ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
15,160	\frac{4}{50}+\frac{4}{50}+\frac{4}{50}+\frac{4}{50}+\frac{3}{50} = \frac{19}{50}
251	1/27 + \frac{1}{51}\cdot 3\cdot 26/27 = 43/459
31,666	A \cdot p + s \cdot A = A \cdot (p + s)
-20,962	-\frac{50}{-90} = \frac59 (-10/(-10))
-23,475	\frac{1}{14}\cdot 5 = \frac{5 / 7}{2}\cdot 1
21,961	\mathbb{Z}_{28} = \left\{0, 2, 27, 4, 3, 5, 1, \dotsm\right\}
45,528	(-1) + 3 (-1) = -4 \neq 1
7,095	\left(2/3\right)^3\times 1/3 = \dfrac{8}{81}
6,318	n \cdot {n + (-1) \choose (-1) + i} = {n \choose i} \cdot i
693	e^{i \times x} = 1 + x \times i/1! + (i \times x) \times (i \times x)/2! \times \cdots
18,315	\left(42 = \omega + 2\omega + \omega3 + \omega4 + \omega5 + \omega6 \Rightarrow 42 = \omega*21\right) \Rightarrow 2 = \omega
-153	\binom{7}{3} = \dfrac{7!}{(7 + 3 \cdot (-1))! \cdot 3!}
707	\frac{\partial}{\partial x} x^n = n \times x^{n + (-1)}
3,326	\frac{y z}{z y} = \frac{z}{z} y/y
-11,553	-3 + 5 - 16 \times i = 2 - 16 \times i
22,179	0 = 1 - \cosh{d}\cdot \cos{d}\Longrightarrow \cos{d} - 1/\cosh{d} = 0
6,618	v_2 - \frac{1}{v_2} = -1/\left(v_1\right) + v_1 \implies -v_1 + v_2 = -\frac{1}{v_1} + \frac{1}{v_2}
35,669	y_0 \cdot c = y_0 \cdot c
7,540	(xy)^{-1} = x^{-1}y^{-1}
19,949	s^2 + s + 1 = (s + 1/2)^2 + \frac{3}{4}
2,604	x^2 - y^2 = (x - y) (x^1 y^0 + x^0 y^1) = (x - y) \left(x + y\right)
35,230	9! = 7!3!2!*3!
8,207	(x - y)\cdot (x^2 + x\cdot y + y \cdot y) = -y^3 + x^3
31,309	\overline{g} + \overline{d} = \overline{g + d}
-1,173	\frac{1}{1/9 \cdot \left(-8\right)} \cdot ((-5) \cdot \tfrac19) = -\frac18 \cdot 9 \cdot (-5/9)
28,697	\frac{1}{6}\times(4.5+9+13.5+18+22.5+27)=15.75
-26,647	4 + 121 \cdot c^4 - 44 \cdot c^2 = (c^2 \cdot 11 + 2 \cdot (-1))^2
-22,269	n^2 + 6n + 7(-1) = (n + (-1)) \left(n + 7\right)
32,548	(2 - 1/3)^2\cdot \frac{4}{9} + (-1 - 1/3)^2\cdot \frac{5}{9} = \frac{20}{9}
32,110	di = ci \Rightarrow c = d
-8,811	27 = 9 \cdot 3
3,609	1 - \tfrac{3}{4}\frac45 = \frac{1}{5}2
30,340	x = 20 + x + 20 \cdot (-1)
20,885	\tfrac{1}{-t + 1} = 1 + t + t \cdot t + \dotsm
4,295	\frac{\partial}{\partial z} e^{zd + \sinh(gz)} = (d + g\cosh(zg)) e^{zd + \sinh(gz)}
26,893	-2 \cdot x \cdot \left(-1/2\right) = x
17,769	\binom{5}{3}\times \binom{2}{2}\times 13\times 12 = 1560
12,878	b \cdot y \cdot H = y \cdot H \cdot b
6,655	x^T D x = (x^T D x)^T = x^T D^T x = -x^T D x
-23,114	\frac138 (-4/3) = -32/9
28,930	x \cdot \binom{\left(-1\right) + x}{\left(-1\right) + l} = l \cdot \binom{x}{l}
-3,971	\frac{1}{n^4} \times n = \dfrac{1}{n \times n \times n \times n} \times n = \dfrac{1}{n \times n \times n}
-21,662	-\frac{5}{11} = -\dfrac{5}{11}
-21,013	\frac{1}{x4 + 4(-1)}\left(-x + 1\right) = -\dfrac{1}{4}\frac{1}{\left(-1\right) + x}*(x + (-1))
8,550	(x + z)q = xq + q*z
-6,251	\dfrac{3}{12 \cdot (-1) + 2 \cdot x} = \frac{1}{2 \cdot (6 \cdot (-1) + x)} \cdot 3
33,923	(y + x) \left(y - x\right) = -x^2 + y^2
27,298	( g_1g_2, h_2h_1) = ( g_1g_2, h_1h_2)
-2,501	(1 + 3) \cdot \sqrt{5} = 4 \cdot \sqrt{5}
25,434	3/z - z/2 = \frac3z - z/2
24,988	1 + \dfrac{1}{\left(x + 1\right) \cdot \left(x + 1\right)} = \frac{1}{(1 + x)^2} \cdot (\left(1 + x\right)^2 + 1)
1,258	((x + (-1))\cdot (x + 4\cdot (-1)))^{1 / 2} = \left(\left(1 - x\right)\cdot (4 - x)\right)^{\dfrac{1}{2}} = (1 - x)^{1 / 2}\cdot (4 - x)^{\frac{1}{2}}
29,374	(n + 1)^2 = n^2 + 2\cdot n + 1
-24,771	\dfrac{7\pi}{12}=\dfrac{3\pi}{12}+\dfrac{4\pi}{12} =\dfrac{\pi}{4}+\dfrac{\pi}{3}
10,956	z/y = 1/(1/z y)
20,961	b/b*b = b
-27,266	\sum_{n=1}^\infty \frac{1}{n} = \sum_{n=1}^\infty \frac{1}{n}\cdot (3 - 2)^n
8,018	1 + x^3 \cdot 4 - x^2 \cdot 3 = -(2 \cdot x^5 - 4 \cdot x^3) + 2 \cdot x^5 - 3 \cdot x^2 + 1
3,596	1 - x^3 = (1 - x) \left(x^2 + 1 + x\right)
28,645	(1 + t)*\left(1 + \frac{1}{1 + t}\right) = 1 + t + 1 = 2 + t
27,063	(a + 1/2)^n + (b + \tfrac{1}{2})^n = \dfrac{1}{2^n} \cdot ((2 \cdot a + 1)^n + \left(2 \cdot b + 1\right)^n)
8,484	651 = \binom{5}{2} \binom{7}{3} + \binom{19}{15} - \binom{5}{1} \binom{13}{9}
-7,503	\dfrac{45}{6} = \dfrac{15}{2}
12,994	D_1 = 1 \implies \mathbb{E}[D_1] = 1
17,557	2\cdot \sin\left(x\right)\cdot \cos(x) = \sin\left(x\cdot 2\right)
20,881	\frac{d}{dx} (z^3 \cdot 4) = 12 \cdot \frac{dz}{dx} \cdot z^2
6,658	\cos\left(B2\right) = (-1) + \cos^2(B)2
2,158	\left\{c, g, b\right\} = \left\{b, c, g\right\}
13,233	\int (1 + \cos(z))^3\,\mathrm{d}z = \frac{1}{4} \cdot (\sin(3 \cdot z)/3 + 6 \cdot \dfrac12 \cdot \sin(2 \cdot z) + 15 \cdot \sin(z) + 10 \cdot z) + c = \sin(3 \cdot z)/12 + \dfrac{1}{4} \cdot 3 \cdot \sin(2 \cdot z) + \frac{\sin(z)}{4} \cdot 15 + \frac{5}{2} \cdot z + c
7,391	1 - q = -q + 1
17,286	\tan(z) = \tan(z + π)
15,248	{{n \choose 2} \choose 2} = {n \choose 4}\cdot 3 + 3{n \choose 3}
-29,331	-2\cdot i + 9 = -i\cdot 2 + 1 + 8
23,805	1 + 3 + \dotsm + 2m + (-1) = m\frac{1}{2}(2m + \left(-1\right) + 1) = m^2
-24,755	\left(2^{1/2} - 6^{1/2}\right)/4 = \cos(7\pi/12)
15,634	359 = 3\cdot 5! + (-1)
12,578	z + z^2 + z^3 + \cdots + z^{n + (-1)} + z^n = \frac{z^{n + 1} - z}{(-1) + z}
1,817	\frac{1}{81}16 = \frac{1}{3^4}2^4
-16,559	12^{1/2}\times 8 = 8\times (4\times 3)^{1/2}
19,370	(e^{i \cdot y})^{-1} = e^{-y \cdot i}
-3,325	(3(-1) + 1 + 4) \sqrt{7} = \sqrt{7}*2
3,492	x^2yz * z + x^5 = (xyz - y * y^2)(y^2 + xz) + x^5 + y^5
32,430	1 - \cos(z + y) = 2\cdot \sin^2(\left(z + y\right)/2) \leq 2\cdot (z^2 + y^2)/4
-2,914	\sqrt{3} - \sqrt{48} + \sqrt{75} = \sqrt{3} - \sqrt{16 \times 3} + \sqrt{25 \times 3}
55,135	89 - 61 = 28
-174	\frac{1}{5! \cdot (9 + 5 \cdot \left(-1\right))!} \cdot 9! = \binom{9}{5}
29,930	x \cdot x + x + 1 = (x - x_1)\cdot (x - x_2) = x^2 - (x_1 + x_2)\cdot x + x_1\cdot x_2
-18,924	7/12 = G_r/(36\cdot \pi)\cdot 36\cdot \pi = G_r
28,362	x \cdot \left(b + a\right) = x \cdot b + a \cdot x
-23,475	5/14 = \frac{5}{2} \cdot \frac17
20,508	503\times 497 = \left(500 + 3\right)\times \left(500 + 3\times (-1)\right) = 500^2 - 3^2 = 250000 + 9\times (-1) = 249991
33,533	\cos{2 \cdot z} = \cos^2{z} - \sin^2{z} = 2 \cdot \cos^2{z} + (-1) = 1 - 2 \cdot \sin^2{z}
11,791	\cos{z} = \cos(\pi*2 - z)
19,477	(-w + k) \cdot (-w + k) = (w - k)^2
-210	\dfrac{1}{(8 + 5 \cdot \left(-1\right))!} \cdot 8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4
22,295	\{\} = ( x_1 + (-1), z') \Rightarrow [x_1,z'] = \{\}
-9,517	18 = 2\times 9
44,540	\pi = \frac{\pi}{2} + \pi/2
-15,944	-\frac{32}{10} = 7\frac{4}{10} - 6/1010
-639	\left(e^{7\pii/6}\right)^5 = e^{\frac{7\pii}{6}15}

15,160

\frac{4}{50}+\frac{4}{50}+\frac{4}{50}+\frac{4}{50}+\frac{3}{50} = \frac{19}{50}

251

1/27 + \frac{1}{51}\cdot 3\cdot 26/27 = 43/459

31,666

A \cdot p + s \cdot A = A \cdot (p + s)

-20,962

-\frac{50}{-90} = \frac59 (-10/(-10))

-23,475

\frac{1}{14}\cdot 5 = \frac{5 / 7}{2}\cdot 1

21,961

\mathbb{Z}_{28} = \left\{0, 2, 27, 4, 3, 5, 1, \dotsm\right\}

45,528

(-1) + 3 (-1) = -4 \neq 1

7,095

\left(2/3\right)^3\times 1/3 = \dfrac{8}{81}

6,318

n \cdot {n + (-1) \choose (-1) + i} = {n \choose i} \cdot i

693

e^{i \times x} = 1 + x \times i/1! + (i \times x) \times (i \times x)/2! \times \cdots

18,315

\left(42 = \omega + 2\omega + \omega*3 + \omega*4 + \omega*5 + \omega*6 \Rightarrow 42 = \omega*21\right) \Rightarrow 2 = \omega

-153

\binom{7}{3} = \dfrac{7!}{(7 + 3 \cdot (-1))! \cdot 3!}

707

\frac{\partial}{\partial x} x^n = n \times x^{n + (-1)}

3,326

\frac{y z}{z y} = \frac{z}{z} y/y

-11,553

-3 + 5 - 16 \times i = 2 - 16 \times i

22,179

0 = 1 - \cosh{d}\cdot \cos{d}\Longrightarrow \cos{d} - 1/\cosh{d} = 0

6,618

v_2 - \frac{1}{v_2} = -1/\left(v_1\right) + v_1 \implies -v_1 + v_2 = -\frac{1}{v_1} + \frac{1}{v_2}

35,669

y_0 \cdot c = y_0 \cdot c

7,540

(xy)^{-1} = x^{-1}y^{-1}

19,949

s^2 + s + 1 = (s + 1/2)^2 + \frac{3}{4}

2,604

x^2 - y^2 = (x - y) (x^1 y^0 + x^0 y^1) = (x - y) \left(x + y\right)

35,230

9! = 7!*3!*2!*3!

8,207

(x - y)\cdot (x^2 + x\cdot y + y \cdot y) = -y^3 + x^3

31,309

\overline{g} + \overline{d} = \overline{g + d}

-1,173

\frac{1}{1/9 \cdot \left(-8\right)} \cdot ((-5) \cdot \tfrac19) = -\frac18 \cdot 9 \cdot (-5/9)

28,697

\frac{1}{6}\times(4.5+9+13.5+18+22.5+27)=15.75

-26,647

4 + 121 \cdot c^4 - 44 \cdot c^2 = (c^2 \cdot 11 + 2 \cdot (-1))^2

-22,269

n^2 + 6n + 7(-1) = (n + (-1)) \left(n + 7\right)

32,548

(2 - 1/3)^2\cdot \frac{4}{9} + (-1 - 1/3)^2\cdot \frac{5}{9} = \frac{20}{9}

32,110

di = ci \Rightarrow c = d

-8,811

27 = 9 \cdot 3

3,609

1 - \tfrac{3}{4}*\frac45 = \frac{1}{5}*2

30,340

x = 20 + x + 20 \cdot (-1)

20,885

\tfrac{1}{-t + 1} = 1 + t + t \cdot t + \dotsm

4,295

\frac{\partial}{\partial z} e^{zd + \sinh(gz)} = (d + g\cosh(zg)) e^{zd + \sinh(gz)}

26,893

-2 \cdot x \cdot \left(-1/2\right) = x

17,769

\binom{5}{3}\times \binom{2}{2}\times 13\times 12 = 1560

12,878

b \cdot y \cdot H = y \cdot H \cdot b

6,655

x^T D x = (x^T D x)^T = x^T D^T x = -x^T D x

-23,114

\frac138 (-4/3) = -32/9

28,930

x \cdot \binom{\left(-1\right) + x}{\left(-1\right) + l} = l \cdot \binom{x}{l}

-3,971

\frac{1}{n^4} \times n = \dfrac{1}{n \times n \times n \times n} \times n = \dfrac{1}{n \times n \times n}

-21,662

-\frac{5}{11} = -\dfrac{5}{11}

-21,013

\frac{1}{x*4 + 4*(-1)}*\left(-x + 1\right) = -\dfrac{1}{4}*\frac{1}{\left(-1\right) + x}*(x + (-1))

8,550

(x + z)*q = x*q + q*z

-6,251

\dfrac{3}{12 \cdot (-1) + 2 \cdot x} = \frac{1}{2 \cdot (6 \cdot (-1) + x)} \cdot 3

33,923

(y + x) \left(y - x\right) = -x^2 + y^2

27,298

( g_1*g_2, h_2*h_1) = ( g_1*g_2, h_1*h_2)

-2,501

(1 + 3) \cdot \sqrt{5} = 4 \cdot \sqrt{5}

25,434

3/z - z/2 = \frac3z - z/2

24,988

1 + \dfrac{1}{\left(x + 1\right) \cdot \left(x + 1\right)} = \frac{1}{(1 + x)^2} \cdot (\left(1 + x\right)^2 + 1)

1,258

((x + (-1))\cdot (x + 4\cdot (-1)))^{1 / 2} = \left(\left(1 - x\right)\cdot (4 - x)\right)^{\dfrac{1}{2}} = (1 - x)^{1 / 2}\cdot (4 - x)^{\frac{1}{2}}

29,374

(n + 1)^2 = n^2 + 2\cdot n + 1

-24,771

\dfrac{7\pi}{12}=\dfrac{3\pi}{12}+\dfrac{4\pi}{12} =\dfrac{\pi}{4}+\dfrac{\pi}{3}

10,956

z/y = 1/(1/z y)

20,961

b/b*b = b

-27,266

\sum_{n=1}^\infty \frac{1}{n} = \sum_{n=1}^\infty \frac{1}{n}\cdot (3 - 2)^n

8,018

1 + x^3 \cdot 4 - x^2 \cdot 3 = -(2 \cdot x^5 - 4 \cdot x^3) + 2 \cdot x^5 - 3 \cdot x^2 + 1

3,596

1 - x^3 = (1 - x) \left(x^2 + 1 + x\right)

28,645

(1 + t)*\left(1 + \frac{1}{1 + t}\right) = 1 + t + 1 = 2 + t

27,063

(a + 1/2)^n + (b + \tfrac{1}{2})^n = \dfrac{1}{2^n} \cdot ((2 \cdot a + 1)^n + \left(2 \cdot b + 1\right)^n)

8,484

651 = \binom{5}{2} \binom{7}{3} + \binom{19}{15} - \binom{5}{1} \binom{13}{9}

-7,503

\dfrac{45}{6} = \dfrac{15}{2}

12,994

D_1 = 1 \implies \mathbb{E}[D_1] = 1

17,557

2\cdot \sin\left(x\right)\cdot \cos(x) = \sin\left(x\cdot 2\right)

20,881

\frac{d}{dx} (z^3 \cdot 4) = 12 \cdot \frac{dz}{dx} \cdot z^2

6,658

\cos\left(B*2\right) = (-1) + \cos^2(B)*2

2,158

\left\{c, g, b\right\} = \left\{b, c, g\right\}

13,233

\int (1 + \cos(z))^3\,\mathrm{d}z = \frac{1}{4} \cdot (\sin(3 \cdot z)/3 + 6 \cdot \dfrac12 \cdot \sin(2 \cdot z) + 15 \cdot \sin(z) + 10 \cdot z) + c = \sin(3 \cdot z)/12 + \dfrac{1}{4} \cdot 3 \cdot \sin(2 \cdot z) + \frac{\sin(z)}{4} \cdot 15 + \frac{5}{2} \cdot z + c

7,391

1 - q = -q + 1

17,286

\tan(z) = \tan(z + π)

15,248

{{n \choose 2} \choose 2} = {n \choose 4}\cdot 3 + 3{n \choose 3}

-29,331

-2\cdot i + 9 = -i\cdot 2 + 1 + 8

23,805

1 + 3 + \dotsm + 2*m + (-1) = m*\frac{1}{2}*(2*m + \left(-1\right) + 1) = m^2

-24,755

\left(2^{1/2} - 6^{1/2}\right)/4 = \cos(7\pi/12)

15,634

359 = 3\cdot 5! + (-1)

12,578

z + z^2 + z^3 + \cdots + z^{n + (-1)} + z^n = \frac{z^{n + 1} - z}{(-1) + z}

1,817

\frac{1}{81}*16 = \frac{1}{3^4}*2^4

-16,559

12^{1/2}\times 8 = 8\times (4\times 3)^{1/2}

19,370

(e^{i \cdot y})^{-1} = e^{-y \cdot i}

-3,325

(3(-1) + 1 + 4) \sqrt{7} = \sqrt{7}*2

3,492

x^2*y*z * z + x^5 = (x*y*z - y * y^2)*(y^2 + x*z) + x^5 + y^5

32,430

1 - \cos(z + y) = 2\cdot \sin^2(\left(z + y\right)/2) \leq 2\cdot (z^2 + y^2)/4

-2,914

\sqrt{3} - \sqrt{48} + \sqrt{75} = \sqrt{3} - \sqrt{16 \times 3} + \sqrt{25 \times 3}

55,135

89 - 61 = 28

-174

\frac{1}{5! \cdot (9 + 5 \cdot \left(-1\right))!} \cdot 9! = \binom{9}{5}

29,930

x \cdot x + x + 1 = (x - x_1)\cdot (x - x_2) = x^2 - (x_1 + x_2)\cdot x + x_1\cdot x_2

-18,924

7/12 = G_r/(36\cdot \pi)\cdot 36\cdot \pi = G_r

28,362

x \cdot \left(b + a\right) = x \cdot b + a \cdot x

-23,475

5/14 = \frac{5}{2} \cdot \frac17

20,508

503\times 497 = \left(500 + 3\right)\times \left(500 + 3\times (-1)\right) = 500^2 - 3^2 = 250000 + 9\times (-1) = 249991

33,533

\cos{2 \cdot z} = \cos^2{z} - \sin^2{z} = 2 \cdot \cos^2{z} + (-1) = 1 - 2 \cdot \sin^2{z}

11,791

\cos{z} = \cos(\pi*2 - z)

19,477

(-w + k) \cdot (-w + k) = (w - k)^2

-210

\dfrac{1}{(8 + 5 \cdot \left(-1\right))!} \cdot 8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4

22,295

\{\} = ( x_1 + (-1), z') \Rightarrow [x_1,z'] = \{\}

-9,517

18 = 2\times 9

44,540

\pi = \frac{\pi}{2} + \pi/2

-15,944

-\frac{32}{10} = 7*\frac{4}{10} - 6/10*10

-639

\left(e^{7*\pi*i/6}\right)^5 = e^{\frac{7*\pi*i}{6}*1*5}