ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-21,202	1/2\cdot 2/3 = 2/6
881	(1 + y)^3 + 1 + 3(y + 1)^2 = 5 + y^3 + y y6 + 9y
35,726	240 = 5!\times 2
1,653	-\frac{256}{9} = -\frac{256}{9}
7,350	(z + 1)\times (z^2 - 6\times z + 13) = z^2 \times z - z \times z\times 5 + z\times 7 + 13
9,557	b_1 (2/9)^{(-1) + 1} = b_1
-6,559	\tfrac{1}{4 \cdot s + 4} \cdot 4 = \frac{4}{4 \cdot \left(1 + s\right)}
40,542	2^{n + (-1)} = \frac{1}{2} \cdot 2^n
12,580	180 = 3 \cdot 5 \cdot 4 \cdot 3
37,613	2^{4 \cdot (-1) + N} + 2^{N + 5 \cdot (-1)} + \cdots + 1 = 2^N \cdot (1/16 + 2 \cdot \left(-5\right) + \cdots + 2^{-N})
1,980	\left(-1\right)^3 \binom{-3}{3} = \binom{5}{3}
18,109	0 = z^2 + 9y^2 - 4z + 18 y + 4 = (z + 2(-1)) \cdot (z + 2(-1)) + 4(-1) + 9(y + 1)^2 + 9(-1) + 4
-7,602	\frac{i\cdot 16 + 11}{-i\cdot 5 + 2} = \frac{i\cdot 5 + 2}{5\cdot i + 2}\cdot \frac{1}{-i\cdot 5 + 2}\cdot \left(11 + 16\cdot i\right)
35,346	\frac{h\cdot x}{h + \frac{I}{n}} - x = \dfrac{1}{h + \frac{1}{n}\cdot I}\cdot (h\cdot x - (h + \frac{I}{n})\cdot x) = \dfrac{(-1)\cdot \frac1n\cdot x}{h + \frac{I}{n}}
3,492	x^2yz^2 + x^5 = (zx + y^2)(xyz - y^3) + x^5 + y^5
17,486	(n + 1) \cdot n/2 = {1 + n \choose 2}
-20,543	\dfrac{1}{(-3) \cdot r} \cdot (\left(-3\right) \cdot r) \cdot (-5/3) = \frac{r \cdot 15}{(-9) \cdot r}
32,668	18 = 27 + \left(-1\right) + 8 \cdot (-1)
-18,358	\frac{x \cdot x - x\cdot 6}{6(-1) + x^2 - 5x} = \dfrac{(x + 6\left(-1\right)) x}{(6\left(-1\right) + x) (1 + x)}
30,722	\frac{2}{2 + 2^{1 / 2}} = 2 - 2^{\frac{1}{2}}
26,803	\overline{(-1) + z} = (-1) + \overline{z}
16,152	-a/b = \dfrac{1}{b}\cdot ((-1)\cdot a) = \frac{a}{(-1)\cdot b}
15,462	x+2=\dfrac{(x-2)(x+2)}{(x-2)}=\dfrac{x^2-4}{x-2}
9,848	gd - ec = gd - c*0 + 0d - ec = gd - ec
-6,705	10^{-1} + \frac{1}{100}*6 = 6/100 + 10/100
5,965	\sum_{n=1}^\infty d \times n \times (2 \times (-1) - 1)^n = \sum_{n=1}^\infty d \times (-3)^n \times n
-2,960	\left(3 + 2\cdot (-1)\right)\cdot \sqrt{3} = \sqrt{3}
27,568	(x + y + A)^2 = x^2 + y^2 + A^2 + 2(xy + xA + Ay)
25,476	1/9 + 2/15 = 11/45
12,036	\cos(\beta + x) = \cos{x}\cos{\beta} - \sin{x}\sin{\beta}
20,611	COV\left(X_1, X_2\right) = COV\left(X_2, X_1\right)
31,303	b \cdot g \cdot b = g = b \cdot g \cdot b
26,298	R_aR_b = R_bR_a
-18,281	\dfrac{1}{n^2 - n \cdot 7 + 12} \cdot (-3 \cdot n + n^2) = \frac{\left(3 \cdot (-1) + n\right) \cdot n}{(4 \cdot (-1) + n) \cdot \left(n + 3 \cdot (-1)\right)}
2,834	W\cdot x^2 = \left(x\cdot W + q\right)\cdot x = x\cdot (x\cdot W + q) + q\cdot x = x^2\cdot W + 2\cdot x\cdot q
16,405	2\times (-1) + y^2 - y = \left(y + 1\right)\times (y + 2\times (-1))
10,114	\mathbb{Var}\left(X + Y\right) = \mathbb{Var}\left(X\right) + \mathbb{Var}\left(Y\right) + 2 \mathbb{Cov}\left(X, Y\right) = \mathbb{Var}\left(X\right) + \mathbb{Var}\left(Y\right)
-26,063	\frac{1}{10}(-24 - 18i + 8i + 6(-1)) = \frac{1}{10}\left(-30 - 10i\right) = -3 - i
-27,382	591 = 10 + 581
13,679	\sqrt{4 \cdot p^2 + 4} = x \Rightarrow 4 + 4 \cdot p^2 = x^2\wedge p^2 = (4 \cdot (-1) + x^2)/4
-3,204	\sqrt{25}\cdot \sqrt{2} + \sqrt{2}\cdot \sqrt{9} = 5\cdot \sqrt{2} + \sqrt{2}\cdot 3
41,046	8^{\dfrac13} = 2
12,844	5!/2! = \frac{1}{(5 + 3(-1))!}5!
34,432	\frac{a + x}{-a + x} = \frac{1}{-a/x + 1} \cdot (1 + a/x)
20,104	72^4 - 72^3 = 72^3(2 + (-1)) = 56
23,194	h \cdot h + 1 - 2\cdot h = (-h + 1)^2
19,301	\\|(1 - w_n)^{\frac12} \cdot c \cdot (1 - w_n)^{\frac{1}{2}}\\|^2 = \\|c - w_n \cdot c\\|^2
31,119	\mathbb{E}[V] = \mathbb{E}[p_H\times V_1 + p_F\times V_2] = p_H\times \mathbb{E}[V_1] + p_F\times \mathbb{E}[V_2]
22,570	d = b\Longrightarrow \{d, b\}
-4,094	\frac79i = 7i/9
-28,796	1 = \pi*2/(2\pi)
-20,527	56/(-48) = -\frac16 \cdot 7 \cdot (-\frac{1}{-8} \cdot 8)
2,472	-z^3 + x^3 = \left(-z + x\right) \cdot (z^2 + x^2 + x \cdot z)
13,506	-(x \times 2 - Q) = Q - 2 \times x
20,357	2 \cdot 3^n = \frac{(-1) + 3^{n + 1}}{3 + (-1)} \Rightarrow -3^n + (-1) = 0
-10,992	\frac{1}{11}\cdot 77 = 7
33,217	a*2 - a = a
11,612	x \cdot x + x = (1/2 + x)^2 - 1/4
13,072	\sum_{r=x}^n r = \sum_{r=1}^n r - \sum_{r=1}^{\left(-1\right) + x} r
15,841	\left\|{B\gamma\gamma^W}\right\| = \left\|{B}\right\|\left\|{\gamma\gamma^W}\right\| = \left\|{B}\right\|\gamma\gamma^W
-2,145	\pi\frac{1}{12}29 = \pi/2 + \pi*23/12
11,592	2^3\cdot 3\cdot 5 = (-1) + 11^2
16,302	t = t\cdot \frac{3}{4} + t/4
50,206	4 + 6 + 10 + 1 = 21
-19,356	1/3 \cdot 7/(1/4) = 4/1 \cdot \dfrac{1}{3} \cdot 7
9,936	(1 + 8)\cdot (4 + 1)\cdot \left(1 + 1\right)\cdot (1 + 1)\cdot (1 + 1) = 360
35,876	3/1 = 3*(-1) + p \Rightarrow p = 6
-20,369	5/5\frac{-r2 + 3\left(-1\right)}{9r + 6} = \frac{1}{30 + 45r}\left(-10r + 15(-1)\right)
5,250	1 + 2^0 + 2^1\cdots2^{n + (-1)} + 2^n = 2*2^n = 2^{n + 1}
14,158	-ny^2 + x^2 = (-y\sqrt{n} + x) (y\sqrt{n} + x)
9,823	1 + 4 \cdot 3^{9/8} \cdot \lambda - 4 \cdot 3^{\frac18} \cdot \lambda = 0 \Rightarrow 8 \cdot 3^{\frac18} \cdot \lambda = -1
24,279	(a^3)^x + x = \left(a^3\right)^x + x^3 - x * x * x + x = (a^x + x) ((a^2)^x - xa^x + x^2) - x * x * x + x
-27,735	\frac{\text{d}}{\text{d}z} (-2\cot{z}) = -2\frac{\text{d}}{\text{d}z} \cot{z} = 2\csc^2{z}
4,514	\cos\left(x\right)\cdot \sin(d) + \sin(x)\cdot \cos(d) = \sin\left(d + x\right)
35,670	1 = \sin{-\frac{3}{2}\cdot \pi}
44,528	\dfrac54 = \frac14\cdot 5
15,539	\mathbb{N} = \left\{\cdots, 2, 0, 1\right\}
-9,458	60 \cdot q + 54 = 2 \cdot 3 \cdot 3 \cdot 3 + q \cdot 2 \cdot 2 \cdot 3 \cdot 5
-19,509	\frac{4}{\frac17}\times \frac13 = 7/1\times \tfrac43
5,235	24\cdot (x + 2)\cdot (x + 1) = 48 + x^2\cdot 24 + 72\cdot x
20,717	s^{x + 1} \gt s^{x + 1} + (-1) = (s^x + (-1))^s \gt s^{(x + \left(-1\right))\cdot s}
39,494	3/36 = \dfrac{1}{6}*3/6
33,378	\sum_{k=1}^m k + \sum_{k=1}^m 1 = \sum_{k=1}^m (k + 1)
10,967	\lambda = 1/(1/\lambda)
-29,730	16x^3 - 21x^2 = \frac{\mathrm{d}}{\mathrm{d}x} (4x^4 - 7x^3 + 3)
-15,786	41/10 = 8\cdot \dfrac{7}{10} - 5\cdot \frac{3}{10}
3,282	\|x\|^{-\alpha}\cdot \sum_{n=1}^∞ b_n = \sum_{n=1}^∞ b_n\cdot \|x\|^{-\alpha}
8,599	b\cdot b\cdot c = b = b\cdot b\cdot c
1,482	\frac{C}{\psi}x = xC/CC/\psi \leq \dfrac{x}{\psi}C/\psi
23,636	\sin\left(z\right) = \cos(z/2)\cdot \sin(z/2)\cdot 2
23,876	(-x)! = (-x + (-1))! (-x)
-9,348	2 \cdot 11 + 11 q = 22 + 11 q
21,518	\frac{9}{16} = 3/4*3/4
21,820	V^x = \frac{1}{V^{-x}}
-5,504	\tfrac{1}{\left(q + 3(-1)\right)(q + 7(-1))}q + \frac{4(3(-1) + q)}{(q + 7(-1))(q + 3\left(-1\right))} + \frac{(q + 7(-1))5}{(7(-1) + q)(q + 3\left(-1\right))} = \frac{1}{(3(-1) + q)\left(q + 7(-1)\right)}\left(q + (3(-1) + q)4 + (7(-1) + q)5\right)
26,782	\frac{d}{d}\cdot g = g/d\cdot d
23,252	\frac{n!}{(n + 2 \cdot \left(-1\right))! \cdot 2!} = {n \choose 2}
27,565	20/60\cdot 10/60\cdot 30/60 = \frac{1}{36}
23,031	\left(2 = \frac{1}{x^2}h^2 \Rightarrow h^2 = 2x^2\right) \Rightarrow h = x\cdot 2^{1 / 2}
27,277	3^{4x + 3} = 3^3(10 + \left(-1\right))^{2x} = 3^2 3(1 + 10(-1))^{2*x}

-21,202

1/2\cdot 2/3 = 2/6

881

(1 + y)^3 + 1 + 3*(y + 1)^2 = 5 + y^3 + y * y*6 + 9*y

35,726

240 = 5!\times 2

1,653

-\frac{256}{9} = -\frac{256}{9}

7,350

(z + 1)\times (z^2 - 6\times z + 13) = z^2 \times z - z \times z\times 5 + z\times 7 + 13

9,557

b_1 (2/9)^{(-1) + 1} = b_1

-6,559

\tfrac{1}{4 \cdot s + 4} \cdot 4 = \frac{4}{4 \cdot \left(1 + s\right)}

40,542

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

12,580

180 = 3 \cdot 5 \cdot 4 \cdot 3

37,613

2^{4 \cdot (-1) + N} + 2^{N + 5 \cdot (-1)} + \cdots + 1 = 2^N \cdot (1/16 + 2 \cdot \left(-5\right) + \cdots + 2^{-N})

1,980

\left(-1\right)^3 \binom{-3}{3} = \binom{5}{3}

18,109

0 = z^2 + 9y^2 - 4z + 18 y + 4 = (z + 2(-1)) \cdot (z + 2(-1)) + 4(-1) + 9(y + 1)^2 + 9(-1) + 4

-7,602

\frac{i\cdot 16 + 11}{-i\cdot 5 + 2} = \frac{i\cdot 5 + 2}{5\cdot i + 2}\cdot \frac{1}{-i\cdot 5 + 2}\cdot \left(11 + 16\cdot i\right)

35,346

\frac{h\cdot x}{h + \frac{I}{n}} - x = \dfrac{1}{h + \frac{1}{n}\cdot I}\cdot (h\cdot x - (h + \frac{I}{n})\cdot x) = \dfrac{(-1)\cdot \frac1n\cdot x}{h + \frac{I}{n}}

3,492

x^2*y*z^2 + x^5 = (z*x + y^2)*(x*y*z - y^3) + x^5 + y^5

17,486

(n + 1) \cdot n/2 = {1 + n \choose 2}

-20,543

\dfrac{1}{(-3) \cdot r} \cdot (\left(-3\right) \cdot r) \cdot (-5/3) = \frac{r \cdot 15}{(-9) \cdot r}

32,668

18 = 27 + \left(-1\right) + 8 \cdot (-1)

-18,358

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

30,722

\frac{2}{2 + 2^{1 / 2}} = 2 - 2^{\frac{1}{2}}

26,803

\overline{(-1) + z} = (-1) + \overline{z}

16,152

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

15,462

x+2=\dfrac{(x-2)(x+2)}{(x-2)}=\dfrac{x^2-4}{x-2}

9,848

gd - ec = gd - c*0 + 0d - ec = gd - ec

-6,705

10^{-1} + \frac{1}{100}*6 = 6/100 + 10/100

5,965

\sum_{n=1}^\infty d \times n \times (2 \times (-1) - 1)^n = \sum_{n=1}^\infty d \times (-3)^n \times n

-2,960

\left(3 + 2\cdot (-1)\right)\cdot \sqrt{3} = \sqrt{3}

27,568

(x + y + A)^2 = x^2 + y^2 + A^2 + 2*(x*y + x*A + A*y)

25,476

1/9 + 2/15 = 11/45

12,036

\cos(\beta + x) = \cos{x}*\cos{\beta} - \sin{x}*\sin{\beta}

20,611

COV\left(X_1, X_2\right) = COV\left(X_2, X_1\right)

31,303

b \cdot g \cdot b = g = b \cdot g \cdot b

26,298

R_a*R_b = R_b*R_a

-18,281

\dfrac{1}{n^2 - n \cdot 7 + 12} \cdot (-3 \cdot n + n^2) = \frac{\left(3 \cdot (-1) + n\right) \cdot n}{(4 \cdot (-1) + n) \cdot \left(n + 3 \cdot (-1)\right)}

2,834

W\cdot x^2 = \left(x\cdot W + q\right)\cdot x = x\cdot (x\cdot W + q) + q\cdot x = x^2\cdot W + 2\cdot x\cdot q

16,405

2\times (-1) + y^2 - y = \left(y + 1\right)\times (y + 2\times (-1))

10,114

\mathbb{Var}\left(X + Y\right) = \mathbb{Var}\left(X\right) + \mathbb{Var}\left(Y\right) + 2 \mathbb{Cov}\left(X, Y\right) = \mathbb{Var}\left(X\right) + \mathbb{Var}\left(Y\right)

-26,063

\frac{1}{10}*(-24 - 18*i + 8*i + 6*(-1)) = \frac{1}{10}*\left(-30 - 10*i\right) = -3 - i

-27,382

591 = 10 + 581

13,679

\sqrt{4 \cdot p^2 + 4} = x \Rightarrow 4 + 4 \cdot p^2 = x^2\wedge p^2 = (4 \cdot (-1) + x^2)/4

-3,204

\sqrt{25}\cdot \sqrt{2} + \sqrt{2}\cdot \sqrt{9} = 5\cdot \sqrt{2} + \sqrt{2}\cdot 3

41,046

8^{\dfrac13} = 2

12,844

5!/2! = \frac{1}{(5 + 3*(-1))!}*5!

34,432

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

20,104

7*2^4 - 7*2^3 = 7*2^3*(2 + (-1)) = 56

23,194

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

19,301

\|(1 - w_n)^{\frac12} \cdot c \cdot (1 - w_n)^{\frac{1}{2}}\|^2 = \|c - w_n \cdot c\|^2

31,119

\mathbb{E}[V] = \mathbb{E}[p_H\times V_1 + p_F\times V_2] = p_H\times \mathbb{E}[V_1] + p_F\times \mathbb{E}[V_2]

22,570

d = b\Longrightarrow \{d, b\}

-4,094

\frac79*i = 7*i/9

-28,796

1 = \pi*2/(2\pi)

-20,527

56/(-48) = -\frac16 \cdot 7 \cdot (-\frac{1}{-8} \cdot 8)

2,472

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

13,506

-(x \times 2 - Q) = Q - 2 \times x

20,357

2 \cdot 3^n = \frac{(-1) + 3^{n + 1}}{3 + (-1)} \Rightarrow -3^n + (-1) = 0

-10,992

\frac{1}{11}\cdot 77 = 7

33,217

a*2 - a = a

11,612

x \cdot x + x = (1/2 + x)^2 - 1/4

13,072

\sum_{r=x}^n r = \sum_{r=1}^n r - \sum_{r=1}^{\left(-1\right) + x} r

15,841

\left|{B*\gamma*\gamma^W}\right| = \left|{B}\right|*\left|{\gamma*\gamma^W}\right| = \left|{B}\right|*\gamma*\gamma^W

-2,145

\pi*\frac{1}{12}*29 = \pi/2 + \pi*23/12

11,592

2^3\cdot 3\cdot 5 = (-1) + 11^2

16,302

t = t\cdot \frac{3}{4} + t/4

50,206

4 + 6 + 10 + 1 = 21

-19,356

1/3 \cdot 7/(1/4) = 4/1 \cdot \dfrac{1}{3} \cdot 7

9,936

(1 + 8)\cdot (4 + 1)\cdot \left(1 + 1\right)\cdot (1 + 1)\cdot (1 + 1) = 360

35,876

3/1 = 3*(-1) + p \Rightarrow p = 6

-20,369

5/5*\frac{-r*2 + 3*\left(-1\right)}{9*r + 6} = \frac{1}{30 + 45*r}*\left(-10*r + 15*(-1)\right)

5,250

1 + 2^0 + 2^1*\cdots*2^{n + (-1)} + 2^n = 2*2^n = 2^{n + 1}

14,158

-ny^2 + x^2 = (-y\sqrt{n} + x) (y\sqrt{n} + x)

9,823

1 + 4 \cdot 3^{9/8} \cdot \lambda - 4 \cdot 3^{\frac18} \cdot \lambda = 0 \Rightarrow 8 \cdot 3^{\frac18} \cdot \lambda = -1

24,279

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

-27,735

\frac{\text{d}}{\text{d}z} (-2\cot{z}) = -2\frac{\text{d}}{\text{d}z} \cot{z} = 2\csc^2{z}

4,514

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

35,670

1 = \sin{-\frac{3}{2}\cdot \pi}

44,528

\dfrac54 = \frac14\cdot 5

15,539

\mathbb{N} = \left\{\cdots, 2, 0, 1\right\}

-9,458

60 \cdot q + 54 = 2 \cdot 3 \cdot 3 \cdot 3 + q \cdot 2 \cdot 2 \cdot 3 \cdot 5

-19,509

\frac{4}{\frac17}\times \frac13 = 7/1\times \tfrac43

5,235

24\cdot (x + 2)\cdot (x + 1) = 48 + x^2\cdot 24 + 72\cdot x

20,717

s^{x + 1} \gt s^{x + 1} + (-1) = (s^x + (-1))^s \gt s^{(x + \left(-1\right))\cdot s}

39,494

3/36 = \dfrac{1}{6}*3/6

33,378

\sum_{k=1}^m k + \sum_{k=1}^m 1 = \sum_{k=1}^m (k + 1)

10,967

\lambda = 1/(1/\lambda)

-29,730

16*x^3 - 21*x^2 = \frac{\mathrm{d}}{\mathrm{d}x} (4*x^4 - 7*x^3 + 3)

-15,786

41/10 = 8\cdot \dfrac{7}{10} - 5\cdot \frac{3}{10}

3,282

|x|^{-\alpha}\cdot \sum_{n=1}^∞ b_n = \sum_{n=1}^∞ b_n\cdot |x|^{-\alpha}

8,599

b\cdot b\cdot c = b = b\cdot b\cdot c

1,482

\frac{C}{\psi}*x = x*C/C*C/\psi \leq \dfrac{x}{\psi}*C/\psi

23,636

\sin\left(z\right) = \cos(z/2)\cdot \sin(z/2)\cdot 2

23,876

(-x)! = (-x + (-1))! (-x)

-9,348

2 \cdot 11 + 11 q = 22 + 11 q

21,518

\frac{9}{16} = 3/4*3/4

21,820

V^x = \frac{1}{V^{-x}}

-5,504

\tfrac{1}{\left(q + 3*(-1)\right)*(q + 7*(-1))}*q + \frac{4*(3*(-1) + q)}{(q + 7*(-1))*(q + 3*\left(-1\right))} + \frac{(q + 7*(-1))*5}{(7*(-1) + q)*(q + 3*\left(-1\right))} = \frac{1}{(3*(-1) + q)*\left(q + 7*(-1)\right)}*\left(q + (3*(-1) + q)*4 + (7*(-1) + q)*5\right)

26,782

\frac{d}{d}\cdot g = g/d\cdot d

23,252

\frac{n!}{(n + 2 \cdot \left(-1\right))! \cdot 2!} = {n \choose 2}

27,565

20/60\cdot 10/60\cdot 30/60 = \frac{1}{36}

23,031

\left(2 = \frac{1}{x^2}h^2 \Rightarrow h^2 = 2x^2\right) \Rightarrow h = x\cdot 2^{1 / 2}

27,277

3^{4*x + 3} = 3^3*(10 + \left(-1\right))^{2*x} = 3^2 * 3*(1 + 10*(-1))^{2*x}