ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-10,389	\frac{100}{60\cdot r + 240\cdot (-1)} = \dfrac{1}{12\cdot (-1) + 3\cdot r}\cdot 5\cdot 20/20
10,812	(x + x^2 + \dots + x^{15}) \cdot (x + x^2 + \dots + x^{15}) = \frac{x \cdot x}{\left(1 - x\right)^2}
-15,529	\frac{1}{r\cdot \frac{1}{\frac{1}{x^4}\cdot r^8}} = \frac{1}{\frac{x^4}{r^8}\cdot r}
27,143	1 + 5 + \dots + 5^n + 5^{n + 1} = \dfrac14 \cdot (5^{n + 1} + (-1)) + 5^{n + 1}
28,056	(-1) + \left\lfloor{y}\right\rfloor = \left\lfloor{(-1) + y}\right\rfloor
19,698	\sin(-π/3) = \sin(4 \cdot π/3)
21,218	\left\|{x\times J}\right\| = \left\|{x\times J}\right\|
1,316	54860400 = 10!\cdot \tfrac{2177}{144}
27,420	(f9 + 80a)^2 - (9a + f)^280 = -80a^2 + f f
6,542	(1+ 1)^n= 2^n
1,526	\tfrac{4}{2\cdot j} = -2\cdot j = -2\cdot e^{\frac{j\cdot \pi}{2}} = 2\cdot e^{((-1)\cdot j\cdot \pi)/2}
28,303	\dfrac{1}{5000} = 2/10000
16,272	(2n + 1) (n2)! (n2 + 2) = (n*2 + 2)!
18,858	( r', i')\cdot ( r, i) \coloneqq ( r\cdot r', i'\cdot r + r'\cdot i)
10,331	-L + a_n = (\sqrt{a_n} + \sqrt{L}) \cdot \left(-\sqrt{L} + \sqrt{a_n}\right)
8,839	W^2 + 2\cdot W + 1 = (1 + W)^2
15,856	\frac{\mathrm{d}}{\mathrm{d}\gamma} \tan{\gamma} = \sec^2{\gamma} = 1 + \tan^2{\gamma}
-14,122	\dfrac{ 16 }{ 10 + 6 } = \dfrac{ 16 }{ (16) } = \dfrac{ 16 }{ 16 } = 1
11,204	ln(2z)=ln(2)+ln(z)
-10,407	-\frac{2(-1) + n}{3 + n}3/3 = -\tfrac{1}{9 + 3n}(6(-1) + 3n)
-4,568	(4 + y)\cdot \left(y + 5\cdot (-1)\right) = y^2 - y + 20\cdot \left(-1\right)
1,400	\frac{1}{-2}(5(-1) + 0) (2 + 0) = 0 + 0 + B\Longrightarrow 5 = B
6,784	r g = g r g = g r
25,687	3/5 = 6/10 \geq 5/10
2,195	\ln(2) = \frac12 + \frac{1}{6} + \frac{1}{30} + \frac{1}{56} + \dotsm \leq 1 + \frac14 + 1/25 + 1/49 + \dotsm
3,364	x = \dfrac{x^2\cdot 3}{x\cdot 3}
3,470	\dfrac{1}{3} 5 - \frac19 6 = 5/3 - 2/3 = \dfrac33 = 1
1,403	1/(4*y) + y = (y^2 + 1/4)/y
2,119	\cos\left(\pi2014/12 - 2\pi83\right) = \cos(\dfrac{\pi2014}{12})
2,167	-s^2 + (1 + s)^2 = 1 + 2 \cdot s
12,178	6 + \sqrt{3} = \left(1 + 2 \cdot \sqrt{3}\right) \cdot \sqrt{3}
26,324	\left(x\times c\right)^3 = c^3\times x^3
37,209	0.999 \cdot \dots = \frac{9 \cdot \frac{1}{10}}{1 - \frac{1}{10}} = 1
-26,571	320 (-1) + x^2\cdot 5 = 5(x^2 + 64 (-1))
54,769	21^3 = 9261
150	\left(z_1,z_2 \geq 0\Longrightarrow z_2*2 = 2z_1\right)\Longrightarrow z_2 = z_1
-20,003	\dfrac44 \cdot (z + 9)/4 = (4 \cdot z + 36)/16
1,987	\frac{1}{n}\cdot (n + (-1)) = -1/n + 1
-9,566	100\% = \dfrac{100}{100} = 1^{-1}
30,862	x\times n + p\times x = \left(n + p\right)\times x
5,104	a + b\omega + h\omega * \omega = a + b\omega - h*(1 + \omega) = a - h + \left(b - h\right) \omega
28,244	(k + 1)! = 1\cdot 2 \dots k\cdot (k + 1) = k! \left(k + 1\right)
9,329	\int \frac1x\,dx = (\ln(x))! = \ln(x)
-18,572	-\frac94 = -\frac14 \cdot 9
9,633	4 + (5^m + (-1))\cdot 5 = (-1) + 5^{m + 1}
2,351	x + y + \lambda = y + \lambda + x
23,015	\frac{6!}{6^6} = \dfrac{5}{324}
31,490	\frac{\mathrm{d}}{\mathrm{d}y} \left(-\csc(y)\right) = \csc(y)*\cot(y)
16,758	(2\cdot k + 1) \cdot (2\cdot k + 1) = 1 + (k^2 + k)\cdot 4
6,625	\dfrac{2\cdot l!}{(l + 3)!} = \dfrac{2}{(l + 1)\cdot (l + 2)\cdot \left(l + 3\right)} \leq \dfrac{2}{l^2}
-1,812	\pi \cdot \frac76 = \pi \cdot 7/6 + 0
17,125	{3 \choose 2} \cdot {5 \choose 2} = 3 \cdot 10 = 30
16,523	\overline{E} \cap (X \cap \overline{G}) = X \cap (\overline{E} \cap \overline{G}) = X \cap \overline{E \cup G}
17,771	4/5 \cdot \frac15 \cdot 4 \cdot 25 = 16
-28,990	4 \cdot \pi/20 = \frac{1}{5} \cdot \pi
-10,500	-8/(t2)2/2 = -\frac{1}{4t}16
-20,966	\dfrac{t10}{(-12) t} = -5/6 \frac{1}{t\left(-2\right)}((-2) t)
21,460	x^4 + 1 = x^4 - 2x^2 + 1 - -2x \cdot x = (x^2 + (-1))^2 - x \cdot x = (x^2 + \left(-1\right) + x) (x^2 + (-1) - x)
-6,732	2/100 + 2/10 = \frac{2}{100} + \frac{20}{100}
-3,258	\sqrt{7}5 = \sqrt{7}(4 + (-1) + 2)
2,117	-((-1) + k) + n + 1 = 2 + n - k
4,795	(l + 10\cdot (-1))/2 = -1.3 \Rightarrow l = 7.4
30,925	\frac{1}{5 \cdot 24} \cdot \left((-1) + 24^2 + 5 \cdot 5\right) = 5
-5,850	\frac{1}{(x + 8) (x + 9)} = \tfrac{1}{72 + x^2 + x*17}
-20,080	6/6\cdot \left(-\frac{9}{n + 9\cdot (-1)}\right) = -\frac{54}{54\cdot (-1) + 6\cdot n}
-15,278	\frac{k\times x^4}{\frac{1}{k^3}\times x \times x} = \frac{k\times x^4}{x \times x\times \frac{1}{k^3}}
71	\left\{x, C\right\} \implies C \cup x = C
27,478	1 = \dfrac{\left(2*0\right)!}{0!^2}
31,610	\left( 0, 2\right) + E2 = ( 2, 0) + E2 = ( 2, 2) + E2 = ( 0, 0) + E2 = E*2
-2,777	10^{\frac{1}{2}}(2 + 4 + (-1)) = 10^{1 / 2}5
6,776	x^4 + 1 = 7 + \left(3 + x^2 - x\right)\cdot (x + (-1))\cdot (x + 2) - 5\cdot x
14,637	\frac{1}{(-1) + \|y\|^m} = \frac{1}{\left(\|y\|^m + (-1)\right)\cdot \|y\|^m} + \frac{1}{\|y\|^m}
-18,671	(-1)*0.3085 + 0.7734 = 0.4649
126	\cos(\theta/2) = \frac{\sin\left(\theta\right)}{2 \cdot \sin\left(\theta/2\right)}
1,713	e^{-N \cdot N} \cdot \frac{\mathrm{d}z}{\mathrm{d}N} - N \cdot z \cdot e^{-N^2} \cdot 2 = \frac{\partial}{\partial N} (e^{-N^2} \cdot z)
15,783	\sqrt{x} - \frac{1}{\sqrt{x}} = (x + \left(-1\right))/(\sqrt{x})
5,510	\tfrac{1}{2^{n + (-1)}} = \frac{1}{2^n}\cdot \left(0\cdot (-1) + 2\right)
4,940	56 = 312 + 256 \cdot (-1)
-5,054	5.8/10 = 5.8/10 \times 10 \times 10 = 5.8 \times 10^1
39,744	0*1/2 = 0
-11,538	6 + 9(-1) - 15 i = -3 - i*15
2,006	\frac{x^k + \left(-1\right)}{x + (-1)} = \frac{1}{x + (-1)}(x + (-1))(1 + x + \cdots + x^{k + (-1)}) = 1 + x + \cdots + x^{k + (-1)}
24,526	x^4 + 1 = (x^2 + x2^{1 / 2} + 1)(1 + x^2 - 2^{1 / 2}*x)
23,588	\sqrt{7}/2\times 2 = \sqrt{7}
35,283	\cos(\pi/8) = \cos(\frac{1/4}{2} \cdot \pi)
-1,150	8/9\cdot 3/2 = \frac{8\cdot 3}{9\cdot 2} = 24/18
30,420	(\dfrac{1}{4}\times (w^2 + h^2))^{\dfrac{1}{2}} = \frac{1}{2}\times (h \times h + w^2)^{1 / 2}
-10,284	-\frac{32}{k \cdot 24} = -\frac{1}{6 \cdot k} \cdot 8 \cdot \frac{4}{4}
-5,641	\dfrac{15q - 60 + 25q - 200 + 45}{15q^2 - 180q + 480} = \dfrac{40q - 215}{15q^2 - 180q + 480}
10,268	-\sin{A} \cdot \sin{B} + \cos{A} \cdot \cos{B} = \cos(A + B)
5,393	540 = {2 \choose 1} \cdot {10 \choose 6} + {10 \choose 7} \cdot {2 \choose 0}
20,605	(x - E)^2 = x - 2\cdot E + E^2 = x - E
-11,079	(x + c)^2 = (x + c) (x + c) = x^2 + 2c x + c^2
11,971	\frac{d}{du} (\frac{e^u}{e^u}) = 0 = e^u - e^{2u}
-6,176	\frac{1}{15 \cdot \left(-1\right) + y^2 + 2 \cdot y} \cdot 4 = \frac{4}{(y + 3 \cdot (-1)) \cdot (y + 5)}
-20,781	\frac{1}{6x + 10 (-1)}1 = \frac{6}{36 x + 60 (-1)}
-7,592	\tfrac{\left(5 \cdot i + 2\right) \cdot \left(11 + i \cdot 16\right)}{(2 + 5 \cdot i) \cdot (2 - 5 \cdot i)} = \frac{1}{2^2 - (-i \cdot 5)^2} \cdot (2 + i \cdot 5) \cdot (11 + 16 \cdot i)
43,319	706 = 2 \times 353
649	-c + (x + j)^2 = -c + x^2 + j \cdot x \cdot 2 + j^2
23,968	sr := sr

-10,389

\frac{100}{60\cdot r + 240\cdot (-1)} = \dfrac{1}{12\cdot (-1) + 3\cdot r}\cdot 5\cdot 20/20

10,812

(x + x^2 + \dots + x^{15}) \cdot (x + x^2 + \dots + x^{15}) = \frac{x \cdot x}{\left(1 - x\right)^2}

-15,529

\frac{1}{r\cdot \frac{1}{\frac{1}{x^4}\cdot r^8}} = \frac{1}{\frac{x^4}{r^8}\cdot r}

27,143

1 + 5 + \dots + 5^n + 5^{n + 1} = \dfrac14 \cdot (5^{n + 1} + (-1)) + 5^{n + 1}

28,056

(-1) + \left\lfloor{y}\right\rfloor = \left\lfloor{(-1) + y}\right\rfloor

19,698

\sin(-π/3) = \sin(4 \cdot π/3)

21,218

\left|{x\times J}\right| = \left|{x\times J}\right|

1,316

54860400 = 10!\cdot \tfrac{2177}{144}

27,420

(f*9 + 80*a)^2 - (9*a + f)^2*80 = -80*a^2 + f * f

6,542

(1+ 1)^n= 2^n

1,526

\tfrac{4}{2\cdot j} = -2\cdot j = -2\cdot e^{\frac{j\cdot \pi}{2}} = 2\cdot e^{((-1)\cdot j\cdot \pi)/2}

28,303

\dfrac{1}{5000} = 2/10000

16,272

(2n + 1) (n*2)! (n*2 + 2) = (n*2 + 2)!

18,858

( r', i')\cdot ( r, i) \coloneqq ( r\cdot r', i'\cdot r + r'\cdot i)

10,331

-L + a_n = (\sqrt{a_n} + \sqrt{L}) \cdot \left(-\sqrt{L} + \sqrt{a_n}\right)

8,839

W^2 + 2\cdot W + 1 = (1 + W)^2

15,856

\frac{\mathrm{d}}{\mathrm{d}\gamma} \tan{\gamma} = \sec^2{\gamma} = 1 + \tan^2{\gamma}

-14,122

\dfrac{ 16 }{ 10 + 6 } = \dfrac{ 16 }{ (16) } = \dfrac{ 16 }{ 16 } = 1

11,204

ln(2z)=ln(2)+ln(z)

-10,407

-\frac{2*(-1) + n}{3 + n}*3/3 = -\tfrac{1}{9 + 3*n}*(6*(-1) + 3*n)

-4,568

(4 + y)\cdot \left(y + 5\cdot (-1)\right) = y^2 - y + 20\cdot \left(-1\right)

1,400

\frac{1}{-2}(5(-1) + 0) (2 + 0) = 0 + 0 + B\Longrightarrow 5 = B

6,784

r g = g r g = g r

25,687

3/5 = 6/10 \geq 5/10

2,195

\ln(2) = \frac12 + \frac{1}{6} + \frac{1}{30} + \frac{1}{56} + \dotsm \leq 1 + \frac14 + 1/25 + 1/49 + \dotsm

3,364

x = \dfrac{x^2\cdot 3}{x\cdot 3}

3,470

\dfrac{1}{3} 5 - \frac19 6 = 5/3 - 2/3 = \dfrac33 = 1

1,403

1/(4*y) + y = (y^2 + 1/4)/y

2,119

\cos\left(\pi*2014/12 - 2*\pi*83\right) = \cos(\dfrac{\pi*2014}{12})

2,167

-s^2 + (1 + s)^2 = 1 + 2 \cdot s

12,178

6 + \sqrt{3} = \left(1 + 2 \cdot \sqrt{3}\right) \cdot \sqrt{3}

26,324

\left(x\times c\right)^3 = c^3\times x^3

37,209

0.999 \cdot \dots = \frac{9 \cdot \frac{1}{10}}{1 - \frac{1}{10}} = 1

-26,571

320 (-1) + x^2\cdot 5 = 5(x^2 + 64 (-1))

54,769

21^3 = 9261

150

\left(z_1,z_2 \geq 0\Longrightarrow z_2*2 = 2z_1\right)\Longrightarrow z_2 = z_1

-20,003

\dfrac44 \cdot (z + 9)/4 = (4 \cdot z + 36)/16

1,987

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

-9,566

100\% = \dfrac{100}{100} = 1^{-1}

30,862

x\times n + p\times x = \left(n + p\right)\times x

5,104

a + b\omega + h\omega * \omega = a + b\omega - h*(1 + \omega) = a - h + \left(b - h\right) \omega

28,244

(k + 1)! = 1\cdot 2 \dots k\cdot (k + 1) = k! \left(k + 1\right)

9,329

\int \frac1x\,dx = (\ln(x))! = \ln(x)

-18,572

-\frac94 = -\frac14 \cdot 9

9,633

4 + (5^m + (-1))\cdot 5 = (-1) + 5^{m + 1}

2,351

x + y + \lambda = y + \lambda + x

23,015

\frac{6!}{6^6} = \dfrac{5}{324}

31,490

\frac{\mathrm{d}}{\mathrm{d}y} \left(-\csc(y)\right) = \csc(y)*\cot(y)

16,758

(2\cdot k + 1) \cdot (2\cdot k + 1) = 1 + (k^2 + k)\cdot 4

6,625

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

-1,812

\pi \cdot \frac76 = \pi \cdot 7/6 + 0

17,125

{3 \choose 2} \cdot {5 \choose 2} = 3 \cdot 10 = 30

16,523

\overline{E} \cap (X \cap \overline{G}) = X \cap (\overline{E} \cap \overline{G}) = X \cap \overline{E \cup G}

17,771

4/5 \cdot \frac15 \cdot 4 \cdot 25 = 16

-28,990

4 \cdot \pi/20 = \frac{1}{5} \cdot \pi

-10,500

-8/(t*2)*2/2 = -\frac{1}{4*t}*16

-20,966

\dfrac{t*10}{(-12) t} = -5/6 \frac{1}{t*\left(-2\right)}((-2) t)

21,460

x^4 + 1 = x^4 - 2x^2 + 1 - -2x \cdot x = (x^2 + (-1))^2 - x \cdot x = (x^2 + \left(-1\right) + x) (x^2 + (-1) - x)

-6,732

2/100 + 2/10 = \frac{2}{100} + \frac{20}{100}

-3,258

\sqrt{7}*5 = \sqrt{7}*(4 + (-1) + 2)

2,117

-((-1) + k) + n + 1 = 2 + n - k

4,795

(l + 10\cdot (-1))/2 = -1.3 \Rightarrow l = 7.4

30,925

\frac{1}{5 \cdot 24} \cdot \left((-1) + 24^2 + 5 \cdot 5\right) = 5

-5,850

\frac{1}{(x + 8) (x + 9)} = \tfrac{1}{72 + x^2 + x*17}

-20,080

6/6\cdot \left(-\frac{9}{n + 9\cdot (-1)}\right) = -\frac{54}{54\cdot (-1) + 6\cdot n}

-15,278

\frac{k\times x^4}{\frac{1}{k^3}\times x \times x} = \frac{k\times x^4}{x \times x\times \frac{1}{k^3}}

71

\left\{x, C\right\} \implies C \cup x = C

27,478

1 = \dfrac{\left(2*0\right)!}{0!^2}

31,610

\left( 0, 2\right) + E*2 = ( 2, 0) + E*2 = ( 2, 2) + E*2 = ( 0, 0) + E*2 = E*2

-2,777

10^{\frac{1}{2}}*(2 + 4 + (-1)) = 10^{1 / 2}*5

6,776

x^4 + 1 = 7 + \left(3 + x^2 - x\right)\cdot (x + (-1))\cdot (x + 2) - 5\cdot x

14,637

\frac{1}{(-1) + |y|^m} = \frac{1}{\left(|y|^m + (-1)\right)\cdot |y|^m} + \frac{1}{|y|^m}

-18,671

(-1)*0.3085 + 0.7734 = 0.4649

126

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

1,713

e^{-N \cdot N} \cdot \frac{\mathrm{d}z}{\mathrm{d}N} - N \cdot z \cdot e^{-N^2} \cdot 2 = \frac{\partial}{\partial N} (e^{-N^2} \cdot z)

15,783

\sqrt{x} - \frac{1}{\sqrt{x}} = (x + \left(-1\right))/(\sqrt{x})

5,510

\tfrac{1}{2^{n + (-1)}} = \frac{1}{2^n}\cdot \left(0\cdot (-1) + 2\right)

4,940

56 = 312 + 256 \cdot (-1)

-5,054

5.8/10 = 5.8/10 \times 10 \times 10 = 5.8 \times 10^1

39,744

0*1/2 = 0

-11,538

6 + 9(-1) - 15 i = -3 - i*15

2,006

\frac{x^k + \left(-1\right)}{x + (-1)} = \frac{1}{x + (-1)}*(x + (-1))*(1 + x + \cdots + x^{k + (-1)}) = 1 + x + \cdots + x^{k + (-1)}

24,526

x^4 + 1 = (x^2 + x*2^{1 / 2} + 1)*(1 + x^2 - 2^{1 / 2}*x)

23,588

\sqrt{7}/2\times 2 = \sqrt{7}

35,283

\cos(\pi/8) = \cos(\frac{1/4}{2} \cdot \pi)

-1,150

8/9\cdot 3/2 = \frac{8\cdot 3}{9\cdot 2} = 24/18

30,420

(\dfrac{1}{4}\times (w^2 + h^2))^{\dfrac{1}{2}} = \frac{1}{2}\times (h \times h + w^2)^{1 / 2}

-10,284

-\frac{32}{k \cdot 24} = -\frac{1}{6 \cdot k} \cdot 8 \cdot \frac{4}{4}

-5,641

\dfrac{15q - 60 + 25q - 200 + 45}{15q^2 - 180q + 480} = \dfrac{40q - 215}{15q^2 - 180q + 480}

10,268

-\sin{A} \cdot \sin{B} + \cos{A} \cdot \cos{B} = \cos(A + B)

5,393

540 = {2 \choose 1} \cdot {10 \choose 6} + {10 \choose 7} \cdot {2 \choose 0}

20,605

(x - E)^2 = x - 2\cdot E + E^2 = x - E

-11,079

(x + c)^2 = (x + c) (x + c) = x^2 + 2c x + c^2

11,971

\frac{d}{du} (\frac{e^u}{e^u}) = 0 = e^u - e^{2u}

-6,176

\frac{1}{15 \cdot \left(-1\right) + y^2 + 2 \cdot y} \cdot 4 = \frac{4}{(y + 3 \cdot (-1)) \cdot (y + 5)}

-20,781

\frac{1}{6x + 10 (-1)}1 = \frac{6}{36 x + 60 (-1)}

-7,592

\tfrac{\left(5 \cdot i + 2\right) \cdot \left(11 + i \cdot 16\right)}{(2 + 5 \cdot i) \cdot (2 - 5 \cdot i)} = \frac{1}{2^2 - (-i \cdot 5)^2} \cdot (2 + i \cdot 5) \cdot (11 + 16 \cdot i)

43,319

706 = 2 \times 353

649

-c + (x + j)^2 = -c + x^2 + j \cdot x \cdot 2 + j^2

23,968

sr := sr