ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
23,031	\left(\frac{1}{b^2} \cdot a^2 = 2 rightarrow a^2 = b^2 \cdot 2\right) rightarrow a = 2^{1/2} \cdot b
-13,343	\dfrac{1}{9 + 3\left(-1\right)}12 = \frac{12}{6} = 12/6 = 2
13,586	{4 \choose 2}\cdot {12 \choose 3} = {23 \choose 3} - {13 \choose 3} - {11 \choose 3}
25,713	x^l = {x \choose x} \cdot x^l
27,240	h - d = (\sqrt{h} - \sqrt{d}) (\sqrt{d} + \sqrt{h})
32,244	3 \cdot a + 3 \cdot x = 3 \cdot (a + x)
19,857	x \cdot x \cdot \alpha^2 = \alpha \cdot \alpha \cdot x^2
13,524	y^2 + 4\cdot y + 3\cdot (-1) = (1 - 2\cdot y)^2 = 1 - 4\cdot y + 4\cdot y^2
36,639	884 + k = 891 rightarrow k = 7
26,715	1 + z^3 = (1 + z)\times (z^2 - z + 1)
24,960	(10*10897 + 3)^n = 108973^n
-11,955	13/15 = \tfrac{1}{12 \cdot \pi} \cdot s \cdot 12 \cdot \pi = s
37,062	x = B \cup G \setminus B \cap G = Bx \cup Gx
7,851	1 + 2^{33} = \left(2^{11} + 1\right)\cdot \left(1 + 2^{22} - 2^{11}\right)
-20,393	\dfrac{1}{70p + 35}(-p14 + 56(-1)) = \frac{-p2 + 8(-1)}{p10 + 5}7/7
9,458	\sin(y) = \frac{1}{1 + \frac{y^2}{23 - y y + \ldots}}*y
22,876	E' = C^c \implies E' = C
14,781	680 = 5^12 2 * 2*17^1
-26,470	y\cdot 16 = y\cdot 8\cdot 2
-12,064	7/9 = \frac{x}{14 \cdot \pi} \cdot 14 \cdot \pi = x
6,279	\sin^2{y} = (2\cos{y} + \sin{y})\left(\sin{y} - \cos{y}*2\right)/5 + 4/5
-23,017	117/(411) = 77/44
-5,709	\frac{4}{4(9 + k)} = \frac{1}{36 + k4}*4
10,188	\sin\left(\tan^{-1}{x}2\right) = \dfrac{x2}{x^2 + 1}
27,553	\binom{5}{3} \binom{20}{1}1918 = 68400
-20,458	\dfrac{1 - n}{1 - n} (-9/2) = \frac{1}{-2n + 2}(9\left(-1\right) + 9n)
15,714	\cos(\tan^{-1}{y}) = \dfrac{1}{\sqrt{1 + y^2}}
23,546	\int x\,dx = \frac{1/2}{x} \cdot x^2 = \frac{x}{2}
1,274	\frac{1}{a} = \dfrac1a := \dfrac1a
48,340	-g = -g
7,957	\cos{C} = \sin{C\cdot 2}/(2\cdot \sin{C})
25,442	\frac{1}{g^2} = (1/g)^2
-20,792	\dfrac{1}{16 - 48 m}((-40) m) = \frac{\left(-5\right) m}{-6m + 2}*8/8
-21,035	\tfrac{42 \cdot (-1) + 7 \cdot x}{28 \cdot (-1) - x \cdot 35} = 7/7 \cdot \frac{1}{-5 \cdot x + 4 \cdot (-1)} \cdot (x + 6 \cdot (-1))
4,773	(x + y)^2 = x \cdot x + y^2 + 2 \cdot x \cdot y \leq 4 \cdot x \cdot y
13,377	\tfrac{z^2}{(z^2 + f^2)^2} = z\frac{z}{(f^2 + z z)^2}
-27,677	\sin\left(2\alpha\right) = 2\cos(\alpha)*\sin(\alpha)
-20,096	\frac{1/7 \cdot 7}{a + 6 \cdot (-1)} = \frac{7}{7 \cdot a + 42 \cdot (-1)}
19,841	-0.5 = 7 - \frac{15}{2} \neq \sqrt{(7 - 15/2)^2} = \|7 - \frac{15}{2}\| = 0.5
29,878	mx^{(-1) + m} = \frac{\partial}{\partial x} x^m
30,401	S_n - \sum_{k=1}^n \mathbb{E}\left(X_k\right) = S_n + S_n - \sum_{k=1}^n \mathbb{E}\left(X_k\right)
14,654	7^2 + 2 * 2 = 53
30,144	2\cdot 3\cdot 3/2 - \frac13\cdot 16 - \frac{1}{3}\cdot 2 = 9 + 6\cdot (-1) = 3
23,668	b + c + c = 2\cdot c + b
-10,399	10 = -5\cdot e + 9\cdot (-1) + 5\cdot \left(-1\right) = -5\cdot e + 14\cdot (-1)
9,126	2 + \frac{1}{2}(n + 2\left(-1\right))(3(-1) + n) = \dfrac12(n^2 - n5 + 10)
12,249	m^4 + 4\cdot n^4 = (m^2)^2 + \left(2\cdot n \cdot n\right)^2 = (m^2 + 2\cdot n^2)^2 - (2\cdot m\cdot n)^2
9,091	\frac{b^m}{z}\cdot z = (b/z\cdot z)^m
11,112	5\cdot 3 - 5\cdot 2 = 5\cdot \left(2\cdot \left(-1\right) + 3\right)
20,442	a^2 + 5 = c^2 + 5 \implies a^2 = c^2
-1,546	\frac{5}{2} = \dfrac{5}{2}
-28,895	\dfrac12 = \tfrac{1}{2 + 3 + 1}*3
-13,821	\frac{1}{6 + 3 \cdot \left(-1\right)} \cdot 24 = 24/3 = \frac{1}{3} \cdot 24 = 8
-15,196	\frac{1}{\frac{1}{t^{25}}\frac{1}{y^5}}t^6 = \frac{t^6t^{25}}{\tfrac{1}{y^5}}1 = t^{6 - -25}y^5 = t^{31}y^5
32,172	\sin{\frac{1}{m}}/\left(1/m\right) = m \times \sin{\frac{1}{m}}
2,932	666^2 + 4667 4667 = 22224445
10,011	-\sin{U} = \sin{-U}
5,147	(1 + 3\cdot x)^4\cdot (1 + 18\cdot x) = \left(x\cdot 15 + 1 + 3\cdot x\right)\cdot (1 + 3\cdot x)^4
49,328	( -z_x, -z_y, 1) = \Phi_y\cdot \Phi_x\Longrightarrow \|\Phi_x\cdot \Phi_y\| = \sqrt{1 + z_x^2 + z_y^2}
14,435	\bar{q}x = f = qf/x*x
-9,472	2\times 2\times 2\times 2\times 3 - 2\times 3\times 3\times 3\times n = 48 - n\times 54
10,045	\frac{1}{1/h} \cdot (-z_l + \frac{1}{h}) = 1 - z_l \cdot h
24,060	D \cup (Y \cap D^c) = (D \cup Y) \cap (D \cup D^c) = (D \cup Y) \cap (D \cup D^c)
20,358	\frac{1}{a^4} + a^4 = (a^2)^2 + (\dfrac{1}{a^2})^2
7,262	\binom{s}{i} = \tfrac{1}{i! (s - i)!}s!
3,462	G = G - eG + eG = \left(1 - e\right) G + eG
24,017	1/27 + \frac{12}{27} + \dfrac{6}{27} = \dfrac{19}{27}
33,921	6 * 6 * 6 + \left(z5\right)^3 = 216 + 125z^3
-2,685	(163)^{1/2} - (93)^{1/2} = -27^{1/2} + 48^{1/2}
-10,150	1^{-1} = 5/5
8,116	(a + k + k \cdot k - a^2)/2 = \frac12 \cdot (\left(k - a\right) \cdot (k + a) + k + a)
34,156	1 - \sin(\pi/2 - z) = 1 - \cos{z} = 2 \cdot \sin^2{\frac{z}{2}}
-25,053	\frac{5}{10} \cdot 4/9 = \frac{20}{90} = 2/9
-10,874	20 = \dfrac{140}{7}
15,247	\frac{\mathrm{d}}{\mathrm{d}z} z^{-\dfrac15} = -\tfrac{1}{5\cdot z^{\frac{6}{5}}}
-212	\frac{10!}{\left(10 + 4\cdot (-1)\right)!} = 10\cdot 9\cdot 8\cdot 7
14,804	\frac{4 / 3}{18} 1 + \dfrac19 = 5/27
24,624	-\frac{1}{2}(8! - 7!) + 9!/(2!*3!) = 12600
22,620	(-b + a)\cdot (a^{\left(-1\right) + n} + a^{n + 2\cdot (-1)}\cdot b + a^{n + 3\cdot \left(-1\right)}\cdot b \cdot b + \dotsm + a^2\cdot b^{3\cdot (-1) + n} + a\cdot b^{n + 2\cdot (-1)} + b^{n + (-1)}) = -b^n + a^n
12,930	\frac13\cdot (\left(-1\right)\cdot 2\cdot \pi) = \frac{\pi}{3} - \pi
-3,406	99^{1/2} + 44^{1/2} = \left(9\cdot 11\right)^{1/2} + \left(4\cdot 11\right)^{1/2}
5,271	\sum_{m=1}^n (-(1 + (-1)) + 1) = \sum_{m=1}^n \left(1 + 0 (-1)\right)
26,290	-(18 + y^2) + y\cdot 3 + 6 = 0 \implies 0 = 12 + y^2 - 3\cdot y
18,909	16 = \frac45\cdot \dfrac{60}{2}\cdot 2/3
34,570	3\cdot a^2 - 2\cdot a\cdot b + b^2\cdot 3 = (a - b)^2 + 2\cdot a \cdot a + 2\cdot b^2
35,443	(-g + y)^2 + c - g \cdot g = y^2 - y \cdot g \cdot 2 + c
12,190	4575 \cdot \left(-1\right) + 12000 = 7425
-17,510	22\times (-1) + 25 = 3
-20,695	\frac{30 (-1) + 5z}{48 \left(-1\right) + 8z} = \frac185 \tfrac{z + 6(-1)}{z + 6(-1)}
10,677	A b = b A
16,503	41^4 + z^4 = -\left(z2\right)^2 + z^4 + 2*2 z z + 2^2
28,578	(\cos(-q + t) - \cos(t + q))/2 = \sin(t) \sin(q)
-9,927	18\% = \dfrac{18}{100} = \frac{1}{50}\times 9
34,899	\left(\frac{1}{y}\right)^2 = \frac{1}{y^2}
30,959	(\left(-1\right) + x)^2 + 1 = 2 + x^2 - x \cdot 2
1,045	\dfrac19\cdot (2^{3^{1 + n}} + 1) = (1 + 2^{3^n}\cdot 2^{3^n}\cdot 2^{3^n})/9
2,294	3^22^53^44^32^1 = 2^63^64 * 4 * 4
16,322	(n + 2) \cdot (2 \cdot n + 3) \cdot (1 + n)/6 = 1 \cdot 1 + \ldots + n^2 + (1 + n)^2
16,067	5 \cdot 5 \cdot 5 = (1^2 + 2^2)\cdot 5^2
-3,058	-\sqrt{4}\sqrt{7} + \sqrt{25}\sqrt{7} = \sqrt{7}5 - 2\sqrt{7}

23,031

\left(\frac{1}{b^2} \cdot a^2 = 2 rightarrow a^2 = b^2 \cdot 2\right) rightarrow a = 2^{1/2} \cdot b

-13,343

\dfrac{1}{9 + 3*\left(-1\right)}*12 = \frac{12}{6} = 12/6 = 2

13,586

{4 \choose 2}\cdot {12 \choose 3} = {23 \choose 3} - {13 \choose 3} - {11 \choose 3}

25,713

x^l = {x \choose x} \cdot x^l

27,240

h - d = (\sqrt{h} - \sqrt{d}) (\sqrt{d} + \sqrt{h})

32,244

3 \cdot a + 3 \cdot x = 3 \cdot (a + x)

19,857

x \cdot x \cdot \alpha^2 = \alpha \cdot \alpha \cdot x^2

13,524

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

36,639

884 + k = 891 rightarrow k = 7

26,715

1 + z^3 = (1 + z)\times (z^2 - z + 1)

24,960

(10*10897 + 3)^n = 108973^n

-11,955

13/15 = \tfrac{1}{12 \cdot \pi} \cdot s \cdot 12 \cdot \pi = s

37,062

x = B \cup G \setminus B \cap G = B*x \cup G*x

7,851

1 + 2^{33} = \left(2^{11} + 1\right)\cdot \left(1 + 2^{22} - 2^{11}\right)

-20,393

\dfrac{1}{70*p + 35}*(-p*14 + 56*(-1)) = \frac{-p*2 + 8*(-1)}{p*10 + 5}*7/7

9,458

\sin(y) = \frac{1}{1 + \frac{y^2}{2*3 - y * y + \ldots}}*y

22,876

E' = C^c \implies E' = C

14,781

680 = 5^1*2 * 2 * 2*17^1

-26,470

y\cdot 16 = y\cdot 8\cdot 2

-12,064

7/9 = \frac{x}{14 \cdot \pi} \cdot 14 \cdot \pi = x

6,279

\sin^2{y} = (2*\cos{y} + \sin{y})*\left(\sin{y} - \cos{y}*2\right)/5 + 4/5

-23,017

11*7/(4*11) = 77/44

-5,709

\frac{4}{4*(9 + k)} = \frac{1}{36 + k*4}*4

10,188

\sin\left(\tan^{-1}{x}*2\right) = \dfrac{x*2}{x^2 + 1}

27,553

\binom{5}{3} \binom{20}{1}*19*18 = 68400

-20,458

\dfrac{1 - n}{1 - n} (-9/2) = \frac{1}{-2n + 2}(9\left(-1\right) + 9n)

15,714

\cos(\tan^{-1}{y}) = \dfrac{1}{\sqrt{1 + y^2}}

23,546

\int x\,dx = \frac{1/2}{x} \cdot x^2 = \frac{x}{2}

1,274

\frac{1}{a} = \dfrac1a := \dfrac1a

48,340

-g = -g

7,957

\cos{C} = \sin{C\cdot 2}/(2\cdot \sin{C})

25,442

\frac{1}{g^2} = (1/g)^2

-20,792

\dfrac{1}{16 - 48 m}((-40) m) = \frac{\left(-5\right) m}{-6m + 2}*8/8

-21,035

\tfrac{42 \cdot (-1) + 7 \cdot x}{28 \cdot (-1) - x \cdot 35} = 7/7 \cdot \frac{1}{-5 \cdot x + 4 \cdot (-1)} \cdot (x + 6 \cdot (-1))

4,773

(x + y)^2 = x \cdot x + y^2 + 2 \cdot x \cdot y \leq 4 \cdot x \cdot y

13,377

\tfrac{z^2}{(z^2 + f^2)^2} = z*\frac{z}{(f^2 + z * z)^2}

-27,677

\sin\left(2*\alpha\right) = 2*\cos(\alpha)*\sin(\alpha)

-20,096

\frac{1/7 \cdot 7}{a + 6 \cdot (-1)} = \frac{7}{7 \cdot a + 42 \cdot (-1)}

19,841

-0.5 = 7 - \frac{15}{2} \neq \sqrt{(7 - 15/2)^2} = |7 - \frac{15}{2}| = 0.5

29,878

mx^{(-1) + m} = \frac{\partial}{\partial x} x^m

30,401

S_n - \sum_{k=1}^n \mathbb{E}\left(X_k\right) = S_n + S_n - \sum_{k=1}^n \mathbb{E}\left(X_k\right)

14,654

7^2 + 2 * 2 = 53

30,144

2\cdot 3\cdot 3/2 - \frac13\cdot 16 - \frac{1}{3}\cdot 2 = 9 + 6\cdot (-1) = 3

23,668

b + c + c = 2\cdot c + b

-10,399

10 = -5\cdot e + 9\cdot (-1) + 5\cdot \left(-1\right) = -5\cdot e + 14\cdot (-1)

9,126

2 + \frac{1}{2}*(n + 2*\left(-1\right))*(3*(-1) + n) = \dfrac12*(n^2 - n*5 + 10)

12,249

m^4 + 4\cdot n^4 = (m^2)^2 + \left(2\cdot n \cdot n\right)^2 = (m^2 + 2\cdot n^2)^2 - (2\cdot m\cdot n)^2

9,091

\frac{b^m}{z}\cdot z = (b/z\cdot z)^m

11,112

5\cdot 3 - 5\cdot 2 = 5\cdot \left(2\cdot \left(-1\right) + 3\right)

20,442

a^2 + 5 = c^2 + 5 \implies a^2 = c^2

-1,546

\frac{5}{2} = \dfrac{5}{2}

-28,895

\dfrac12 = \tfrac{1}{2 + 3 + 1}*3

-13,821

\frac{1}{6 + 3 \cdot \left(-1\right)} \cdot 24 = 24/3 = \frac{1}{3} \cdot 24 = 8

-15,196

\frac{1}{\frac{1}{t^{25}}*\frac{1}{y^5}}*t^6 = \frac{t^6*t^{25}}{\tfrac{1}{y^5}}*1 = t^{6 - -25}*y^5 = t^{31}*y^5

32,172

\sin{\frac{1}{m}}/\left(1/m\right) = m \times \sin{\frac{1}{m}}

2,932

666^2 + 4667 4667 = 22224445

10,011

-\sin{U} = \sin{-U}

5,147

(1 + 3\cdot x)^4\cdot (1 + 18\cdot x) = \left(x\cdot 15 + 1 + 3\cdot x\right)\cdot (1 + 3\cdot x)^4

49,328

( -z_x, -z_y, 1) = \Phi_y\cdot \Phi_x\Longrightarrow |\Phi_x\cdot \Phi_y| = \sqrt{1 + z_x^2 + z_y^2}

14,435

\bar{q}*x = f = q*f/x*x

-9,472

2\times 2\times 2\times 2\times 3 - 2\times 3\times 3\times 3\times n = 48 - n\times 54

10,045

\frac{1}{1/h} \cdot (-z_l + \frac{1}{h}) = 1 - z_l \cdot h

24,060

D \cup (Y \cap D^c) = (D \cup Y) \cap (D \cup D^c) = (D \cup Y) \cap (D \cup D^c)

20,358

\frac{1}{a^4} + a^4 = (a^2)^2 + (\dfrac{1}{a^2})^2

7,262

\binom{s}{i} = \tfrac{1}{i! (s - i)!}s!

3,462

G = G - eG + eG = \left(1 - e\right) G + eG

24,017

1/27 + \frac{12}{27} + \dfrac{6}{27} = \dfrac{19}{27}

33,921

6 * 6 * 6 + \left(z*5\right)^3 = 216 + 125*z^3

-2,685

(16*3)^{1/2} - (9*3)^{1/2} = -27^{1/2} + 48^{1/2}

-10,150

1^{-1} = 5/5

8,116

(a + k + k \cdot k - a^2)/2 = \frac12 \cdot (\left(k - a\right) \cdot (k + a) + k + a)

34,156

1 - \sin(\pi/2 - z) = 1 - \cos{z} = 2 \cdot \sin^2{\frac{z}{2}}

-25,053

\frac{5}{10} \cdot 4/9 = \frac{20}{90} = 2/9

-10,874

20 = \dfrac{140}{7}

15,247

\frac{\mathrm{d}}{\mathrm{d}z} z^{-\dfrac15} = -\tfrac{1}{5\cdot z^{\frac{6}{5}}}

-212

\frac{10!}{\left(10 + 4\cdot (-1)\right)!} = 10\cdot 9\cdot 8\cdot 7

14,804

\frac{4 / 3}{18} 1 + \dfrac19 = 5/27

24,624

-\frac{1}{2}(8! - 7!) + 9!/(2!*3!) = 12600

22,620

(-b + a)\cdot (a^{\left(-1\right) + n} + a^{n + 2\cdot (-1)}\cdot b + a^{n + 3\cdot \left(-1\right)}\cdot b \cdot b + \dotsm + a^2\cdot b^{3\cdot (-1) + n} + a\cdot b^{n + 2\cdot (-1)} + b^{n + (-1)}) = -b^n + a^n

12,930

\frac13\cdot (\left(-1\right)\cdot 2\cdot \pi) = \frac{\pi}{3} - \pi

-3,406

99^{1/2} + 44^{1/2} = \left(9\cdot 11\right)^{1/2} + \left(4\cdot 11\right)^{1/2}

5,271

\sum_{m=1}^n (-(1 + (-1)) + 1) = \sum_{m=1}^n \left(1 + 0 (-1)\right)

26,290

-(18 + y^2) + y\cdot 3 + 6 = 0 \implies 0 = 12 + y^2 - 3\cdot y

18,909

16 = \frac45\cdot \dfrac{60}{2}\cdot 2/3

34,570

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

35,443

(-g + y)^2 + c - g \cdot g = y^2 - y \cdot g \cdot 2 + c

12,190

4575 \cdot \left(-1\right) + 12000 = 7425

-17,510

22\times (-1) + 25 = 3

-20,695

\frac{30 (-1) + 5z}{48 \left(-1\right) + 8z} = \frac185 \tfrac{z + 6(-1)}{z + 6(-1)}

10,677

A b = b A

16,503

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

28,578

(\cos(-q + t) - \cos(t + q))/2 = \sin(t) \sin(q)

-9,927

18\% = \dfrac{18}{100} = \frac{1}{50}\times 9

34,899

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

30,959

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

1,045

\dfrac19\cdot (2^{3^{1 + n}} + 1) = (1 + 2^{3^n}\cdot 2^{3^n}\cdot 2^{3^n})/9

2,294

3^2*2^5*3^4*4^3*2^1 = 2^6*3^6*4 * 4 * 4

16,322

(n + 2) \cdot (2 \cdot n + 3) \cdot (1 + n)/6 = 1 \cdot 1 + \ldots + n^2 + (1 + n)^2

16,067

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

-3,058

-\sqrt{4}*\sqrt{7} + \sqrt{25}*\sqrt{7} = \sqrt{7}*5 - 2*\sqrt{7}