ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-24,758	\tan\left(\dfrac{17\pi}{12}\right) = \dfrac{\sqrt{3}+1}{\sqrt{3}-1}
5,694	3 + \sqrt{3 + \sqrt{3 + ...}} = B \Rightarrow (B + 3 (-1))^2 = B
31,764	(3 + z) \cdot (z + 3 \cdot (-1)) = z^2 + 9 \cdot \left(-1\right)
19,324	\sqrt{2} = \left(2^{1/4}\right)^2
17,780	5^n*5^3 = 5^{n + 3}
36,120	44 + 32 \cdot (-1) = 12
6,520	\frac{\binom{13}{4}}{\binom{13}{4} + \binom{13}{3} \cdot 39} = 5/83
-29,344	(h - b)*(h + b) = -b^2 + h^2
12,992	(C_1 + C_2)^2 = C_1^2 + C_2^2 + C_2 C_1 + C_2 C_1
8,009	\frac32\times 1/2\times 3 = 9/4
16,374	\sin(\frac{\pi}{4}) = \sin(\frac{3\pi}{4}) = \frac122^{1 / 2}
33,656	5!/3! = \frac{5!}{(2\left(-1\right) + 5)!}
2,491	21^2 = (a^2 + b^2)^2 = a^4 + b^4 + 2a^2b^2 = a^4 + b^4 + 22 2
33,829	\dfrac{\frac{1}{z}}{\sin{z}}\cdot \sin{z} = \frac{1}{z}
16,651	\sin{\pi/4} - \sin{(\pi*(-1))/4} = \sqrt{2}
-1,096	\frac{5\cdot 1/9}{1/5\cdot 8} = \dfrac58\cdot 5/9
17,023	0 = (y^3 + (-1))\cdot (y^6 + y^3 + 1) = y^9 + \left(-1\right)
29,022	x \cdot x + 3 \cdot v \cdot v = (2 \cdot v)^2 + (x - v)^2 + v \cdot 2 \cdot \left(-v + x\right)
16,295	\dfrac{1}{m!(x - m)!}x! = \binom{x}{m}
-20,480	-1/4 (-10/(-10)) = \frac{10}{-40}
2,917	\left(\left(f_1 \lt f_2 rightarrow f_2 + f_1 < f_2 + f_2\right) rightarrow f_2 + f_1 \lt 2f_2\right) rightarrow f_2 > \frac{1}{2}(f_1 + f_2)
39,661	m = x^2 \implies x = m^{\dfrac{1}{2}}
22,696	3^{70}*(\left(\frac23\right)^{70} + 1) = 2^{70} + 3^{70}
22,305	x\cdot 6 = 0 \implies 0 = x
13,335	10 = 5\cdot 2 = 5\cdot \left(3 + (-1)\right) = 5\cdot (\left(3^4\right)^{1/4} + \left(-1\right))
39,297	\left(10 + 6\cdot (-1)\right)^k = 4^k \geq 2^k
6,071	k + \left(-1\right) = 2k + 2\left(-1\right) - k + (-1)
-29,871	\frac{\mathrm{d}}{\mathrm{d}y} (5\cdot y^4) = 5\cdot \frac{\mathrm{d}}{\mathrm{d}y} y^4 = 5\cdot 4\cdot y^3 = 20\cdot y^3
454	Q + 27 \cdot (-1) = (9 + Q^{\frac{1}{3} \cdot 2} + 3 \cdot Q^{1/3}) \cdot (3 \cdot (-1) + Q^{1/3})
-3,422	\left(3 + 2\right) \sqrt{10} = \sqrt{10}*5
30,203	2 = 6\left(-1\right) + 2 + 0 + 1 + 6 + 2(-1) + 0*(-1) + 1
30,803	4^{x + (-1)} \cdot 4 = 4^x
40,845	1.5 = \dfrac14\cdot 6
9,951	(-a + x) \cdot \left(a^2 + x^2 + a \cdot x\right) = -a^3 + x^2 \cdot x
25,992	\frac{1}{1 - 9/10} + \left(-1\right) = 10 + \left(-1\right) = 9
11,192	7/12 \cdot \dfrac{6}{11} = \dfrac{42}{132}
18,906	h\cdot i = (h\cdot i)^a = h^a\cdot i^a
6,624	-1 = \xi^k \implies \xi^{k*2} = 1
11,487	9 \cdot \sqrt{3} + \sqrt{2} \cdot 11 = (\sqrt{2} + \sqrt{3})^3
-19,461	7/2\frac89 = 71/2/(1/8*9)
3,668	\dfrac{41/5}{8}x^2 = x^2/10
-6,351	\frac{1}{5\cdot \left(10 + r\right)} = \frac{1}{r\cdot 5 + 50}
16,282	(z + 1)\cdot \left(z^2 - z + 1\right) = z \cdot z^2 + 1
5,507	6^3 + 2^3 + 3^3 = 1^3 + 5 5 5 + 5^3
3,929	k6 + 2 = 32*k + 2
14,291	2a + (-1) = a a - (a + \left(-1\right))^2
1,868	h_2/(h_1 x) = \dfrac{h_2}{xh_1}
25,222	\|4 + 5 (-1)\| = 1 < 2
-6,305	\frac{1}{(z + 6) (9\left(-1\right) + z)} = \frac{1}{z^2 - 3z + 54 (-1)}
45,659	(\sqrt{17} - 1)/2 = -\frac{1}{2} + \sqrt{17}/2
-8,091	\dfrac{-23 - i15}{-5 - i}\frac{-5 + i}{i - 5} = \frac{-23 - i*15}{-5 - i}
-20,954	\frac{-x\cdot 5 + (-1)}{(-1) - 5\cdot x}\cdot (-\dfrac{2}{7}) = \frac{10\cdot x + 2}{7\cdot (-1) - 35\cdot x}
-18,509	-5 = 3\cdot \left(2\cdot x + (-1)\right) = 6\cdot x + 3\cdot (-1) = 6\cdot x + 3\cdot \left(-1\right)
5,947	-2 \cdot x = 6 \cdot e^{(-5 - 2 \cdot x)/2} = 6 \cdot \dfrac{1}{e^\frac{5}{2}} \cdot e^{-x}
7,071	15 + g \cdot 100 \cdot 5 + 5 \cdot e \cdot 10 = 100 g + e \cdot 10 + 1 + g \cdot 100 + 10 e + 2 + \cdots + 100 g + e \cdot 10 + 5
-6,389	\frac{2}{m \cdot 5 + 45} = \frac{1}{5 \cdot \left(m + 9\right)} \cdot 2
665	x * x - 3x + 2(-1) = (x + (-1))(x + 2(-1)) = (2x + 2\left(-1\right))*(\frac{x}{2} + (-1))
31,914	\sqrt{y} \times 21 \times y^2 = y^{5/2} \times 21
11,317	z^{\frac12} \cdot z^3 = z^{7/2}
6,093	\left(a - b\right)\cdot \left(b + a\right) = a^2 - b^2
-1,604	\pi\cdot \frac{1}{3}\cdot 5 = \pi\cdot 2 - \frac{\pi}{3}
1,266	3^2\cdot 3^{z + 2} = 3^{z + 4}
-17,425	38 \times (-1) + 44 = 6
1,911	2\cdot 84/17\cdot 1/2\cdot 17/2 = 42
25,456	1 + 2^0 + 2^1 + 2^2...2^n = 2^{n + 1}
-9,299	-77 z^2 = -zz711
19,834	(\alpha \cdot \alpha)^3 = \alpha^3 \cdot \alpha^3 = (\alpha + 1)^2 = \alpha^2 + 1
-260	\binom{8}{6} = \frac{8!}{(8 + 6(-1))! \cdot 6!}
35,046	-8/3 = -\frac{1}{3} \cdot 8
-24,906	8 \times 7 \times 6 \times 5 \times 4 = \frac{1}{(8 + 5 \times \left(-1\right))!} \times 8! = 6720
16,988	Axy = yAx = y^SAx = (A^Sy)^Sx
-23,286	4/7 \cdot \frac{3}{8} = 3/14
4,866	n - 2 \times j + j = n - j
39,052	h \cdot k = k \cdot h
39,615	2 + 2 + 3 + 0 = 7
30,107	0 = x^{15}\cdot 31 - x^{10} + 32 \Rightarrow 0 = (1 + x^{15})\cdot 31 - x^{10} + (-1)
5,398	\frac{2}{\sqrt{H} + \sqrt{2 + H}} = -\sqrt{H} + \sqrt{2 + H}
547	-\frac25 = -\frac152
1,123	-i\frac{1 - (-1)^{24}}{1 - i} = -i\frac{1}{1 - i}(1 + (-1)) = 0
21,268	7142 = 21000\cdot \left(-1\right) + 28142
35,634	8 \cdot (7 \cdot (-1) + a \cdot 8 + h \cdot 4) + a - h \cdot 7 = 4 \Rightarrow h = \frac15 \cdot (-13 \cdot a + 12)
2,456	x^l + \theta + \left(-1\right) = (\theta + (-1))\cdot (x^{l + (-1)} + \cdots + x + 1)
16,055	\int 1*2\pi p\,\text{d}p = 2\pi p^2/2 = \pi p^2
-635	(e^{\pi i*17/12})^3 = e^{3\frac{i\pi}{12}17}
5,283	\frac{1}{2} = 0 + 0 + 1/2
21,457	\left(j = c^{\tfrac12} \Rightarrow (c^{1/2})^2 = j^2\right) \Rightarrow c = j \times j
3,253	\|z \cdot w\|^2 = z \cdot w \cdot \overline{z \cdot w} = \|z\|^2 \cdot \|w\|^2
-1,773	\frac{\pi}{3} = 2 \cdot \pi - 5/3 \cdot \pi
-23,545	\frac{\frac{3}{7}}{2} \cdot 1 = \dfrac{1}{14} \cdot 3
2,353	\cos(\alpha - \beta) = \cos{\alpha}\cdot \cos{\beta} + \sin{\alpha}\cdot \sin{\beta}
-18,258	\frac{z}{(z + 6\cdot (-1))\cdot (6\cdot (-1) + z)}\cdot (6\cdot (-1) + z) = \frac{1}{36 + z \cdot z - 12\cdot z}\cdot (z^2 - 6\cdot z)
-4,764	\frac{1}{2 + x^2 - 3x}(7(-1) + 6x) = \tfrac{1}{x + \left(-1\right)} + \dfrac{5}{x + 2(-1)}
15,710	0 = 8 z - (z + y)^3 = 8 y - (z + y)^3
-9,367	x^2\cdot 8 - x\cdot 12 = -x\cdot 2\cdot 2\cdot 3 + 2\cdot 2\cdot 2\cdot x\cdot x
-20,896	-\frac{6}{-20} = -2/\left(-2\right) \cdot \frac{3}{10}
22,412	0 = x^2\cdot a + x\cdot f_2 + f_1 \Rightarrow x = \frac{1}{a\cdot 2}\cdot (-f_2 +- \sqrt{-4\cdot f_1\cdot a + f_2^2})
-7,657	(-20 - 95i + 8i + 38(-1))/29 = \frac{1}{29}(-58 - 87i) = -2 - 3i
4,252	l = g\cdot \mu/g\Longrightarrow g\cdot \mu = l\cdot g
-27,936	\frac{d}{dx} (2\sec(x)) = 2d/dx \sec(x) = 2\sec(x) \tan(x)
5,208	1^2 + 3\cdot 4^2 = 49 = 7^2

-24,758

\tan\left(\dfrac{17\pi}{12}\right) = \dfrac{\sqrt{3}+1}{\sqrt{3}-1}

5,694

3 + \sqrt{3 + \sqrt{3 + ...}} = B \Rightarrow (B + 3 (-1))^2 = B

31,764

(3 + z) \cdot (z + 3 \cdot (-1)) = z^2 + 9 \cdot \left(-1\right)

19,324

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

17,780

5^n*5^3 = 5^{n + 3}

36,120

44 + 32 \cdot (-1) = 12

6,520

\frac{\binom{13}{4}}{\binom{13}{4} + \binom{13}{3} \cdot 39} = 5/83

-29,344

(h - b)*(h + b) = -b^2 + h^2

12,992

(C_1 + C_2)^2 = C_1^2 + C_2^2 + C_2 C_1 + C_2 C_1

8,009

\frac32\times 1/2\times 3 = 9/4

16,374

\sin(\frac{\pi}{4}) = \sin(\frac{3*\pi}{4}) = \frac12*2^{1 / 2}

33,656

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

2,491

21^2 = (a^2 + b^2)^2 = a^4 + b^4 + 2*a^2*b^2 = a^4 + b^4 + 2*2 * 2

33,829

\dfrac{\frac{1}{z}}{\sin{z}}\cdot \sin{z} = \frac{1}{z}

16,651

\sin{\pi/4} - \sin{(\pi*(-1))/4} = \sqrt{2}

-1,096

\frac{5\cdot 1/9}{1/5\cdot 8} = \dfrac58\cdot 5/9

17,023

0 = (y^3 + (-1))\cdot (y^6 + y^3 + 1) = y^9 + \left(-1\right)

29,022

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

16,295

\dfrac{1}{m!*(x - m)!}*x! = \binom{x}{m}

-20,480

-1/4 (-10/(-10)) = \frac{10}{-40}

2,917

\left(\left(f_1 \lt f_2 rightarrow f_2 + f_1 < f_2 + f_2\right) rightarrow f_2 + f_1 \lt 2f_2\right) rightarrow f_2 > \frac{1}{2}(f_1 + f_2)

39,661

m = x^2 \implies x = m^{\dfrac{1}{2}}

22,696

3^{70}*(\left(\frac23\right)^{70} + 1) = 2^{70} + 3^{70}

22,305

x\cdot 6 = 0 \implies 0 = x

13,335

10 = 5\cdot 2 = 5\cdot \left(3 + (-1)\right) = 5\cdot (\left(3^4\right)^{1/4} + \left(-1\right))

39,297

\left(10 + 6\cdot (-1)\right)^k = 4^k \geq 2^k

6,071

k + \left(-1\right) = 2k + 2\left(-1\right) - k + (-1)

-29,871

\frac{\mathrm{d}}{\mathrm{d}y} (5\cdot y^4) = 5\cdot \frac{\mathrm{d}}{\mathrm{d}y} y^4 = 5\cdot 4\cdot y^3 = 20\cdot y^3

454

Q + 27 \cdot (-1) = (9 + Q^{\frac{1}{3} \cdot 2} + 3 \cdot Q^{1/3}) \cdot (3 \cdot (-1) + Q^{1/3})

-3,422

\left(3 + 2\right) \sqrt{10} = \sqrt{10}*5

30,203

2 = 6*\left(-1\right) + 2 + 0 + 1 + 6 + 2*(-1) + 0*(-1) + 1

30,803

4^{x + (-1)} \cdot 4 = 4^x

40,845

1.5 = \dfrac14\cdot 6

9,951

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

25,992

\frac{1}{1 - 9/10} + \left(-1\right) = 10 + \left(-1\right) = 9

11,192

7/12 \cdot \dfrac{6}{11} = \dfrac{42}{132}

18,906

h\cdot i = (h\cdot i)^a = h^a\cdot i^a

6,624

-1 = \xi^k \implies \xi^{k*2} = 1

11,487

9 \cdot \sqrt{3} + \sqrt{2} \cdot 11 = (\sqrt{2} + \sqrt{3})^3

-19,461

7/2*\frac89 = 7*1/2/(1/8*9)

3,668

\dfrac{4*1/5}{8}*x^2 = x^2/10

-6,351

\frac{1}{5\cdot \left(10 + r\right)} = \frac{1}{r\cdot 5 + 50}

16,282

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

5,507

6^3 + 2^3 + 3^3 = 1^3 + 5 5 5 + 5^3

3,929

k*6 + 2 = 3*2*k + 2

14,291

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

1,868

h_2/(h_1 x) = \dfrac{h_2}{xh_1}

25,222

|4 + 5 (-1)| = 1 < 2

-6,305

\frac{1}{(z + 6) (9\left(-1\right) + z)} = \frac{1}{z^2 - 3z + 54 (-1)}

45,659

(\sqrt{17} - 1)/2 = -\frac{1}{2} + \sqrt{17}/2

-8,091

\dfrac{-23 - i*15}{-5 - i}*\frac{-5 + i}{i - 5} = \frac{-23 - i*15}{-5 - i}

-20,954

\frac{-x\cdot 5 + (-1)}{(-1) - 5\cdot x}\cdot (-\dfrac{2}{7}) = \frac{10\cdot x + 2}{7\cdot (-1) - 35\cdot x}

-18,509

-5 = 3\cdot \left(2\cdot x + (-1)\right) = 6\cdot x + 3\cdot (-1) = 6\cdot x + 3\cdot \left(-1\right)

5,947

-2 \cdot x = 6 \cdot e^{(-5 - 2 \cdot x)/2} = 6 \cdot \dfrac{1}{e^\frac{5}{2}} \cdot e^{-x}

7,071

15 + g \cdot 100 \cdot 5 + 5 \cdot e \cdot 10 = 100 g + e \cdot 10 + 1 + g \cdot 100 + 10 e + 2 + \cdots + 100 g + e \cdot 10 + 5

-6,389

\frac{2}{m \cdot 5 + 45} = \frac{1}{5 \cdot \left(m + 9\right)} \cdot 2

665

x * x - 3*x + 2*(-1) = (x + (-1))*(x + 2*(-1)) = (2*x + 2*\left(-1\right))*(\frac{x}{2} + (-1))

31,914

\sqrt{y} \times 21 \times y^2 = y^{5/2} \times 21

11,317

z^{\frac12} \cdot z^3 = z^{7/2}

6,093

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

-1,604

\pi\cdot \frac{1}{3}\cdot 5 = \pi\cdot 2 - \frac{\pi}{3}

1,266

3^2\cdot 3^{z + 2} = 3^{z + 4}

-17,425

38 \times (-1) + 44 = 6

1,911

2\cdot 84/17\cdot 1/2\cdot 17/2 = 42

25,456

1 + 2^0 + 2^1 + 2^2*...*2^n = 2^{n + 1}

-9,299

-77 z^2 = -zz*7*11

19,834

(\alpha \cdot \alpha)^3 = \alpha^3 \cdot \alpha^3 = (\alpha + 1)^2 = \alpha^2 + 1

-260

\binom{8}{6} = \frac{8!}{(8 + 6(-1))! \cdot 6!}

35,046

-8/3 = -\frac{1}{3} \cdot 8

-24,906

8 \times 7 \times 6 \times 5 \times 4 = \frac{1}{(8 + 5 \times \left(-1\right))!} \times 8! = 6720

16,988

A*x*y = y*A*x = y^S*A*x = (A^S*y)^S*x

-23,286

4/7 \cdot \frac{3}{8} = 3/14

4,866

n - 2 \times j + j = n - j

39,052

h \cdot k = k \cdot h

39,615

2 + 2 + 3 + 0 = 7

30,107

0 = x^{15}\cdot 31 - x^{10} + 32 \Rightarrow 0 = (1 + x^{15})\cdot 31 - x^{10} + (-1)

5,398

\frac{2}{\sqrt{H} + \sqrt{2 + H}} = -\sqrt{H} + \sqrt{2 + H}

547

-\frac25 = -\frac152

1,123

-i\frac{1 - (-1)^{24}}{1 - i} = -i\frac{1}{1 - i}(1 + (-1)) = 0

21,268

7142 = 21000\cdot \left(-1\right) + 28142

35,634

8 \cdot (7 \cdot (-1) + a \cdot 8 + h \cdot 4) + a - h \cdot 7 = 4 \Rightarrow h = \frac15 \cdot (-13 \cdot a + 12)

2,456

x^l + \theta + \left(-1\right) = (\theta + (-1))\cdot (x^{l + (-1)} + \cdots + x + 1)

16,055

\int 1*2\pi p\,\text{d}p = 2\pi p^2/2 = \pi p^2

-635

(e^{\pi i*17/12})^3 = e^{3\frac{i\pi}{12}17}

5,283

\frac{1}{2} = 0 + 0 + 1/2

21,457

\left(j = c^{\tfrac12} \Rightarrow (c^{1/2})^2 = j^2\right) \Rightarrow c = j \times j

3,253

|z \cdot w|^2 = z \cdot w \cdot \overline{z \cdot w} = |z|^2 \cdot |w|^2

-1,773

\frac{\pi}{3} = 2 \cdot \pi - 5/3 \cdot \pi

-23,545

\frac{\frac{3}{7}}{2} \cdot 1 = \dfrac{1}{14} \cdot 3

2,353

\cos(\alpha - \beta) = \cos{\alpha}\cdot \cos{\beta} + \sin{\alpha}\cdot \sin{\beta}

-18,258

\frac{z}{(z + 6\cdot (-1))\cdot (6\cdot (-1) + z)}\cdot (6\cdot (-1) + z) = \frac{1}{36 + z \cdot z - 12\cdot z}\cdot (z^2 - 6\cdot z)

-4,764

\frac{1}{2 + x^2 - 3x}(7(-1) + 6x) = \tfrac{1}{x + \left(-1\right)} + \dfrac{5}{x + 2(-1)}

15,710

0 = 8 z - (z + y)^3 = 8 y - (z + y)^3

-9,367

x^2\cdot 8 - x\cdot 12 = -x\cdot 2\cdot 2\cdot 3 + 2\cdot 2\cdot 2\cdot x\cdot x

-20,896

-\frac{6}{-20} = -2/\left(-2\right) \cdot \frac{3}{10}

22,412

0 = x^2\cdot a + x\cdot f_2 + f_1 \Rightarrow x = \frac{1}{a\cdot 2}\cdot (-f_2 +- \sqrt{-4\cdot f_1\cdot a + f_2^2})

-7,657

(-20 - 95*i + 8*i + 38*(-1))/29 = \frac{1}{29}*(-58 - 87*i) = -2 - 3*i

4,252

l = g\cdot \mu/g\Longrightarrow g\cdot \mu = l\cdot g

-27,936

\frac{d}{dx} (2\sec(x)) = 2d/dx \sec(x) = 2\sec(x) \tan(x)

5,208

1^2 + 3\cdot 4^2 = 49 = 7^2