ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
21,341	\frac{1}{15}\cdot 8 = \frac15 + 1/3
-29,105	-35 = 5 \left(-7\right)
24,501	M \cdot M/M = M
26,556	y = x rightarrow e^y = e^x
-15,299	\frac{1}{1/k\frac{1}{q^5}}k * k * k = \frac{k^3}{\frac{1}{k}*\tfrac{1}{q^5}}
30,332	0.1 = 0.011111*\dots
5,977	-x * x + z^2 = 1 rightarrow \sqrt{1 + x^2} = z
-6,118	10/10\cdot \frac{2}{(l + 6\cdot (-1))\cdot \left(l + 10\right)} = \frac{1}{10\cdot (l + 6\cdot (-1))\cdot (l + 10)}\cdot 20
23,434	\dfrac{2}{1 + 2} = \frac{1}{3}2 \gt 1/2
4,380	(w_2 + w_1) (x + T) = (x + T) w_1 + (T + x) w_2
21,169	y \cdot y \cdot y - 3 \cdot y \cdot y - y + (-1) = (y + (-1))^3 - 3 \cdot y + 1 - y + (-1) = (y + (-1))^3 - 4 \cdot (y + (-1)) + 4 \cdot \left(-1\right)
-23,064	\frac{1}{4}\cdot 7 = \frac12\cdot 7 / 2
-2,900	2 \cdot \sqrt{13} = (3 \cdot (-1) + 5) \cdot \sqrt{13}
39,229	(-1)\cdot (-1) + 3 = 4
-10,759	15 = 96 - 20x + 24 = -20x + 120
-3,867	\frac{p^4 \cdot 90}{10 \cdot p} = 90/10 \cdot \frac1p \cdot p^4
-2,896	-7^{\frac{1}{2}} + 4^{\dfrac{1}{2}}\cdot 7^{\frac{1}{2}} = -7^{1 / 2} + 2\cdot 7^{\dfrac{1}{2}}
31,828	\gamma \cdot \alpha + x \cdot \alpha = (x + \gamma) \cdot \alpha
23,441	\frac{-x^{n + 1} + 1}{-x + 1} = 1 + x + x^2 + \cdots + x^n
12,741	1 + 1/2 + 1/4 + \dotsm + \frac{1}{2^m} = -\frac{1}{2^m} + 2
5,137	X * X - U^2 = (-U + X)*(U + X)
37,562	{n \choose i}\cdot i = n\cdot {\left(-1\right) + n \choose (-1) + i}
-4,295	\frac{k^2\cdot 40}{k^3\cdot 36} = 40/36\cdot \frac{1}{k^3}\cdot k \cdot k
4,295	\frac{\partial}{\partial z} e^{z\times b + \sinh(h\times z)} = e^{\sinh(z\times h) + b\times z}\times \left(h\times \cosh(h\times z) + b\right)
-20,558	\frac{9 \times (-1) - 9 \times k}{3 \times k + 6 \times (-1)} = \frac{1}{k + 2 \times (-1)} \times (3 \times (-1) - 3 \times k) \times \frac{3}{3}
-6,102	\frac{1}{8(-1) + 2x} = \dfrac{1}{2(4(-1) + x)}
-11,765	64/49 = (\frac178) (\frac17*8)
-927	\frac72 = 7/2
-1,196	\frac{6}{15} = \frac{6 \cdot \frac13}{15 \cdot 1/3} = 2/5
-4,612	\tfrac{19 \cdot (-1) + 5 \cdot z}{5 \cdot (-1) + z^2 - 4 \cdot z} = \frac{4}{1 + z} + \frac{1}{z + 5 \cdot (-1)}
8,883	\frac{16}{81} = (2/3)^4 = 1/\frac{81}{16}
-1,374	\dfrac{1}{7/4 \cdot 7} = 1/7 \cdot 4/7
-15,927	8/10 - 8 \cdot 9/10 = -64/10
11,351	(l + 1)/2 - \frac{l}{2} = \frac{1}{2} = l/2 - \frac{1}{2}\cdot (l + \left(-1\right))
16,695	(5 + 2 + 1 + 1) \times (3 + 1 + 1) \times \left(2 + 1\right) \times (1 + 1) = 270
13,243	2 \cdot 2^w = 2^{1 + w}
-19,630	\frac{1/2}{2} \cdot 3 = \frac{1}{2 \cdot 2/3}
18,305	(2^{20})^{10} = 2^{20}2^{20} \ldots2^{20}
31,239	\dfrac{1}{\theta^2} = \mathbb{E}\left(\frac{1}{\theta^2}\right)
15,355	\frac{1}{(u + \left(-1\right))\cdot \left(u + 1\right)} = \frac{1}{u^2 + (-1)}
25,698	468 = 3^2 \times 2^2 \times 13
13,767	(-3\cdot 10^4 + 31415)\cdot 10 + 9 = 14159
874	\frac{\mathrm{d}W}{\mathrm{d}X} = \dfrac{1}{X + W} (3 X - W) = \frac{3 - W/X}{1 + \frac1X W}
19,583	\frac{c^m}{3} = c^m/3 = -c\cdot c^m = -c^{m + 1}
17,903	100 = 89 + 1 \cdot 23 + 4(-1) + 5 + 6(-1) + 7(-1)
48,319	-\sin{\theta} = \sin{-\theta}
37,070	\sin{y}\cos{y}2 = \sin{2*y}
8,102	2 = \left(\dfrac13\cdot (d \cdot d \cdot d + h^3 + c^3)\right)^{1/3} \geq \frac{1}{3}\cdot \left(d + h + c\right)
13,020	(x + 6) \cdot (x + 4 \cdot (-1)) = x^2 + x \cdot 2 + 24 \cdot \left(-1\right)
1,246	\sin(-x)/((-1) x) = \frac{\sin(x)}{x}
-23,611	5/6 \cdot \dfrac59 = \dfrac{1}{54} \cdot 25
25,254	\binom{n}{k} = \binom{(-1) + n}{k + (-1)} \cdot n/k
4,286	m_2n + m_1n = m_2n + nm_1
7,328	\tfrac{1}{1 - x x} = 1 + x^2 + x^4 + x^6 ...
-6,725	\frac{9}{100} + 8/10 = 80/100 + \dfrac{1}{100}9
34,389	6 + x \cdot 6 = x \cdot 6 + 3 \cdot 2
94	B\cdot B + B\cdot B + B\cdot B = 3\cdot B^2 = 3\cdot B\cdot B > B
34,766	1/(63\cdot 32) = \frac{1}{2016}
12,935	4 \cdot y^3 - 7 \cdot y + 3 \cdot (-1) = (y + 1) \cdot \left(g_1 \cdot y^2 + g_2 \cdot y + c\right) = g_1 \cdot y^3 + g_2 \cdot y^2 + c \cdot y + g_1 \cdot y^2 + g_2 \cdot y + c
-1,589	\pi \frac{7}{4} = -\frac14\pi + 2\pi
21,253	\dfrac{5}{60} rightarrow 1/2\cdot 3/5
1,196	\tan{A} = \frac{\sqrt{3} \times 1/2}{\left(-1\right) \times \frac12} = \sin{A}/\cos{A}
32,548	\tfrac{20}{9} = \frac{1}{9} \cdot 4 \cdot (-\frac13 + 2)^2 + 5/9 \cdot \left(-1/3 - 1\right)^2
10,760	2\alpha + 1 + x2 + 1 = (\alpha + x + 1)*2
-2,612	\sqrt{2} \sqrt{9} + \sqrt{2} \sqrt{16} = 4\sqrt{2} + 3\sqrt{2}
22,169	X^2-8X+25=(X-\alpha^2)(X-\beta^2)=X^2-(\alpha^2+\beta^2)X+\alpha^2\beta^2
32,265	b \cdot 2 \cdot a = -(b^2 + a^2) + (a + b) \cdot (a + b)
15,046	\frac{z^2}{z + 2 \cdot \left(-1\right)} = \dfrac{z^2}{2 \cdot (-1) + z}
33,318	\frac{1}{\sqrt{x}} \cdot \sin(x) = \sqrt{x} \cdot \sin(x)/x
7,883	\frac{l + 1}{2^{l + 1}} = \dfrac{1}{2^{l + 1}}(3(-1) + l*2 + 4 - l)
-5,822	\frac{1}{(2(-1) + k)2}3 = \dfrac{3}{4(-1) + k*2}
11,764	\frac{dy}{dz} = \frac{3z^2}{3y^2} = \frac{1}{y^2}z z
6,965	\frac{1}{2(-1) + x}(yN + x^3 - xNy - 2x^2 + 2yx) = \tfrac{-Ny + 4y}{x + 2(-1)} + x^2 - yN + y*2
27,828	y^2 \cdot h + h = y + y \cdot h \cdot h \implies y = h
23,061	r^{\dfrac13} + 2*(-1) = d \Rightarrow r = (d + 2)^3
20,152	sxr = rsx
8,585	b\cdot x = 1/(b\cdot x) = \frac{1}{x\cdot b} = x\cdot b
-27,037	\sum_{n=1}^\infty \frac{1}{n6^n} \left(1 + 5\right)^n(n + 2) = \sum_{n=1}^\infty \frac{6^n}{n*6^n} (n + 2) = \sum_{n=1}^\infty (n + 2)/n
-7,123	4/10\frac392/8 = 1/30
-23,377	\dfrac{9}{32} = 3/8\cdot \frac{3}{4}
28,727	2^{k + 2} + 4\cdot (-1) = (2^{k + 1} + 2\cdot \left(-1\right))\cdot 2
43,272	\binom{13 + 3 + \left(-1\right)}{3 + (-1)} = \binom{15}{2}
-20,335	\tfrac{1}{7\cdot x + 14\cdot (-1)}\cdot \left(-4\cdot x + 8\right) = \dfrac{x + 2\cdot (-1)}{x + 2\cdot (-1)}\cdot (-\frac{4}{7})
-8,803	48\cdot \pi = 30\cdot \pi + 9\cdot \pi + 9\cdot \pi
6,131	\cos(\dfrac{1}{4}\cdot \pi + z - \dfrac14\cdot \pi) = \cos\left(z\right)
-1,979	\pi/3 + \pi/6 = \frac{\pi}{2}
-19,384	\frac{7\cdot \frac{1}{4}}{3\cdot 1/4} = 7/4\cdot 4/3
3,674	24\cdot \mathrm{i} - 7 = \left(4\cdot \mathrm{i} + 3\right)^2
3,075	1 - z*2 \lt (-1) - z \Rightarrow 2 < z
-20,769	-5/6 \frac{6x + 10 (-1)}{10 \left(-1\right) + 6x} = \frac{50 - x \cdot 30}{36 x + 60 (-1)}
3,204	(\sqrt{a + z_1})^2 = (\sqrt{F + z_2})^2 \Rightarrow z_2 + F = z_1 + a
-6,207	\frac{4\cdot q}{(q + 3\cdot (-1))\cdot \left(q + 8\right)} = \frac{4\cdot q}{q^2 + q\cdot 5 + 24\cdot \left(-1\right)}
2,241	p - 1/2 = p - \tfrac12\left(p + 1\right) = (p + (-1))/2
35,149	-a \cdot 2 + 5 + 2 \cdot (-1) = -2 \cdot a + 3
13,966	\cos(\operatorname{asin}(x)) = \left(1 - x^2\right)^{\tfrac{1}{2}}
11,347	f_2 + f_3 + \dotsm + f_l + f_{l + 1} = f_2 + f_3 + \dotsm + f_l + f_{l + 1}
-6,995	\frac19 \cdot 4 = 4/9
7,740	\cos^2{\theta} = (\cos{2\theta} + 1)/2
35,343	-((-1) + t) = 1 - t
-578	\left(e^{i\cdot \pi/3}\right)^{14} = e^{\frac{i\cdot \pi}{3}\cdot 14}

21,341

\frac{1}{15}\cdot 8 = \frac15 + 1/3

-29,105

-35 = 5 \left(-7\right)

24,501

M \cdot M/M = M

26,556

y = x rightarrow e^y = e^x

-15,299

\frac{1}{1/k*\frac{1}{q^5}}*k * k * k = \frac{k^3}{\frac{1}{k}*\tfrac{1}{q^5}}

30,332

0.1 = 0.011111*\dots

5,977

-x * x + z^2 = 1 rightarrow \sqrt{1 + x^2} = z

-6,118

10/10\cdot \frac{2}{(l + 6\cdot (-1))\cdot \left(l + 10\right)} = \frac{1}{10\cdot (l + 6\cdot (-1))\cdot (l + 10)}\cdot 20

23,434

\dfrac{2}{1 + 2} = \frac{1}{3}2 \gt 1/2

4,380

(w_2 + w_1) (x + T) = (x + T) w_1 + (T + x) w_2

21,169

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

-23,064

\frac{1}{4}\cdot 7 = \frac12\cdot 7 / 2

-2,900

2 \cdot \sqrt{13} = (3 \cdot (-1) + 5) \cdot \sqrt{13}

39,229

(-1)\cdot (-1) + 3 = 4

-10,759

15 = 96 - 20*x + 24 = -20*x + 120

-3,867

\frac{p^4 \cdot 90}{10 \cdot p} = 90/10 \cdot \frac1p \cdot p^4

-2,896

-7^{\frac{1}{2}} + 4^{\dfrac{1}{2}}\cdot 7^{\frac{1}{2}} = -7^{1 / 2} + 2\cdot 7^{\dfrac{1}{2}}

31,828

\gamma \cdot \alpha + x \cdot \alpha = (x + \gamma) \cdot \alpha

23,441

\frac{-x^{n + 1} + 1}{-x + 1} = 1 + x + x^2 + \cdots + x^n

12,741

1 + 1/2 + 1/4 + \dotsm + \frac{1}{2^m} = -\frac{1}{2^m} + 2

5,137

X * X - U^2 = (-U + X)*(U + X)

37,562

{n \choose i}\cdot i = n\cdot {\left(-1\right) + n \choose (-1) + i}

-4,295

\frac{k^2\cdot 40}{k^3\cdot 36} = 40/36\cdot \frac{1}{k^3}\cdot k \cdot k

4,295

\frac{\partial}{\partial z} e^{z\times b + \sinh(h\times z)} = e^{\sinh(z\times h) + b\times z}\times \left(h\times \cosh(h\times z) + b\right)

-20,558

\frac{9 \times (-1) - 9 \times k}{3 \times k + 6 \times (-1)} = \frac{1}{k + 2 \times (-1)} \times (3 \times (-1) - 3 \times k) \times \frac{3}{3}

-6,102

\frac{1}{8*(-1) + 2*x} = \dfrac{1}{2*(4*(-1) + x)}

-11,765

64/49 = (\frac17*8) * (\frac17*8)

-927

\frac72 = 7/2

-1,196

\frac{6}{15} = \frac{6 \cdot \frac13}{15 \cdot 1/3} = 2/5

-4,612

\tfrac{19 \cdot (-1) + 5 \cdot z}{5 \cdot (-1) + z^2 - 4 \cdot z} = \frac{4}{1 + z} + \frac{1}{z + 5 \cdot (-1)}

8,883

\frac{16}{81} = (2/3)^4 = 1/\frac{81}{16}

-1,374

\dfrac{1}{7/4 \cdot 7} = 1/7 \cdot 4/7

-15,927

8/10 - 8 \cdot 9/10 = -64/10

11,351

(l + 1)/2 - \frac{l}{2} = \frac{1}{2} = l/2 - \frac{1}{2}\cdot (l + \left(-1\right))

16,695

(5 + 2 + 1 + 1) \times (3 + 1 + 1) \times \left(2 + 1\right) \times (1 + 1) = 270

13,243

2 \cdot 2^w = 2^{1 + w}

-19,630

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

18,305

(2^{20})^{10} = 2^{20}*2^{20} \ldots*2^{20}

31,239

\dfrac{1}{\theta^2} = \mathbb{E}\left(\frac{1}{\theta^2}\right)

15,355

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

25,698

468 = 3^2 \times 2^2 \times 13

13,767

(-3\cdot 10^4 + 31415)\cdot 10 + 9 = 14159

874

\frac{\mathrm{d}W}{\mathrm{d}X} = \dfrac{1}{X + W} (3 X - W) = \frac{3 - W/X}{1 + \frac1X W}

19,583

\frac{c^m}{3} = c^m/3 = -c\cdot c^m = -c^{m + 1}

17,903

100 = 89 + 1 \cdot 23 + 4(-1) + 5 + 6(-1) + 7(-1)

48,319

-\sin{\theta} = \sin{-\theta}

37,070

\sin{y}*\cos{y}*2 = \sin{2*y}

8,102

2 = \left(\dfrac13\cdot (d \cdot d \cdot d + h^3 + c^3)\right)^{1/3} \geq \frac{1}{3}\cdot \left(d + h + c\right)

13,020

(x + 6) \cdot (x + 4 \cdot (-1)) = x^2 + x \cdot 2 + 24 \cdot \left(-1\right)

1,246

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

-23,611

5/6 \cdot \dfrac59 = \dfrac{1}{54} \cdot 25

25,254

\binom{n}{k} = \binom{(-1) + n}{k + (-1)} \cdot n/k

4,286

m_2*n + m_1*n = m_2*n + n*m_1

7,328

\tfrac{1}{1 - x x} = 1 + x^2 + x^4 + x^6 ...

-6,725

\frac{9}{100} + 8/10 = 80/100 + \dfrac{1}{100}9

34,389

6 + x \cdot 6 = x \cdot 6 + 3 \cdot 2

94

B\cdot B + B\cdot B + B\cdot B = 3\cdot B^2 = 3\cdot B\cdot B > B

34,766

1/(63\cdot 32) = \frac{1}{2016}

12,935

4 \cdot y^3 - 7 \cdot y + 3 \cdot (-1) = (y + 1) \cdot \left(g_1 \cdot y^2 + g_2 \cdot y + c\right) = g_1 \cdot y^3 + g_2 \cdot y^2 + c \cdot y + g_1 \cdot y^2 + g_2 \cdot y + c

-1,589

\pi \frac{7}{4} = -\frac14\pi + 2\pi

21,253

\dfrac{5}{60} rightarrow 1/2\cdot 3/5

1,196

\tan{A} = \frac{\sqrt{3} \times 1/2}{\left(-1\right) \times \frac12} = \sin{A}/\cos{A}

32,548

\tfrac{20}{9} = \frac{1}{9} \cdot 4 \cdot (-\frac13 + 2)^2 + 5/9 \cdot \left(-1/3 - 1\right)^2

10,760

2*\alpha + 1 + x*2 + 1 = (\alpha + x + 1)*2

-2,612

\sqrt{2} \sqrt{9} + \sqrt{2} \sqrt{16} = 4\sqrt{2} + 3\sqrt{2}

22,169

X^2-8X+25=(X-\alpha^2)(X-\beta^2)=X^2-(\alpha^2+\beta^2)X+\alpha^2\beta^2

32,265

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

15,046

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

33,318

\frac{1}{\sqrt{x}} \cdot \sin(x) = \sqrt{x} \cdot \sin(x)/x

7,883

\frac{l + 1}{2^{l + 1}} = \dfrac{1}{2^{l + 1}}*(3*(-1) + l*2 + 4 - l)

-5,822

\frac{1}{(2*(-1) + k)*2}*3 = \dfrac{3}{4*(-1) + k*2}

11,764

\frac{dy}{dz} = \frac{3*z^2}{3*y^2} = \frac{1}{y^2}*z * z

6,965

\frac{1}{2*(-1) + x}*(y*N + x^3 - x*N*y - 2*x^2 + 2*y*x) = \tfrac{-N*y + 4*y}{x + 2*(-1)} + x^2 - y*N + y*2

27,828

y^2 \cdot h + h = y + y \cdot h \cdot h \implies y = h

23,061

r^{\dfrac13} + 2*(-1) = d \Rightarrow r = (d + 2)^3

20,152

s*x*r = r*s*x

8,585

b\cdot x = 1/(b\cdot x) = \frac{1}{x\cdot b} = x\cdot b

-27,037

\sum_{n=1}^\infty \frac{1}{n*6^n} \left(1 + 5\right)^n*(n + 2) = \sum_{n=1}^\infty \frac{6^n}{n*6^n} (n + 2) = \sum_{n=1}^\infty (n + 2)/n

-7,123

4/10*\frac39*2/8 = 1/30

-23,377

\dfrac{9}{32} = 3/8\cdot \frac{3}{4}

28,727

2^{k + 2} + 4\cdot (-1) = (2^{k + 1} + 2\cdot \left(-1\right))\cdot 2

43,272

\binom{13 + 3 + \left(-1\right)}{3 + (-1)} = \binom{15}{2}

-20,335

\tfrac{1}{7\cdot x + 14\cdot (-1)}\cdot \left(-4\cdot x + 8\right) = \dfrac{x + 2\cdot (-1)}{x + 2\cdot (-1)}\cdot (-\frac{4}{7})

-8,803

48\cdot \pi = 30\cdot \pi + 9\cdot \pi + 9\cdot \pi

6,131

\cos(\dfrac{1}{4}\cdot \pi + z - \dfrac14\cdot \pi) = \cos\left(z\right)

-1,979

\pi/3 + \pi/6 = \frac{\pi}{2}

-19,384

\frac{7\cdot \frac{1}{4}}{3\cdot 1/4} = 7/4\cdot 4/3

3,674

24\cdot \mathrm{i} - 7 = \left(4\cdot \mathrm{i} + 3\right)^2

3,075

1 - z*2 \lt (-1) - z \Rightarrow 2 < z

-20,769

-5/6 \frac{6x + 10 (-1)}{10 \left(-1\right) + 6x} = \frac{50 - x \cdot 30}{36 x + 60 (-1)}

3,204

(\sqrt{a + z_1})^2 = (\sqrt{F + z_2})^2 \Rightarrow z_2 + F = z_1 + a

-6,207

\frac{4\cdot q}{(q + 3\cdot (-1))\cdot \left(q + 8\right)} = \frac{4\cdot q}{q^2 + q\cdot 5 + 24\cdot \left(-1\right)}

2,241

p - 1/2 = p - \tfrac12\left(p + 1\right) = (p + (-1))/2

35,149

-a \cdot 2 + 5 + 2 \cdot (-1) = -2 \cdot a + 3

13,966

\cos(\operatorname{asin}(x)) = \left(1 - x^2\right)^{\tfrac{1}{2}}

11,347

f_2 + f_3 + \dotsm + f_l + f_{l + 1} = f_2 + f_3 + \dotsm + f_l + f_{l + 1}

-6,995

\frac19 \cdot 4 = 4/9

7,740

\cos^2{\theta} = (\cos{2\theta} + 1)/2

35,343

-((-1) + t) = 1 - t

-578

\left(e^{i\cdot \pi/3}\right)^{14} = e^{\frac{i\cdot \pi}{3}\cdot 14}