ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
12,413	(y + (-1)) (y^2 + y + 1) = y \cdot y \cdot y + (-1)
21,318	\frac{5!}{3} \cdot \dfrac{1}{2} = 20
33,381	3744 = 3796 + 52 \cdot \left(-1\right)
21,040	\lim_{t \to \infty} \|2 \cdot (-1) + t\|/t = \lim_{t \to \infty} \left(t + 2 \cdot (-1)\right)/t
4,101	(x + 1)\cdot (2\cdot (-1) - x^2 + x\cdot 2) = -x^3 + x^2 + 2\cdot (-1)
-4,770	\dfrac{15\cdot (-1) + 6\cdot y}{y^2 - y\cdot 5 + 4} = \frac{3}{4\cdot (-1) + y} + \frac{1}{(-1) + y}\cdot 3
7,818	\left(c + b\right)\cdot a = a\cdot c + a\cdot b
3,592	\cos(30)\cdot \sin(C)\cdot H\cdot 2 = x_3 - x_1 \Rightarrow \sin(C)\cdot H = \frac{1}{3^{1 / 2}}\cdot (-x_1 + x_3)
4	x = E_1\cdot E_2 rightarrow E_2\cdot E_1 = x
34,873	{94 \choose 3} = 134044
50,878	4\cdot x = 3\cdot x + x \geq 1 + x
2,156	\sqrt{64}/(\sqrt{8}) = \sqrt{\dfrac18 \cdot 64} = \sqrt{8} = 2 \cdot \sqrt{2}
39,178	\left(\tan{2\cdot x} - \sin{4\cdot x} = 0 \implies \sin{x\cdot 4} = \tan{2\cdot x}\right) \implies \dfrac{\sin{2\cdot x}}{\cos{2\cdot x}} = \sin{x\cdot 4}
18,092	\sin^{26}(x) - \sin^{24}(x) = \sin\left(6 \cdot x + 4 \cdot x\right) \cdot \sin(6 \cdot x - 4 \cdot x) = \sin(2 \cdot x) \cdot \sin(10 \cdot x)
25,422	3 = \frac{2!}{2!}2!/2! \binom{3}{2}
-13,214	-16.8/\left(-0.21\right) = 80
1,999	{l + q + (-1) \choose l} = {l + q + (-1) \choose q + \left(-1\right)}
4,244	\left\|{E \cdot E^T}\right\| = 0 = \left\|{E^T \cdot E}\right\|
23,492	0 = \cos(\tfrac{1}{2^{0 + 1}}*\pi)
19,116	\frac{2}{x + 2} = \frac{2}{x + (-1) + 3} = \frac{2}{3} \times \frac{2}{(x + (-1))/3 + 1}
34,298	2 \cdot 3 + 2 = 8
6,357	3\cdot y^2 - 7\cdot y + 2 = \left(y\cdot 3 + (-1)\right)\cdot (2\cdot (-1) + y)
9,107	\frac{l^2 + 9(-1)}{4(-1) + l} = \frac{7}{l + 4*\left(-1\right)} + l + 4
39,721	\frac12*\|-1\| = 1/2
-2,511	7^{1/2} = 7^{1/2} \cdot \left(3 + 2 \cdot (-1)\right)
29,273	A^2\cdot B^2 = (A\cdot B)^2
5,868	3/4\cdot \tfrac{2}{5}\cdot \frac{4}{7}\cdot 7/10\cdot \frac69\cdot 5/8\cdot 3/6 = \frac{3\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2}{4\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5}
12,132	\sin(\dfrac{3}{2} \cdot \pi - E) = -\cos(E)
21,633	\dfrac14*2 / 5 = \dfrac{1}{10}
8,687	\dfrac{14}{12} = 7/6
-26,046	(d - h)\cdot (h + d) = -h^2 + d^2
14,332	\tfrac{a - b}{a + b} = -2 \cdot \frac{b}{a + b} + 1
13,323	1 - \frac{15!D_{15}}{15!^2}1 = 1 - D_{15}/15!
-9,154	-15 \cdot z^3 = -z \cdot z \cdot 3 \cdot 5 \cdot z
27,375	\overline{q} + \overline{x} = \overline{q + x}
-10,609	\frac{1}{12x + 8}12 = \frac{3}{2 + 3x}4/4
-20,344	\frac55 \cdot \frac{\left(-1\right) + 5 \cdot x}{4 \cdot x} = \tfrac{1}{x \cdot 20} \cdot \left(x \cdot 25 + 5 \cdot (-1)\right)
-9,156	-a\times 7\times 7\times a + 2\times 5\times 7\times a = -a^2\times 49 + a\times 70
14,637	\dfrac{1}{\|a\|^n + (-1)} = \frac{1}{\|a\|^n\cdot (\|a\|^n + \left(-1\right))} + \dfrac{1}{\|a\|^n}
8,587	\frac{(\sqrt{24})^2}{8^2} = \tfrac{24}{64} = 6/16
16,087	\frac{1}{1 + x} \cdot (x \cdot 6 + 17) = 7 \implies 6 \cdot x + 17 = 7 \cdot x + 7
5,988	y * y = \|y\|^2 > 4\Longrightarrow \|y\| \gt \sqrt{4} = 2
37,585	n!n + n! = (n + 1)n! = \left(n + 1\right)!
-11,768	16^{-\frac14} = (1/16)^{\frac14} = 1/2
30,788	e^y = 1 + y + y^2/2 + \cdots
-3,649	\frac{1}{q \cdot 7} \cdot 6 = \dfrac{6 / 7}{q} \cdot 1
15,881	i = j \implies j - i = 0
-9,236	-x2335 - 2225 = 40 (-1) - x90
14,838	\gamma\cdot \alpha = \alpha\cdot \gamma
27,847	b_1 b_2 + a_1 a_2 - a_1 b_2 - a_2 b_1 = (-b_2 + a_2) (a_1 - b_1)
21,386	3^{n + 2} = 3^n\cdot 3 \cdot 3
36,765	B^{-2.6}B^4B^7/B = B^{7.4}
50,822	0 = 0 = 1 + (-1) + 1 + (-1) + 1 + (-1)
42,436	100/10! = \tfrac{1}{36288}
16,214	\cos(f)\cdot \sin(g) + \sin(f)\cdot \cos(g) = \sin(g + f)
16,428	y^3 + (-1) = (4 + y)\times (y^2 - y\times 4 + 3) + 13\times ((-1) + y)
26,767	\cos(k\theta) = \cos(-k\theta)
-5,695	\tfrac{4}{4\cdot (k + 9)\cdot (1 + k)} = \frac{1/4\cdot 4}{(k + 9)\cdot \left(1 + k\right)}
20,795	n = \binom{x + n + 2 \cdot (-1)}{x + \left(-1\right)} = \binom{x + n + 3 \cdot (-1)}{x + 2 \cdot \left(-1\right)} + \binom{x + n + 3 \cdot (-1)}{x + (-1)}
-15,276	\frac{1}{d^8\cdot \frac{1}{x^{10}}}\cdot d = \dfrac{1}{\frac{1}{d}\cdot (\frac{d^4}{x^5})^2}
18,053	E(X_F) E(X_A) = E(X_F X_A)
17,830	\cos{\pi/4} = 1/(\sqrt{2}) = \frac{\sqrt{2}}{2}
17,276	n \cdot L \cdot Y + a \cdot L \cdot Q = (n \cdot Y + Q \cdot a) \cdot L
-19,686	14/9 = 2 \cdot 7/(9)
11,085	\dfrac{1}{2*5}30 + (-1) = 30/10 + (-1) = 3 + (-1) = 2
4,370	\frac{\mathrm{d}}{\mathrm{d}y} (y\cdot e^y) = y\cdot e^y + e^y = y\cdot e^y + e^y
22,644	\|x\|^{t + 2(-1)} x = \eta \Rightarrow x = \eta \|\eta\|^{\tfrac{2 - t}{t + (-1)}}
30,095	a_i \cdot N = a_i \cdot N
23,698	\dfrac{1}{6^3} \cdot {6 \choose 3} = \frac{1}{216} \cdot 20 = 5/54
-22,424	125^{\frac23} = (125^{\frac13})^2 = 5^2 = 5*5 = 25
54,050	4^{23} = 70368744177664
29,280	2\pi/51.25 = \frac{2\pi}{5}1\tfrac{5}{4} = \pi/2
-9,122	-p\cdot 20 = -2\cdot 2\cdot 5 p
21,494	\frac{4 \cdot \dfrac15}{3 \cdot 1/6} = \frac{8}{5} = 1.6
21,385	x = \frac{1}{-(1 - 1/x) + 1}
27,975	C^2C = CC^2
21,530	\binom{k}{1} + 2*\binom{k}{2} = k^2
47,947	2^{40}=(2^{10})^4=1048576^2=1099511627776
-9,317	-x\cdot 35 + x^2\cdot 5 = -x\cdot 5\cdot 7 + x\cdot 5\cdot x
31,104	1328/2450 = 1 - \frac{1}{49}\cdot 33\cdot \frac{34}{50}
4,617	2*(3y - z) = 6y - 2z
22,867	\cos^2(x) = \frac{1+\cos(2x)}{2}
-30,034	z^{(-1) + \varepsilon} \varepsilon = \frac{\mathrm{d}}{\mathrm{d}z} z^\varepsilon
23,335	x^2 \cdot D^2 - 6 \cdot x \cdot D + 9 = \left(x \cdot D\right)^2 - 7 \cdot x \cdot D + 9 = (x \cdot D - 3.5)^2 - 3.25
5,283	0 + 0 + 1/2 = \dfrac{1}{2}
2,366	(x + 2)/((-1)\cdot x) = \frac{y}{(-1)\cdot (y + 4\cdot (-1))}\Longrightarrow 2\cdot x + 4 = y
-19,689	\frac{1}{8}\cdot 27 = 3\cdot 9/\left(8\right)
27,191	-a/b = -a/b = ((-1)\cdot a)/b = \frac{1}{\left(-1\right)\cdot b}\cdot a
32,424	56/825 = 0.4\dfrac{8}{4 + 8}0.4*\frac{8 + (-1)}{4 + 8 + (-1)}
9,045	\cos{3x} = \cos{x \cdot 3}
873	e^z = e^z = 1 + z + \frac{1}{2}z^2 + \dfrac{1}{6}z \cdot z \cdot z + \dots
15,632	-b + (b + (-1)) d = db - d - b
-6,250	\frac{4}{3 \times y + 9} = \frac{1}{3 \times (y + 3)} \times 4
9,105	1^5 + 2^5 + 3^5 + 4^5 + 1^7 + 2^7 + 3^7 + 4^7 = (4 + 1 + 2 + 3)^4\cdot 2
-6,228	\frac{3 \cdot y}{\left(y + (-1)\right) \cdot \left(4 + y\right)} \cdot 25/25 = \frac{75 \cdot y}{25 \cdot (\left(-1\right) + y) \cdot (y + 4)} \cdot 1
-11,935	6.208\cdot 0.001 = \frac{6.208}{1000}
38,454	\|\left( 2, -4, 1\right) \cdot \left( -4, 5, -1\right)\| = \|-8 + 20 \cdot \left(-1\right) + (-1)\| = 29
4,307	\|x + (-1)\|2 = \|-2x + 2\|
9,531	\dfrac{1}{1/b \cdot h} = b/h
29,995	2 \cdot 2^3/3 + \frac12 \cdot 2 \cdot 2 - \frac{14}{6} \cdot 1 = 5.3 + 2 - 2.3 = 5

12,413

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

21,318

\frac{5!}{3} \cdot \dfrac{1}{2} = 20

33,381

3744 = 3796 + 52 \cdot \left(-1\right)

21,040

\lim_{t \to \infty} |2 \cdot (-1) + t|/t = \lim_{t \to \infty} \left(t + 2 \cdot (-1)\right)/t

4,101

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

-4,770

\dfrac{15\cdot (-1) + 6\cdot y}{y^2 - y\cdot 5 + 4} = \frac{3}{4\cdot (-1) + y} + \frac{1}{(-1) + y}\cdot 3

7,818

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

3,592

\cos(30)\cdot \sin(C)\cdot H\cdot 2 = x_3 - x_1 \Rightarrow \sin(C)\cdot H = \frac{1}{3^{1 / 2}}\cdot (-x_1 + x_3)

4

x = E_1\cdot E_2 rightarrow E_2\cdot E_1 = x

34,873

{94 \choose 3} = 134044

50,878

4\cdot x = 3\cdot x + x \geq 1 + x

2,156

\sqrt{64}/(\sqrt{8}) = \sqrt{\dfrac18 \cdot 64} = \sqrt{8} = 2 \cdot \sqrt{2}

39,178

\left(\tan{2\cdot x} - \sin{4\cdot x} = 0 \implies \sin{x\cdot 4} = \tan{2\cdot x}\right) \implies \dfrac{\sin{2\cdot x}}{\cos{2\cdot x}} = \sin{x\cdot 4}

18,092

\sin^{26}(x) - \sin^{24}(x) = \sin\left(6 \cdot x + 4 \cdot x\right) \cdot \sin(6 \cdot x - 4 \cdot x) = \sin(2 \cdot x) \cdot \sin(10 \cdot x)

25,422

3 = \frac{2!}{2!}2!/2! \binom{3}{2}

-13,214

-16.8/\left(-0.21\right) = 80

1,999

{l + q + (-1) \choose l} = {l + q + (-1) \choose q + \left(-1\right)}

4,244

\left|{E \cdot E^T}\right| = 0 = \left|{E^T \cdot E}\right|

23,492

0 = \cos(\tfrac{1}{2^{0 + 1}}*\pi)

19,116

\frac{2}{x + 2} = \frac{2}{x + (-1) + 3} = \frac{2}{3} \times \frac{2}{(x + (-1))/3 + 1}

34,298

2 \cdot 3 + 2 = 8

6,357

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

9,107

\frac{l^2 + 9*(-1)}{4*(-1) + l} = \frac{7}{l + 4*\left(-1\right)} + l + 4

39,721

\frac12*|-1| = 1/2

-2,511

7^{1/2} = 7^{1/2} \cdot \left(3 + 2 \cdot (-1)\right)

29,273

A^2\cdot B^2 = (A\cdot B)^2

5,868

3/4\cdot \tfrac{2}{5}\cdot \frac{4}{7}\cdot 7/10\cdot \frac69\cdot 5/8\cdot 3/6 = \frac{3\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2}{4\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5}

12,132

\sin(\dfrac{3}{2} \cdot \pi - E) = -\cos(E)

21,633

\dfrac14*2 / 5 = \dfrac{1}{10}

8,687

\dfrac{14}{12} = 7/6

-26,046

(d - h)\cdot (h + d) = -h^2 + d^2

14,332

\tfrac{a - b}{a + b} = -2 \cdot \frac{b}{a + b} + 1

13,323

1 - \frac{15!*D_{15}}{15!^2}*1 = 1 - D_{15}/15!

-9,154

-15 \cdot z^3 = -z \cdot z \cdot 3 \cdot 5 \cdot z

27,375

\overline{q} + \overline{x} = \overline{q + x}

-10,609

\frac{1}{12*x + 8}*12 = \frac{3}{2 + 3*x}*4/4

-20,344

\frac55 \cdot \frac{\left(-1\right) + 5 \cdot x}{4 \cdot x} = \tfrac{1}{x \cdot 20} \cdot \left(x \cdot 25 + 5 \cdot (-1)\right)

-9,156

-a\times 7\times 7\times a + 2\times 5\times 7\times a = -a^2\times 49 + a\times 70

14,637

\dfrac{1}{|a|^n + (-1)} = \frac{1}{|a|^n\cdot (|a|^n + \left(-1\right))} + \dfrac{1}{|a|^n}

8,587

\frac{(\sqrt{24})^2}{8^2} = \tfrac{24}{64} = 6/16

16,087

\frac{1}{1 + x} \cdot (x \cdot 6 + 17) = 7 \implies 6 \cdot x + 17 = 7 \cdot x + 7

5,988

y * y = |y|^2 > 4\Longrightarrow |y| \gt \sqrt{4} = 2

37,585

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

-11,768

16^{-\frac14} = (1/16)^{\frac14} = 1/2

30,788

e^y = 1 + y + y^2/2 + \cdots

-3,649

\frac{1}{q \cdot 7} \cdot 6 = \dfrac{6 / 7}{q} \cdot 1

15,881

i = j \implies j - i = 0

-9,236

-x*2*3*3*5 - 2*2*2*5 = 40 (-1) - x*90

14,838

\gamma\cdot \alpha = \alpha\cdot \gamma

27,847

b_1 b_2 + a_1 a_2 - a_1 b_2 - a_2 b_1 = (-b_2 + a_2) (a_1 - b_1)

21,386

3^{n + 2} = 3^n\cdot 3 \cdot 3

36,765

B^{-2.6}*B^4*B^7/B = B^{7.4}

50,822

0 = 0 = 1 + (-1) + 1 + (-1) + 1 + (-1)

42,436

100/10! = \tfrac{1}{36288}

16,214

\cos(f)\cdot \sin(g) + \sin(f)\cdot \cos(g) = \sin(g + f)

16,428

y^3 + (-1) = (4 + y)\times (y^2 - y\times 4 + 3) + 13\times ((-1) + y)

26,767

\cos(k\theta) = \cos(-k\theta)

-5,695

\tfrac{4}{4\cdot (k + 9)\cdot (1 + k)} = \frac{1/4\cdot 4}{(k + 9)\cdot \left(1 + k\right)}

20,795

n = \binom{x + n + 2 \cdot (-1)}{x + \left(-1\right)} = \binom{x + n + 3 \cdot (-1)}{x + 2 \cdot \left(-1\right)} + \binom{x + n + 3 \cdot (-1)}{x + (-1)}

-15,276

\frac{1}{d^8\cdot \frac{1}{x^{10}}}\cdot d = \dfrac{1}{\frac{1}{d}\cdot (\frac{d^4}{x^5})^2}

18,053

E(X_F) E(X_A) = E(X_F X_A)

17,830

\cos{\pi/4} = 1/(\sqrt{2}) = \frac{\sqrt{2}}{2}

17,276

n \cdot L \cdot Y + a \cdot L \cdot Q = (n \cdot Y + Q \cdot a) \cdot L

-19,686

14/9 = 2 \cdot 7/(9)

11,085

\dfrac{1}{2*5}30 + (-1) = 30/10 + (-1) = 3 + (-1) = 2

4,370

\frac{\mathrm{d}}{\mathrm{d}y} (y\cdot e^y) = y\cdot e^y + e^y = y\cdot e^y + e^y

22,644

|x|^{t + 2(-1)} x = \eta \Rightarrow x = \eta |\eta|^{\tfrac{2 - t}{t + (-1)}}

30,095

a_i \cdot N = a_i \cdot N

23,698

\dfrac{1}{6^3} \cdot {6 \choose 3} = \frac{1}{216} \cdot 20 = 5/54

-22,424

125^{\frac23} = (125^{\frac13})^2 = 5^2 = 5*5 = 25

54,050

4^{23} = 70368744177664

29,280

2\pi/5*1.25 = \frac{2\pi}{5}1*\tfrac{5}{4} = \pi/2

-9,122

-p\cdot 20 = -2\cdot 2\cdot 5 p

21,494

\frac{4 \cdot \dfrac15}{3 \cdot 1/6} = \frac{8}{5} = 1.6

21,385

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

27,975

C^2*C = C*C^2

21,530

\binom{k}{1} + 2*\binom{k}{2} = k^2

47,947

2^{40}=(2^{10})^4=1048576^2=1099511627776

-9,317

-x\cdot 35 + x^2\cdot 5 = -x\cdot 5\cdot 7 + x\cdot 5\cdot x

31,104

1328/2450 = 1 - \frac{1}{49}\cdot 33\cdot \frac{34}{50}

4,617

2*(3y - z) = 6y - 2z

22,867

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

-30,034

z^{(-1) + \varepsilon} \varepsilon = \frac{\mathrm{d}}{\mathrm{d}z} z^\varepsilon

23,335

x^2 \cdot D^2 - 6 \cdot x \cdot D + 9 = \left(x \cdot D\right)^2 - 7 \cdot x \cdot D + 9 = (x \cdot D - 3.5)^2 - 3.25

5,283

0 + 0 + 1/2 = \dfrac{1}{2}

2,366

(x + 2)/((-1)\cdot x) = \frac{y}{(-1)\cdot (y + 4\cdot (-1))}\Longrightarrow 2\cdot x + 4 = y

-19,689

\frac{1}{8}\cdot 27 = 3\cdot 9/\left(8\right)

27,191

-a/b = -a/b = ((-1)\cdot a)/b = \frac{1}{\left(-1\right)\cdot b}\cdot a

32,424

56/825 = 0.4*\dfrac{8}{4 + 8}*0.4*\frac{8 + (-1)}{4 + 8 + (-1)}

9,045

\cos{3x} = \cos{x \cdot 3}

873

e^z = e^z = 1 + z + \frac{1}{2}z^2 + \dfrac{1}{6}z \cdot z \cdot z + \dots

15,632

-b + (b + (-1)) d = db - d - b

-6,250

\frac{4}{3 \times y + 9} = \frac{1}{3 \times (y + 3)} \times 4

9,105

1^5 + 2^5 + 3^5 + 4^5 + 1^7 + 2^7 + 3^7 + 4^7 = (4 + 1 + 2 + 3)^4\cdot 2

-6,228

\frac{3 \cdot y}{\left(y + (-1)\right) \cdot \left(4 + y\right)} \cdot 25/25 = \frac{75 \cdot y}{25 \cdot (\left(-1\right) + y) \cdot (y + 4)} \cdot 1

-11,935

6.208\cdot 0.001 = \frac{6.208}{1000}

38,454

|\left( 2, -4, 1\right) \cdot \left( -4, 5, -1\right)| = |-8 + 20 \cdot \left(-1\right) + (-1)| = 29

4,307

|x + (-1)|*2 = |-2*x + 2|

9,531

\dfrac{1}{1/b \cdot h} = b/h

29,995

2 \cdot 2^3/3 + \frac12 \cdot 2 \cdot 2 - \frac{14}{6} \cdot 1 = 5.3 + 2 - 2.3 = 5