ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
26,427	-15/2 + 7 = -1/2
18,142	\frac{1}{2}(1 + \cos{2y}) = \cos^2{y}
-1,592	-7/6 \cdot \pi + \frac{11}{12} \cdot \pi = -\frac{\pi}{4}
47,342	\tan{\frac{\pi}{4}} = 1
17,192	lx\beta = \betalx
-4,843	6.30 \times 10^{-1} = {6.30 \times 10^{-1}} \times 10^{2} = 6.30\times 10^{1}
-10,769	5/5 \dfrac{1}{8 (-1) + 10 q} \left(q\cdot 3 + 2\right) = \frac{1}{50 q + 40 (-1)} (15 q + 10)
41,949	E_1 \cap (B \cup E_2) = (B \cap E_1) \cup (E_1 \cap E_2) = E_2 \cup \left(B \cap E_1\right)
-9,188	20 - n \cdot 24 = 2 \cdot 2 \cdot 5 - n \cdot 2 \cdot 2 \cdot 2 \cdot 3
7,157	10 \cdot 32 = 320
293	20 - 24 \cdot z + 5 \cdot z = 7 \Rightarrow 20 - 19 \cdot z = 7
19,455	\frac{40}{120} = \frac{1}{3} = \frac{2}{6}
-3,096	3^{1 / 2}\cdot 5 + 4\cdot 3^{1 / 2} = 3^{1 / 2}\cdot 25^{1 / 2} + 16^{1 / 2}\cdot 3^{\frac{1}{2}}
-1,418	\frac{10}{14} = 10\cdot \tfrac{1}{2}/(14\cdot 1/2) = \tfrac{5}{7}
54,688	\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\implies \frac{1}{x^5}+\frac{1}{y^5}+\frac{1}{z^5}=\frac{1}{(x+y+z)^5}
3,845	y + 2 = 4 + y + 2(-1) = 4\left(1 + (y + 2(-1))/4\right)
1,477	4\cdot l + 2 = (2\cdot l + 1)\cdot 2 + 0
10,650	R = q^3\Longrightarrow R^{1/3} = q
-10,560	\dfrac33 \cdot (-\frac{10}{20 \cdot \left(-1\right) + z \cdot 5}) = -\frac{30}{z \cdot 15 + 60 \cdot \left(-1\right)}
-27,503	2\cdot 2\cdot 3\cdot w\cdot 5 = w\cdot 60
45,659	-\frac{1}{2} + \frac{\sqrt{17}}{2} = (-1 + \sqrt{17})/2
17,192	n\alpha\beta = \betan\alpha
-7,626	\tfrac{1}{2 - i}(i13 - 1) \tfrac{2 + i}{i + 2} = \dfrac{-1 + i13}{-i + 2}
25,684	\binom{a}{-b + a} = \binom{a}{b}
8,818	\frac{1}{250}\cdot 203 = \dfrac{27}{30^4}\cdot 30\cdot 29\cdot 28
-29,565	\frac1z \times (4 + 2 \times z^4 + 5 \times z) = \frac4z + 2 \times z^4/z + \frac{5}{z} \times z
28,526	\frac{2}{2} \times 2 = 2
9,535	(\frac{20}{5})^{\dfrac{15}{10}} = 8
4,040	(\left(-1\right) + q) \cdot \left((-1) + x\right) + (-1) = x \cdot q - x - q
47,401	2^{2 \cdot 2^2} = 256
30,723	\left(1.75 - 0.5 \cdot y = -y + 2 \implies 0.25 = 0.5 \cdot y\right) \implies y = 0.5
-10,111	-\tfrac{19}{20}\cdot \left(-\frac{7}{8}\right) = ((-19)\cdot (-7))/(20\cdot 8) = \frac{1}{160}\cdot 133
29,083	\dfrac{10}{x} = 20/(2x)
23,200	2 \cdot 2 \cdot 2^3 = 32
21,481	f \cdot e_2 \cdot e_1 = e_2 \cdot e_1 \cdot f = e_1 \cdot f \cdot e_2
-20,343	\frac{p\cdot 3}{30\cdot (-1) + 12\cdot p} = 3/3\cdot \tfrac{1}{4\cdot p + 10\cdot (-1)}\cdot p
3,487	x\cdot (-\frac{x}{x} + 1) = 0 \implies 1 = x/x
16,576	(a + b)^3 = b \cdot b^2 + a^3 + 3\cdot a^2\cdot b + 3\cdot a\cdot b^2
-22,047	\dfrac72 = \dfrac{1}{10}*35
25,137	2 = \sqrt{(-1) + 2} \cdot 2
9,356	\cos(\theta + \alpha) = \cos{\theta}\cdot \cos{\alpha} - \sin{\alpha}\cdot \sin{\theta}
22,243	a^2 - 4 \cdot a + 4 = (a + 2 \cdot (-1))^2
21,862	\left(V \cdot A = x\Longrightarrow x/A = A \cdot V/A\right)\Longrightarrow V \cdot A/A = x/A
-746	-\pi20 + 121/6\pi = \pi/6
32,661	4k + 2 = 2(2k + 1)
-19,003	1/9 = \tfrac{C_x}{36 \pi}\cdot 36 \pi = C_x
37,743	1 = (1 - 2^{1 / 2})^{2 \cdot 0}
35,606	\tan{\beta} = \dfrac{\sin{\beta}}{\cos{\beta}} \implies \tan{\beta}
18,112	( a', e) \left( a, e\right) = ( aa', e)
20,821	x^2 - f^2 = (x - f) \cdot (x + f)
-1,755	\pi\cdot \tfrac{5}{4} - \dfrac{\pi}{3} = 11/12\cdot \pi
30,986	\binom{x}{t} = \binom{x}{x - t}
54,430	\sin\left(c - \dfrac{3 \pi}{2}\right) = \sin(-(\dfrac{3 \pi}{2} - c)) = -\sin(\frac{\pi}{2} 3 - c) = \cos(c) = \cos(c)
783	6!/(2!\cdot 3!) = {3 \choose 2}\cdot {1 \choose 1}\cdot {6 \choose 3}
12,787	f^2 = \left(6 + f\right) \left(12 + f\right)\Longrightarrow f = -4
-28,407	x^2 - 14 \cdot x + 58 = x^2 - 14 \cdot x + 49 + 9 = (x + 7 \cdot (-1))^2 + 9 = \left(x \cdot (-7)\right)^2 + 3^2
-11,977	26/45 = G/(18\cdot \pi)\cdot 18\cdot \pi = G
29,847	16^{\frac{3}{4}} = (16^3)^{\tfrac14} = (2^{4^3})^{\frac14} = 2^3
2,785	5 \cdot \pi \cdot r^2 = 9 \cdot \pi \cdot r^2 - 4 \cdot r^2 \cdot \pi
-4,445	-\tfrac{2}{1 + z} - \frac{1}{4 + z}\cdot 3 = \frac{1}{z^2 + z\cdot 5 + 4}\cdot (11\cdot (-1) - 5\cdot z)
29,937	49 = ( a + d, a + d) = 16 + 2 \cdot ( a, d) + 25
-11,527	-12 i - 16 + 0(-1) = -16 - 12 i
28,581	\frac{k + \left(-1\right)}{k + 1} = \frac{k + 1 + 2(-1)}{k + 1} = 1 - \frac{1}{k + 1}2
-21,025	\tfrac{15 \cdot y + 21 \cdot (-1)}{-y \cdot 3 + 18 \cdot (-1)} = \dfrac{1}{6 \cdot (-1) - y} \cdot (y \cdot 5 + 7 \cdot (-1)) \cdot 3/3
-20,448	\dfrac{12 \cdot (-1) + 4 \cdot z}{z \cdot 4 + 12} = \frac44 \cdot \dfrac{z + 3 \cdot (-1)}{z + 3}
-6,556	\frac{1}{z^2 - 12 \cdot z + 27} \cdot 3 = \frac{3}{\left(3 \cdot \left(-1\right) + z\right) \cdot \left(9 \cdot (-1) + z\right)}
-4,519	\frac{20\cdot (-1) + 8\cdot z}{z^2 - 6\cdot z + 5} = \dfrac{1}{5\cdot (-1) + z}\cdot 5 + \dfrac{3}{z + (-1)}
5,943	(10^3 - 9^2 * 9*2 + 8^3)/6 = 9
-21,161	\dfrac{1}{4} = \dfrac{2}{8}
6,199	k + (-1) = k\cdot 2 - k + (-1)
26,764	2\sqrt{3} = p/r + (-1) = \left(p - r\right)/r
33,955	1/28 + 1/14 + \frac{1}{7} + 1/4 + \dfrac{1}{2} = \frac{1}{28}(1 + 2 + 4 + 7 + 14) = 1
11,097	\tan^3(\tan^{-1}(x)) = \tan(\tan(\tan(\tan^{-1}(x)))) = \tan(\tan(x)) = \tan^2(x)
6,612	w_1x + w_2e + w_3f = (w_3 + w_1 + w_2)f + w_1(-f + x - e + f) + (e - f)(w_1 + w_2)
966	5 \cdot (z + 2 \cdot (-1)) = 10 \cdot (-1) + 5 \cdot z
26,155	\tan{f} = -\tan(\pi - f)
8,790	4 \cdot (x - d)^2 = 4 \cdot d^2 - d \cdot x \cdot 8 + 4 \cdot x \cdot x
-28,869	0.01\cdot 20/(60\cdot 0.01) = \frac{1}{3}
11,095	x^2 + a^2 + x \cdot a \cdot 2 = (a + x)^2
390	\frac{\mathrm{d}}{\mathrm{d}z} (e^{2z + (-1)} + 1) = 2e^{z*2 + (-1)}
31,280	W = X \implies W = X
31,497	\N = \left\{2, 0, 1, \dots\right\}
38,435	\sqrt{124} = \sqrt{2^2\cdot 31} = \sqrt{2^2} \sqrt{31} = 2\sqrt{31}
16,481	(1 + x)^2 + 24 = 25 + x^2 + 2 \cdot x
21,932	x^5 = n\times x^2 - m\times x^3 = n\times x^2 - m\times n + m \times m\times x
2,280	(-z + y)^2 = z^2 \cdot z \implies 4 - z^3 + y^2 - 2zy = 0
21,978	\left(a + b \times i\right) \times (m + n \times i) = a \times m - b \times n + (a \times n + b \times m) \times i = a \times m - b \times n + i
21,270	2 \times (-1) + x_2 = x_2
11,563	x2 + y3 + 4\beta = 24 \Rightarrow y3 = -\beta4 + 24 - x2
5,122	\tan{x} = \frac{\sin{x}}{\cos{x}} = \sqrt{1 - \cos^2{x}}/\cos{x}
-5,159	0.81 \times 10^4 = 10^{6 + 2 \times (-1)} \times 0.81
-20,071	\frac{36 + 32\cdot y}{27 + y\cdot 24} = \frac{y\cdot 8 + 9}{8\cdot y + 9}\cdot 4/3
-22,283	70 (-1) + y^2 - 3y = (y + 7) (y + 10 \left(-1\right))
-29,877	d/dy y^l = y^{(-1) + l} l
51,372	2 + 2 + 2 + 2 + 2 + 4 = 14
21,961	\left\{1, 3, 27, 5, 4, 0, 2, \dotsm\right\} = \mathbf{Z}_{28}
-10,628	-\frac{30}{100 + q\cdot 60} = \frac{10}{10}\cdot (-\frac{3}{q\cdot 6 + 10})
-1,646	13/12 \times \pi = 0 + 13/12 \times \pi
22,688	n! = n(n + (-1))! = n(n + (-1)) \left(n + 2(-1)\right)!
-20,024	\tfrac{r\cdot 18 + 72}{-r\cdot 4 + 16\cdot \left(-1\right)} = \dfrac{1}{8\cdot (-1) - 2\cdot r}\cdot (-2\cdot r + 8\cdot (-1))\cdot (-\frac{1}{2}\cdot 9)

26,427

-15/2 + 7 = -1/2

18,142

\frac{1}{2}*(1 + \cos{2*y}) = \cos^2{y}

-1,592

-7/6 \cdot \pi + \frac{11}{12} \cdot \pi = -\frac{\pi}{4}

47,342

\tan{\frac{\pi}{4}} = 1

17,192

l*x*\beta = \beta*l*x

-4,843

6.30 \times 10^{-1} = {6.30 \times 10^{-1}} \times 10^{2} = 6.30\times 10^{1}

-10,769

5/5 \dfrac{1}{8 (-1) + 10 q} \left(q\cdot 3 + 2\right) = \frac{1}{50 q + 40 (-1)} (15 q + 10)

41,949

E_1 \cap (B \cup E_2) = (B \cap E_1) \cup (E_1 \cap E_2) = E_2 \cup \left(B \cap E_1\right)

-9,188

20 - n \cdot 24 = 2 \cdot 2 \cdot 5 - n \cdot 2 \cdot 2 \cdot 2 \cdot 3

7,157

10 \cdot 32 = 320

293

20 - 24 \cdot z + 5 \cdot z = 7 \Rightarrow 20 - 19 \cdot z = 7

19,455

\frac{40}{120} = \frac{1}{3} = \frac{2}{6}

-3,096

3^{1 / 2}\cdot 5 + 4\cdot 3^{1 / 2} = 3^{1 / 2}\cdot 25^{1 / 2} + 16^{1 / 2}\cdot 3^{\frac{1}{2}}

-1,418

\frac{10}{14} = 10\cdot \tfrac{1}{2}/(14\cdot 1/2) = \tfrac{5}{7}

54,688

\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{x+y+z}\implies \frac{1}{x^5}+\frac{1}{y^5}+\frac{1}{z^5}=\frac{1}{(x+y+z)^5}

3,845

y + 2 = 4 + y + 2(-1) = 4\left(1 + (y + 2(-1))/4\right)

1,477

4\cdot l + 2 = (2\cdot l + 1)\cdot 2 + 0

10,650

R = q^3\Longrightarrow R^{1/3} = q

-10,560

\dfrac33 \cdot (-\frac{10}{20 \cdot \left(-1\right) + z \cdot 5}) = -\frac{30}{z \cdot 15 + 60 \cdot \left(-1\right)}

-27,503

2\cdot 2\cdot 3\cdot w\cdot 5 = w\cdot 60

45,659

-\frac{1}{2} + \frac{\sqrt{17}}{2} = (-1 + \sqrt{17})/2

17,192

n*\alpha*\beta = \beta*n*\alpha

-7,626

\tfrac{1}{2 - i}(i*13 - 1) \tfrac{2 + i}{i + 2} = \dfrac{-1 + i*13}{-i + 2}

25,684

\binom{a}{-b + a} = \binom{a}{b}

8,818

\frac{1}{250}\cdot 203 = \dfrac{27}{30^4}\cdot 30\cdot 29\cdot 28

-29,565

\frac1z \times (4 + 2 \times z^4 + 5 \times z) = \frac4z + 2 \times z^4/z + \frac{5}{z} \times z

28,526

\frac{2}{2} \times 2 = 2

9,535

(\frac{20}{5})^{\dfrac{15}{10}} = 8

4,040

(\left(-1\right) + q) \cdot \left((-1) + x\right) + (-1) = x \cdot q - x - q

47,401

2^{2 \cdot 2^2} = 256

30,723

\left(1.75 - 0.5 \cdot y = -y + 2 \implies 0.25 = 0.5 \cdot y\right) \implies y = 0.5

-10,111

-\tfrac{19}{20}\cdot \left(-\frac{7}{8}\right) = ((-19)\cdot (-7))/(20\cdot 8) = \frac{1}{160}\cdot 133

29,083

\dfrac{10}{x} = 20/(2x)

23,200

2 \cdot 2 \cdot 2^3 = 32

21,481

f \cdot e_2 \cdot e_1 = e_2 \cdot e_1 \cdot f = e_1 \cdot f \cdot e_2

-20,343

\frac{p\cdot 3}{30\cdot (-1) + 12\cdot p} = 3/3\cdot \tfrac{1}{4\cdot p + 10\cdot (-1)}\cdot p

3,487

x\cdot (-\frac{x}{x} + 1) = 0 \implies 1 = x/x

16,576

(a + b)^3 = b \cdot b^2 + a^3 + 3\cdot a^2\cdot b + 3\cdot a\cdot b^2

-22,047

\dfrac72 = \dfrac{1}{10}*35

25,137

2 = \sqrt{(-1) + 2} \cdot 2

9,356

\cos(\theta + \alpha) = \cos{\theta}\cdot \cos{\alpha} - \sin{\alpha}\cdot \sin{\theta}

22,243

a^2 - 4 \cdot a + 4 = (a + 2 \cdot (-1))^2

21,862

\left(V \cdot A = x\Longrightarrow x/A = A \cdot V/A\right)\Longrightarrow V \cdot A/A = x/A

-746

-\pi*20 + 121/6*\pi = \pi/6

32,661

4k + 2 = 2(2k + 1)

-19,003

1/9 = \tfrac{C_x}{36 \pi}\cdot 36 \pi = C_x

37,743

1 = (1 - 2^{1 / 2})^{2 \cdot 0}

35,606

\tan{\beta} = \dfrac{\sin{\beta}}{\cos{\beta}} \implies \tan{\beta}

18,112

( a', e) \left( a, e\right) = ( aa', e)

20,821

x^2 - f^2 = (x - f) \cdot (x + f)

-1,755

\pi\cdot \tfrac{5}{4} - \dfrac{\pi}{3} = 11/12\cdot \pi

30,986

\binom{x}{t} = \binom{x}{x - t}

54,430

\sin\left(c - \dfrac{3 \pi}{2}\right) = \sin(-(\dfrac{3 \pi}{2} - c)) = -\sin(\frac{\pi}{2} 3 - c) = \cos(c) = \cos(c)

783

6!/(2!\cdot 3!) = {3 \choose 2}\cdot {1 \choose 1}\cdot {6 \choose 3}

12,787

f^2 = \left(6 + f\right) \left(12 + f\right)\Longrightarrow f = -4

-28,407

x^2 - 14 \cdot x + 58 = x^2 - 14 \cdot x + 49 + 9 = (x + 7 \cdot (-1))^2 + 9 = \left(x \cdot (-7)\right)^2 + 3^2

-11,977

26/45 = G/(18\cdot \pi)\cdot 18\cdot \pi = G

29,847

16^{\frac{3}{4}} = (16^3)^{\tfrac14} = (2^{4^3})^{\frac14} = 2^3

2,785

5 \cdot \pi \cdot r^2 = 9 \cdot \pi \cdot r^2 - 4 \cdot r^2 \cdot \pi

-4,445

-\tfrac{2}{1 + z} - \frac{1}{4 + z}\cdot 3 = \frac{1}{z^2 + z\cdot 5 + 4}\cdot (11\cdot (-1) - 5\cdot z)

29,937

49 = ( a + d, a + d) = 16 + 2 \cdot ( a, d) + 25

-11,527

-12 i - 16 + 0(-1) = -16 - 12 i

28,581

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

-21,025

\tfrac{15 \cdot y + 21 \cdot (-1)}{-y \cdot 3 + 18 \cdot (-1)} = \dfrac{1}{6 \cdot (-1) - y} \cdot (y \cdot 5 + 7 \cdot (-1)) \cdot 3/3

-20,448

\dfrac{12 \cdot (-1) + 4 \cdot z}{z \cdot 4 + 12} = \frac44 \cdot \dfrac{z + 3 \cdot (-1)}{z + 3}

-6,556

\frac{1}{z^2 - 12 \cdot z + 27} \cdot 3 = \frac{3}{\left(3 \cdot \left(-1\right) + z\right) \cdot \left(9 \cdot (-1) + z\right)}

-4,519

\frac{20\cdot (-1) + 8\cdot z}{z^2 - 6\cdot z + 5} = \dfrac{1}{5\cdot (-1) + z}\cdot 5 + \dfrac{3}{z + (-1)}

5,943

(10^3 - 9^2 * 9*2 + 8^3)/6 = 9

-21,161

\dfrac{1}{4} = \dfrac{2}{8}

6,199

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

26,764

2\sqrt{3} = p/r + (-1) = \left(p - r\right)/r

33,955

1/28 + 1/14 + \frac{1}{7} + 1/4 + \dfrac{1}{2} = \frac{1}{28}(1 + 2 + 4 + 7 + 14) = 1

11,097

\tan^3(\tan^{-1}(x)) = \tan(\tan(\tan(\tan^{-1}(x)))) = \tan(\tan(x)) = \tan^2(x)

6,612

w_1*x + w_2*e + w_3*f = (w_3 + w_1 + w_2)*f + w_1*(-f + x - e + f) + (e - f)*(w_1 + w_2)

966

5 \cdot (z + 2 \cdot (-1)) = 10 \cdot (-1) + 5 \cdot z

26,155

\tan{f} = -\tan(\pi - f)

8,790

4 \cdot (x - d)^2 = 4 \cdot d^2 - d \cdot x \cdot 8 + 4 \cdot x \cdot x

-28,869

0.01\cdot 20/(60\cdot 0.01) = \frac{1}{3}

11,095

x^2 + a^2 + x \cdot a \cdot 2 = (a + x)^2

390

\frac{\mathrm{d}}{\mathrm{d}z} (e^{2*z + (-1)} + 1) = 2*e^{z*2 + (-1)}

31,280

W = X \implies W = X

31,497

\N = \left\{2, 0, 1, \dots\right\}

38,435

\sqrt{124} = \sqrt{2^2\cdot 31} = \sqrt{2^2} \sqrt{31} = 2\sqrt{31}

16,481

(1 + x)^2 + 24 = 25 + x^2 + 2 \cdot x

21,932

x^5 = n\times x^2 - m\times x^3 = n\times x^2 - m\times n + m \times m\times x

2,280

(-z + y)^2 = z^2 \cdot z \implies 4 - z^3 + y^2 - 2zy = 0

21,978

\left(a + b \times i\right) \times (m + n \times i) = a \times m - b \times n + (a \times n + b \times m) \times i = a \times m - b \times n + i

21,270

2 \times (-1) + x_2 = x_2

11,563

x*2 + y*3 + 4*\beta = 24 \Rightarrow y*3 = -\beta*4 + 24 - x*2

5,122

\tan{x} = \frac{\sin{x}}{\cos{x}} = \sqrt{1 - \cos^2{x}}/\cos{x}

-5,159

0.81 \times 10^4 = 10^{6 + 2 \times (-1)} \times 0.81

-20,071

\frac{36 + 32\cdot y}{27 + y\cdot 24} = \frac{y\cdot 8 + 9}{8\cdot y + 9}\cdot 4/3

-22,283

70 (-1) + y^2 - 3y = (y + 7) (y + 10 \left(-1\right))

-29,877

d/dy y^l = y^{(-1) + l} l

51,372

2 + 2 + 2 + 2 + 2 + 4 = 14

21,961

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

-10,628

-\frac{30}{100 + q\cdot 60} = \frac{10}{10}\cdot (-\frac{3}{q\cdot 6 + 10})

-1,646

13/12 \times \pi = 0 + 13/12 \times \pi

22,688

n! = n*(n + (-1))! = n*(n + (-1)) \left(n + 2(-1)\right)!

-20,024

\tfrac{r\cdot 18 + 72}{-r\cdot 4 + 16\cdot \left(-1\right)} = \dfrac{1}{8\cdot (-1) - 2\cdot r}\cdot (-2\cdot r + 8\cdot (-1))\cdot (-\frac{1}{2}\cdot 9)