ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
5,662	\dfrac{1}{y^{1/2}}((-3) y) = -3y^{1/2}
1,474	4n^2 + (-1) = (2n + (-1)) (2n + 1)
-564	\left(e^{\pi\cdot i}\right)^{15} = e^{15\cdot i\cdot \pi}
8,715	g \cdot g \cdot e = e = g \cdot g \cdot e
-4,142	\frac{1}{l^3} \cdot l \cdot l \cdot l \cdot \dfrac{1}{40} \cdot 8 = \frac{1}{40 \cdot l^3} \cdot 8 \cdot l^3
-29,007	4 = \left(4\cdot (-1) + 12\right)/2
-20,859	\frac{1}{25\times p + 5\times (-1)}\times (-30\times p + 6) = \frac{1}{5\times p + (-1)}\times \left(5\times p + (-1)\right)\times (-\frac{6}{5})
-12,347	2^{1 / 2}\cdot 6 = 72^{\dfrac{1}{2}}
27,695	\dfrac{1}{x \cdot d} = \dfrac{1}{x \cdot d}
-12,541	31 = 90 + 59\cdot (-1)
17,337	b = h \cdot b = b \cdot h
29,511	\left\lfloor{\frac15(7 + (-1))}\right\rfloor + 1 = 2
6,976	\frac{1}{Y\cdot \dfrac{1}{X}} = X/Y
22,012	2^{1 + k} + 2\cdot \left(-1\right) = 2\cdot ((-1) + 2^k)
36,419	-x + x \coloneqq -x + x
35,082	\dfrac{1}{-y^2 + 1} = \tfrac12 \cdot (\dfrac{1}{1 - y} + \frac{1}{1 + y})
10,331	(-\sqrt{x} + \sqrt{a_m})\cdot (\sqrt{a_m} + \sqrt{x}) = -x + a_m
-2,626	-\sqrt{6} + \sqrt{24} = \sqrt{4\cdot 6} - \sqrt{6}
32,919	1/L + 1/M + 1/x = \frac{1}{LMx}(LM + xM + xL)
16,860	\int\limits_{-\infty}^\infty ...\,\mathrm{d}z = 2 \cdot \int_0^\infty ...\,\mathrm{d}z
30,546	-4 \cdot x + 12 = 4 \cdot x \cdot x - (x^2 + x + 3 \cdot (-1)) \cdot 4
-1,260	-20/18 = \frac{1}{18 \cdot 1/2}((-20) \cdot 1/2) = -10/9
36,245	-(8 \cdot 9^5 + 9^6) + 9 \cdot 10^5 = -103833
-28,798	150 = \pi2/(\pi2*\frac{1}{150})
38,983	x + x = x \cdot 2
-15,966	-10 \cdot \frac{3}{10} + 7 \cdot 7/10 = 19/10
-4,233	k^2 \cdot 7/6 = 7 \cdot k^2/6
19,541	-9 = 4 + 2 * 2 + (-1)^29 - 42 + 18*\left(-1\right)
13,277	\|(-c_k\cdot t + h_k\cdot M)/\left(M\cdot c_k\right)\| = \|\frac{h_k}{c_k} - \frac{1}{M}\cdot t\|
-2,429	12 \cdot 6^{1/2} = (3 + 4 + 5) \cdot 6^{1/2}
-9,286	-x77 + 37 = -x49 + 21
-26,552	2\times y^2 - 40\times y + 200 = 2\times \left(y \times y - 20\times y + 100\right) = 2\times \left(y + 10\times (-1)\right)^2
14,971	8 \cdot x = 4 \cdot x + 2 \cdot x \cdot 2
-27,734	\frac{\mathrm{d}}{\mathrm{d}x} \csc{x} = -\csc{x} \cot{x}
-11,955	13/15 = \tfrac{1}{12\pi}s12\pi = s
-20,653	-7/8 \frac{1}{-5\varepsilon + 2(-1)}\left(-5\varepsilon + 2(-1)\right) = \frac{14 + 35 \varepsilon}{-40 \varepsilon + 16 \left(-1\right)}
-22,805	\frac{60}{4 \cdot 12} \cdot 1 = \frac{1}{48} \cdot 60
10,420	5^{2n} + (-1) = \left(5^n + 1\right) \left(5^n + (-1)\right)
-20,339	-\frac{36}{-8} = 9/2 \cdot (-\frac{4}{-4})
8,821	a*\sin{\pi/2} + \cos{\pi/2} = a
-11,009	112/8 = 14
27,041	(b + a) \times (b + a) - b \times a \times 2 = a \times a + b^2
12,189	\left(-5\right)^2 + (-3) * (-3) + 1^2 = 35 = 5((-5)(-3) - 5 - 3)
7,423	11 = (z + \sigma + y) \cdot 6\Longrightarrow z + \sigma + y = \dfrac{11}{6}
29,030	\csc{\theta} = \frac{1}{\sin{\theta}} = \dfrac{1}{(1 - \cos^2{\theta})^{\tfrac{1}{2}}}
15,310	-A_1B_1 + (A_0 + A_1)\left(B_0 + B_1\right) - B_0A_0 = A_1B_0 + B_1*A_0
-20,018	\frac{s\cdot 5 + 30}{45\cdot (-1) + 5\cdot s} = \frac55\cdot \frac{s + 6}{s + 9\cdot \left(-1\right)}
22,130	\binom{1}{1}\times \binom{19}{2}/\left(\binom{20}{3}\right) = \frac{1}{20}\times 3
4,601	-2 \cdot x \cdot e^{-x^2} = (-2 + 4 \cdot x^2) \cdot e^{-x^2}
33,017	\frac{1}{2^{1 / 2}} = \dfrac{2^{\frac{1}{2}}}{2}
-5,036	18.0\cdot 10^{5 + 2} = 18\cdot 10^7
25,998	5 + \left(z + (-1)\right)^4 + z^2\cdot 3 - z\cdot 6 = (z + (-1))^4 + 3\cdot (z + (-1))^2 + 2
41,682	2^{23} + (-1) = 47 \cdot 178481
-20,293	\frac{9 - 6 \cdot \delta}{\delta + 5 \cdot (-1)} \cdot 5/5 = \frac{45 - 30 \cdot \delta}{\delta \cdot 5 + 25 \cdot (-1)}
20,760	\sin(2\cdot \theta) = \cos(\dfrac{\pi}{2} - 2\cdot \theta)
52,112	770 = {5 \choose 4} \cdot {7 \choose 1} + {7 \choose 2} \cdot {5 \choose 3} + {7 \choose 3} \cdot {5 \choose 2} + {7 \choose 4} \cdot {5 \choose 1}
37,155	(2k)^2 = (2n+25)^2-549\Rightarrow (2n+25)^2-(2k)^2 = 549 = 3^3\cdot 61
1,662	-2\cdot \frac{x}{d^3} = 2\cdot x/d\cdot \frac{1}{d}\cdot x
14,616	0 = t rightarrow 0 = t
34,516	y^2 \cdot 2 - y \cdot 6 = (-y \cdot 3 + y^2) \cdot 2
18,704	-(1 + i)2 = \left(-2\right) (-1) - (2 + i)2
17,020	-\frac{1}{e^y + 1} + 1 = \frac{e^y}{1 + e^y}
9,556	\mathbb{E}[\|X\| + \|Y\|] = \mathbb{E}[\|X\|] + \mathbb{E}[\|Y\|]
26,739	273/999 = \frac{91}{333}
-19,518	\dfrac{1}{5} \cdot 7 \cdot \frac19 \cdot 4 = \frac{\frac{7}{5}}{1/4 \cdot 9} \cdot 1
-23,113	-\frac{1}{27} \cdot 4 = \frac13 \cdot (\frac{1}{9} \cdot (-4))
6,359	\frac{48}{50} \cdot \dfrac{3}{51} = 144/2550
-1,650	\frac94\cdot \pi = 4/3\cdot \pi + \frac{11}{12}\cdot \pi
-20,344	\dfrac{1}{5}\cdot 5\cdot \frac{1}{4\cdot z}\cdot ((-1) + 5\cdot z) = \left(5\cdot (-1) + 25\cdot z\right)/(20\cdot z)
14,872	19 = 3^3 - 2 2 2
19,156	1/\left((-1)\cdot b\right) = \frac{1}{(-1)\cdot b} = \frac{\frac{1}{-1}}{b} = -1/b = -\frac1b
729	27 + \|Y\cdot E\| = 27 + \|E\cdot Y\|
37	4\cdot a\cdot d + \left(a - d\right)^2 = (d + a)^2
17,481	(b + f)^2 = b^2 + f^2 + 2bf
24,603	\frac{1}{-(-1) \cdot x^2 + 1} = \frac{1}{1 + x^2}
12,492	\dfrac1y\cdot 2 = \dfrac{2}{y^2}\cdot y
19,043	\int (y + 2\cdot (-1))\cdot e^y\,\mathrm{d}y = (y + 2\cdot (-1))\cdot e^y - \int e^y\,\mathrm{d}y = \left(y + 2\cdot (-1)\right)\cdot e^y - e^y = y\cdot e^y - 3\cdot e^y
2,719	5 = 0 \implies 1 = 0
26,608	10 \cdot z \cdot z + z \cdot 20 + 10 = 10 \cdot (z + 1)^2
29,700	-y^2 + x^2 = (-y + x)*(x + y)
13,028	q^2 - yq*2 + 1 = 0 \Rightarrow q = y ± \sqrt{y^2 + (-1)}
9,457	(x \cdot 2 + 5) (2 + x \cdot 5) = 10 x^2 + 29 x + 10
30,866	1/3 = \dfrac{1}{6}\cdot 2
-13,952	3 + 8 \cdot 6 - 5 = 3 + 48 - 5 = 51 - 5 = 51 + 5 \cdot (-1) = 46
785	\dfrac23\cdot \tfrac{1}{5}/4 = \frac{1}{30}
-1,868	\pi \cdot \frac{19}{12} - \dfrac{1}{12} \cdot 19 \cdot \pi = 0
-18,639	-\frac{7}{22} = -\frac{7}{22}
-4,046	\frac{1}{2} \cdot 9 = \frac{1}{2} \cdot 9
-21,039	\frac17 \cdot 7 \cdot (x + 5 \cdot (-1))/8 = (7 \cdot x + 35 \cdot (-1))/56
13,610	2^{\frac{5}{12}} = 1.334839 \ldots \approx 4/3
-4,262	\frac{111/12}{a^2} = \frac{11}{12a * a}
5,431	t^5 + t^3 + t^2 + 1 = (t^2 + 1) \cdot (1 + t^2 - t) \cdot (t + 1)
151	0 \lt (\sqrt{h_1} - \sqrt{h_2})^2 = h_1 - 2\sqrt{h_1h_2} + h_2 = h_1 + 2*(-1) + h_2
20,527	72/625 = \frac{5\binom{6}{2}4!}{5^6}*1
-19,388	4/5 \cdot \frac{1}{1} \cdot 8 = \dfrac{1}{5} \cdot 4/\left(1/8\right)
-19,010	\tfrac78 = \frac{A_r}{9\pi}9*\pi = A_r
11,798	L = L \cdot 2 \implies L = 0
-2,384	\left(-6\right)^3 = \left(-6\right) \times (-6) \times \left(-6\right) = 36 \times (-6) = -216
2,088	1 + y^3 = (1 + y^2 - y) \cdot (1 + y)
-22,189	\left(p + 8\right)\cdot \left(p + 7\cdot (-1)\right) = p^2 + p + 56\cdot \left(-1\right)

5,662

\dfrac{1}{y^{1/2}}((-3) y) = -3y^{1/2}

1,474

4n^2 + (-1) = (2n + (-1)) (2n + 1)

-564

\left(e^{\pi\cdot i}\right)^{15} = e^{15\cdot i\cdot \pi}

8,715

g \cdot g \cdot e = e = g \cdot g \cdot e

-4,142

\frac{1}{l^3} \cdot l \cdot l \cdot l \cdot \dfrac{1}{40} \cdot 8 = \frac{1}{40 \cdot l^3} \cdot 8 \cdot l^3

-29,007

4 = \left(4\cdot (-1) + 12\right)/2

-20,859

\frac{1}{25\times p + 5\times (-1)}\times (-30\times p + 6) = \frac{1}{5\times p + (-1)}\times \left(5\times p + (-1)\right)\times (-\frac{6}{5})

-12,347

2^{1 / 2}\cdot 6 = 72^{\dfrac{1}{2}}

27,695

\dfrac{1}{x \cdot d} = \dfrac{1}{x \cdot d}

-12,541

31 = 90 + 59\cdot (-1)

17,337

b = h \cdot b = b \cdot h

29,511

\left\lfloor{\frac15(7 + (-1))}\right\rfloor + 1 = 2

6,976

\frac{1}{Y\cdot \dfrac{1}{X}} = X/Y

22,012

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

36,419

-x + x \coloneqq -x + x

35,082

\dfrac{1}{-y^2 + 1} = \tfrac12 \cdot (\dfrac{1}{1 - y} + \frac{1}{1 + y})

10,331

(-\sqrt{x} + \sqrt{a_m})\cdot (\sqrt{a_m} + \sqrt{x}) = -x + a_m

-2,626

-\sqrt{6} + \sqrt{24} = \sqrt{4\cdot 6} - \sqrt{6}

32,919

1/L + 1/M + 1/x = \frac{1}{L*M*x}*(L*M + x*M + x*L)

16,860

\int\limits_{-\infty}^\infty ...\,\mathrm{d}z = 2 \cdot \int_0^\infty ...\,\mathrm{d}z

30,546

-4 \cdot x + 12 = 4 \cdot x \cdot x - (x^2 + x + 3 \cdot (-1)) \cdot 4

-1,260

-20/18 = \frac{1}{18 \cdot 1/2}((-20) \cdot 1/2) = -10/9

36,245

-(8 \cdot 9^5 + 9^6) + 9 \cdot 10^5 = -103833

-28,798

150 = \pi*2/(\pi*2*\frac{1}{150})

38,983

x + x = x \cdot 2

-15,966

-10 \cdot \frac{3}{10} + 7 \cdot 7/10 = 19/10

-4,233

k^2 \cdot 7/6 = 7 \cdot k^2/6

19,541

-9 = 4 + 2 * 2 + (-1)^2*9 - 4*2 + 18*\left(-1\right)

13,277

|(-c_k\cdot t + h_k\cdot M)/\left(M\cdot c_k\right)| = |\frac{h_k}{c_k} - \frac{1}{M}\cdot t|

-2,429

12 \cdot 6^{1/2} = (3 + 4 + 5) \cdot 6^{1/2}

-9,286

-x*7*7 + 3*7 = -x*49 + 21

-26,552

2\times y^2 - 40\times y + 200 = 2\times \left(y \times y - 20\times y + 100\right) = 2\times \left(y + 10\times (-1)\right)^2

14,971

8 \cdot x = 4 \cdot x + 2 \cdot x \cdot 2

-27,734

\frac{\mathrm{d}}{\mathrm{d}x} \csc{x} = -\csc{x} \cot{x}

-11,955

13/15 = \tfrac{1}{12*\pi}*s*12*\pi = s

-20,653

-7/8 \frac{1}{-5\varepsilon + 2(-1)}\left(-5\varepsilon + 2(-1)\right) = \frac{14 + 35 \varepsilon}{-40 \varepsilon + 16 \left(-1\right)}

-22,805

\frac{60}{4 \cdot 12} \cdot 1 = \frac{1}{48} \cdot 60

10,420

5^{2n} + (-1) = \left(5^n + 1\right) \left(5^n + (-1)\right)

-20,339

-\frac{36}{-8} = 9/2 \cdot (-\frac{4}{-4})

8,821

a*\sin{\pi/2} + \cos{\pi/2} = a

-11,009

112/8 = 14

27,041

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

12,189

\left(-5\right)^2 + (-3) * (-3) + 1^2 = 35 = 5*((-5)*(-3) - 5 - 3)

7,423

11 = (z + \sigma + y) \cdot 6\Longrightarrow z + \sigma + y = \dfrac{11}{6}

29,030

\csc{\theta} = \frac{1}{\sin{\theta}} = \dfrac{1}{(1 - \cos^2{\theta})^{\tfrac{1}{2}}}

15,310

-A_1*B_1 + (A_0 + A_1)*\left(B_0 + B_1\right) - B_0*A_0 = A_1*B_0 + B_1*A_0

-20,018

\frac{s\cdot 5 + 30}{45\cdot (-1) + 5\cdot s} = \frac55\cdot \frac{s + 6}{s + 9\cdot \left(-1\right)}

22,130

\binom{1}{1}\times \binom{19}{2}/\left(\binom{20}{3}\right) = \frac{1}{20}\times 3

4,601

-2 \cdot x \cdot e^{-x^2} = (-2 + 4 \cdot x^2) \cdot e^{-x^2}

33,017

\frac{1}{2^{1 / 2}} = \dfrac{2^{\frac{1}{2}}}{2}

-5,036

18.0\cdot 10^{5 + 2} = 18\cdot 10^7

25,998

5 + \left(z + (-1)\right)^4 + z^2\cdot 3 - z\cdot 6 = (z + (-1))^4 + 3\cdot (z + (-1))^2 + 2

41,682

2^{23} + (-1) = 47 \cdot 178481

-20,293

\frac{9 - 6 \cdot \delta}{\delta + 5 \cdot (-1)} \cdot 5/5 = \frac{45 - 30 \cdot \delta}{\delta \cdot 5 + 25 \cdot (-1)}

20,760

\sin(2\cdot \theta) = \cos(\dfrac{\pi}{2} - 2\cdot \theta)

52,112

770 = {5 \choose 4} \cdot {7 \choose 1} + {7 \choose 2} \cdot {5 \choose 3} + {7 \choose 3} \cdot {5 \choose 2} + {7 \choose 4} \cdot {5 \choose 1}

37,155

(2k)^2 = (2n+25)^2-549\Rightarrow (2n+25)^2-(2k)^2 = 549 = 3^3\cdot 61

1,662

-2\cdot \frac{x}{d^3} = 2\cdot x/d\cdot \frac{1}{d}\cdot x

14,616

0 = t rightarrow 0 = t

34,516

y^2 \cdot 2 - y \cdot 6 = (-y \cdot 3 + y^2) \cdot 2

18,704

-(1 + i)*2 = \left(-2\right) (-1) - (2 + i)*2

17,020

-\frac{1}{e^y + 1} + 1 = \frac{e^y}{1 + e^y}

9,556

\mathbb{E}[|X| + |Y|] = \mathbb{E}[|X|] + \mathbb{E}[|Y|]

26,739

273/999 = \frac{91}{333}

-19,518

\dfrac{1}{5} \cdot 7 \cdot \frac19 \cdot 4 = \frac{\frac{7}{5}}{1/4 \cdot 9} \cdot 1

-23,113

-\frac{1}{27} \cdot 4 = \frac13 \cdot (\frac{1}{9} \cdot (-4))

6,359

\frac{48}{50} \cdot \dfrac{3}{51} = 144/2550

-1,650

\frac94\cdot \pi = 4/3\cdot \pi + \frac{11}{12}\cdot \pi

-20,344

\dfrac{1}{5}\cdot 5\cdot \frac{1}{4\cdot z}\cdot ((-1) + 5\cdot z) = \left(5\cdot (-1) + 25\cdot z\right)/(20\cdot z)

14,872

19 = 3^3 - 2 2 2

19,156

1/\left((-1)\cdot b\right) = \frac{1}{(-1)\cdot b} = \frac{\frac{1}{-1}}{b} = -1/b = -\frac1b

729

27 + |Y\cdot E| = 27 + |E\cdot Y|

37

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

17,481

(b + f)^2 = b^2 + f^2 + 2*b*f

24,603

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

12,492

\dfrac1y\cdot 2 = \dfrac{2}{y^2}\cdot y

19,043

\int (y + 2\cdot (-1))\cdot e^y\,\mathrm{d}y = (y + 2\cdot (-1))\cdot e^y - \int e^y\,\mathrm{d}y = \left(y + 2\cdot (-1)\right)\cdot e^y - e^y = y\cdot e^y - 3\cdot e^y

2,719

5 = 0 \implies 1 = 0

26,608

10 \cdot z \cdot z + z \cdot 20 + 10 = 10 \cdot (z + 1)^2

29,700

-y^2 + x^2 = (-y + x)*(x + y)

13,028

q^2 - yq*2 + 1 = 0 \Rightarrow q = y ± \sqrt{y^2 + (-1)}

9,457

(x \cdot 2 + 5) (2 + x \cdot 5) = 10 x^2 + 29 x + 10

30,866

1/3 = \dfrac{1}{6}\cdot 2

-13,952

3 + 8 \cdot 6 - 5 = 3 + 48 - 5 = 51 - 5 = 51 + 5 \cdot (-1) = 46

785

\dfrac23\cdot \tfrac{1}{5}/4 = \frac{1}{30}

-1,868

\pi \cdot \frac{19}{12} - \dfrac{1}{12} \cdot 19 \cdot \pi = 0

-18,639

-\frac{7}{22} = -\frac{7}{22}

-4,046

\frac{1}{2} \cdot 9 = \frac{1}{2} \cdot 9

-21,039

\frac17 \cdot 7 \cdot (x + 5 \cdot (-1))/8 = (7 \cdot x + 35 \cdot (-1))/56

13,610

2^{\frac{5}{12}} = 1.334839 \ldots \approx 4/3

-4,262

\frac{11*1/12}{a^2} = \frac{11}{12*a * a}

5,431

t^5 + t^3 + t^2 + 1 = (t^2 + 1) \cdot (1 + t^2 - t) \cdot (t + 1)

151

0 \lt (\sqrt{h_1} - \sqrt{h_2})^2 = h_1 - 2*\sqrt{h_1*h_2} + h_2 = h_1 + 2*(-1) + h_2

20,527

72/625 = \frac{5*\binom{6}{2}*4!}{5^6}*1

-19,388

4/5 \cdot \frac{1}{1} \cdot 8 = \dfrac{1}{5} \cdot 4/\left(1/8\right)

-19,010

\tfrac78 = \frac{A_r}{9*\pi}*9*\pi = A_r

11,798

L = L \cdot 2 \implies L = 0

-2,384

\left(-6\right)^3 = \left(-6\right) \times (-6) \times \left(-6\right) = 36 \times (-6) = -216

2,088

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

-22,189

\left(p + 8\right)\cdot \left(p + 7\cdot (-1)\right) = p^2 + p + 56\cdot \left(-1\right)