ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-20,518	\frac{s}{90 \cdot s} \cdot 20 = \frac{10 \cdot s}{s \cdot 10} \cdot 2/9
8,106	1/2 = \tfrac13 + \dfrac{1}{6}
2,314	\cos{x} = (e^{ix} + e^{-ix})/2 = \frac{1}{2e^{ix}}(e^{2ix} + 1)
15,394	\frac{7^{55}}{5^{72}} = (\frac{7^3}{5^4})^{18}\cdot 7
27,556	6\cdot 11 = 66 = \dfrac{11}{2}\cdot (11 + 1)
7,747	\|-x_m + x_{m + 1}\| = \|x_m - x_{m + 1}\|
19,234	((-1) + B)\cdot \left(1 + B^4 + B^3 + B^2 + B\right) = B^5 + (-1)
-22,637	-4/7\cdot \frac{5}{7} = \dfrac{1}{7\cdot 7}\cdot ((-4)\cdot 5) = -20/49 = -20/49
19,221	\cos{x} = \sin\left(x + π/2\right)
12,348	0.5\cdot y + x = 1.75 \Rightarrow 1.75 - 0.5\cdot y = x
-22,371	(p + 7 \cdot (-1)) \cdot \left(6 + p\right) = p^2 - p + 42 \cdot (-1)
6,175	0 = x^4 - 2 \cdot x^3 - 2 \cdot x + 1 = (x^2 - (1 + 3^{1/2}) \cdot x + 1) \cdot (x^2 - \left(1 - 3^{1/2}\right) \cdot x + 1)
8,138	\left(5 + y5 = 20 \Rightarrow 15 = 5y\right) \Rightarrow 3 = y
5,821	-\dfrac{55}{25}\cdot 25 + 80 + 25\cdot \left(-1\right) = 0
13,536	3 \cdot x \cdot x + x + 24 = -\frac{-287}{12} + 3 \cdot x \cdot x + x + 1/12
-8,104	21 = \frac{42}{2} \times 1
-4,427	\frac{8(-1) - 5x}{x^2 + 3*x + 2} = -\tfrac{2}{x + 2} - \frac{3}{x + 1}
-23,678	\frac{12}{35} = \dfrac{1}{7} \cdot 6 \cdot 2/5
25,521	(9 + x) (9(-1) + x) = 81 (-1) + x^2
10,208	\binom{g}{h} = \binom{g}{g - h}
-26,673	8x^2-18x-5=(4x+1)(2x-5)
-30,339	5 \cdot (-1) + 9 = 4
6,190	\frac16\cdot (1 + n)\cdot (1 + n + 1)\cdot (1 + (n + 1)\cdot 2) = 1^2 + 2^2 + 3 \cdot 3 + \ldots + n^2 + (n + 1) \cdot (n + 1)
28,679	\frac{5!}{2!}25 = 600
-9,880	\phantom{ \dfrac{1}{1} \times -\dfrac{1}{1} \times -\dfrac{3}{5}} = \dfrac{1 \times -1 \times -3}{1 \times 1 \times 5} = \dfrac{3}{5}
14,203	x = x^1 = x^{0 + 1} = x^0 x^1 = xx = x \cdot x
14,750	-y^2 + x^2 = \left(-y + x\right)\cdot (x + y)
6,353	(m + k) h = -(-m - k) h = -(-m h + -k h) = --m h - -k h = mh + kh
-23,127	-\dfrac23 = -\dfrac{2}{3}
23,895	9025 - 190 \cdot s + s^2 = (95 - s) \cdot (95 - s)
11,222	\tfrac{1}{z + 1} = \dfrac{1}{(-1)*\left(-1\right) + z}
14,553	0 = 2 \cdot Z \cdot B_2^4 + 2 \cdot B_1 \cdot B_2^4 - 26 \cdot (Z \cdot B_2^2 + B_1 \cdot B_2^2) + 145 \geq \left(Z \cdot B_2^2 + B_1 \cdot B_2^2\right) \cdot \left(Z \cdot B_2^2 + B_1 \cdot B_2^2\right) - 38 \cdot (Z \cdot B_2^2 + B_1 \cdot B_2^2) + 1 + 18^2
11,910	\cos(\frac{\pi}{2} - w) = \sin{w}
25,411	6(\frac{5}{4}) (\frac{5}{4}) = 75/8 \lt 10
-28,766	\frac{1}{y + 2}\cdot \left((-1) + y^3\right) = y \cdot y - 2\cdot y + 4 - \frac{9}{y + 2}
-4,761	\dfrac{1}{z + 1}4 - \frac{5}{3(-1) + z} = \dfrac{17(-1) - z}{3(-1) + z^2 - z*2}
-5,467	\dfrac{1}{(7\cdot \left(-1\right) + k)\cdot 2} = \frac{1}{14\cdot (-1) + k\cdot 2}
-22,181	\tfrac{10}{2} = 5
-17,200	\dfrac{1}{\cos^2(\theta)} \times \cos^2(\theta) = \frac{1}{\cos^2(\theta)} \times (-\sin^2(\theta) + 1)
7,524	\pi3/8 + \pi = \dfrac{11\pi}{8}*1
27,254	7 - \left(2 + 1\right)*2/2 = 4
20,481	(a \cdot c)^n = a^n \cdot c^n = 1 \implies c^{-n} = a^n
-18,398	\frac{1}{9x + x^2}(63 + x^2 + x\cdot 16) = \frac{(9 + x) (7 + x)}{x\cdot (x + 9)}
-21,585	\cos(-\frac{4}{3}\pi) = -0.5
35,779	2 \cdot (k + 1) = 2 + k \cdot 2
18,663	\binom{n}{2} + \binom{n}{0} + \binom{n}{1} = \frac12\cdot (n^2 + n + 2)
24,830	z \cdot 7 - z \cdot 4 = 3 \cdot z
-20,362	\frac{7}{14 - 63 r} = \frac{7 \cdot \frac{1}{7}}{-9r + 2}
-480	\pi = 15 \cdot \pi - \pi \cdot 14
14,409	h\cdot x^n = x^n\cdot h
15,120	\sqrt{2}\sqrt{2\pi} = 2*\sqrt{\pi}
10,672	-2\cdot \sin^2{\frac{t}{2}} + 1 = \cos{t}
-578	e^{\pi\cdot i/3\cdot 14} = (e^{\pi\cdot i/3})^{14}
15,365	6 \cdot (-1) + n \cdot 3 = n + 4 \cdot (-1) + n + (-1) + n + \left(-1\right)
19,459	2^{2\cdot n + 2\cdot (-1)} = \left(2^{n + (-1)}\right)^2
11,741	-t + x_0 + i \cdot 2 = x_0 + i - t - i
28,177	\|z + \left(-1\right)\| = \|z + \left(-1\right) + 0 \cdot (-1)\| \leq \|z + (-1)\| + 0
2,818	-\tfrac{\cos^{1 + j}(y)}{j + 1} = \sin(y)\times \cos^j\left(y\right)
-19,098	\frac{1}{45}2 = A_s/(36\pi)36\pi = A_s
6,950	(10\cdot c + b) \cdot (10\cdot c + b) = 100\cdot c \cdot c + 20\cdot c\cdot b + b^2 = 10\cdot (10\cdot c^2 + 2\cdot c\cdot b) + b \cdot b
-19,511	\dfrac{\tfrac17 \cdot 9}{3 \cdot \dfrac{1}{8}} = 8/3 \cdot \frac97
-7,542	(-28 + 4i + 21i + 3)/25 = \left(-25 + 25*i\right)/25 = -1 + i
-2,547	\sqrt{4} \sqrt{6} + \sqrt{16} \sqrt{6} = 4\sqrt{6} + 2\sqrt{6}
-19,558	\frac72 \dfrac{1}{5} = \frac{7}{2 \cdot 5} = \dfrac{7}{10}
21,015	-hx + nx = x*(n - h)
-5,273	10^{9 + 5 \cdot (-1)} \cdot 5.3 = 10^4 \cdot 5.3
28,972	X\cdot X^n = X^{1 + n}
29,587	1 + 3(-1) = 6(-1) + 4
3,579	0 = 24 + 4\cdot y \Rightarrow -6 = y
1,780	2\cdot b\cdot x + 2\cdot y = 4 \implies y = 2 - x\cdot b
4,858	\dfrac{x + 1}{(-1) + x} = \frac{1}{\left(-1\right) + x}2 + 1
-26,554	-(3x)^2 + 10^2 = (-3x + 10) (x*3 + 10)
-8,370	-3 \times -2 = 6
13,578	\mathbb{E}((-\mathbb{E}(X) + X)^2) = -\mathbb{E}(X) \cdot \mathbb{E}(X) + \mathbb{E}(X^2)
-645	\frac{209}{12} \pi - 16 \pi = \frac{17}{12} \pi
-20,812	\frac{70}{56(-1) - 14t} = \dfrac177\dfrac{10}{8(-1) - t2}
13,910	\frac{16}{3} = \tfrac{1}{4}*16 + 16/12
-2,063	-\pi/4 = -\pi \cdot \frac23 + \frac{5}{12} \cdot \pi
37,263	2+7+9=18
11,832	\left(x * x - 4x + 13\right)\left(x + 2\right)(1 + x) = 26 + x^4 - x^3 + 3x * x + 31*x
18,323	r \cdot (J + w) = J \cdot r + r \cdot w
13,182	( x^2, x\cdot z) = ( x \cdot x, x) \cap ( x^2, z) = x \cap ( x^2, z)
47,481	6 + 25 + 28 + 21 + 10 + 1 = 91
36,739	4^{2n} = 16^n
2,661	-(180 - B - Y) + 180 = B + Y
-6,384	\frac{1}{\left(8(-1) + q\right)*3} = \frac{1}{3q + 24 (-1)}
-3,269	\sqrt{6} \cdot \sqrt{9} + \sqrt{6} \cdot \sqrt{16} = \sqrt{6} \cdot 4 + \sqrt{6} \cdot 3
13,659	1 + y^2 = 10001 - (100 - y) (y + 100)
27,055	2036 = (503 + 499 \cdot (-1)) \cdot 509
4,063	\lambda^{-b} = \frac{1}{\lambda^b}
18,484	2 \times 6 =12
-20,310	\tfrac{1}{2(-1) + t}(2(-1) + t)(-5/6) = \dfrac{1}{6t + 12(-1)}(-t5 + 10)
49,817	910 = 182 \cdot 5
9,339	-\alpha \gt \beta \Rightarrow \alpha < -\beta
-30,112	\frac{\mathrm{d}}{\mathrm{d}z} z^k = k\cdot z^{k + \left(-1\right)}
14,198	\tan{x} = E/10\Longrightarrow 10*\tan{x} = E
29,442	-(-q - x) = x - q + q \cdot 2
20,303	\mathbb{E}\left((Q - \mathbb{E}\left(Q\right)) \cdot (-\mathbb{E}\left(x\right) + x)\right) = -\mathbb{E}\left(x\right) \cdot \mathbb{E}\left(Q\right) + \mathbb{E}\left(x \cdot Q\right)
26,738	3 + 11 + 19 = 33
3,034	k^3 = 12\cdot k - 5\cdot k^2 = 12\cdot k - 5\cdot (12 - 5\cdot k) = 37 - 60\cdot k

-20,518

\frac{s}{90 \cdot s} \cdot 20 = \frac{10 \cdot s}{s \cdot 10} \cdot 2/9

8,106

1/2 = \tfrac13 + \dfrac{1}{6}

2,314

\cos{x} = (e^{ix} + e^{-ix})/2 = \frac{1}{2e^{ix}}(e^{2ix} + 1)

15,394

\frac{7^{55}}{5^{72}} = (\frac{7^3}{5^4})^{18}\cdot 7

27,556

6\cdot 11 = 66 = \dfrac{11}{2}\cdot (11 + 1)

7,747

|-x_m + x_{m + 1}| = |x_m - x_{m + 1}|

19,234

((-1) + B)\cdot \left(1 + B^4 + B^3 + B^2 + B\right) = B^5 + (-1)

-22,637

-4/7\cdot \frac{5}{7} = \dfrac{1}{7\cdot 7}\cdot ((-4)\cdot 5) = -20/49 = -20/49

19,221

\cos{x} = \sin\left(x + π/2\right)

12,348

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

-22,371

(p + 7 \cdot (-1)) \cdot \left(6 + p\right) = p^2 - p + 42 \cdot (-1)

6,175

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

8,138

\left(5 + y*5 = 20 \Rightarrow 15 = 5*y\right) \Rightarrow 3 = y

5,821

-\dfrac{55}{25}\cdot 25 + 80 + 25\cdot \left(-1\right) = 0

13,536

3 \cdot x \cdot x + x + 24 = -\frac{-287}{12} + 3 \cdot x \cdot x + x + 1/12

-8,104

21 = \frac{42}{2} \times 1

-4,427

\frac{8*(-1) - 5*x}{x^2 + 3*x + 2} = -\tfrac{2}{x + 2} - \frac{3}{x + 1}

-23,678

\frac{12}{35} = \dfrac{1}{7} \cdot 6 \cdot 2/5

25,521

(9 + x) (9(-1) + x) = 81 (-1) + x^2

10,208

\binom{g}{h} = \binom{g}{g - h}

-26,673

8x^2-18x-5=(4x+1)(2x-5)

-30,339

5 \cdot (-1) + 9 = 4

6,190

\frac16\cdot (1 + n)\cdot (1 + n + 1)\cdot (1 + (n + 1)\cdot 2) = 1^2 + 2^2 + 3 \cdot 3 + \ldots + n^2 + (n + 1) \cdot (n + 1)

28,679

\frac{5!}{2!}*2*5 = 600

-9,880

\phantom{ \dfrac{1}{1} \times -\dfrac{1}{1} \times -\dfrac{3}{5}} = \dfrac{1 \times -1 \times -3}{1 \times 1 \times 5} = \dfrac{3}{5}

14,203

x = x^1 = x^{0 + 1} = x^0 x^1 = xx = x \cdot x

14,750

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

6,353

(m + k) h = -(-m - k) h = -(-m h + -k h) = --m h - -k h = mh + kh

-23,127

-\dfrac23 = -\dfrac{2}{3}

23,895

9025 - 190 \cdot s + s^2 = (95 - s) \cdot (95 - s)

11,222

\tfrac{1}{z + 1} = \dfrac{1}{(-1)*\left(-1\right) + z}

14,553

0 = 2 \cdot Z \cdot B_2^4 + 2 \cdot B_1 \cdot B_2^4 - 26 \cdot (Z \cdot B_2^2 + B_1 \cdot B_2^2) + 145 \geq \left(Z \cdot B_2^2 + B_1 \cdot B_2^2\right) \cdot \left(Z \cdot B_2^2 + B_1 \cdot B_2^2\right) - 38 \cdot (Z \cdot B_2^2 + B_1 \cdot B_2^2) + 1 + 18^2

11,910

\cos(\frac{\pi}{2} - w) = \sin{w}

25,411

6*(\frac{5}{4}) * (\frac{5}{4}) = 75/8 \lt 10

-28,766

\frac{1}{y + 2}\cdot \left((-1) + y^3\right) = y \cdot y - 2\cdot y + 4 - \frac{9}{y + 2}

-4,761

\dfrac{1}{z + 1}*4 - \frac{5}{3*(-1) + z} = \dfrac{17*(-1) - z}{3*(-1) + z^2 - z*2}

-5,467

\dfrac{1}{(7\cdot \left(-1\right) + k)\cdot 2} = \frac{1}{14\cdot (-1) + k\cdot 2}

-22,181

\tfrac{10}{2} = 5

-17,200

\dfrac{1}{\cos^2(\theta)} \times \cos^2(\theta) = \frac{1}{\cos^2(\theta)} \times (-\sin^2(\theta) + 1)

7,524

\pi*3/8 + \pi = \dfrac{11*\pi}{8}*1

27,254

7 - \left(2 + 1\right)*2/2 = 4

20,481

(a \cdot c)^n = a^n \cdot c^n = 1 \implies c^{-n} = a^n

-18,398

\frac{1}{9x + x^2}(63 + x^2 + x\cdot 16) = \frac{(9 + x) (7 + x)}{x\cdot (x + 9)}

-21,585

\cos(-\frac{4}{3}\pi) = -0.5

35,779

2 \cdot (k + 1) = 2 + k \cdot 2

18,663

\binom{n}{2} + \binom{n}{0} + \binom{n}{1} = \frac12\cdot (n^2 + n + 2)

24,830

z \cdot 7 - z \cdot 4 = 3 \cdot z

-20,362

\frac{7}{14 - 63 r} = \frac{7 \cdot \frac{1}{7}}{-9r + 2}

-480

\pi = 15 \cdot \pi - \pi \cdot 14

14,409

h\cdot x^n = x^n\cdot h

15,120

\sqrt{2}*\sqrt{2*\pi} = 2*\sqrt{\pi}

10,672

-2\cdot \sin^2{\frac{t}{2}} + 1 = \cos{t}

-578

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

15,365

6 \cdot (-1) + n \cdot 3 = n + 4 \cdot (-1) + n + (-1) + n + \left(-1\right)

19,459

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

11,741

-t + x_0 + i \cdot 2 = x_0 + i - t - i

28,177

|z + \left(-1\right)| = |z + \left(-1\right) + 0 \cdot (-1)| \leq |z + (-1)| + 0

2,818

-\tfrac{\cos^{1 + j}(y)}{j + 1} = \sin(y)\times \cos^j\left(y\right)

-19,098

\frac{1}{45}*2 = A_s/(36*\pi)*36*\pi = A_s

6,950

(10\cdot c + b) \cdot (10\cdot c + b) = 100\cdot c \cdot c + 20\cdot c\cdot b + b^2 = 10\cdot (10\cdot c^2 + 2\cdot c\cdot b) + b \cdot b

-19,511

\dfrac{\tfrac17 \cdot 9}{3 \cdot \dfrac{1}{8}} = 8/3 \cdot \frac97

-7,542

(-28 + 4*i + 21*i + 3)/25 = \left(-25 + 25*i\right)/25 = -1 + i

-2,547

\sqrt{4} \sqrt{6} + \sqrt{16} \sqrt{6} = 4\sqrt{6} + 2\sqrt{6}

-19,558

\frac72 \dfrac{1}{5} = \frac{7}{2 \cdot 5} = \dfrac{7}{10}

21,015

-h*x + n*x = x*(n - h)

-5,273

10^{9 + 5 \cdot (-1)} \cdot 5.3 = 10^4 \cdot 5.3

28,972

X\cdot X^n = X^{1 + n}

29,587

1 + 3(-1) = 6(-1) + 4

3,579

0 = 24 + 4\cdot y \Rightarrow -6 = y

1,780

2\cdot b\cdot x + 2\cdot y = 4 \implies y = 2 - x\cdot b

4,858

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

-26,554

-(3x)^2 + 10^2 = (-3x + 10) (x*3 + 10)

-8,370

-3 \times -2 = 6

13,578

\mathbb{E}((-\mathbb{E}(X) + X)^2) = -\mathbb{E}(X) \cdot \mathbb{E}(X) + \mathbb{E}(X^2)

-645

\frac{209}{12} \pi - 16 \pi = \frac{17}{12} \pi

-20,812

\frac{70}{56*(-1) - 14*t} = \dfrac17*7*\dfrac{10}{8*(-1) - t*2}

13,910

\frac{16}{3} = \tfrac{1}{4}*16 + 16/12

-2,063

-\pi/4 = -\pi \cdot \frac23 + \frac{5}{12} \cdot \pi

37,263

2+7+9=18

11,832

\left(x * x - 4*x + 13\right)*\left(x + 2\right)*(1 + x) = 26 + x^4 - x^3 + 3*x * x + 31*x

18,323

r \cdot (J + w) = J \cdot r + r \cdot w

13,182

( x^2, x\cdot z) = ( x \cdot x, x) \cap ( x^2, z) = x \cap ( x^2, z)

47,481

6 + 25 + 28 + 21 + 10 + 1 = 91

36,739

4^{2n} = 16^n

2,661

-(180 - B - Y) + 180 = B + Y

-6,384

\frac{1}{\left(8(-1) + q\right)*3} = \frac{1}{3q + 24 (-1)}

-3,269

\sqrt{6} \cdot \sqrt{9} + \sqrt{6} \cdot \sqrt{16} = \sqrt{6} \cdot 4 + \sqrt{6} \cdot 3

13,659

1 + y^2 = 10001 - (100 - y) (y + 100)

27,055

2036 = (503 + 499 \cdot (-1)) \cdot 509

4,063

\lambda^{-b} = \frac{1}{\lambda^b}

18,484

2 \times 6 =12

-20,310

\tfrac{1}{2*(-1) + t}*(2*(-1) + t)*(-5/6) = \dfrac{1}{6*t + 12*(-1)}*(-t*5 + 10)

49,817

910 = 182 \cdot 5

9,339

-\alpha \gt \beta \Rightarrow \alpha < -\beta

-30,112

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

14,198

\tan{x} = E/10\Longrightarrow 10*\tan{x} = E

29,442

-(-q - x) = x - q + q \cdot 2

20,303

\mathbb{E}\left((Q - \mathbb{E}\left(Q\right)) \cdot (-\mathbb{E}\left(x\right) + x)\right) = -\mathbb{E}\left(x\right) \cdot \mathbb{E}\left(Q\right) + \mathbb{E}\left(x \cdot Q\right)

26,738

3 + 11 + 19 = 33

3,034

k^3 = 12\cdot k - 5\cdot k^2 = 12\cdot k - 5\cdot (12 - 5\cdot k) = 37 - 60\cdot k