ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-9,266	-y237 = -42y
-4,764	\frac{7\cdot (-1) + 6\cdot x}{x^2 - 3\cdot x + 2} = \dfrac{1}{x + 2\cdot (-1)}\cdot 5 + \frac{1}{x + (-1)}
19,778	\dfrac{1}{\phi^2} = \frac{\phi^2 - \phi}{\phi \cdot \phi} = 1 - (\phi^2 - \phi)/\phi = 2 - \phi
47,973	1/100 = 92/93\cdot 94/95\cdot 97/98\cdot 99/100\cdot 98/99\cdot \frac{96}{97}\cdot 95/96\cdot 93/94\cdot \frac{1}{92}\cdot 91/91
29,820	3 = z z z \Rightarrow 3 (-1) + z^3 = 0
-3,797	9/4 \cdot y^2 = y^2 \cdot 9/4
3,467	5/7\cdot 7 = 5
26,383	35 = 50 + 3\cdot \left(-1\right) + 12\cdot (-1)
-2,660	\sqrt{11}\sqrt{16} + \sqrt{11} = \sqrt{11} + 4\sqrt{11}
5,977	z^2 - q^2 = 1 rightarrow \sqrt{q^2 + 1} = z
11,487	\sqrt{2} \cdot 11 + 9\sqrt{3} = (\sqrt{3} + \sqrt{2})^3
27,820	cx + x = (-1) - c3 \Rightarrow c = -\frac{x + 1}{x + 3}
5,287	(x^4 + x^28 + 3x^2)/12 = 11/12 x x + \dfrac{x^4}{12}
14,680	C\cdot A + A\cdot D + B\cdot C + D\cdot B = (A + B)\cdot (D + C)
13,658	111111111 \cdot \left(-1\right) + 123456789 = 999999999 + 987654321 \cdot (-1)
456	2D'^2 + D \cdot D + D' D\cdot 3 = (D + D'\cdot 2) \left(D' + D\right)
35,175	{i \choose x} = {i + (-1) \choose x + (-1)} + {i + (-1) \choose x} = {i + (-1) \choose x + (-1)} + {i + 2 \left(-1\right) \choose x + \left(-1\right)} + {i + 2 \left(-1\right) \choose x}
-6,564	\frac{j}{j^2 - j \cdot 10 + 21} = \dfrac{1}{(j + 3 \cdot (-1)) \cdot (7 \cdot (-1) + j)} \cdot j
17,527	\sin(2\times \pi/3 + \pi/3) = \sin\left(\pi\right) = 0
29,276	(\sqrt{2} + 3)^3 = 45 + 29 \cdot \sqrt{2}
20,332	\|g_m\|^{\frac1m} \leq 1/t = 1/t\Longrightarrow t^{-m} \geq \|g_m\|
39,305	1 - (\frac12)^i = 1 - \dfrac{1}{2^i} = \tfrac{1}{2^i}(2^i + (-1)) = \frac{1}{2^{m - x}}(2^{m - x} + (-1)) \Rightarrow \tfrac{-(1/2)^i + 1}{2^{\left(-1\right) + x}} = \frac{1}{2^{m + (-1)}}\left(2^{m - x} + (-1)\right)
16,961	h^N h^z = h^{N + z}
-2,308	-2/17 + \frac{3}{17} = \frac{1}{17}
-2,171	\pi35/12 = \dfrac123\pi + \frac{1}{12}17*\pi
32,878	(1 + z) \cdot (z + (-1)) = (-1) + z^2
49,755	\sin(2^n) = \sin(2\pi \dfrac{2^n}{2\pi}) = \sin(2\pi \frac{1}{\pi}2^{n + (-1)})
27,966	\dfrac{\sqrt{3}\cdot 16}{9} 1 = \dfrac{16}{3 \sqrt{3}}
1,421	1/((-1)\cdot b) = \frac{1}{(-1)\cdot b} + \frac0b = \frac{1}{(-1)\cdot b} + \frac1b\cdot (1 - 1) = \dfrac{1}{(-1)\cdot b} + 1/b - \frac{1}{b}
17,919	1/2 - 1/8 = 3/8
-2,664	\sqrt{3}\cdot \left(3 + 4 + 5\right) = \sqrt{3}\cdot 12
-4,324	\dfrac{c^4}{10 \cdot c^2} = \tfrac{1}{10} \cdot \frac{1}{c^2} \cdot c^4
-19,576	\dfrac45\cdot \frac15\cdot 2 = 1/5\cdot 2/\left(5\cdot 1/4\right)
-3,099	3 \cdot \sqrt{13} = \sqrt{13} \cdot \left(4 + (-1)\right)
12,252	2^{f_2} \cdot 2^{f_1} = 2^{f_1 + f_2}
6,747	\tfrac{1}{(n - x)!}\cdot n! = n\cdot (n + (-1))\cdot \dots\cdot (n - x + 1)
29,804	(-\frac{1}{1 + y} + \frac{1}{\left(-1\right) + y}) \frac{y^2}{2} = \dfrac{y^2}{(-1) + y^2}
-5,000	10^917.4 = 10^{6 + 3}17.4
3,810	V^3 = -V + (-1) = 4 \times V + 4
18,357	(12 \cdot 2^{1/2} + 19) \cdot (19 - 12 \cdot 2^{1/2}) = 73
4,314	\tfrac{1}{(2 \times x + (-1)) \times (2 \times x + (-1))} = \frac{1}{(2 \times (x - 1/2))^2} = \frac{1/4}{(x - 1/2)^2}
2,698	\left(\frac2x + 1 - x\right)^5 = \frac{1}{x^5}(-x^2 + 2 + x)^5
25,357	(x + Q^{1/2}y - C^{1/2}z)(zC^{1/2} + x + Q^{1/2}y) = x^2 + Qy^2 + 2Q^{1/2}yx - z z*C
-15,799	-6/105 + 4/106 = -6/10
31,194	\sin{2z} = 2\sin{z} \cos{z} + 0(-1) = 2\sin{z} \cos{z}
-17,808	41 = 56 + 15 (-1)
-4,047	\frac{28}{7} \cdot \frac{y^5}{y^2} = \frac{y^5 \cdot 28}{7 \cdot y^2}
9,086	\int_1^2 \tfrac{1}{2 \cdot w}\,dw = (\int_1^2 {1/w}\,dw)/2
31,414	72 = \binom{4}{2}\cdot 2\cdot 3!
4,162	x \cdot b = \frac{1}{2} \cdot (x^2 + b^2 - (-b + x)^2)
48,651	16*(-1) + 17 = 1
16,181	\operatorname{E}(Z) - \operatorname{E}(V) = \operatorname{E}(Z - V)
11,831	d \cdot E = d \cdot E
24,425	2 \cdot ((-1) + x) \cdot \left(1 + x\right) = 2 \cdot \left(-1\right) + 2 \cdot x^2
8,740	h_2 d_2 + d_1 h_1 = h_1 d_1 + d_2 h_2
34,595	B \cdot C^m = B \cdot C^m
11,287	\sin(\frac{5\pi}{2})=\sin(\frac{\pi}{2})=1
27,459	Ax=\lambda x \implies A^{-1}Ax=A^{-1}\lambda x
18,420	2 \cdot (-1) + y \cdot y + y = (y + 2) \cdot ((-1) + y)
23,592	\frac{5}{77} = 5/21\cdot \dfrac{1}{22}\cdot 6
21,984	-\frac12 \cdot \pi = (\left(-1\right) \cdot \pi)/2
-22,404	-4 = 6 (-1) + 2
31,502	\frac{1}{y} \cdot (y \cdot y + y) = y + 1
27,098	\cos{\frac{\pi}{2}} = \cos{3*\pi/2} = 0
32,190	\left(1 + y + \dots + y^4\right)^{1 + n} = (1 + y + \dots + y^4)^n\cdot (1 + y + \dots + y^4)
7,979	x + 2 (-1) > 0 \implies 2 < x
33,752	25 = f^2\Longrightarrow \sqrt{f \cdot f} = \sqrt{25}
-1,776	-5/3\cdot \pi = \pi/4 - \dfrac{1}{12}\cdot 23\cdot \pi
-2,010	-\frac{2}{3}\pi = \dfrac{1}{12}\pi - \frac{3}{4}*\pi
46,783	1111 = 11 \cdot 101
16,312	\dfrac{n^2}{2^{-\sqrt{n}}} = n^2*2^{\sqrt{n}}
20,011	\frac{2}{((-1) + z) (1 + z)} = \frac{1}{((-1) + z) (1 + z)} (-(\left(-1\right) + z) + z + 1)
1,186	(2\tfrac13)^2 + (2\frac{2}{3})^2 = \dfrac{20}{9} \gt 2
19,751	\frac{\pi}{2^{\frac{1}{2}}} = 2^{1 / 2} \cdot \pi/2
4,867	s^4 - 20s^2 + 19 = (s s + 19\left(-1\right))((-1) + s * s)
29,410	3 + \frac12 = \frac{7}{2}
-19	-5 - 3 = -8
30,324	(h + b)\cdot (1 + h\cdot \chi) = h^2\cdot \chi + h\cdot (1 + b\cdot \chi) + b
-3,138	\sqrt{25 \cdot 2} + \sqrt{16 \cdot 2} + \sqrt{9 \cdot 2} = \sqrt{32} + \sqrt{18} + \sqrt{50}
16,428	(z^2 - z \cdot 4 + 3) \cdot (4 + z) + 13 \cdot (z + (-1)) = (-1) + z^3
28,917	0 = -\frac35 + 3/5
17,433	x^i a_i = x^i + (a_i + (-1)) x^i
28,224	y \cdot 2 + 5 \cdot y = 7 \cdot y
17,747	\dfrac{1}{xx^V} = \frac{1}{xx^V}
24,103	{n \choose r} = \dfrac{1}{r!\cdot (n - r)!}\cdot n!
9,497	ca + af = (c + f)*a
10,525	\pi\cdot 2 - \frac{\pi}{6} = \frac{\pi}{6}\cdot 11
1,268	\left(\mu^2 + n^2\right)^2 - (\mu^2 - n^2) \cdot (\mu^2 - n^2) = (2\mu n) \cdot (2\mu n) = 2 \cdot 2\mu^2 n \cdot n
15,742	\dfrac{1}{m^{d_2 - d_1}} = \frac{m^{d_1}}{m^{d_2}}
38,655	\sqrt{h} \cdot \sqrt{b} = \sqrt{h \cdot b} = \sqrt{h \cdot b}
38,423	(z + 1)(z + 3) = (z + 1)z + (z + 3)3 = z^2 + z + 3z + 9 = z^2 + 4*z + 9
52,561	680\times 211=143480=199\times 721+1
4,951	(n \cdot 2 + 1)^2 + ((-1) + 2 \cdot n)^2 = 2 + 8 \cdot n^2
-446	e^{\pi\cdot i\cdot 4/3\cdot 16} = \left(e^{4\cdot \pi\cdot i/3}\right)^{16}
46,439	2\cdot 126 = 252
23,282	C/(E_2) + A/(E_1) = (CE_1 + E_2 A)/(E_2 E_1)
-3,414	(3 + 2 + 5) \times 11^{1 / 2} = 11^{\frac{1}{2}} \times 10
-10,389	\frac{100}{60\cdot t + 240\cdot (-1)} = \frac{5}{t\cdot 3 + 12\cdot (-1)}\cdot 20/20
-30,322	1 = 3\cdot (-1) + 4
-17,594	84 + 66 \cdot (-1) = 18

-9,266

-y*2*3*7 = -42*y

-4,764

\frac{7\cdot (-1) + 6\cdot x}{x^2 - 3\cdot x + 2} = \dfrac{1}{x + 2\cdot (-1)}\cdot 5 + \frac{1}{x + (-1)}

19,778

\dfrac{1}{\phi^2} = \frac{\phi^2 - \phi}{\phi \cdot \phi} = 1 - (\phi^2 - \phi)/\phi = 2 - \phi

47,973

1/100 = 92/93\cdot 94/95\cdot 97/98\cdot 99/100\cdot 98/99\cdot \frac{96}{97}\cdot 95/96\cdot 93/94\cdot \frac{1}{92}\cdot 91/91

29,820

3 = z z z \Rightarrow 3 (-1) + z^3 = 0

-3,797

9/4 \cdot y^2 = y^2 \cdot 9/4

3,467

5/7\cdot 7 = 5

26,383

35 = 50 + 3\cdot \left(-1\right) + 12\cdot (-1)

-2,660

\sqrt{11}*\sqrt{16} + \sqrt{11} = \sqrt{11} + 4*\sqrt{11}

5,977

z^2 - q^2 = 1 rightarrow \sqrt{q^2 + 1} = z

11,487

\sqrt{2} \cdot 11 + 9\sqrt{3} = (\sqrt{3} + \sqrt{2})^3

27,820

c*x + x = (-1) - c*3 \Rightarrow c = -\frac{x + 1}{x + 3}

5,287

(x^4 + x^2*8 + 3x^2)/12 = 11/12 x * x + \dfrac{x^4}{12}

14,680

C\cdot A + A\cdot D + B\cdot C + D\cdot B = (A + B)\cdot (D + C)

13,658

111111111 \cdot \left(-1\right) + 123456789 = 999999999 + 987654321 \cdot (-1)

456

2D'^2 + D \cdot D + D' D\cdot 3 = (D + D'\cdot 2) \left(D' + D\right)

35,175

{i \choose x} = {i + (-1) \choose x + (-1)} + {i + (-1) \choose x} = {i + (-1) \choose x + (-1)} + {i + 2 \left(-1\right) \choose x + \left(-1\right)} + {i + 2 \left(-1\right) \choose x}

-6,564

\frac{j}{j^2 - j \cdot 10 + 21} = \dfrac{1}{(j + 3 \cdot (-1)) \cdot (7 \cdot (-1) + j)} \cdot j

17,527

\sin(2\times \pi/3 + \pi/3) = \sin\left(\pi\right) = 0

29,276

(\sqrt{2} + 3)^3 = 45 + 29 \cdot \sqrt{2}

20,332

|g_m|^{\frac1m} \leq 1/t = 1/t\Longrightarrow t^{-m} \geq |g_m|

39,305

1 - (\frac12)^i = 1 - \dfrac{1}{2^i} = \tfrac{1}{2^i}(2^i + (-1)) = \frac{1}{2^{m - x}}(2^{m - x} + (-1)) \Rightarrow \tfrac{-(1/2)^i + 1}{2^{\left(-1\right) + x}} = \frac{1}{2^{m + (-1)}}\left(2^{m - x} + (-1)\right)

16,961

h^N h^z = h^{N + z}

-2,308

-2/17 + \frac{3}{17} = \frac{1}{17}

-2,171

\pi*35/12 = \dfrac12*3*\pi + \frac{1}{12}*17*\pi

32,878

(1 + z) \cdot (z + (-1)) = (-1) + z^2

49,755

\sin(2^n) = \sin(2\pi \dfrac{2^n}{2\pi}) = \sin(2\pi \frac{1}{\pi}2^{n + (-1)})

27,966

\dfrac{\sqrt{3}\cdot 16}{9} 1 = \dfrac{16}{3 \sqrt{3}}

1,421

1/((-1)\cdot b) = \frac{1}{(-1)\cdot b} + \frac0b = \frac{1}{(-1)\cdot b} + \frac1b\cdot (1 - 1) = \dfrac{1}{(-1)\cdot b} + 1/b - \frac{1}{b}

17,919

1/2 - 1/8 = 3/8

-2,664

\sqrt{3}\cdot \left(3 + 4 + 5\right) = \sqrt{3}\cdot 12

-4,324

\dfrac{c^4}{10 \cdot c^2} = \tfrac{1}{10} \cdot \frac{1}{c^2} \cdot c^4

-19,576

\dfrac45\cdot \frac15\cdot 2 = 1/5\cdot 2/\left(5\cdot 1/4\right)

-3,099

3 \cdot \sqrt{13} = \sqrt{13} \cdot \left(4 + (-1)\right)

12,252

2^{f_2} \cdot 2^{f_1} = 2^{f_1 + f_2}

6,747

\tfrac{1}{(n - x)!}\cdot n! = n\cdot (n + (-1))\cdot \dots\cdot (n - x + 1)

29,804

(-\frac{1}{1 + y} + \frac{1}{\left(-1\right) + y}) \frac{y^2}{2} = \dfrac{y^2}{(-1) + y^2}

-5,000

10^9*17.4 = 10^{6 + 3}*17.4

3,810

V^3 = -V + (-1) = 4 \times V + 4

18,357

(12 \cdot 2^{1/2} + 19) \cdot (19 - 12 \cdot 2^{1/2}) = 73

4,314

\tfrac{1}{(2 \times x + (-1)) \times (2 \times x + (-1))} = \frac{1}{(2 \times (x - 1/2))^2} = \frac{1/4}{(x - 1/2)^2}

2,698

\left(\frac2x + 1 - x\right)^5 = \frac{1}{x^5}(-x^2 + 2 + x)^5

25,357

(x + Q^{1/2}*y - C^{1/2}*z)*(z*C^{1/2} + x + Q^{1/2}*y) = x^2 + Q*y^2 + 2*Q^{1/2}*y*x - z * z*C

-15,799

-6/10*5 + 4/10*6 = -6/10

31,194

\sin{2z} = 2\sin{z} \cos{z} + 0(-1) = 2\sin{z} \cos{z}

-17,808

41 = 56 + 15 (-1)

-4,047

\frac{28}{7} \cdot \frac{y^5}{y^2} = \frac{y^5 \cdot 28}{7 \cdot y^2}

9,086

\int_1^2 \tfrac{1}{2 \cdot w}\,dw = (\int_1^2 {1/w}\,dw)/2

31,414

72 = \binom{4}{2}\cdot 2\cdot 3!

4,162

x \cdot b = \frac{1}{2} \cdot (x^2 + b^2 - (-b + x)^2)

48,651

16*(-1) + 17 = 1

16,181

\operatorname{E}(Z) - \operatorname{E}(V) = \operatorname{E}(Z - V)

11,831

d \cdot E = d \cdot E

24,425

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

8,740

h_2 d_2 + d_1 h_1 = h_1 d_1 + d_2 h_2

34,595

B \cdot C^m = B \cdot C^m

11,287

\sin(\frac{5\pi}{2})=\sin(\frac{\pi}{2})=1

27,459

Ax=\lambda x \implies A^{-1}Ax=A^{-1}\lambda x

18,420

2 \cdot (-1) + y \cdot y + y = (y + 2) \cdot ((-1) + y)

23,592

\frac{5}{77} = 5/21\cdot \dfrac{1}{22}\cdot 6

21,984

-\frac12 \cdot \pi = (\left(-1\right) \cdot \pi)/2

-22,404

-4 = 6 (-1) + 2

31,502

\frac{1}{y} \cdot (y \cdot y + y) = y + 1

27,098

\cos{\frac{\pi}{2}} = \cos{3*\pi/2} = 0

32,190

\left(1 + y + \dots + y^4\right)^{1 + n} = (1 + y + \dots + y^4)^n\cdot (1 + y + \dots + y^4)

7,979

x + 2 (-1) > 0 \implies 2 < x

33,752

25 = f^2\Longrightarrow \sqrt{f \cdot f} = \sqrt{25}

-1,776

-5/3\cdot \pi = \pi/4 - \dfrac{1}{12}\cdot 23\cdot \pi

-2,010

-\frac{2}{3}*\pi = \dfrac{1}{12}*\pi - \frac{3}{4}*\pi

46,783

1111 = 11 \cdot 101

16,312

\dfrac{n^2}{2^{-\sqrt{n}}} = n^2*2^{\sqrt{n}}

20,011

\frac{2}{((-1) + z) (1 + z)} = \frac{1}{((-1) + z) (1 + z)} (-(\left(-1\right) + z) + z + 1)

1,186

(2*\tfrac13)^2 + (2*\frac{2}{3})^2 = \dfrac{20}{9} \gt 2

19,751

\frac{\pi}{2^{\frac{1}{2}}} = 2^{1 / 2} \cdot \pi/2

4,867

s^4 - 20*s^2 + 19 = (s * s + 19*\left(-1\right))*((-1) + s * s)

29,410

3 + \frac12 = \frac{7}{2}

-19

-5 - 3 = -8

30,324

(h + b)\cdot (1 + h\cdot \chi) = h^2\cdot \chi + h\cdot (1 + b\cdot \chi) + b

-3,138

\sqrt{25 \cdot 2} + \sqrt{16 \cdot 2} + \sqrt{9 \cdot 2} = \sqrt{32} + \sqrt{18} + \sqrt{50}

16,428

(z^2 - z \cdot 4 + 3) \cdot (4 + z) + 13 \cdot (z + (-1)) = (-1) + z^3

28,917

0 = -\frac35 + 3/5

17,433

x^i a_i = x^i + (a_i + (-1)) x^i

28,224

y \cdot 2 + 5 \cdot y = 7 \cdot y

17,747

\dfrac{1}{xx^V} = \frac{1}{xx^V}

24,103

{n \choose r} = \dfrac{1}{r!\cdot (n - r)!}\cdot n!

9,497

c*a + a*f = (c + f)*a

10,525

\pi\cdot 2 - \frac{\pi}{6} = \frac{\pi}{6}\cdot 11

1,268

\left(\mu^2 + n^2\right)^2 - (\mu^2 - n^2) \cdot (\mu^2 - n^2) = (2\mu n) \cdot (2\mu n) = 2 \cdot 2\mu^2 n \cdot n

15,742

\dfrac{1}{m^{d_2 - d_1}} = \frac{m^{d_1}}{m^{d_2}}

38,655

\sqrt{h} \cdot \sqrt{b} = \sqrt{h \cdot b} = \sqrt{h \cdot b}

38,423

(z + 1)*(z + 3) = (z + 1)*z + (z + 3)*3 = z^2 + z + 3*z + 9 = z^2 + 4*z + 9

52,561

680\times 211=143480=199\times 721+1

4,951

(n \cdot 2 + 1)^2 + ((-1) + 2 \cdot n)^2 = 2 + 8 \cdot n^2

-446

e^{\pi\cdot i\cdot 4/3\cdot 16} = \left(e^{4\cdot \pi\cdot i/3}\right)^{16}

46,439

2\cdot 126 = 252

23,282

C/(E_2) + A/(E_1) = (CE_1 + E_2 A)/(E_2 E_1)

-3,414

(3 + 2 + 5) \times 11^{1 / 2} = 11^{\frac{1}{2}} \times 10

-10,389

\frac{100}{60\cdot t + 240\cdot (-1)} = \frac{5}{t\cdot 3 + 12\cdot (-1)}\cdot 20/20

-30,322

1 = 3\cdot (-1) + 4

-17,594

84 + 66 \cdot (-1) = 18