ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
14,912	\|y\| = \sqrt{y\overline{y}} = \sqrt{\overline{y}y} = \|\overline{y}\|
-7,108	6/15\times 6/14 = \frac{6}{35}
1,773	-x \times Y = Y \times x - x \times Y - Y \times x = (Y, x)
15,237	\tan(\dfrac12π - x) = \frac{\cos(x)}{\sin(x)} = \dfrac{1}{\tan(x)}
8,899	p^4 + p^2 + p^2 + p^2 + p^2 - p^3 - p^3 - p^3 - p^3 = 4p p - p * p * p*4 + p^4
-4,202	\tfrac{110}{44}\frac{a^4}{a} = \dfrac{110a^4}{a*44}
25,315	x! = x \cdot ((-1) + x) \cdot \left(x + 2 \cdot (-1)\right) \cdot \cdots \cdot 3 \cdot 2
8,324	x = -(1 + x)2 + x3 + 2
15,126	\left(g\cdot \frac{d}{g}\right)^l = g\cdot \frac1g\cdot d^l
20,357	2\cdot 3^n = \frac{1}{3 + (-1)}\cdot (3^{n + 1} + (-1)) \Rightarrow -3^n + (-1) = 0
-15,927	\dfrac{8}{10} - 8\cdot \frac{1}{10}\cdot 9 = -64/10
22,378	EAh = AhE
32,852	\frac{d}{dz} \arcsin(z) = \dfrac{1}{(1 - z \cdot z)^{1/2}}
28,227	\tfrac{1}{l + 2 \cdot (-1)} \cdot (l + 1) = 1 + \frac{3}{2 \cdot (-1) + l}
17,879	\sin(\pi/6) = \sin(5 \times \pi/6) = 1/2
32,171	\\|xd\\|_1 = \\|xd\\|_1
343	a \cdot a \cdot a - f^3 = (f^2 + a^2 + a \cdot f) \cdot (a - f)
42,230	41 = 67 + 22\cdot (-1) + 6\cdot \left(-1\right) + 2
723	(f^2 + f \cdot h + h^2) \cdot (f - h) = f^3 - h^3
12,353	-x\cdot z\cdot 2 + x^2\cdot 4 + y^2\cdot 9 + z \cdot z - 6\cdot x\cdot y - 3\cdot z\cdot y = (-z + x)^2 + \left(x - 3\cdot y\right)^2 + x^2\cdot 2 - 3\cdot y\cdot z
31,382	y = \frac{6}{\left(2(-1) + x\right)^3} \Rightarrow 6/y = (x + 2(-1))^3
-26,171	\frac{1}{3}21 + (-1) - \frac1212 = 7 + (-1) + 6*(-1) = 0
21,849	\sqrt{\sin^2{q} + \cos^2{q}}fqa = aq*f
11,803	\frac{k*2}{k^4}1 = \frac{2}{k^3}
22,717	\frac13(7 + 11) = 18/3 = 6
-20,847	\tfrac{6 \cdot m}{m \cdot 6} \cdot 1 \cdot (-8/3) = \frac{1}{18 \cdot m} \cdot (\left(-48\right) \cdot m)
16,284	4 = \binom{0 + 4 + (-1)}{0}*\binom{4}{3}
32,260	x^2 \cdot y^2 = (y \cdot x)^2
26,948	z^{k + \left(-1\right)} \cdot k = \frac{\partial}{\partial z} z^k
18,946	2z^2 + 2z = 2z \cdot (1 + z)
22,592	\sum_{k=1}^\infty \frac{k}{k^3} = \sum_{k=1}^\infty \frac{1}{k \cdot k}
30,059	\sum_{n=1}^{\infty} \frac{n}{(n+1)!}=\sum_{n=1}^{\infty} \frac{n+1-1}{(n+1)!}
28,551	(\sqrt{X}\cdot \sqrt{F})^2 = \sqrt{F}\cdot \sqrt{X}\cdot \sqrt{X}\cdot \sqrt{F}
28,620	\frac{7}{40} = \frac{3\cdot 7}{120}
-499	e^{\frac14\cdot \pi\cdot i\cdot 17} = (e^{\tfrac{\pi}{4}\cdot i})^{17}
10,199	-b + h = 0 \Rightarrow b = h
245	y = \dfrac{1}{z + 2} \Rightarrow z = \tfrac1y(1 - 2y)
-9,356	-2 \cdot 5 \cdot 5 + 2 \cdot 5 \cdot p = p \cdot 10 + 50 \cdot (-1)
26,711	1 - x + x^2 = -(-\frac{x^2}{2} + x) + 1 + \frac12x x
5,018	\frac{1}{((-1) + m)\cdot \left(2\cdot \left(-1\right) + m\right)}\cdot (4\cdot (-1) + m)\cdot (m + 3\cdot \left(-1\right)) = \frac{1}{{m + (-1) \choose 2}}\cdot {3\cdot (-1) + m \choose 2}
52,612	\frac{\sin(A)\cdot\cos(B) + \sin(B)\cdot\cos(A)}{\cos(A)\cdot\cos(B)} = \tan(A) + \tan(B)
14,368	\left\{\cdots, \left( 3, 1\right), ( 2, 0), ( 4, 2)\right\} = 2
19,554	\sqrt{x^2 + 4(-1)} - x = \frac{x^2 + 4(-1) - x^2}{\sqrt{x^2 + 4(-1)} + x} = -\frac{1}{\sqrt{x^2 + 4(-1)} + x}4
36,544	53 = 272 + 1 + 2(-1)
-159	\frac{6!}{(6 + 4 \cdot (-1))! \cdot 4!} = {6 \choose 4}
11,423	\tan\left(\arctan(x) + \arctan(x^3)\right) = \frac{x + x^3}{1 - x^4} = \frac{x}{1 - x^2}
35,344	z_2 = -z_1 + 2\Longrightarrow -z_1 = z_2
20,034	x2 - e2 = 2 (x - e)
2,074	\sin(\pi - x \cdot \pi) = \sin(\pi \cdot x)
28,437	\dfrac{1}{x\cdot h} = \frac{1}{h\cdot x}
7,329	I_J\cdot d^2 + I_J\cdot c^2 = (c \cdot c + d^2)\cdot I_J
18,688	\sqrt{5} = \frac{l}{n}2 + (-1) = \frac1n(2l - n)
11,639	x\cdot 8 + 5\cdot (-1) = -3\cdot x \cdot x + 8\cdot x + 1 - 6 - 3\cdot x^2
35,055	e^{x \cdot 5} \cdot e^{x \cdot 5} = e^{x \cdot 10}
12,753	\sqrt{q - \sqrt{q + \sqrt{q \dotsm}}} = \left((-1) + \sqrt{1 + (q + \left(-1\right)) \cdot 4}\right)/2
-1,864	-\pi \cdot \frac{11}{6} + \pi/12 = -\pi \cdot \frac14 \cdot 7
20,397	\frac{1}{n^2}\cdot (\left(-1\right) + n^2) = -\frac{1}{n \cdot n} + 1
28,921	\frac{(-1) + x}{(-1) + x^n} = \frac{1}{(x^n + \left(-1\right))\cdot \frac{1}{x + (-1)}}
-1,589	-\pi/4 + 2\pi = 7/4\pi
-24,089	\tfrac{1}{6 + 10}32 = \frac{32}{16} = \tfrac{1}{16}32 = 2
25,530	(1 + x)^3 = 3\cdot (1^1 + 2^2 + \dots + x^2) + 3\cdot (1 + 2 + \dots + x) + x + 1
16,115	a \cdot 5 + b \cdot 5^{1 / 2} = 5^{1 / 2} \cdot (a \cdot 5^{1 / 2} + b)
8,415	h * hk = (2h) * (2h)k*0.25
4,363	1 + \cos{x} = 1 + 2\times \cos^2{\tfrac{x}{2}} + \left(-1\right) = 2\times \cos^2{x/2}
19,655	\left((-1) + r\right)\cdot r\cdot (2\cdot (-1) + r)! = r!
18,110	(-3 \cdot a + 5)^2 - 4 \cdot (a \cdot a^2 - 2 \cdot a^2 - 2 \cdot a + 4) = -4 \cdot a^3 + 17 \cdot a^2 - 22 \cdot a + 9 = (9 - 4 \cdot a) \cdot (a + (-1))^2
-14,778	84 = \tfrac{420}{5}
34,787	\binom{(-1) + n}{n - r} = \binom{(-1) + n}{r + (-1)}
4,170	\sin(3x) = \sin(x + 2x) = \sin(x) \cos\left(2x\right) + \cos(x) \sin(2x)
26,224	16 + 12*\left(-1\right) = 4
-2,739	13^{1/2}\left(3 + 4\right) = 13^{1/2}7
29,193	4(a \cdot a - mg \cdot g) = (a \cdot 2)^2 - (2g)^2 m
-8,417	\dfrac{1}{-3}*3 = -1
-9,156	-a \cdot 7 \cdot 7 a + 2 \cdot 5 \cdot 7 a = 70 a - 49 a^2
4,635	1 + 25 (-1) + 100 + 100 (-1) + 25 + \left(-1\right) = 0
24,864	(x^2)^{1 / 2} = (x^2)^{\frac{1}{2}} = x^{2/2} = x
-6,143	\tfrac{-r5 + 7}{r^22 + 20 r + 18} = \frac{1}{18 + 2r^2 + r20}\left(r + 9 - r2 + 2(-1) - 4r\right)
6,742	(-1)^3 = (-1)^{6/2} = ((-1)^6)^{1/2} = 1^{\dfrac{1}{2}} = 1
13,738	\frac{1}{3} = (\dfrac{2}{3})^3 + \left(\frac{1}{3}\right) * (\dfrac{1}{3})^2
14,096	m^r = {m \choose 0} m^r
25,761	\frac98 = 3\frac14/(21/3)
-21,657	-\tfrac{4}{11} = -4/11
-20,684	-\frac{3}{7} \frac{1}{(-6) q}(q(-6)) = \frac{18 q}{(-1)42 q}
37,089	768 = 2^8 \cdot 3
-2,491	\sqrt{8}+\sqrt{50}-\sqrt{18} = \sqrt{4 \cdot 2}+\sqrt{25 \cdot 2}-\sqrt{9 \cdot 2}
27,568	(x + y + z)^2 = x^2 + y \cdot y + z \cdot z + (x \cdot y + z \cdot x + y \cdot z) \cdot 2
-5,231	0.18 \cdot 10^{(-5) \cdot (-1) - 5} = 10^0 \cdot 0.18
13,677	0 = \frac{13 + y\cdot 4}{5 + 2y^2 + y\cdot 7} \Rightarrow 0 = 13 + 4y
9,319	\theta_1 + 2\piz = 2\theta_2 - \theta_1 \Rightarrow z\pi + \theta_1 = \theta_2
26,555	\frac{dY}{dX} = -\frac{1}{3X + 4Y}(2X + 3Y) = -\frac{2 + 3\dfrac{1}{X}Y}{3 + 4Y/X}
15,769	\dfrac{1}{6}\left(y^2 + y\right)(2 + y) = \binom{2 + y}{3}
-12,281	\dfrac34 = \frac{r}{16 \cdot \pi} \cdot 16 \cdot \pi = r
24,587	\left(k + 2\right)*(k + 1)! = (k + 2)!
21,352	k\cdot z = k\cdot z
32,900	\left(e^{\frac{1}{2}\cdot t}\right)^2 = e^{\frac{t}{2}}\cdot e^{\dfrac{t}{2}} = e^t
21,816	99 z = 13 \Rightarrow z = 13/99
10,056	(3n + 3\left(-1\right))/3 + \frac{1}{3}(3n^2 + 3n + 1) = \frac{1}{3}(3n^2 + 6n + 2(-1)) = n^2 + 2n - 2/3
3,301	\left(x + 1 = x \Rightarrow x \cdot x = x^2 + x \cdot 2 + 1\right) \Rightarrow x \cdot 2 + 1 = 0
19,148	-2 \cdot e^{\dfrac13 \cdot (i \cdot (-1) \cdot \pi)} = -2 \cdot e^{\dfrac{1}{3} \cdot (i \cdot \pi \cdot (-1))}
41,378	\|a \cdot g\| = \|g \cdot a\|

14,912

|y| = \sqrt{y*\overline{y}} = \sqrt{\overline{y}*y} = |\overline{y}|

-7,108

6/15\times 6/14 = \frac{6}{35}

1,773

-x \times Y = Y \times x - x \times Y - Y \times x = (Y, x)

15,237

\tan(\dfrac12π - x) = \frac{\cos(x)}{\sin(x)} = \dfrac{1}{\tan(x)}

8,899

p^4 + p^2 + p^2 + p^2 + p^2 - p^3 - p^3 - p^3 - p^3 = 4*p * p - p * p * p*4 + p^4

-4,202

\tfrac{110}{44}*\frac{a^4}{a} = \dfrac{110*a^4}{a*44}

25,315

x! = x \cdot ((-1) + x) \cdot \left(x + 2 \cdot (-1)\right) \cdot \cdots \cdot 3 \cdot 2

8,324

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

15,126

\left(g\cdot \frac{d}{g}\right)^l = g\cdot \frac1g\cdot d^l

20,357

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

-15,927

\dfrac{8}{10} - 8\cdot \frac{1}{10}\cdot 9 = -64/10

22,378

E*A*h = A*h*E

32,852

\frac{d}{dz} \arcsin(z) = \dfrac{1}{(1 - z \cdot z)^{1/2}}

28,227

\tfrac{1}{l + 2 \cdot (-1)} \cdot (l + 1) = 1 + \frac{3}{2 \cdot (-1) + l}

17,879

\sin(\pi/6) = \sin(5 \times \pi/6) = 1/2

32,171

\|xd\|_1 = \|xd\|_1

343

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

42,230

41 = 67 + 22\cdot (-1) + 6\cdot \left(-1\right) + 2

723

(f^2 + f \cdot h + h^2) \cdot (f - h) = f^3 - h^3

12,353

-x\cdot z\cdot 2 + x^2\cdot 4 + y^2\cdot 9 + z \cdot z - 6\cdot x\cdot y - 3\cdot z\cdot y = (-z + x)^2 + \left(x - 3\cdot y\right)^2 + x^2\cdot 2 - 3\cdot y\cdot z

31,382

y = \frac{6}{\left(2(-1) + x\right)^3} \Rightarrow 6/y = (x + 2(-1))^3

-26,171

\frac{1}{3}*21 + (-1) - \frac12*12 = 7 + (-1) + 6*(-1) = 0

21,849

\sqrt{\sin^2{q} + \cos^2{q}}*f*q*a = a*q*f

11,803

\frac{k*2}{k^4}1 = \frac{2}{k^3}

22,717

\frac13(7 + 11) = 18/3 = 6

-20,847

\tfrac{6 \cdot m}{m \cdot 6} \cdot 1 \cdot (-8/3) = \frac{1}{18 \cdot m} \cdot (\left(-48\right) \cdot m)

16,284

4 = \binom{0 + 4 + (-1)}{0}*\binom{4}{3}

32,260

x^2 \cdot y^2 = (y \cdot x)^2

26,948

z^{k + \left(-1\right)} \cdot k = \frac{\partial}{\partial z} z^k

18,946

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

22,592

\sum_{k=1}^\infty \frac{k}{k^3} = \sum_{k=1}^\infty \frac{1}{k \cdot k}

30,059

\sum_{n=1}^{\infty} \frac{n}{(n+1)!}=\sum_{n=1}^{\infty} \frac{n+1-1}{(n+1)!}

28,551

(\sqrt{X}\cdot \sqrt{F})^2 = \sqrt{F}\cdot \sqrt{X}\cdot \sqrt{X}\cdot \sqrt{F}

28,620

\frac{7}{40} = \frac{3\cdot 7}{120}

-499

e^{\frac14\cdot \pi\cdot i\cdot 17} = (e^{\tfrac{\pi}{4}\cdot i})^{17}

10,199

-b + h = 0 \Rightarrow b = h

245

y = \dfrac{1}{z + 2} \Rightarrow z = \tfrac1y(1 - 2y)

-9,356

-2 \cdot 5 \cdot 5 + 2 \cdot 5 \cdot p = p \cdot 10 + 50 \cdot (-1)

26,711

1 - x + x^2 = -(-\frac{x^2}{2} + x) + 1 + \frac12*x * x

5,018

\frac{1}{((-1) + m)\cdot \left(2\cdot \left(-1\right) + m\right)}\cdot (4\cdot (-1) + m)\cdot (m + 3\cdot \left(-1\right)) = \frac{1}{{m + (-1) \choose 2}}\cdot {3\cdot (-1) + m \choose 2}

52,612

\frac{\sin(A)\cdot\cos(B) + \sin(B)\cdot\cos(A)}{\cos(A)\cdot\cos(B)} = \tan(A) + \tan(B)

14,368

\left\{\cdots, \left( 3, 1\right), ( 2, 0), ( 4, 2)\right\} = 2

19,554

\sqrt{x^2 + 4(-1)} - x = \frac{x^2 + 4(-1) - x^2}{\sqrt{x^2 + 4(-1)} + x} = -\frac{1}{\sqrt{x^2 + 4(-1)} + x}4

36,544

53 = 27*2 + 1 + 2*(-1)

-159

\frac{6!}{(6 + 4 \cdot (-1))! \cdot 4!} = {6 \choose 4}

11,423

\tan\left(\arctan(x) + \arctan(x^3)\right) = \frac{x + x^3}{1 - x^4} = \frac{x}{1 - x^2}

35,344

z_2 = -z_1 + 2\Longrightarrow -z_1 = z_2

20,034

x*2 - e*2 = 2 (x - e)

2,074

\sin(\pi - x \cdot \pi) = \sin(\pi \cdot x)

28,437

\dfrac{1}{x\cdot h} = \frac{1}{h\cdot x}

7,329

I_J\cdot d^2 + I_J\cdot c^2 = (c \cdot c + d^2)\cdot I_J

18,688

\sqrt{5} = \frac{l}{n}2 + (-1) = \frac1n(2l - n)

11,639

x\cdot 8 + 5\cdot (-1) = -3\cdot x \cdot x + 8\cdot x + 1 - 6 - 3\cdot x^2

35,055

e^{x \cdot 5} \cdot e^{x \cdot 5} = e^{x \cdot 10}

12,753

\sqrt{q - \sqrt{q + \sqrt{q \dotsm}}} = \left((-1) + \sqrt{1 + (q + \left(-1\right)) \cdot 4}\right)/2

-1,864

-\pi \cdot \frac{11}{6} + \pi/12 = -\pi \cdot \frac14 \cdot 7

20,397

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

28,921

\frac{(-1) + x}{(-1) + x^n} = \frac{1}{(x^n + \left(-1\right))\cdot \frac{1}{x + (-1)}}

-1,589

-\pi/4 + 2*\pi = 7/4*\pi

-24,089

\tfrac{1}{6 + 10}*32 = \frac{32}{16} = \tfrac{1}{16}*32 = 2

25,530

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

16,115

a \cdot 5 + b \cdot 5^{1 / 2} = 5^{1 / 2} \cdot (a \cdot 5^{1 / 2} + b)

8,415

h * h*k = (2*h) * (2*h)*k*0.25

4,363

1 + \cos{x} = 1 + 2\times \cos^2{\tfrac{x}{2}} + \left(-1\right) = 2\times \cos^2{x/2}

19,655

\left((-1) + r\right)\cdot r\cdot (2\cdot (-1) + r)! = r!

18,110

(-3 \cdot a + 5)^2 - 4 \cdot (a \cdot a^2 - 2 \cdot a^2 - 2 \cdot a + 4) = -4 \cdot a^3 + 17 \cdot a^2 - 22 \cdot a + 9 = (9 - 4 \cdot a) \cdot (a + (-1))^2

-14,778

84 = \tfrac{420}{5}

34,787

\binom{(-1) + n}{n - r} = \binom{(-1) + n}{r + (-1)}

4,170

\sin(3x) = \sin(x + 2x) = \sin(x) \cos\left(2x\right) + \cos(x) \sin(2x)

26,224

16 + 12*\left(-1\right) = 4

-2,739

13^{1/2}*\left(3 + 4\right) = 13^{1/2}*7

29,193

4(a \cdot a - mg \cdot g) = (a \cdot 2)^2 - (2g)^2 m

-8,417

\dfrac{1}{-3}*3 = -1

-9,156

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

4,635

1 + 25 (-1) + 100 + 100 (-1) + 25 + \left(-1\right) = 0

24,864

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

-6,143

\tfrac{-r*5 + 7}{r^2*2 + 20 r + 18} = \frac{1}{18 + 2r^2 + r*20}\left(r + 9 - r*2 + 2(-1) - 4r\right)

6,742

(-1)^3 = (-1)^{6/2} = ((-1)^6)^{1/2} = 1^{\dfrac{1}{2}} = 1

13,738

\frac{1}{3} = (\dfrac{2}{3})^3 + \left(\frac{1}{3}\right) * (\dfrac{1}{3})^2

14,096

m^r = {m \choose 0} m^r

25,761

\frac98 = 3*\frac14/(2*1/3)

-21,657

-\tfrac{4}{11} = -4/11

-20,684

-\frac{3}{7} \frac{1}{(-6) q}(q*(-6)) = \frac{18 q}{(-1)*42 q}

37,089

768 = 2^8 \cdot 3

-2,491

\sqrt{8}+\sqrt{50}-\sqrt{18} = \sqrt{4 \cdot 2}+\sqrt{25 \cdot 2}-\sqrt{9 \cdot 2}

27,568

(x + y + z)^2 = x^2 + y \cdot y + z \cdot z + (x \cdot y + z \cdot x + y \cdot z) \cdot 2

-5,231

0.18 \cdot 10^{(-5) \cdot (-1) - 5} = 10^0 \cdot 0.18

13,677

0 = \frac{13 + y\cdot 4}{5 + 2y^2 + y\cdot 7} \Rightarrow 0 = 13 + 4y

9,319

\theta_1 + 2*\pi*z = 2*\theta_2 - \theta_1 \Rightarrow z*\pi + \theta_1 = \theta_2

26,555

\frac{dY}{dX} = -\frac{1}{3*X + 4*Y}*(2*X + 3*Y) = -\frac{2 + 3*\dfrac{1}{X}*Y}{3 + 4*Y/X}

15,769

\dfrac{1}{6}*\left(y^2 + y\right)*(2 + y) = \binom{2 + y}{3}

-12,281

\dfrac34 = \frac{r}{16 \cdot \pi} \cdot 16 \cdot \pi = r

24,587

\left(k + 2\right)*(k + 1)! = (k + 2)!

21,352

k\cdot z = k\cdot z

32,900

\left(e^{\frac{1}{2}\cdot t}\right)^2 = e^{\frac{t}{2}}\cdot e^{\dfrac{t}{2}} = e^t

21,816

99 z = 13 \Rightarrow z = 13/99

10,056

(3*n + 3*\left(-1\right))/3 + \frac{1}{3}*(3*n^2 + 3*n + 1) = \frac{1}{3}*(3*n^2 + 6*n + 2*(-1)) = n^2 + 2*n - 2/3

3,301

\left(x + 1 = x \Rightarrow x \cdot x = x^2 + x \cdot 2 + 1\right) \Rightarrow x \cdot 2 + 1 = 0

19,148

-2 \cdot e^{\dfrac13 \cdot (i \cdot (-1) \cdot \pi)} = -2 \cdot e^{\dfrac{1}{3} \cdot (i \cdot \pi \cdot (-1))}

41,378

|a \cdot g| = |g \cdot a|