ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
11,620	5 + \sqrt{30} \cdot 0 = \sqrt{5} \cdot \sqrt{5}
-1,035	2 * 2 = 2*2 = 4
-1,727	-\pi\frac56 = -\frac76\pi + \frac{1}{3}*\pi
-16,797	7 = 7\cdot 5\cdot k + 7\cdot 4 = 35\cdot k + 28 = 35\cdot k + 28
-27,485	11\cdot c\cdot c\cdot c\cdot 2 = c^3\cdot 22
-18,083	31 = 17 \cdot (-1) + 48
49,613	\frac{1}{-x + 3} \cdot (3/2 + \frac12 \cdot 3 \cdot x) = -\frac12 \cdot 3 + \dfrac{2}{-\dfrac{x}{3} + 1}
-10,622	\frac{15(-1) + x15}{36 + 24x} = \frac{5(-1) + x5}{12 + x8}*3/3
11,621	0 = z^4 + 6z^2 + 25 = (z^2 + 5)^2 - 4z^2 = (z * z - 2z + 5)(z^2 + 2*z + 5)
14,621	-7 \cdot 17 + 2 \cdot 60 = 17 \cdot (-1) + 2 \cdot \left(-17 \cdot 3 + 60\right)
12,348	x + y \cdot 0.5 = 1.75 \Rightarrow x = 1.75 - y \cdot 0.5
24,209	z^{1 + k} = z^k \cdot z
27,985	\|z + (-1)\| = \|1 - z\| \geq \|1\| - \|z\| = 1 - \|z\| \Rightarrow \|z\| \geq \frac{1}{4} \cdot 3
9,336	\left(\left(2 = a + 1/a \Rightarrow 0 = a^2 - 2\cdot a + 1\right) \Rightarrow \left(a + (-1)\right) \cdot \left(a + (-1)\right) = 0\right) \Rightarrow 1 = a
19,830	X \backslash E + z = X \cap \overline{E + z} = X \cap \overline{E} + z
23,311	\left(1 + 2 rightarrow k_1 \cos(E) + k_2 \sin(E) = 1\right) rightarrow \left(\cos(E) k_1\right)^2 = (-\sin(E) k_2 + 1)^2
14,676	45(-1) + 25 = 36(-1) + 16
8,699	(a \cdot x) \cdot (a \cdot x) = (x \cdot a) \cdot (x \cdot a)
-29,195	-1 \cdot 2 + 4\cdot 1 = 2
3,037	\tfrac{\sin{x^5}}{x^5}*x^4 = \frac{\sin{x^5}}{x}
12,956	1 - \frac{2 \cdot c}{a - b + c} \cdot 1 = \frac{1}{a - b + c} \cdot (-c \cdot 2 + a - b + c)
6,189	128\times 512 = 2^7\times 2^9 = 2^{16} = 65536
-20,980	\frac{90 (-1) - s*9}{10 s + 100} = \frac{10 + s}{s + 10} (-\frac{9}{10})
-2,660	\sqrt{11} \cdot 4 + \sqrt{11} = \sqrt{16} \cdot \sqrt{11} + \sqrt{11}
-26,907	\sum_{x=1}^∞ \frac{(-5)^x}{x\cdot 5^x} = \sum_{x=1}^∞ \frac{(-1)^x\cdot 5^x}{x\cdot 5^x} = \sum_{x=1}^∞ (-1)^x/x
24,056	\tfrac{1}{\frac13 + 1/4} = \frac{12}{7}
14,552	\frac{2 + 2\cdot \sin(X)}{\cos(X)\cdot (1 + \sin(X))} = 2/\cos(X) = 2\cdot \sec(X)
41,730	4 + 6\times 166 = 1000
15,997	A - C = 0 \Rightarrow -C^2 + A^2 = 0
36,996	\|g\cdot x\| = \|g\cdot x\|
6,677	9 \cdot (-1) + 2 \cdot d = 2 \cdot d + 1 + 10 \cdot (-1)
-5,246	1.35 \times 10 = \frac{13.5}{10^6} \times 1 = \dfrac{1.35}{10^5}
33,927	2 \cdot \left(c - x\right) = -(x - c) + c - x
34,883	(y + h)\cdot \overline{y + h} = y\cdot \bar{y} + y\cdot \bar{h} + \bar{y}\cdot h + h\cdot \bar{h} = \|y\|^2 + 2\cdot \operatorname{re}{(y\cdot \bar{h})} + \|h\|^2
-11,886	1.255/10 = 1.255\cdot 0.1
8,295	\sqrt{6 + \sqrt{20}} = \sqrt{6 + 2\cdot \sqrt{5}}
-22,769	\frac{70}{30} = \frac{10}{10 \cdot 3} \cdot 7
7,466	\sin(\pi4 + \pi2) = \sin{\pi*4} + \sin{2\pi}
-23,180	-6 = -1/2 \times 12
-20,934	\dfrac{-r \cdot 7 + 2}{2 - r \cdot 7} (-1^{-1}) = \frac{2(-1) + 7r}{2 - 7r}
9,365	\frac{z + 1}{z + (-1)} = \frac{1}{\left(z + (-1)\right) \cdot 1/z} \cdot \tfrac{1}{z} \cdot (z + 1) = \frac{1 + 1/z}{1 - 1/z}
31,311	x f_1 f_2 x = x f_1 f_2 x
52,345	100 \cdot 10.233 = \left\{1023.3\right\} = 0.3
50,596	7^1 = 2^1 + 5^1
-3,354	\sqrt{13}(3 + 2 + 4) = \sqrt{13}9
28,778	23k + 5 = 23k + 4 + 1 = 2(3k + 2) + 1
1,247	z + \left(-1\right) = I\Longrightarrow 1 + I = z
3,101	x\cdot K = 1/(x\cdot K) = 1/\left(K\cdot x\right) = K\cdot x
35,846	1 + e^{2i} = 1 + e^{i} \cdot e^{i} = 1 + e^i \cdot i \sin(1) + e^i \cdot \cos(1)
15,023	\frac{1}{x^2 - y^2} \cdot (x - y) = \frac{1}{y + x}
6,562	g^3 - b^3 = (g^2 + b\cdot g + b^2)\cdot (g - b)
4,185	(1 - \frac{1}{2})^{-\tfrac{3}{2}} = 2^{\frac12\cdot 3} = \sqrt{8}
15,680	\dfrac{\sin^r(\\|x\\|)}{\\|x\\|^r} \\|x\\|^{r + (-1)} = \frac{1}{\\|x\\|}\sin^r(\\|x\\|)
29,362	t + 3 = \dfrac{1}{3\cdot (-1) + t}\cdot \left(t^2 + 9\cdot \left(-1\right)\right)
-18,943	\frac{5}{6} = \frac{1}{9 \cdot \pi} \cdot A_s \cdot 9 \cdot \pi = A_s
8,512	a x + x b = (a + b) x
27,938	\sqrt{h} = h^{\frac{1}{2}}
21,836	2*27 + 9\left(-1\right) = 45
-18,803	\frac{5}{5} \cdot y = y
-4,410	z^2 + 3z + 4(-1) = (z + (-1))*(z + 4)
16,306	\|A - y\cdot I\| = \|(A - y\cdot I)^Z\| = \|A^Z - y\cdot I\|
16,174	A^2 - Z^2 = (A - Z) \cdot (A + Z)
18,087	(e + \left(-1\right)) \cdot ((-1) + g) = 70 \implies \left[g, e\right] = [2, 71], \left[3, 36\right], \left[6,15\right]
-15,100	\frac{1}{r^4\frac{1}{k^{16}}}k^4 = \frac{k^4}{\tfrac{1}{\frac{1}{r^4}*k^{16}}}
-6,468	\frac{3}{16 + z \cdot 2} = \frac{1}{\left(z + 8\right) \cdot 2} \cdot 3
1,287	\frac{1}{-61^3 + 1049^3}(1823^3 - 1699^3) = 1
31,553	(6 + 7 + 12 + 15 + 19)^3 = 19^4 + 6^4 + 7^4 + 12^4 + 15^4
6,298	2 \cdot \dfrac26 \cdot 3 \cdot \frac{6 + 3 \cdot (-1)}{6 + (-1)} = \frac15 \cdot 6 = 1.2
3,829	t - x + (-1) = t - x + 1
1,004	\frac{1}{1 - y^2} = \frac{1}{2 (y + 1)} + \frac{1}{2*(-y + 1)}
11,664	F * F^2 + 3F^2 H + 3H^2 F + H * H^2 = (H + F)^3
20,330	r^0 - r^l = 1 - r^l
3,314	\beta \cdot s - s^2 = -(s - \frac{\beta}{2}) \cdot (s - \frac{\beta}{2}) + \frac14 \cdot \beta \cdot \beta
11,242	\tfrac{9}{48} + 3/54 = 3/16 + 1/18 = \dots
4,392	(-b^2 + c * c)x = x(c - b)*\left(c + b\right)
-6,086	\frac{1}{40 + x4}5 = \frac{1}{(10 + x)4}5
2,230	0 = (g + b - d)\cdot z \cdot z + 2\cdot (g + b)\cdot z + g + b + d = (g + b)\cdot \left(z + 1\right) \cdot \left(z + 1\right) - d\cdot (z^2 + (-1))
-22,272	a^2 + 2 \times a + 15 \times (-1) = (a + 3 \times (-1)) \times (a + 5)
15,573	\dfrac{2 + 7}{1 + 3} = 9/4
13,517	\alpha^9\times \alpha\times \alpha^3 = \alpha^{13}
-2,771	(4 + 1)\cdot 5^{1 / 2} = 5\cdot 5^{1 / 2}
14,690	-10 \cdot x = 1 \pm \left(4 \cdot x^4 - 4 \cdot x^2 + 1\right)^{1/2} = 1 \pm ((2 \cdot x^2 + (-1)) \cdot (2 \cdot x^2 + (-1)))^{1/2}
11,781	84\cdot 256^3 + 101\cdot 256^2 + 256^1\cdot 115 + 256^0\cdot 116 = 1415934836
29,522	\frac{1}{\left(y - a\right) (y - b)} = \frac{1}{b - a}\left(-\frac{1}{y - a} + \dfrac{1}{-b + y}\right)
1,033	n = \dfrac12\cdot \left((-1) + 1 + 2\cdot n\right)
25,923	a^{b \cdot \psi} = (a^b)^\psi = (a^\psi)^b
-14,236	\dfrac{12}{9 + 3} = \tfrac{12}{12} = 12/12 = 1
-10,376	-12 = -2y + 12 + 10(-1) = -2*y + 2
41,768	\sum_{n=1}^\infty nx^{n-1}=\sum_{n=0}^\infty(n+1)x^n=\sum_{n=0}^\infty nx^n+\sum_{n=0}^\infty x^n
21,165	\left(\tfrac1x\right)^{1/5} = (\frac{1}{x})^{1/5} = x^{-1/5}
2,736	0 + 2 = C\Longrightarrow C = 2
8,159	3 = (\dfrac{1}{2}*\sqrt{12})^2
15,574	(b + a)^2 = a \cdot a + 2 \cdot a \cdot b + b^2
31,627	tx^2/y = \frac{\left(tx\right)^2}{y*t}
6,875	(O_l)/(\pi_l) = O_l/(\pi_l)
12,360	(k + 1)^3 - k + 1 - k^3 + k = k\cdot 3 + k^2\cdot 3
16,801	334*4=144
36,509	55 = 5^2 + 1 \times 1 + 2^2 + 3 \times 3 + 4^2
26,708	2^{\frac34} = \left(2^3\right)^{1/4}
7,139	\frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2}} = 1 + \frac{x}{\sqrt{x^2 + 1}}

11,620

5 + \sqrt{30} \cdot 0 = \sqrt{5} \cdot \sqrt{5}

-1,035

2 * 2 = 2*2 = 4

-1,727

-\pi*\frac56 = -\frac76*\pi + \frac{1}{3}*\pi

-16,797

7 = 7\cdot 5\cdot k + 7\cdot 4 = 35\cdot k + 28 = 35\cdot k + 28

-27,485

11\cdot c\cdot c\cdot c\cdot 2 = c^3\cdot 22

-18,083

31 = 17 \cdot (-1) + 48

49,613

\frac{1}{-x + 3} \cdot (3/2 + \frac12 \cdot 3 \cdot x) = -\frac12 \cdot 3 + \dfrac{2}{-\dfrac{x}{3} + 1}

-10,622

\frac{15*(-1) + x*15}{36 + 24*x} = \frac{5*(-1) + x*5}{12 + x*8}*3/3

11,621

0 = z^4 + 6*z^2 + 25 = (z^2 + 5)^2 - 4*z^2 = (z * z - 2*z + 5)*(z^2 + 2*z + 5)

14,621

-7 \cdot 17 + 2 \cdot 60 = 17 \cdot (-1) + 2 \cdot \left(-17 \cdot 3 + 60\right)

12,348

x + y \cdot 0.5 = 1.75 \Rightarrow x = 1.75 - y \cdot 0.5

24,209

z^{1 + k} = z^k \cdot z

27,985

|z + (-1)| = |1 - z| \geq |1| - |z| = 1 - |z| \Rightarrow |z| \geq \frac{1}{4} \cdot 3

9,336

\left(\left(2 = a + 1/a \Rightarrow 0 = a^2 - 2\cdot a + 1\right) \Rightarrow \left(a + (-1)\right) \cdot \left(a + (-1)\right) = 0\right) \Rightarrow 1 = a

19,830

X \backslash E + z = X \cap \overline{E + z} = X \cap \overline{E} + z

23,311

\left(1 + 2 rightarrow k_1 \cos(E) + k_2 \sin(E) = 1\right) rightarrow \left(\cos(E) k_1\right)^2 = (-\sin(E) k_2 + 1)^2

14,676

45*(-1) + 25 = 36*(-1) + 16

8,699

(a \cdot x) \cdot (a \cdot x) = (x \cdot a) \cdot (x \cdot a)

-29,195

-1 \cdot 2 + 4\cdot 1 = 2

3,037

\tfrac{\sin{x^5}}{x^5}*x^4 = \frac{\sin{x^5}}{x}

12,956

1 - \frac{2 \cdot c}{a - b + c} \cdot 1 = \frac{1}{a - b + c} \cdot (-c \cdot 2 + a - b + c)

6,189

128\times 512 = 2^7\times 2^9 = 2^{16} = 65536

-20,980

\frac{90 (-1) - s*9}{10 s + 100} = \frac{10 + s}{s + 10} (-\frac{9}{10})

-2,660

\sqrt{11} \cdot 4 + \sqrt{11} = \sqrt{16} \cdot \sqrt{11} + \sqrt{11}

-26,907

\sum_{x=1}^∞ \frac{(-5)^x}{x\cdot 5^x} = \sum_{x=1}^∞ \frac{(-1)^x\cdot 5^x}{x\cdot 5^x} = \sum_{x=1}^∞ (-1)^x/x

24,056

\tfrac{1}{\frac13 + 1/4} = \frac{12}{7}

14,552

\frac{2 + 2\cdot \sin(X)}{\cos(X)\cdot (1 + \sin(X))} = 2/\cos(X) = 2\cdot \sec(X)

41,730

4 + 6\times 166 = 1000

15,997

A - C = 0 \Rightarrow -C^2 + A^2 = 0

36,996

|g\cdot x| = |g\cdot x|

6,677

9 \cdot (-1) + 2 \cdot d = 2 \cdot d + 1 + 10 \cdot (-1)

-5,246

1.35 \times 10 = \frac{13.5}{10^6} \times 1 = \dfrac{1.35}{10^5}

33,927

2 \cdot \left(c - x\right) = -(x - c) + c - x

34,883

(y + h)\cdot \overline{y + h} = y\cdot \bar{y} + y\cdot \bar{h} + \bar{y}\cdot h + h\cdot \bar{h} = |y|^2 + 2\cdot \operatorname{re}{(y\cdot \bar{h})} + |h|^2

-11,886

1.255/10 = 1.255\cdot 0.1

8,295

\sqrt{6 + \sqrt{20}} = \sqrt{6 + 2\cdot \sqrt{5}}

-22,769

\frac{70}{30} = \frac{10}{10 \cdot 3} \cdot 7

7,466

\sin(\pi*4 + \pi*2) = \sin{\pi*4} + \sin{2\pi}

-23,180

-6 = -1/2 \times 12

-20,934

\dfrac{-r \cdot 7 + 2}{2 - r \cdot 7} (-1^{-1}) = \frac{2(-1) + 7r}{2 - 7r}

9,365

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

31,311

x f_1 f_2 x = x f_1 f_2 x

52,345

100 \cdot 10.233 = \left\{1023.3\right\} = 0.3

50,596

7^1 = 2^1 + 5^1

-3,354

\sqrt{13}*(3 + 2 + 4) = \sqrt{13}*9

28,778

2*3*k + 5 = 2*3*k + 4 + 1 = 2*(3*k + 2) + 1

1,247

z + \left(-1\right) = I\Longrightarrow 1 + I = z

3,101

x\cdot K = 1/(x\cdot K) = 1/\left(K\cdot x\right) = K\cdot x

35,846

1 + e^{2i} = 1 + e^{i} \cdot e^{i} = 1 + e^i \cdot i \sin(1) + e^i \cdot \cos(1)

15,023

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

6,562

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

4,185

(1 - \frac{1}{2})^{-\tfrac{3}{2}} = 2^{\frac12\cdot 3} = \sqrt{8}

15,680

\dfrac{\sin^r(\|x\|)}{\|x\|^r} \|x\|^{r + (-1)} = \frac{1}{\|x\|}\sin^r(\|x\|)

29,362

t + 3 = \dfrac{1}{3\cdot (-1) + t}\cdot \left(t^2 + 9\cdot \left(-1\right)\right)

-18,943

\frac{5}{6} = \frac{1}{9 \cdot \pi} \cdot A_s \cdot 9 \cdot \pi = A_s

8,512

a x + x b = (a + b) x

27,938

\sqrt{h} = h^{\frac{1}{2}}

21,836

2*27 + 9\left(-1\right) = 45

-18,803

\frac{5}{5} \cdot y = y

-4,410

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

16,306

16,174

A^2 - Z^2 = (A - Z) \cdot (A + Z)

18,087

(e + \left(-1\right)) \cdot ((-1) + g) = 70 \implies \left[g, e\right] = [2, 71], \left[3, 36\right], \left[6,15\right]

-15,100

\frac{1}{r^4*\frac{1}{k^{16}}}*k^4 = \frac{k^4}{\tfrac{1}{\frac{1}{r^4}*k^{16}}}

-6,468

\frac{3}{16 + z \cdot 2} = \frac{1}{\left(z + 8\right) \cdot 2} \cdot 3

1,287

\frac{1}{-61^3 + 1049^3}(1823^3 - 1699^3) = 1

31,553

(6 + 7 + 12 + 15 + 19)^3 = 19^4 + 6^4 + 7^4 + 12^4 + 15^4

6,298

2 \cdot \dfrac26 \cdot 3 \cdot \frac{6 + 3 \cdot (-1)}{6 + (-1)} = \frac15 \cdot 6 = 1.2

3,829

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

1,004

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

11,664

F * F^2 + 3F^2 H + 3H^2 F + H * H^2 = (H + F)^3

20,330

r^0 - r^l = 1 - r^l

3,314

\beta \cdot s - s^2 = -(s - \frac{\beta}{2}) \cdot (s - \frac{\beta}{2}) + \frac14 \cdot \beta \cdot \beta

11,242

\tfrac{9}{48} + 3/54 = 3/16 + 1/18 = \dots

4,392

(-b^2 + c * c)*x = x*(c - b)*\left(c + b\right)

-6,086

\frac{1}{40 + x*4}*5 = \frac{1}{(10 + x)*4}*5

2,230

0 = (g + b - d)\cdot z \cdot z + 2\cdot (g + b)\cdot z + g + b + d = (g + b)\cdot \left(z + 1\right) \cdot \left(z + 1\right) - d\cdot (z^2 + (-1))

-22,272

a^2 + 2 \times a + 15 \times (-1) = (a + 3 \times (-1)) \times (a + 5)

15,573

\dfrac{2 + 7}{1 + 3} = 9/4

13,517

\alpha^9\times \alpha\times \alpha^3 = \alpha^{13}

-2,771

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

14,690

-10 \cdot x = 1 \pm \left(4 \cdot x^4 - 4 \cdot x^2 + 1\right)^{1/2} = 1 \pm ((2 \cdot x^2 + (-1)) \cdot (2 \cdot x^2 + (-1)))^{1/2}

11,781

84\cdot 256^3 + 101\cdot 256^2 + 256^1\cdot 115 + 256^0\cdot 116 = 1415934836

29,522

\frac{1}{\left(y - a\right) (y - b)} = \frac{1}{b - a}\left(-\frac{1}{y - a} + \dfrac{1}{-b + y}\right)

1,033

n = \dfrac12\cdot \left((-1) + 1 + 2\cdot n\right)

25,923

a^{b \cdot \psi} = (a^b)^\psi = (a^\psi)^b

-14,236

\dfrac{12}{9 + 3} = \tfrac{12}{12} = 12/12 = 1

-10,376

-12 = -2*y + 12 + 10*(-1) = -2*y + 2

41,768

\sum_{n=1}^\infty nx^{n-1}=\sum_{n=0}^\infty(n+1)x^n=\sum_{n=0}^\infty nx^n+\sum_{n=0}^\infty x^n

21,165

\left(\tfrac1x\right)^{1/5} = (\frac{1}{x})^{1/5} = x^{-1/5}

2,736

0 + 2 = C\Longrightarrow C = 2

8,159

3 = (\dfrac{1}{2}*\sqrt{12})^2

15,574

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

31,627

t*x^2/y = \frac{\left(t*x\right)^2}{y*t}

6,875

(O_l)/(\pi_l) = O_l/(\pi_l)

12,360

(k + 1)^3 - k + 1 - k^3 + k = k\cdot 3 + k^2\cdot 3

16,801

3*3*4*4=144

36,509

55 = 5^2 + 1 \times 1 + 2^2 + 3 \times 3 + 4^2

26,708

2^{\frac34} = \left(2^3\right)^{1/4}

7,139

\frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2}} = 1 + \frac{x}{\sqrt{x^2 + 1}}