Vary your spin speed to find the minimum!
Marble Vortex Curl a sheet of paper into a cone, leaving a small hole open at the bottom. Place a marble in the cone and find the speed you need to circle the cone in order to keep the marble in the cone. NOTE: This is an excellent demonstration of satellites. The satellite is the marble and the cone apex is the earth. If the marble moves too fast, it will fly out of the cone (which is equivalent to the satellite flying out of orbit and into space). If the marble speed is too slow, it will fall into the bottom of the cone (translation: satellite crashes into earth). There is a very specific speed the satellite must maintain to remain in orbit.
Ping Pong Curves Attach a clear, plastic cup to the end of a long dowel so that the bottom of the cup rests along the length of the dowel, near the end (when the dowel is lying flat on the ground, the cup points up). Insert a ping pong ball in the cup and grab the free end of the stick with your hand. Swing it partway through a circle and suddenly STOP. The ball should pop out of the cup in a line tangent to your circle at the point you stopped. Why does the ball not continue in a circle or stay in the cup?
Answer: An object in motion (the ball) wants to stay in motion (a straight line) and is free to do so when you stopped. Initially, it goes in a straight line tangent to your arc, but then gravity takes over and down it goes to the floor.
Cork Accelerometer Fill an empty soda bottle to the top with water. Modify the soda bottle cap as follows: attach a string 8-10″ long to a clean wine cork. Hot glue the free end of the string to the inside of the cap. Place the cork and string inside the bottle and screw on the top (try to eliminate the air bubbles). The cork should be free to bob around when you hold the bottle upside-down.
To use the accelerometer: invert the bottle and try to make the cork move about. Remember – it is measuring acceleration, which is the change in speed. It will only move when your speed changes.
Roller Coaster Physics This is the best way to learn about physics. All you need is a handful of marbles, several pieces of ¾” foam pipe insulation, a few rolls of masking tape, and a crowd of participants.
To make the roller coasters, you’ll need foam pipe insulation, which is sold by the six-foot increments at the hardware store. You’ll be slicing them in half lengthwise, so each piece makes twelve feet of track. It comes in all sizes, so bring your marbles when you select the size. The ¾” size fits most marbles, but if you’re using ball bearings or shooter marbles, try it out at the store. (At the very least you’ll get smiles and interest from the hardware store sales people.) lit most of the track lengthwise (the hard way) with scissors. You’ll find it is already sliced on one side, so this makes your task easier. Leave a few pieces uncut to become “tunnels” for later roller coasters.
The next step is to join your track together before adding all the features like loops and curves. Join two tracks together in butt-joint fashion and press a piece of masking tape lengthwise along both the inside and the underside of the track. A third piece of tape should go around the entire joint circumferentially. Make this connection as smooth as possible, as your high-speed marble roller coaster will tend to fly off the track at the slightest bump.
Roller Coaster Maneuvers Loops Swing the track around in a complete circle and attach the outside of the track to chairs, table legs, and hard floors with tape to secure in place.
Loops take a bit of speed to make it through, so have your partner hold it while you test it out before taping. Start with smaller loops and increase in size to match your entrance velocity into the loop. Loops can be used to slow a marble down if speed is a problem.
Camel-Backs Make a hill out of track in an upside-down U-shape. Good for show, especially if you get the hill height just right so the marble comes off the track slightly, then back on without missing a beat.
Whirly-Birds Take a loop and make it horizontal. Great around poles and posts, but just keep the bank angle steep enough and the marble speed fast enough so it doesn’t fly off track.
Corkscrew Start with a basic loop, then spread apart the entrance and exit points. The further apart they get, the more fun it becomes. Corkscrews usually require more speed than loops of the same size.
Jump Track A major show-off feature that requires very rigid entrance and exit points on the track. Use a lot of tape and incline the entrance (end of the track) slightly while declining the exit (beginning of new track piece).
Pretzel The cream of the crop in maneuvers. Make a very loose knot that resembles a pretzel. Bank angles and speed are the most critical, with rigid track positioning a close second. If you’re having trouble, make the pretzel smaller and try again. You can bank the track at any angle because the foam is so soft. Use lots of tape and a firm surface (bookcases, chairs, etc).
Troubleshooting Marbles will fly everywhere, so make sure you have a lot of extras! If your marble is not following your track, look very carefully for the point of departure – where it flies off.
oDoes the track change position with the weight of the marble, making it fly off course? Make the track more rigid by taping it to a surface.
oIs the marble jumping over the track wall? Increase your bank angle (the amount of twist the track makes along its length).
oDoes your marble just fall out of the loop? Increase your marble speed by starting at a higher position.
oWhen all else fails and your marble still won’t stay on the track, make it a tunnel section by taping another piece on top the main track. Spiral-wrap the tape along the length of both pieces to secure them together.
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# FIXED: Excel Formulas Explained
Excel is an amazing tool, but let's be honest, sometimes it can feel like deciphering hieroglyphics. And even if you're skilled with building formulas, you've no doubt encountered those frustrating error messages that seem to pop up for no reason. After years of using Excel, I've found myself in exasperating situations many times, but I decided to invest some time learning how to fix those errors and improve my skills. If you're someone who wants to learn how to use Excel without the headaches, keep reading, as I explain common formula errors and how to fix them.
## The Key Essentials of Excel Formulas
Before diving into common errors, it's essential to master the basics. Excel understands a formula as any cell starting with the equal sign (=). Starting off with a cell reference (B2), Excel will apply the formula to all the cells beneath it. As you create more complex formulas, keep in mind the order in which Excel calculates the values. Below is the order in which Excel performs calculations:
1. Mathematical operators (such as * and /)
2. Calculation inside parentheses
3. Exponents (using the ^ symbol)
4. Multiplication and division
Now that we've covered the essentials, let's move on to tackling common formula errors.
## #REF! Error
This error occurs when a cell reference is invalid, typically as a result of deleting or pasting cells. The formula you're trying to access is no longer valid, and Excel tells you with the #REF! Error message. Luckily, fixing this error is straightforward- just update the cell reference to match your new cell or range.
## #DIV/0! Error
You've probably encountered this error even without using Excel- division by zero is impossible. In Excel, it manifests in the form of #DIV/0 and shows when a formula tries to divide a number by zero. I always try to avoid dividing by direct cell references, and instead, use a named range that gets updated automatically. But in case you can't avoid it, the easy fix is to add an IF statement that checks the cell you're dividing, and if it's zero, it returns a value of your choice, typically 0 or "N/A."
## #NAME? Error
This error usually shows up when the formula references a cell or range that doesn't exist. You might have deleted cells, or Excel didn't recognize the reference. The fix for this error is typically to check for typos or ensuring the reference is correct.
## #VALUE! Error
This error arises typically when a formula refers to text when it expects a number, or when it has too many arguments. Excel tries to perform a calculation that doesn't make sense, and hence the #VALUE! Error message. To fix this error, check the inputs and make sure they are valid. Formatting a cell as text can also cause this error, so ensure that you set the formatting as number, and the data type is correct.
## #NUM! Error
This error usually happens when you have invalid numerical values in a formula. For example, suppose a formula includes a function that doesn't work with negative numbers, and the function's input is negative, then Excel displays #NUM! Error. The easy fix for this error is to ensure that the inputs to a formula are valid. If you're using a formula to return a date or a time, always ensure the inputs are in the right format.
## The Bottom Line
As frustrating as Excel errors can be, learning to fix them will boost your productivity and help you focus on more important things. By following these simple steps, you can avoid errors and unlock Excel's true power. Whether you're using the program for personal or professional use, it's always possible to improve your skills. So take the time, be patient with yourself, and get ready to take on more complex worksheets- the world of Excel formulas is waiting for you!
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A three-section ladder network appears in Fig. No.1. The reason for the terminology is quite obvious for the repetitive structure. Basically two approaches are used to solve ladder networks.
The assigned notation for the current through the final branch is $I_6$:
$$I_6 = {V_4 \over R5 + R6}$$ $$= {V_4 \over 1 Ω + 2 Ω} = {V_4 \over 3 Ω}$$ or $$V_4 = (3 Ω)I_6$$ so that $$I_4 = {V_4 \over R4} = (3 Ω)I_6 6 Ω = 0.5 I_6$$ and $$I_3 = I_4 + I_6 = 0.5 I_6 + I_6 = 1.5 I_6$$ $$V_3 = I_3 R3 = (1.5 I_6)(4 Ω) = (6 Ω)I_6$$ Also, $$V_2 = V_3 + V_4 = (6 Ω)I_6 + (3 Ω)I_6 = (9 Ω)I_6$$ so that $$I_2 ={V_2 \over R2} = {(9 Ω)I_6 \over 6 Ω }= 1.5 I_6$$ and $$I_S = I_2 + I_3 = 1.5 I_6 + 1.5 I_6 = 3 I_6$$ with $$V_1 = I_1 R1 = I_S R1 = (5 Ω) I_s$$ so that $$E = V_1 + V_2 = (5 Ω)I_s + (9 Ω)I_6$$ $$= (5 Ω)(3 I_6) + (9 Ω)I_6 = (24 Ω)I_6$$ and $$I_6 = {E \over 24 Ω} = {240 V \over 24 Ω} = 10 A$$ with $$V_6 = I_6 R6 = (10 A)(2 Ω) = 20 V$$
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Elementary Algebra
To find a formula for the annual interest rate, we start from the interest formula. The interest formula is: I = P $\times$ r $\times$ t. To find the annual interest rate, or r, we divide I by P $\times$ t So, r = $\frac{I}{P\ \times\ t}$ t = 1 month = $\frac{1}{12}$ (Because one month is $\frac{1}{12}$ of a year.) P = 2725 I = 38.15 Substitute these values in the formula to obtain: r = $\frac{38.15}{2725\ \times\ \frac{1}{12}}$ r = $\frac{38.15}{227.0833}$ r = 0.168 = 16.8% The annual interest rate charged is 16.8%.
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Visitors Online: 0 | Monday 09th December 2019
CBSE Guess > Papers > Important Questions > Class X > 2014 > Mathematics > Math-index > Mathematics By Mr. Ajai Kumar Shukla CBSE CLASS X Mathematics CLASS X MATHS (H.W) Find the value of k, for which given value is a zero of the given quadratic polynomial (a) (x2+2kx-3);x = -1/2 (b)x2+4ax-k; x= -a Verify that -1, 1, 2 are zeros of a cubic polynomial x3 - 2x2 - x+2 & verify the relationship between the zeros & its coefficients. Form a quadratic polynomial whose (i) zeros are 2 & -3(ii) zeros are -4/5 & 1/3. Solve the equations 15x -6y = 30 ; 17x + 10y =118 Solve the equations ax + by = c; bx – ay = 0 A fraction becomes 9/11, if 2 is added to both the numerator & denominator. If 3 is added to both the numerator & denominator it becomes 5/6. Find the fraction. Solve(By cross multiplication) 2/u + 3/v = 13 ; 5/u – 4/v = -2 Find the values of p & q for which the following system has infinite solutions. 2x + 3y = 7 ; (p + q)x + (2p – q)y = 21. I am three times as old as my son. Five years later, I shall be two and a half times as old as my son. How old I am and how old is my son? A and B are friends and their ages differ by two years. A’s father D is twice as old as A, & B is twice as old as his sister C. The ages of D and C differ by 40 years. Find the ages of A and B? Five years hence father’s age will be three times age of his son. Five years ago father was seven times as old as his son. Find their present ages. Five years ago, Neeta was thrice as old as Gita. Ten years later, Neeta will be twice as old Gita. How old are Gita & Neeta now? If two zeroes of the polynomials are x4 - 6x3 - 26x2 + 138x - 35 are 2±√3, find the other zeroes. On dividing x3 - 3x2 + x + 2 by polynomials g(x), the quotient & remainder were x - 2 & - 2x+4 respectively. Find g(x). If the polynomials x4 + 2x3 + 8x2 + 12x + 18 is divided by another polynomial x2+ 5, the remainder comes out to be p x +q. Find the values of p and q . Prove that √2 is not a rational number. Find the values of m and n for which the following system of equations has infinitely many solutions: 3x+4y = 12; (m + n)x +2(m-n)y=5m-1 Solve for x & y: x/a +y/b =1 ; a(x-a) – b( a + b)=2a2+b2 Solve for x & y: b x/a +ay/b = a2+b2; x+ y =2ab. Solve for x & y: x/a – y/b = a-b; ax +by = a3 + b3 Solve for x & y: 3(2x + y ) = 7xy ; 3(x +3y) = 11xy. Solve for x & y: 3/(x + y) + 2/(x – y) =2 ; 9/(x +y) + 4/(x- y) = 1;(x + y) ≠0 (x - y) ≠0 A two digit number is obtained by either multiplying the sum of the digits by 8 & adding 1, or by multiplying the difference of the digits by 13 & adding 2. Find the number. How many such numbers are there? The difference between two numbers is 15 &the difference between their squares is 465. Find the numbers. In a rectangle if length is increased by 7 units & breadth is decreased by 3 units or if length is decreased by 7 units & breadth is increased by 5 units, in both the cases the area remains same. Find the dimensions of the rectangle. Also find the area of the rectangle. A fraction is such that if the numerator is multiplied by 3 & denominator is reduced by 3, we get 18/11, but if the numerator is increased by 8 & denominator is doubled, we get 2/5. Find the fraction. Solve(By Cross-Multiplication) (a – b) x + (a+ b) y = a2 - 2ab - b2 ; (a + b)( x + y) = a2 + b2 CBSE Important Questions Class X Mathematics By : Mr. Ajai Kumar Shukla Submitted By : Mr. Ajai Kumar Shukla, RACHNA TUTORIALS, 6/1003 JANKIPURAM VISTAR LUCKNOW Mobile : +91 - 9415467421 Email Id : [email protected]
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https://avalanchespaces.com/qa/quick-answer-how-many-calories-do-i-burn-per-hour-at-rest.html
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# Quick Answer: How Many Calories Do I Burn Per Hour At Rest?
## At what age do women’s metabolism slow down?
As we age, our metabolism slows and the rate at which we break down food decreases by 10 percent each decade after age 20.
Metabolism is the amount of energy (calories) your body uses to maintain itself..
## Do resting calories count towards weight loss?
Because resting energy expenditure accounts for 60% to 75% of the calories you burn each day, any increase in resting energy expenditure is extremely important to your weight-loss effort.
## How many calories are burned walking 30 mins?
Walking moderately for 15 minutes per day burned 36 calories per day, where as moderate walking for 30 minutes burned only 85 calories per day. Walking briskly resulted in an increase in the calories burned.
## How do you calculate calories burned per hour?
Here’s your equation: MET value multiplied by weight in kilograms tells you calories burned per hour (MET*weight in kg=calories/hour). If you only want to know how many calories you burned in a half hour, divide that number by two. If you want to know about 15 minutes, divide that number by four.
## Will my metabolism ever speed up?
While you can’t really speed up metabolism, you can, unfortunately, slow it.
## How many calories do you burn in a day without exercise?
‘ If you jiggle your leg, tap your foot, or twirl a pen, you’re burning a small number of calories that can add up over the course of a day or week. In fact, one study found that fidgeting or other non-exercise movement (which was more common among lean than obese individuals) could burn up to 350 calories a day.
## How many calories does my body burn at rest?
This means that, at rest, they’ll burn approximately 1,829.8 calories in a day (equation: 66 + (6.2 x 180) + (12.7 x 72) – (6.76 x 40) = 1,829.8). For females, use the following equation: 655.1 + (4.35 x weight) + (4.7 x height) – (4.7 x age) = BMR for females.
## What are signs of slow metabolism?
Signs of a slow metabolismYou have gas.You crave sugar.You keep gaining weight.It’s tough to lose weight.You always feel bloated.You have hypothyroidism.You easily develop cellulite.Your blood sugar is too high.More items…•Jan 22, 2021
## What calorie deficit Do I need to lose 2 pounds a week?
Generally to lose 1 to 2 pounds a week, you need to burn 500 to 1,000 calories more than you consume each day, through a lower calorie diet and regular physical activity.
## How many calories should I eat a day to lose weight?
When trying to lose weight, a general rule of thumb is to reduce your calorie intake to 500 fewer calories than your body needs to maintain your current weight. This will help you lose about 1 pound (0.45 kg) of body weight per week.
## What foods slow down your metabolism?
The latter, found in foods like butter, pork products, chicken thighs, cookies, and more, may be responsible for slowing metabolism. “The American diet used to be balanced in both omega-6 and omega-3 fatty acids,” says Zuckerbrot.
## What exercise burns the most calories in 30 minutes?
Calories burned in 30 minutes: Generally, running is the best calorie-burning exercise. But if you don’t have enough time to go on a run, you can shorten your workout into high-intensity sprints. Your body will rapidly burn calories to fuel your workout.
## How do I calculate my resting calorie burn?
Resting Metabolic Rate (RMR) equations: (RMR) kcal/day: (males) = 9.99 x weight (kg) + 6.25 x height (cm) – 4.92 x age(years) + 5; (RMR) kcal/day: (females) = 9.99 x weight(kg) + 6.25 x height (cm) – 4.92 x age (years) – 161.
## Do you subtract exercise from calories?
Subtracting exercise calories from total calories consumed gives the impression that you can eat more. Most adults do not need to eat back their exercise calories because they are doing moderate activities, like walking, biking, swimming, weight-lifting, etc.
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# How to Measure Propane Level in a Tank Without a Gauge
Written by mary lougee
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Propane tanks without a gauge can be a disappointment when planning a large backyard barbecue for family and friends. Most outdoor cooks know about how much propane it takes to cook all foods on the grill during a certain time period. If the propane tank does not have a gauge, a determination needs to be made whether to buy a new tank or hope that there is enough fuel to complete a meal. There are two easy methods to determine the propane level in a tank; one is the observation/temperature method, and the other is the weight method.
Skill level:
Moderately Easy
### Things you need
• Small pot
• 2 cups water
• Scale
• Pen and paper
## Observation/Temperature Method
1. 1
Place 2 cups of water in a small pot.
2. 2
Put the pot on a stove burner and bring it to a boil.
3. 3
Remove the pot and pour the boiling water on one side of the propane tank near the top of the tank.
4. 4
Feel the tank on the side where the boiling water ran down the tank. The area that is warm at the top will meet an area that is cooler farther down the tank side. The point where the temperature changes to much cooler is the height of the propane level in the tank. This method gives a visual idea of a tank's contents as half-full, one-quarter full, and so forth.
## Weight Method
1. 1
Place the propane tank on a bathroom scale. Record the total weight on a piece of paper with a pen.
2. 2
Subtract the empty propane tank weight from the current weight. This number is on the tank after the letters "TW" for "empty tank weight." For example, if the total current weight of the tank is 13.6 Kilogram and the tank has "TW18" on the side, subtract 18 from 30 to get 5.44 Kilogram of propane remaining in the tank. A standard 20-pound tank for a gas grill containing 5.44 Kilogram of propane is 60-percent full.
3. 3
Convert pounds into gallons by dividing the number of remaining pounds of propane by 4.23. In this example, 5.44 Kilogram divided by 1.92 Kilogram per gallon of propane equals 2.84 gallons of propane remaining in the tank.
#### Tips and warnings
• Pouring boiling water on a propane tank will warm the exterior of a tank that contains air, while the bottom portion that contains propane will not warm.
• A gallon of propane weighs about 1.92 Kilogram.
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##### Alice got a 12% raise in salary from last year, this year she is earning \$26,950, how much last year
label Algebra
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5
The cost of living last year went up 12%. Fortunately, Alice Swanson got a 12% raise in her salary from last year. This year she is earning \$26,950. How much did she make last year?
Oct 7th, 2015
Thank you for the opportunity to help you with your question!
let x be the amount of money Alice earned. after an increase by 12%, then the salary was \$26950. This gives an equation of;
x+ (12/100)x=\$26950
x+0.12x=\$26950
1.12x= \$26950
divide both sides by 1.12 to remain with x on the left side;
(1.12x)/1.12=\$26950/1.12
x= \$24,062.5 is what Alice made the previous year.
Please take a moment to rate my work. Thank you.
Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 7th, 2015
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Oct 7th, 2015
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Oct 22nd, 2017
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https://trollingpowersolution.com/how-to-calculate-battery-life/
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# How to Calculate Battery Life (Experts Guide) | Trolling Power Solution
Boating and fishing are no doubt fun and relaxing activities you can do on your weekends or days off. It is an activity where you can get away from the hustle and bustle of city life and think of just pure fun and fulfilling water action. However, since you use boating and fishing devices, motors and equipment such as a marine battery with a definite useful life per activity, it is important for you to know how long it will last on water, for you to keep safe and avoid getting stranded on water. Thus, this article will teach how to calculate battery life, so that you can prepare ahead of time before you do your boating and fishing activities.
In addition, we will also share you knowledge on how long does a marine battery last, how long does a deep cycle battery last and how to tell if a battery is dead, just to ensure every water activity you embark on is fun, enjoyable and stress-free.
## How to Calculate Battery Life
Calculating your marine or deep cycle’s battery’s life is not as difficult as you think it is. You just have to take note of your marine battery’s amperage hour rating and amperage draw. Then, divide the battery’s amperage hour with its amperage draw.
Use this as an example: If you have a marine battery with an amperage hour rating of 120, and its amperage draw is 30, its battery life or running time is approximately 4 hours. Here is the calculation:
120 amp hour/30 amp draw= 4-hour running time
However, in some cases, what you see is not exactly what you get, because your marine battery’s life is still affected by one uncontrollable factor, and that is the weather conditions while you are doing your water activities.
If you are sailing through calm water conditions, then the calculations above will serve as your reliable reference. However, if you mostly sail on strong current water conditions, then most likely you will change your boat’s speed, which will then affect your battery’s running time. Thus, calculation adjustment or approximation should be done.
#### How to Tell if Battery is Bad
The first thing you can do to know if your battery is bad is to simply have a thorough look on it. You can tell if a battery is bad if you can see leaks, discoloration, plastic cracks, case bumps or a broken terminal in your battery. Furthermore, these signs are not just indications of a bad battery, but these are also signs that the battery is actually dangerous to use, because it might cause short circuit, or worse, explosion.
The second step you can take to know if your battery is bad is to take its voltage reading. If the battery is fully charged, and its voltage reading is 0, the battery might have experienced short circuit or it may have a dead cell.
The last step to confirm if your battery is really a bad battery is to load test it by using a voltmeter. If your battery’s voltage quickly goes down to 0, then, your battery might really be a bad battery.
#### How to Tell if a Battery is Dead
Nowadays, it is easier for you to tell if your marine or deep cycle battery is dead by using voltmeters or hydrometers. Thus, all you need to do to know if your battery is dead is by performing a load test on it. If your marine battery has a capacity of 12 volts and the voltmeter says it has 9V, most probably it has a dead cell, because a healthy and useful 12V battery should be around 13V when at rest.
In addition, you can also tell if your battery is low or dead if you test it, and it shows very low voltage levels.
#### How Many Amp Hours in a Marine Battery
A battery’s available amp hours serve as a guide for you to know how long you can use the battery before it goes discharged. Usually, marine or deep cycle batteries available in the market nowadays have labels indicating how much amp hours they have. However, to give you an idea, a battery with a low amp charge of around 2-4A is usable and functional for around 8-12 hours. However, it still depends on the battery’s current condition.
You can also use this as an example to know how many amp hours your marine battery has. Let’s say your marine battery has 210 amp hours, and it is given that it has a 20-hour discharge rate. Thus, divide 210 amp hours by 20 discharge rate hours, and it will give you 10.5 amp hours of available power until the battery’s power goes down to 0.
However, it is not advisable to discharge your marine battery all the way down to 0, because it can affect the performance of the battery. Thus, it is recommended that you just discharge 50% of the battery’s capacity.
#### How Long Will a Deep Cycle Battery Last?
A deep cycle battery lasts and is fully functional for around 2-3 years when taken with great care. Its useful life can even extend if you do not expose it on high temperatures or hot weather conditions because usually, deep cycle batteries last longer when they are exposed to low or cold temperatures.
In addition, your deep cycle battery has the high tendency to last longer if you avoid discharging it all the way to 0. Thus, it is always best and highly recommended to use only a half portion of your battery’s capacity to prolong its useful life as much as possible.
Conclusion
So, how to calculate battery life?
With this essential information, you know have a deeper knowledge on how your battery can be of best use to you during your boating and fishing activity. Thus, just remember all of these, and rest assured you will have a convenient and stress-free experience in every water activity. Happy boating!
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