ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
35,914	\frac92 = \dfrac92
-7,511	7 = \tfrac{1}{9}63
20,899	\frac{1}{1 + x \cdot x} \cdot (x^4 + x^3 + 8 \cdot x^2 + d \cdot x + h) = x^2 + x + 7 + \frac{1}{1 + x^2} \cdot (7 \cdot \left(-1\right) + x \cdot (d + (-1)) + h)
1,353	\sin(z) = \frac{1}{i \cdot 2} \cdot (-e^{-z \cdot i} + e^{i \cdot z})\Longrightarrow 2 \cdot \sin(z) \cdot i = e^{i \cdot z} - e^{-z \cdot i}
-20,925	\frac{x \cdot \left(-5\right)}{\left(-5\right) \cdot x} \cdot (-\frac{10}{7}) = \dfrac{50 \cdot x}{(-1) \cdot 35 \cdot x}
-3,743	14 = 2\cdot 7
35,873	1 = e^{i0} = e^{\dfrac{i3\pi}{2}} = -i
-23,189	\dfrac{1}{2}*4 = 2
-12,332	2\sqrt{6} = \sqrt{24}
-3,044	3\cdot 7^{1/2} + 2\cdot 7^{1/2} = 7^{1/2}\cdot 4^{1/2} + 7^{1/2}\cdot 9^{1/2}
25,617	3 \cdot 3 \cdot 3 + 19\cdot 2^3 + 2018 = 2197 = 13^3
35,051	2 * 2^2/32 = 2^{3 + 5*\left(-1\right)} = 1/4
-5,561	\frac{15 \cdot \left(t + 10 \cdot (-1)\right)}{9 \cdot (10 \cdot (-1) + t) \cdot (2 + t)} = \frac{5}{\left(2 + t\right) \cdot 3} \cdot \dfrac{1}{3 \cdot (10 \cdot (-1) + t)} \cdot (t \cdot 3 - 30)
35,781	d^2 + 3 \cdot a^2 - 2 \cdot 3^{1 / 2} \cdot a \cdot d = 0 \Rightarrow a \cdot 3^{\frac{1}{2}} = d
27,063	\frac{1}{2^k} \cdot ((1 + 2 \cdot x)^k + (1 + 2 \cdot b)^k) = \left(1/2 + x\right)^k + (b + \frac{1}{2})^k
-20,447	-\frac{1}{2 + k} \cdot 7 \cdot 8/8 = -\frac{56}{k \cdot 8 + 16}
-17,146	5 = 5 \times r + 5 \times (-8) = 5 \times r - 40 = 5 \times r + 40 \times (-1)
3,747	\left(1 + n^22 - 4n\right)3 = 3 + n n6 - n12
33,554	m + 2 (-1) + 2 = m
-7,740	-\dfrac{8i}{-4} + \dfrac{12}{-4} = \frac{1}{-4}\left(-8*i + 12\right)
16,014	g_2 \cdot g_1 \cdot y = g_1 \cdot g_2 \cdot y
1,967	{10 + 4 + \left(-1\right) \choose 4 + (-1)} = 286
10,796	B'\cdot J = J\cdot B'
5,572	(x_1 + x_2)^2 = x_1^2 + x_2^2 + 2!x_1x_2
-6,304	\frac{5x}{x \cdot x + 81 \left(-1\right)} = \dfrac{5x}{(x + 9) \left(x + 9(-1)\right)}
-29,343	(-c_1 + c_2) (c_2 + c_1) = c_2^2 - c_1^2
36,560	\frac12 + \frac{1}{4} + 1/8 = 7/8
23,968	q\cdot r := r\cdot q
2,953	(1 + z) (z + 4) = 4 + z^2 + z \cdot 5
-6,811	210 = 3\cdot 7\cdot 10
49,148	-(c + z) = -(c + z) = -c - z
-1,740	0 + \pi \cdot 5/4 = 5/4 \cdot \pi
-4,620	\frac{8\cdot \left(-1\right) + 2\cdot x}{x \cdot x - 4\cdot x + 3} = \frac{3}{x + (-1)} - \dfrac{1}{3\cdot (-1) + x}
21,978	\left(a + bi\right) \left(x + ni\right) = ax - bn + (an + bx) i = ax - bn + i
523	\cos{p\cdot 2} = -\sin^2{p}\cdot 2 + 1
3,425	\frac{1}{y} + 0\cdot (-1) = 1/y
31,762	1 = x + 2 \cdot \nu + z = ( 1, 2, 1) \cdot ( x, \nu, z)
5,767	\left(-1\right) + x^3 = (x^2 + x + 1) (\left(-1\right) + x)
14,490	\left\|{F + B}\right\|\left\|{F - B}\right\| = \left\|{F + B}\right\|\left\|{F^X - B^X}\right\| = \left\|{(F + B)*(F^X - B^X)}\right\|
25,544	\cos(x) = \frac{\sin(x\cdot 2)}{2\cdot \sin\left(x\right)}
18,129	\dfrac{1^{-1}\cdot 1^{-1}}{3^4} = \frac{1^{-1}\cdot 1^{-1}}{3^4} = \frac{1^{-1}}{3^4} = \frac{1}{3^4} = \frac{1}{81}
7,514	n - \sum_{k=1}^n \cos(kx) = \sum_{k=1}^n (1 - \cos(kx)) = 2\sum_{k=1}^n \sin^2(kx/2)
-9,146	x\cdot 16 + 32 = 2\cdot 2\cdot 2\cdot 2\cdot x + 2\cdot 2\cdot 2\cdot 2\cdot 2
-18,253	\tfrac{-7\cdot s + s^2}{14 + s^2 - 9\cdot s} = \frac{s}{(7\cdot (-1) + s)\cdot (s + 2\cdot (-1))}\cdot (s + 7\cdot (-1))
-19,276	\frac{1}{\frac{2}{9} \cdot 5} = 9 \cdot \tfrac{1}{2}/5
23,091	\left(l * l * l - l^2 + l^2 - l + 1\right) (l * l + l + 1) = (l * l + l + 1) (l * l * l - l + 1)
30,879	\sin{\theta2} = \sin{\theta} \cos{\theta}2
-3,322	\sqrt{13}\times ((-1) + 3 + 5) = 7\times \sqrt{13}
4,948	\left((-1) + x\right) \cdot l + l = l \cdot x
6,395	s\times c\times x = c\times s\times x
14,772	4 = y - z \cdot 2 \Rightarrow 2z = y + 4(-1)
-20,700	\frac{8\cdot y + 8}{y\cdot 8 + 48\cdot \left(-1\right)} = \frac{8}{8}\cdot \dfrac{1 + y}{y + 6\cdot \left(-1\right)}
5,462	2^{\tfrac13} + 3^{\frac{1}{3}} = 2^{\dfrac{1}{3}} + 3^{1/3}
11,752	x_k := x_k - x
-4,486	\frac{-6\cdot z + 18}{5\cdot (-1) + z^2 - z\cdot 4} = -\frac{2}{5\cdot (-1) + z} - \dfrac{4}{1 + z}
13,544	-(z + 1) + x + 1 = x - z
855	\cos^n\left(x + \pi\cdot 2\right) = \cos^n(x)
-10,437	\frac{1}{s \cdot 80} \cdot 10 = \frac{2}{s \cdot 16} \cdot 5/5
10,760	x*2 + 1 + 2\beta + 1 = 2(x + \beta + 1)
-7,177	0 = \frac{1}{5}2 \cdot 0
19,059	y^2 + 5 y + 2 = \left(y + 1\right)^2 = y^2 + 2 y + 1 \Rightarrow y = -\frac{1}{3}
22,653	2 - 10^4 = 2 + 10000\left(-1\right) = -9998 = -9.99810^3 = -\dfrac{10}{1000} = -10^4
10,597	\|L\| \|f\| = \|L f\|
3,573	1101870 = 1484 \cdot 1485/2
46,254	\tfrac{1}{2^{2\cdot k}\cdot k!^2}\cdot (2\cdot k)! = \frac{1}{(2^2)^k}\cdot \frac{1}{k!\cdot (2\cdot k - k)!}\cdot (2\cdot k)! = \frac{1}{4^k}\cdot \binom{2\cdot k}{k}
21,041	\min{\cdots,1/2} = \min{\cdots,\frac12}
19,613	h10 + h h = (5 + h + 5(-1)) (5 + 5 + h)
-7,390	\frac25\dfrac{3}{6}4/7*5/8 = 1/14
-2,340	\frac{1}{11}\cdot 6 - 4/11 = \tfrac{1}{11}\cdot 2
-20,830	7/7\cdot \tfrac{8}{a + 5\cdot (-1)} = \frac{56}{35\cdot (-1) + a\cdot 7}
10,240	x \cdot x \cdot 4 = (x \cdot 2) \cdot (x \cdot 2)
1,865	\frac{1}{{15 \choose 3}}154 = 22/65
14,326	\tfrac{1}{3} = (1 - a)/3 \Rightarrow a = 0
-8,036	\frac{27 - i5}{5 - i2} = \dfrac{2i + 5}{i2 + 5}\frac{27 - i5}{5 - 2*i}
-14,227	(10 + 4 - 84)10 = \left(10 + 4 + 32(-1)\right)10 = (10 - 28)10 = (10 + 28(-1))10 = (-18)10 = (-18)*10 = -180
18,350	\sin(\dfrac{1}{q}\pi)/(\pi \frac1q) \pi = q\sin(\frac{\pi}{q})
16,990	l = \left\{l, ..., 1, 2\right\}
39,553	100 + 1.04*\left(-2500\right) = -2500
488	ac + bc + c^2 = (a + b + c) c = abc
-18,255	\frac{1}{p^2 + p \cdot 9 + 8}(p \cdot p - p + 2\left(-1\right)) = \frac{\left(p + 2\left(-1\right)\right) (p + 1)}{(1 + p) (p + 8)}
-5,082	10^318.0 = 1810^{2 + 1}
-20,878	\frac{-y\cdot 70 + 21\cdot \left(-1\right)}{7\cdot y + 7\cdot (-1)} = \frac{-10\cdot y + 3\cdot (-1)}{y + \left(-1\right)}\cdot \frac77
-9,184	k^3\cdot 40 = k\cdot k\cdot 2\cdot 2\cdot 2\cdot 5\cdot k
-10,269	\frac{6 (-1) + q*9}{12 (-1) + 24 q} = \tfrac{3 q + 2 (-1)}{8 q + 4 (-1)} \frac33
-15,845	-\frac{74}{10} = -\frac{1}{10} \cdot 9 \cdot 9 + 7/10
-5,039	10^{0(-1) + 5}0.73 = 10^50.73
-26,043	(-30 - 120i - 50i + 200)/34 = \tfrac{1}{34}(170 - 170i) = 5 - 5*i
-1,216	-3/2 \cdot (-\frac{1}{2}) = \frac{(-1) \cdot \frac{1}{2}}{\frac{1}{3} \cdot (-2)}
4,393	31250 = 5^2\times 5^3\times \binom{5}{3}
33,249	30 = \frac{5!}{2!*2!}
38,352	-80 = -2^{5 + (-1)}*5
-471	(e^{5 \cdot \pi \cdot i/12})^{12} = e^{12 \cdot \tfrac{1}{12} \cdot 5 \cdot \pi \cdot i}
-15,081	\frac{1}{r^4 \cdot \frac{x^{12}}{r^9}} = \frac{1}{\frac{1}{r^9 \cdot \frac{1}{x^{12}}} \cdot r^4}
9,624	\left(\frac{600}{\pi \cdot 4}\right)^{1/3} = \frac{150^{1/3}}{\pi^{1/3}}
-7,952	\tfrac{1}{29} \cdot (-65 + 90 \cdot i + 26 \cdot i + 36) = \left(-29 + 116 \cdot i\right)/29 = -1 + 4 \cdot i
18,654	1 + 2\cdot \left(-1\right) + 3\cdot \left(-1\right) + 4 = 0 + (-1) + 2 + 3 + 4\cdot (-1) = 0
34,407	{9 + r \choose 9} = {r + 10 + (-1) \choose (-1) + 10}
2,865	(\frac{1}{(z + 1)!}\cdot z!)^3 = \dfrac{1}{\left(z + 1\right) \cdot (z + 1)^2}
23,152	\frac{4}{3!^2}\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5 = 1680
21,035	i \cdot e^{x \cdot i} = d/dx e^{i \cdot x}

35,914

\frac92 = \dfrac92

-7,511

7 = \tfrac{1}{9}63

20,899

\frac{1}{1 + x \cdot x} \cdot (x^4 + x^3 + 8 \cdot x^2 + d \cdot x + h) = x^2 + x + 7 + \frac{1}{1 + x^2} \cdot (7 \cdot \left(-1\right) + x \cdot (d + (-1)) + h)

1,353

\sin(z) = \frac{1}{i \cdot 2} \cdot (-e^{-z \cdot i} + e^{i \cdot z})\Longrightarrow 2 \cdot \sin(z) \cdot i = e^{i \cdot z} - e^{-z \cdot i}

-20,925

\frac{x \cdot \left(-5\right)}{\left(-5\right) \cdot x} \cdot (-\frac{10}{7}) = \dfrac{50 \cdot x}{(-1) \cdot 35 \cdot x}

-3,743

14 = 2\cdot 7

35,873

1 = e^{i*0} = e^{\dfrac{i*3\pi}{2}} = -i

-23,189

\dfrac{1}{2}*4 = 2

-12,332

2\sqrt{6} = \sqrt{24}

-3,044

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

25,617

3 \cdot 3 \cdot 3 + 19\cdot 2^3 + 2018 = 2197 = 13^3

35,051

2 * 2^2/32 = 2^{3 + 5*\left(-1\right)} = 1/4

-5,561

\frac{15 \cdot \left(t + 10 \cdot (-1)\right)}{9 \cdot (10 \cdot (-1) + t) \cdot (2 + t)} = \frac{5}{\left(2 + t\right) \cdot 3} \cdot \dfrac{1}{3 \cdot (10 \cdot (-1) + t)} \cdot (t \cdot 3 - 30)

35,781

d^2 + 3 \cdot a^2 - 2 \cdot 3^{1 / 2} \cdot a \cdot d = 0 \Rightarrow a \cdot 3^{\frac{1}{2}} = d

27,063

\frac{1}{2^k} \cdot ((1 + 2 \cdot x)^k + (1 + 2 \cdot b)^k) = \left(1/2 + x\right)^k + (b + \frac{1}{2})^k

-20,447

-\frac{1}{2 + k} \cdot 7 \cdot 8/8 = -\frac{56}{k \cdot 8 + 16}

-17,146

5 = 5 \times r + 5 \times (-8) = 5 \times r - 40 = 5 \times r + 40 \times (-1)

3,747

\left(1 + n^2*2 - 4*n\right)*3 = 3 + n * n*6 - n*12

33,554

m + 2 (-1) + 2 = m

-7,740

-\dfrac{8*i}{-4} + \dfrac{12}{-4} = \frac{1}{-4}*\left(-8*i + 12\right)

16,014

g_2 \cdot g_1 \cdot y = g_1 \cdot g_2 \cdot y

1,967

{10 + 4 + \left(-1\right) \choose 4 + (-1)} = 286

10,796

B'\cdot J = J\cdot B'

5,572

(x_1 + x_2)^2 = x_1^2 + x_2^2 + 2!*x_1*x_2

-6,304

\frac{5x}{x \cdot x + 81 \left(-1\right)} = \dfrac{5x}{(x + 9) \left(x + 9(-1)\right)}

-29,343

(-c_1 + c_2) (c_2 + c_1) = c_2^2 - c_1^2

36,560

\frac12 + \frac{1}{4} + 1/8 = 7/8

23,968

q\cdot r := r\cdot q

2,953

(1 + z) (z + 4) = 4 + z^2 + z \cdot 5

-6,811

210 = 3\cdot 7\cdot 10

49,148

-(c + z) = -(c + z) = -c - z

-1,740

0 + \pi \cdot 5/4 = 5/4 \cdot \pi

-4,620

\frac{8\cdot \left(-1\right) + 2\cdot x}{x \cdot x - 4\cdot x + 3} = \frac{3}{x + (-1)} - \dfrac{1}{3\cdot (-1) + x}

21,978

\left(a + bi\right) \left(x + ni\right) = ax - bn + (an + bx) i = ax - bn + i

523

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

3,425

\frac{1}{y} + 0\cdot (-1) = 1/y

31,762

1 = x + 2 \cdot \nu + z = ( 1, 2, 1) \cdot ( x, \nu, z)

5,767

\left(-1\right) + x^3 = (x^2 + x + 1) (\left(-1\right) + x)

14,490

\left|{F + B}\right|*\left|{F - B}\right| = \left|{F + B}\right|*\left|{F^X - B^X}\right| = \left|{(F + B)*(F^X - B^X)}\right|

25,544

\cos(x) = \frac{\sin(x\cdot 2)}{2\cdot \sin\left(x\right)}

18,129

\dfrac{1^{-1}\cdot 1^{-1}}{3^4} = \frac{1^{-1}\cdot 1^{-1}}{3^4} = \frac{1^{-1}}{3^4} = \frac{1}{3^4} = \frac{1}{81}

7,514

n - \sum_{k=1}^n \cos(k*x) = \sum_{k=1}^n (1 - \cos(k*x)) = 2*\sum_{k=1}^n \sin^2(k*x/2)

-9,146

x\cdot 16 + 32 = 2\cdot 2\cdot 2\cdot 2\cdot x + 2\cdot 2\cdot 2\cdot 2\cdot 2

-18,253

\tfrac{-7\cdot s + s^2}{14 + s^2 - 9\cdot s} = \frac{s}{(7\cdot (-1) + s)\cdot (s + 2\cdot (-1))}\cdot (s + 7\cdot (-1))

-19,276

\frac{1}{\frac{2}{9} \cdot 5} = 9 \cdot \tfrac{1}{2}/5

23,091

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

30,879

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

-3,322

\sqrt{13}\times ((-1) + 3 + 5) = 7\times \sqrt{13}

4,948

\left((-1) + x\right) \cdot l + l = l \cdot x

6,395

s\times c\times x = c\times s\times x

14,772

4 = y - z \cdot 2 \Rightarrow 2z = y + 4(-1)

-20,700

\frac{8\cdot y + 8}{y\cdot 8 + 48\cdot \left(-1\right)} = \frac{8}{8}\cdot \dfrac{1 + y}{y + 6\cdot \left(-1\right)}

5,462

2^{\tfrac13} + 3^{\frac{1}{3}} = 2^{\dfrac{1}{3}} + 3^{1/3}

11,752

x_k := x_k - x

-4,486

\frac{-6\cdot z + 18}{5\cdot (-1) + z^2 - z\cdot 4} = -\frac{2}{5\cdot (-1) + z} - \dfrac{4}{1 + z}

13,544

-(z + 1) + x + 1 = x - z

855

\cos^n\left(x + \pi\cdot 2\right) = \cos^n(x)

-10,437

\frac{1}{s \cdot 80} \cdot 10 = \frac{2}{s \cdot 16} \cdot 5/5

10,760

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

-7,177

0 = \frac{1}{5}2 \cdot 0

19,059

y^2 + 5 y + 2 = \left(y + 1\right)^2 = y^2 + 2 y + 1 \Rightarrow y = -\frac{1}{3}

22,653

2 - 10^4 = 2 + 10000*\left(-1\right) = -9998 = -9.998*10^3 = -\dfrac{10}{1000} = -10^4

10,597

|L| |f| = |L f|

3,573

1101870 = 1484 \cdot 1485/2

46,254

\tfrac{1}{2^{2\cdot k}\cdot k!^2}\cdot (2\cdot k)! = \frac{1}{(2^2)^k}\cdot \frac{1}{k!\cdot (2\cdot k - k)!}\cdot (2\cdot k)! = \frac{1}{4^k}\cdot \binom{2\cdot k}{k}

21,041

\min{\cdots,1/2} = \min{\cdots,\frac12}

19,613

h*10 + h * h = (5 + h + 5(-1)) (5 + 5 + h)

-7,390

\frac25*\dfrac{3}{6}*4/7*5/8 = 1/14

-2,340

\frac{1}{11}\cdot 6 - 4/11 = \tfrac{1}{11}\cdot 2

-20,830

7/7\cdot \tfrac{8}{a + 5\cdot (-1)} = \frac{56}{35\cdot (-1) + a\cdot 7}

10,240

x \cdot x \cdot 4 = (x \cdot 2) \cdot (x \cdot 2)

1,865

\frac{1}{{15 \choose 3}}154 = 22/65

14,326

\tfrac{1}{3} = (1 - a)/3 \Rightarrow a = 0

-8,036

\frac{27 - i*5}{5 - i*2} = \dfrac{2*i + 5}{i*2 + 5}*\frac{27 - i*5}{5 - 2*i}

-14,227

(10 + 4 - 8*4)*10 = \left(10 + 4 + 32*(-1)\right)*10 = (10 - 28)*10 = (10 + 28*(-1))*10 = (-18)*10 = (-18)*10 = -180

18,350

\sin(\dfrac{1}{q}\pi)/(\pi \frac1q) \pi = q\sin(\frac{\pi}{q})

16,990

l = \left\{l, ..., 1, 2\right\}

39,553

100 + 1.04*\left(-2500\right) = -2500

488

ac + bc + c^2 = (a + b + c) c = abc

-18,255

\frac{1}{p^2 + p \cdot 9 + 8}(p \cdot p - p + 2\left(-1\right)) = \frac{\left(p + 2\left(-1\right)\right) (p + 1)}{(1 + p) (p + 8)}

-5,082

10^3*18.0 = 18*10^{2 + 1}

-20,878

\frac{-y\cdot 70 + 21\cdot \left(-1\right)}{7\cdot y + 7\cdot (-1)} = \frac{-10\cdot y + 3\cdot (-1)}{y + \left(-1\right)}\cdot \frac77

-9,184

k^3\cdot 40 = k\cdot k\cdot 2\cdot 2\cdot 2\cdot 5\cdot k

-10,269

\frac{6 (-1) + q*9}{12 (-1) + 24 q} = \tfrac{3 q + 2 (-1)}{8 q + 4 (-1)} \frac33

-15,845

-\frac{74}{10} = -\frac{1}{10} \cdot 9 \cdot 9 + 7/10

-5,039

10^{0(-1) + 5}*0.73 = 10^5*0.73

-26,043

(-30 - 120*i - 50*i + 200)/34 = \tfrac{1}{34}*(170 - 170*i) = 5 - 5*i

-1,216

-3/2 \cdot (-\frac{1}{2}) = \frac{(-1) \cdot \frac{1}{2}}{\frac{1}{3} \cdot (-2)}

4,393

31250 = 5^2\times 5^3\times \binom{5}{3}

33,249

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

38,352

-80 = -2^{5 + (-1)}*5

-471

(e^{5 \cdot \pi \cdot i/12})^{12} = e^{12 \cdot \tfrac{1}{12} \cdot 5 \cdot \pi \cdot i}

-15,081

\frac{1}{r^4 \cdot \frac{x^{12}}{r^9}} = \frac{1}{\frac{1}{r^9 \cdot \frac{1}{x^{12}}} \cdot r^4}

9,624

\left(\frac{600}{\pi \cdot 4}\right)^{1/3} = \frac{150^{1/3}}{\pi^{1/3}}

-7,952

\tfrac{1}{29} \cdot (-65 + 90 \cdot i + 26 \cdot i + 36) = \left(-29 + 116 \cdot i\right)/29 = -1 + 4 \cdot i

18,654

1 + 2\cdot \left(-1\right) + 3\cdot \left(-1\right) + 4 = 0 + (-1) + 2 + 3 + 4\cdot (-1) = 0

34,407

{9 + r \choose 9} = {r + 10 + (-1) \choose (-1) + 10}

2,865

(\frac{1}{(z + 1)!}\cdot z!)^3 = \dfrac{1}{\left(z + 1\right) \cdot (z + 1)^2}

23,152

\frac{4}{3!^2}\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5 = 1680

21,035

i \cdot e^{x \cdot i} = d/dx e^{i \cdot x}