ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
17,525	22 - 4\cdot (16 + 21\cdot (k + (-1)) + (-1)) = 42 - 84\cdot k = 21\cdot (2 - 4\cdot k)
21,040	\lim_{t \to \infty} \frac1t \cdot (t + 2 \cdot (-1)) = \lim_{t \to \infty} \|t + 2 \cdot (-1)\|/t
18,556	\mathbb{P}(F) \mathbb{P}(A) + 1 - \mathbb{P}(A) - \mathbb{P}(F) = (1 - \mathbb{P}\left(F\right)) (-\mathbb{P}(A) + 1)
21,496	x \cdot x + 2 \cdot x + 1 = \left(x + 1\right)^2
-7,732	\left(40 + 16 i + 100 i + 40 (-1)\right)/29 = \frac{1}{29} (0 + 116 i) = 4 i
2,964	\sin(x + z) = \cos(z) \cdot \sin\left(x\right) + \sin\left(z\right) \cdot \cos(x)
28,143	3 + x^2 - 3x = \left(\left(-1\right) + x\right)(x + (-1)) - x + 2*(-1)
39,548	18.75 = \frac{1}{4} \cdot 75
54,605	1^{\frac{1}{5}} = 1
3,306	9^x\cdot 8 = 9^{1 + x} - 9^x
17,953	\left(q = h\cdot 120 - 205 q \Rightarrow h\cdot 120 = q\cdot 206\right) \Rightarrow h = q\cdot 103/60
-16,346	\sqrt{4 \cdot 7} \cdot 10 = 10 \sqrt{28}
46,173	\frac{1}{30}\cdot 96 = 3.2
21,576	c\cdot g = \dfrac{1}{2}\cdot (c\cdot g + g\cdot c) = g\cdot c
-3,097	\sqrt{6} \cdot 9 = (5 + 4) \sqrt{6}
3,060	r \cdot r \cdot (-\frac{1}{8 \cdot r \cdot r} + 1) + \frac14 = r^2 + 1/8
3,591	\sin\left(J \cdot 2\right) = \sin\left(J\right) \cdot \cos(J) \cdot 2
10,859	\cos{v \cdot x} = \cos{-x \cdot v}
-12,841	\frac{3}{4} = \tfrac{1}{24} \cdot 18
-20,867	\frac{80 \cdot k}{80 + 8 \cdot k} = \frac{10}{k + 10} \cdot k \cdot \frac18 \cdot 8
-23,704	\frac23\dfrac13 = 2/9
-27,731	d/dy (14\cdot \sin{y}) = \cos{y}\cdot 14
32,427	\frac11*0 = \frac02
5,878	\|Fu\| = \|F\|\|u\|
1,598	\binom{m}{m - k} = \frac{m!}{(m - k)!\cdot (m - m - k)!} = \frac{1}{(m - k)!\cdot k!}\cdot m! = \binom{m}{k}
-4,171	\frac{8}{7 \cdot t^4} = \frac{8}{t^4} \cdot 1/7
37,857	-2 = 2 \cdot (-1) + 0
17,837	\frac{1}{x^3 + (-1)} = \left(-\dfrac{x + 2}{x^2 + x + 1} + \frac{1}{(-1) + x}\right)/3
22,622	y \geq z \implies y = z
-19,688	\frac{24}{8} = 8*3/(8)
-563	(e^{i\cdot \pi/2})^{10} = e^{\dfrac{\pi\cdot i}{2}\cdot 10}
3,211	\left(7 = (-1) + 8 \implies 8\cdot M + \left(-1\right) = 7^{1 + n\cdot 2}\right) \implies 1 + 7^{1 + 2\cdot n} = M\cdot 8
-6,710	2/100 + \frac{60}{100} = \dfrac{1}{10}6 + 2/100
8,072	\left(1 + \|z_2\|\right) \|z_1\| = (1 + \|z_1\|) \|z_2\|\Longrightarrow \|z_2\| = \|z_1\|
11,212	{-1 \choose r} = (\left(-1\right)\left(-2\dotsm*(-1 - r + 1)\right))/r! = (-1)^r
26,037	\sqrt{2} \pi/2 = \frac{1}{\sqrt{2}}\pi
20,481	(a\times f)^n = a^n\times f^n = 1 \Rightarrow f^{-n} = a^n
245	\frac{1}{2 + z} = y \implies (1 - 2\cdot y)/y = z
16,133	2860 = {4 \choose 1} \cdot {3 \choose 3} \cdot {13 \choose 4}
23,523	\frac{7}{161} = 1/23
23,592	\frac{1}{21}5*6/22 = \frac{1}{77}5
15,629	\left\lceil{\frac{10}{5 + (-1)}}\right\rceil = \left\lceil{\dfrac{10}{4}}\right\rceil = \left\lceil{2.5}\right\rceil = 3
27,597	\frac{1}{2^{\frac{1}{n}}} = \tfrac{h}{x}\Longrightarrow \frac{x}{h} = 2^{1/n}
-30,652	5 \cdot (-1) - 5 \cdot \lambda^2 = -5 \cdot (1 + \lambda^2)
19,836	1 + y + y^2 + y^3\times \dotsm = \dfrac{1}{-y + 1}
-7,593	(22 + 32 \cdot i + 55 \cdot i + 80 \cdot \left(-1\right))/29 = (-58 + 87 \cdot i)/29 = -2 + 3 \cdot i
29,551	a^4 \cdot z = z = a^3 \cdot z
37,365	\operatorname{atan}\left(-4\right) = x \Rightarrow \tan(x) = -4
7,577	\frac{1}{y^2 + 1}(2y^2 + y) = 1 + \frac{y^2 + y + (-1)}{y^2 + 1} = 1 + \frac{2y^2 + 2y + 2(-1)}{2\left(y * y + 1\right)}
8,855	\frac{1}{13}*\left(3^{x + 1} + 3^x + 3^{(-1) + x}\right) = 3^{(-1) + x}
32,686	6y'^2 v + 3v^2 y'' + 6x = 0 \Rightarrow vy' * y'2 + v^2 y'' + x2 = 0
22,571	\lim_{l \to \infty} \sin\left(y_l\right) = 0 \neq 1 = \lim_{l \to \infty} \sin\left(y_l\right)
35,278	(-X)^m = X^m = -X^m
-5,012	3.6\times 10^{2 + 0} = 10 \times 10\times 3.6
18,469	\tfrac{1}{(1 - x)^2}\cdot (1 + x^2 + x) = \frac{\left(1 - x\right)^2 + 3\cdot x}{(1 - x)^2} = 1 + \dfrac{3}{(1 - x)^2}\cdot x
1,520	\sqrt{625/2 + \dfrac{1}{2}\cdot 625} = \sqrt{625} = 25
-10,719	\dfrac{3}{3}\cdot \left(-\dfrac{1}{16\cdot \left(-1\right) + s\cdot 4}\cdot 6\right) = -\frac{18}{s\cdot 12 + 48\cdot (-1)}
15,050	128^{1/2} = (11^2 + 7)^{1/2} \approx 11 + \dfrac{7}{22}
26,402	\frac{3}{2} \cdot \frac{1}{2}/2 = 3/8
10,185	a^{b \times b} = a^{b^2}
15,877	(x \cdot x - x \cdot z + z^2) \cdot (x + z) = z \cdot z \cdot z + x^3
9,896	\dfrac{1}{-\dfrac{1}{f} + \frac{1}{f - 1/b}} = -f + bff
8,815	x \lt y \implies \frac{x}{2} \lt \frac{y}{2}
54,750	799 = 17\cdot 47
-30,547	15/30 = 30/60 = \dfrac{60}{120} = \frac{1}{2}
12,951	1 + x^4 = (x^3 - x^2 + x + (-1))*(1 + x) + 2
-8,363	-\frac{1}{-4}*24 = 6
17,756	2^{\frac{1}{2}} = 1.414213562373 \times \dots
-5,573	\frac{2\cdot z}{z^2 - z + 2\cdot (-1)}\cdot 1 = \tfrac{2\cdot z}{(1 + z)\cdot \left(z + 2\cdot (-1)\right)}
874	\frac{\partial}{\partial Z_2} Z_1 = \frac{1}{Z_2 + Z_1}\cdot (3\cdot Z_2 - Z_1) = \dfrac{1}{1 + Z_1/(Z_2)}\cdot (3 - \dfrac{1}{Z_2}\cdot Z_1)
-4,681	(1 + z) (z + 5) = z^2 + 6z + 5
15,010	m + {m \choose 2} \cdot 2 = m \cdot m
21,014	z^3 + z*3 + 2 (-1) = 6 (-1) + \left(z + 1\right) (z^2 - z + 4)
29,919	\frac{r_3}{(x + (-1))^2} + \frac{1}{(-1) + x} \cdot r_2 = \frac{r_3 + r_2 \cdot (\left(-1\right) + x)}{((-1) + x) \cdot ((-1) + x)}
2,336	\frac{10}{24} = (6 + 4)/24
-2,888	\sqrt{3} \cdot (1 + 4) = \sqrt{3} \cdot 5
2,324	y^VAy = (y^VAy)^V = y^VA^Vy = -y^VAy \Rightarrow yAy^V = 0
-26,544	100 - 9\cdot z \cdot z = 10^2 - (z\cdot 3) \cdot (z\cdot 3)
-5,785	\frac{1}{20 + 4 \cdot y} \cdot 2 = \tfrac{2}{4 \cdot \left(y + 5\right)}
44,865	\overline{0} = \overline{0} \overline{4}
25,876	-\dfrac{1}{x_n \cdot 2} \cdot \left(2 \cdot \left(-1\right) + x_n^2\right) + x_n = (x_n + \frac{2}{x_n})/2
11,891	\frac{Z}{b} \cdot a^{1/2} = \frac{Z \cdot x}{( x^2 - a, b)} \cdot 1 = \frac{Z}{x^2 - a} \cdot \frac{1}{b \cdot x}
38,771	7 = -\frac{10}{2} + 12
-9,123	2 \cdot 2 \cdot 3 \cdot 3 \cdot p - 2 \cdot 2 \cdot 3 = 36 \cdot p + 12 \cdot (-1)
10,440	\binom{n + 3 + (-1)}{3 + (-1)} = \binom{n + 2}{2} = \tfrac12 \cdot \left(n + 1\right) \cdot \left(n + 2\right)
5,965	\sum_{n=1}^\infty c \cdot (2 \cdot (-1) - 1)^n \cdot n = \sum_{n=1}^\infty c \cdot n \cdot (-3)^n
-1,822	\pi\cdot \dfrac{1}{3}\cdot 4 = \pi\cdot 13/12 + \dfrac{\pi}{4}
10,775	\frac{1}{x + (-1)} \cdot (x^2 \cdot x + (-1)) = x^2 + x + 1
-16,492	8 \cdot 275^{1/2} = 8 \cdot (25 \cdot 11)^{1/2}
6,311	\sin(x + u) = \sin{u} \cos{x} + \sin{x} \cos{u}
41,105	1000\cdot \left(-1\right) + 1 = -999
9,844	( x + w, x + w) = \left\{w\right\} \implies 0 = [x,w]
4,685	6/y \leq -8 \implies y \geq -\frac{6}{8} = -\frac143
6,783	z + 2\times (-1) = 4\times (z + 2) = 4\times z + 8
37,731	E^2 = E^2
-23,150	-4 = 3 (-4/3)
-3,605	\tfrac{40\cdot n^2}{32\cdot n} = 40/32\cdot \dfrac{n^2}{n}
16,780	\frac{e^{-9} 9^9}{9!} = \frac{e^{-9} 9^8 * 9}{8! * 9}
40,547	\cos{4x} = \cos^2{2x} - \sin^2{2x} = 2\cos^2{2*x} + (-1)
874	\frac{\mathrm{d}x}{\mathrm{d}U} = \frac{3\cdot U - x}{U + x} = \tfrac{1}{1 + \dfrac{x}{U}}\cdot \left(3 - \frac1U\cdot x\right)

17,525

22 - 4\cdot (16 + 21\cdot (k + (-1)) + (-1)) = 42 - 84\cdot k = 21\cdot (2 - 4\cdot k)

21,040

\lim_{t \to \infty} \frac1t \cdot (t + 2 \cdot (-1)) = \lim_{t \to \infty} |t + 2 \cdot (-1)|/t

18,556

\mathbb{P}(F) \mathbb{P}(A) + 1 - \mathbb{P}(A) - \mathbb{P}(F) = (1 - \mathbb{P}\left(F\right)) (-\mathbb{P}(A) + 1)

21,496

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

-7,732

\left(40 + 16 i + 100 i + 40 (-1)\right)/29 = \frac{1}{29} (0 + 116 i) = 4 i

2,964

\sin(x + z) = \cos(z) \cdot \sin\left(x\right) + \sin\left(z\right) \cdot \cos(x)

28,143

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

39,548

18.75 = \frac{1}{4} \cdot 75

54,605

1^{\frac{1}{5}} = 1

3,306

9^x\cdot 8 = 9^{1 + x} - 9^x

17,953

\left(q = h\cdot 120 - 205 q \Rightarrow h\cdot 120 = q\cdot 206\right) \Rightarrow h = q\cdot 103/60

-16,346

\sqrt{4 \cdot 7} \cdot 10 = 10 \sqrt{28}

46,173

\frac{1}{30}\cdot 96 = 3.2

21,576

c\cdot g = \dfrac{1}{2}\cdot (c\cdot g + g\cdot c) = g\cdot c

-3,097

\sqrt{6} \cdot 9 = (5 + 4) \sqrt{6}

3,060

r \cdot r \cdot (-\frac{1}{8 \cdot r \cdot r} + 1) + \frac14 = r^2 + 1/8

3,591

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

10,859

\cos{v \cdot x} = \cos{-x \cdot v}

-12,841

\frac{3}{4} = \tfrac{1}{24} \cdot 18

-20,867

\frac{80 \cdot k}{80 + 8 \cdot k} = \frac{10}{k + 10} \cdot k \cdot \frac18 \cdot 8

-23,704

\frac23\dfrac13 = 2/9

-27,731

d/dy (14\cdot \sin{y}) = \cos{y}\cdot 14

32,427

\frac11*0 = \frac02

5,878

|F*u| = |F|*|u|

1,598

\binom{m}{m - k} = \frac{m!}{(m - k)!\cdot (m - m - k)!} = \frac{1}{(m - k)!\cdot k!}\cdot m! = \binom{m}{k}

-4,171

\frac{8}{7 \cdot t^4} = \frac{8}{t^4} \cdot 1/7

37,857

-2 = 2 \cdot (-1) + 0

17,837

\frac{1}{x^3 + (-1)} = \left(-\dfrac{x + 2}{x^2 + x + 1} + \frac{1}{(-1) + x}\right)/3

22,622

y \geq z \implies y = z

-19,688

\frac{24}{8} = 8*3/(8)

-563

(e^{i\cdot \pi/2})^{10} = e^{\dfrac{\pi\cdot i}{2}\cdot 10}

3,211

\left(7 = (-1) + 8 \implies 8\cdot M + \left(-1\right) = 7^{1 + n\cdot 2}\right) \implies 1 + 7^{1 + 2\cdot n} = M\cdot 8

-6,710

2/100 + \frac{60}{100} = \dfrac{1}{10}6 + 2/100

8,072

\left(1 + |z_2|\right) |z_1| = (1 + |z_1|) |z_2|\Longrightarrow |z_2| = |z_1|

11,212

{-1 \choose r} = (\left(-1\right)*\left(-2*\dotsm*(-1 - r + 1)\right))/r! = (-1)^r

26,037

\sqrt{2} \pi/2 = \frac{1}{\sqrt{2}}\pi

20,481

(a\times f)^n = a^n\times f^n = 1 \Rightarrow f^{-n} = a^n

245

\frac{1}{2 + z} = y \implies (1 - 2\cdot y)/y = z

16,133

2860 = {4 \choose 1} \cdot {3 \choose 3} \cdot {13 \choose 4}

23,523

\frac{7}{161} = 1/23

23,592

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

15,629

\left\lceil{\frac{10}{5 + (-1)}}\right\rceil = \left\lceil{\dfrac{10}{4}}\right\rceil = \left\lceil{2.5}\right\rceil = 3

27,597

\frac{1}{2^{\frac{1}{n}}} = \tfrac{h}{x}\Longrightarrow \frac{x}{h} = 2^{1/n}

-30,652

5 \cdot (-1) - 5 \cdot \lambda^2 = -5 \cdot (1 + \lambda^2)

19,836

1 + y + y^2 + y^3\times \dotsm = \dfrac{1}{-y + 1}

-7,593

(22 + 32 \cdot i + 55 \cdot i + 80 \cdot \left(-1\right))/29 = (-58 + 87 \cdot i)/29 = -2 + 3 \cdot i

29,551

a^4 \cdot z = z = a^3 \cdot z

37,365

\operatorname{atan}\left(-4\right) = x \Rightarrow \tan(x) = -4

7,577

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

8,855

\frac{1}{13}*\left(3^{x + 1} + 3^x + 3^{(-1) + x}\right) = 3^{(-1) + x}

32,686

6y'^2 v + 3v^2 y'' + 6x = 0 \Rightarrow vy' * y'*2 + v^2 y'' + x*2 = 0

22,571

\lim_{l \to \infty} \sin\left(y_l\right) = 0 \neq 1 = \lim_{l \to \infty} \sin\left(y_l\right)

35,278

(-X)^m = X^m = -X^m

-5,012

3.6\times 10^{2 + 0} = 10 \times 10\times 3.6

18,469

\tfrac{1}{(1 - x)^2}\cdot (1 + x^2 + x) = \frac{\left(1 - x\right)^2 + 3\cdot x}{(1 - x)^2} = 1 + \dfrac{3}{(1 - x)^2}\cdot x

1,520

\sqrt{625/2 + \dfrac{1}{2}\cdot 625} = \sqrt{625} = 25

-10,719

\dfrac{3}{3}\cdot \left(-\dfrac{1}{16\cdot \left(-1\right) + s\cdot 4}\cdot 6\right) = -\frac{18}{s\cdot 12 + 48\cdot (-1)}

15,050

128^{1/2} = (11^2 + 7)^{1/2} \approx 11 + \dfrac{7}{22}

26,402

\frac{3}{2} \cdot \frac{1}{2}/2 = 3/8

10,185

a^{b \times b} = a^{b^2}

15,877

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

9,896

\dfrac{1}{-\dfrac{1}{f} + \frac{1}{f - 1/b}} = -f + b*f*f

8,815

x \lt y \implies \frac{x}{2} \lt \frac{y}{2}

54,750

799 = 17\cdot 47

-30,547

15/30 = 30/60 = \dfrac{60}{120} = \frac{1}{2}

12,951

1 + x^4 = (x^3 - x^2 + x + (-1))*(1 + x) + 2

-8,363

-\frac{1}{-4}*24 = 6

17,756

2^{\frac{1}{2}} = 1.414213562373 \times \dots

-5,573

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

874

\frac{\partial}{\partial Z_2} Z_1 = \frac{1}{Z_2 + Z_1}\cdot (3\cdot Z_2 - Z_1) = \dfrac{1}{1 + Z_1/(Z_2)}\cdot (3 - \dfrac{1}{Z_2}\cdot Z_1)

-4,681

(1 + z) (z + 5) = z^2 + 6z + 5

15,010

m + {m \choose 2} \cdot 2 = m \cdot m

21,014

z^3 + z*3 + 2 (-1) = 6 (-1) + \left(z + 1\right) (z^2 - z + 4)

29,919

\frac{r_3}{(x + (-1))^2} + \frac{1}{(-1) + x} \cdot r_2 = \frac{r_3 + r_2 \cdot (\left(-1\right) + x)}{((-1) + x) \cdot ((-1) + x)}

2,336

\frac{10}{24} = (6 + 4)/24

-2,888

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

2,324

y^V*A*y = (y^V*A*y)^V = y^V*A^V*y = -y^V*A*y \Rightarrow y*A*y^V = 0

-26,544

100 - 9\cdot z \cdot z = 10^2 - (z\cdot 3) \cdot (z\cdot 3)

-5,785

\frac{1}{20 + 4 \cdot y} \cdot 2 = \tfrac{2}{4 \cdot \left(y + 5\right)}

44,865

\overline{0} = \overline{0} \overline{4}

25,876

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

11,891

\frac{Z}{b} \cdot a^{1/2} = \frac{Z \cdot x}{( x^2 - a, b)} \cdot 1 = \frac{Z}{x^2 - a} \cdot \frac{1}{b \cdot x}

38,771

7 = -\frac{10}{2} + 12

-9,123

2 \cdot 2 \cdot 3 \cdot 3 \cdot p - 2 \cdot 2 \cdot 3 = 36 \cdot p + 12 \cdot (-1)

10,440

\binom{n + 3 + (-1)}{3 + (-1)} = \binom{n + 2}{2} = \tfrac12 \cdot \left(n + 1\right) \cdot \left(n + 2\right)

5,965

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

-1,822

\pi\cdot \dfrac{1}{3}\cdot 4 = \pi\cdot 13/12 + \dfrac{\pi}{4}

10,775

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

-16,492

8 \cdot 275^{1/2} = 8 \cdot (25 \cdot 11)^{1/2}

6,311

\sin(x + u) = \sin{u} \cos{x} + \sin{x} \cos{u}

41,105

1000\cdot \left(-1\right) + 1 = -999

9,844

( x + w, x + w) = \left\{w\right\} \implies 0 = [x,w]

4,685

6/y \leq -8 \implies y \geq -\frac{6}{8} = -\frac143

6,783

z + 2\times (-1) = 4\times (z + 2) = 4\times z + 8

37,731

E^2 = E^2

-23,150

-4 = 3 (-4/3)

-3,605

\tfrac{40\cdot n^2}{32\cdot n} = 40/32\cdot \dfrac{n^2}{n}

16,780

\frac{e^{-9} 9^9}{9!} = \frac{e^{-9} 9^8 * 9}{8! * 9}

40,547

\cos{4*x} = \cos^2{2*x} - \sin^2{2*x} = 2*\cos^2{2*x} + (-1)

874

\frac{\mathrm{d}x}{\mathrm{d}U} = \frac{3\cdot U - x}{U + x} = \tfrac{1}{1 + \dfrac{x}{U}}\cdot \left(3 - \frac1U\cdot x\right)