ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
5,634	\frac{1}{x^2} \times \left(i + \left(-1\right)\right) = 1/x \times ((-1) + i)/x
2,933	(X + 3)^3 = X^3 + 9X * X + 27 X + 27 = \left(X^2 + 5X + 7\right) \left(X + 4\right) + (-1)
-1,372	-\frac{1}{72} \cdot 18 = \frac{1}{72 \cdot 1/18} \cdot ((-18) \cdot \dfrac{1}{18}) = -\frac{1}{4}
-29,514	\frac{1}{(4\cdot (-1) + 5)!}\cdot 5! = 120
7,608	t^2\cdot p^2 = (p\cdot t)^2
9,883	1 = 1^{1/2} = ((-1)^2)^{1/2} = (\left(-1\right)^{1/2})^2 = i^2 = -1
22,817	4 + 4 + 3\times \left(-1\right) = 5
9,294	1 + q^4 - 2\cdot q \cdot q = (1 - q \cdot q) \cdot (1 - q \cdot q)
16,785	-(c + b) + d = d - b - c
15,985	2^l \cdot (2^{m - l} + (-1)) = -2^l + 2^m
-7,107	\frac{5}{66} = 5/11 \cdot \frac{1}{12} \cdot 2
-10,019	96\% = 96/100 = 0.96
5,869	-B\cdot C + C\cdot A = (-B + A)\cdot C
-20,021	\frac{1}{4r + 4(-1)}(-9r + 9) = \frac{(-1) + r}{(-1) + r}*(-9/4)
2,299	\frac{\sqrt{4 - 4z + z^2}}{z + 2(-1)} = \frac{1}{z + 2(-1)}\sqrt{(z + 2\left(-1\right))^2} = \frac{\|z + 2\left(-1\right)\|}{z + 2*(-1)}
-10,513	-\frac{24}{15 + r \cdot 6} = 3/3 \cdot (-\dfrac{1}{2 \cdot r + 5} \cdot 8)
-6,342	\frac{1}{(z + 6\cdot (-1))\cdot \left(z + 7\cdot \left(-1\right)\right)}\cdot 4 = \tfrac{4}{z^2 - z\cdot 13 + 42}
-18,979	\frac{2}{9} = \dfrac{A_s}{36\times \pi}\times 36\times \pi = A_s
-19,998	\frac91\cdot \frac{-9\cdot l + 3}{-l\cdot 9 + 3} = \frac{1}{3 - l\cdot 9}\cdot (-l\cdot 81 + 27)
-23,275	24\% = -0.01\cdot 76 + 100\%
-9,127	-20 \cdot z + 20 \cdot z^2 = z \cdot z \cdot 2 \cdot 2 \cdot 5 - z \cdot 2 \cdot 2 \cdot 5
19,260	\frac{84}{49 \cdot (-1) + 1} = -7/4
13,315	\frac{1}{4 \times 3} = 1/12
8,663	\frac{1}{2^{10}}116 = \frac{1}{256}29 \approx 0.11328125
-22,339	a^2 - 4a - 5 = (a - 5)(a + 1)
-7,744	\left(4 + i\cdot 6\right)/\left(-2\right) = \frac{6}{-2}\cdot i + \frac{4}{-2}
-18,646	-\frac{1}{11}\cdot 28 = -28/11
-20,026	\frac{1}{48 \cdot (-1) + 48 \cdot p} \cdot (56 \cdot p + 56 \cdot (-1)) = \dfrac{7}{6} \cdot \tfrac{8 \cdot p + 8 \cdot (-1)}{8 \cdot \left(-1\right) + p \cdot 8}
19,713	\sin\left(0\right) \cdot \cos(0) = 0
7,488	\sum_{n=1}^\infty \dfrac{n}{n \cdot n} = \sum_{n=1}^\infty \frac{1}{n}
15,127	exp(j + z) = exp(j) exp(z)
-2,514	(1 + 4) \cdot \sqrt{10} = \sqrt{10} \cdot 5
-15,213	\frac{1}{a^4 \cdot k^4} \cdot a^{15} = \frac{1}{k^4 \cdot a^4} \cdot a^5 \cdot (a^5)^2
34,277	b_2\cdot b_1\cdot b_3 = b_1\cdot b_3\cdot b_2
3,997	-Y^2 + A^2 = (A - Y) \cdot (A + Y)
-19,392	\frac{4}{3} \cdot \frac56 = \frac{1/3 \cdot 4}{6 \cdot \frac{1}{5}}
-2,882	-2\cdot \sqrt{6} + 3\cdot \sqrt{6} = -\sqrt{6}\cdot \sqrt{4} + \sqrt{6}\cdot \sqrt{9}
29,125	\cos^2{\psi_1} + \cos^2{\psi_2} + \cos^2{\psi_3} = 3/2 = \sin^2{\psi_1} + \sin^2{\psi_2} + \sin^2{\psi_3}
3,574	\left\|{F \cdot A + x}\right\| = \left\|{x + A \cdot F}\right\|
-19,276	\frac{9}{5} 1/2 = 1/\left(5*2/9\right)
-16,003	-6/106 + 9\frac{1}{10}*4 = 0
19,817	\left(-1\right)^{b\cdot d} = (-1)^{d\cdot b}
32,229	\frac{1}{2} (\cos(y\cdot 2) + 1) = \cos^2(y)
-25,012	\frac{1}{25}\cdot 33 = 1.32
1,030	36 \cdot (-1) + z^2 = (6 + z) \cdot (z + 6 \cdot (-1))
-2,414	\sqrt{6} \cdot (2 + 3) = \sqrt{6} \cdot 5
-10,250	-\dfrac{1}{100} \times 43 = -0.43
15,880	x_ex_d = x_dx_e
7,729	8 = \frac{4!}{1^1\cdot 1!\cdot 3^1\cdot 1!}
21,028	\sin{2 E} = \sin{E} \cos{E}*2
8,897	n \lt 0 \Rightarrow \|n\| - n = \|n\|\times 2
16,816	1=n/n=1/n + ... + 1/n = (1/n)(1+...+1)\to 0
4,042	\sqrt{(1/u)^2 + 1}\cdot u = \sqrt{((\frac{1}{u})^2 + 1)\cdot u^2} = \sqrt{1 + u^2}
-5,460	\dfrac{1}{l^2\cdot 2 + 98\cdot \left(-1\right)}\cdot (l\cdot 6 + 42\cdot \left(-1\right) - l + 7\cdot (-1) + 4\cdot (-1)) = \dfrac{53\cdot (-1) + 5\cdot l}{2\cdot l^2 + 98\cdot (-1)}
17,876	(2 \cdot a^2 + 1)^2 + \left(-1\right) = 4 \cdot a^4 + 4 \cdot a^2 = (2 \cdot a^2)^2 + (2 \cdot a)^2
10,287	1 = \frac{c}{c} \Rightarrow c = 1/(\frac{1}{c})
27,986	2 = \dfrac{1}{3 + 2(-1)}2
-3,318	3^{1 / 2} + 9^{\frac{1}{2}} \cdot 3^{\dfrac{1}{2}} = 3^{\dfrac{1}{2}} + 3^{1 / 2} \cdot 3
13,737	9y^2 + 36 (-1) = 3y \cdot y - 6^2 = (3y + 6) (3y + 6\left(-1\right)) = 3(y + 2) (3y + 6(-1)) = 9(y + 2) (y + 2(-1))
19,039	\sin{x} = \sin{2 \cdot x} = \dots = \sin{100 \cdot x}
21,807	d_2^2\cdot d_1\cdot d_2^2 = \frac{1}{d_2\cdot d_1\cdot d_2} = d_1\cdot d_2\cdot d_1
6,859	\left(0 + (-1)\right)^2 + (0 + 2 \times (-1))^2 + (1 + 2 \times (-1))^2 = 6
-9,252	-3\cdot 2\cdot 2 - t\cdot 2\cdot 2\cdot 5 = -20\cdot t + 12\cdot (-1)
12,460	z^23 + 6z + 1 = 2(-1) + 3(z + 1)^2
8,373	x + 2 = k\cdot x\cdot 2 + 1\Longrightarrow (2\cdot k + (-1))\cdot x = 1
44,148	\frac{\mathrm{d}x}{\mathrm{d}t} = \frac{\mathrm{d}x^1 + 1}{\mathrm{d}x^1 + 1}
-16,546	\sqrt{8} \cdot 9 = 9 \cdot \sqrt{4 \cdot 2}
17,263	-z^3 + x^3 = (x - z)\cdot (x^2 + x\cdot z + z^2)
22,723	(1 + k)^2 - k^2 = 1 + 2\cdot k
27,481	1 + r^3 - r^2 + r^2 - r + r + (-1) = r^3
11,431	\vartheta x = \vartheta x
13,597	\dfrac{48}{{8 \choose 3}}1 = 6/7
4,033	(2^4)^2 + (2.2^{4.2})^6 + 2^{12} = (2^4)^2 + (2.2^{4.2})^6 + (2^6)^2 = (2^4 + 2^6)^2
30,106	1 - \frac{1}{2^4} = \left(1 - \dfrac{1}{2^2}\right)\cdot \left(\frac{1}{2^2} + 1\right)
25,750	k = \frac{1}{2}\cdot (k\cdot 2 + 0)
28,682	\lim_{x \to 0^+} \sin(\frac1x) = \lim_{x \to \infty} \sin\left(x\right)
-7,792	(-10 + i20)/5 = i20/5 - \frac{10}{5}
7,845	x \cdot x^{m + (-1)} \cdot G = x \cdot x^{m + (-1)} \cdot G = x^m \cdot G
16,064	x/s = \frac{x}{1} \cdot 1/s
12,054	-\cos(A + B) + \cos(A - B) = \sin(A)\cdot \sin(B)\cdot 2
25,116	a^2 + 2\cdot a\cdot b + b \cdot b = \left(b + a\right)^2
25,460	3^m - 3^{m + (-1)} = 3^{m + (-1)}\left(3 + (-1)\right) = 23^{m + (-1)}
25,350	-7/4 = 1/4 - 2
37,530	(\sqrt{21} \cdot \sqrt{17})^2 = 21 \cdot 17 = (-1) \cdot 6 = 5
18,429	5 \cdot k + 5 = (1 + k) \cdot 5
-12,319	2^2 \cdot 3 = 12
463	X^2 + (-1) = (X + (-1)) \cdot \left(X + 1\right)
2,923	0 = 3K - 1 rightarrow K = \dfrac{1}{3}
31,226	1 - (1 - p)\cdot (1 - c) = 1 - 1 - p - c + p\cdot c = p + c - c\cdot p
-27,746	\frac{\text{d}}{\text{d}x} (4 \cdot \tan{x}) = 4 \cdot \frac{\text{d}}{\text{d}x} \tan{x} = 4 \cdot \sec^2{x}
7,039	\cos(\pi/2 - a) = \sin a
5,449	s\cdot v_1 + s\cdot v_2 = (v_1 + v_2)\cdot s
-2,054	\frac{\pi}{4} + 7/12 \pi = 5/6 \pi
16,138	x^2/x^3 = 1/x
19,288	1 + 2 + 3 + ... + 21 = \dfrac{21\cdot 22}{2} = 231
-8,061	\frac{-2 + i \cdot 5}{-2 + 5 \cdot i} \cdot \frac{26 + 7 \cdot i}{-2 - 5 \cdot i} = \frac{26 + i \cdot 7}{-2 - 5 \cdot i}
-8,302	-4 = -\dfrac{1}{3} \cdot 12
-30,576	70 (-1) + 40 x = (7(-1) + 4x)*10
22,793	xs + \left(-1\right) - x + s + 2(-1) = xs - x - s + 1 = (x + (-1)) \left(s + (-1)\right)
-23,129	-3/8 = \dfrac{3}{4} \cdot (-1/2)

5,634

\frac{1}{x^2} \times \left(i + \left(-1\right)\right) = 1/x \times ((-1) + i)/x

2,933

(X + 3)^3 = X^3 + 9X * X + 27 X + 27 = \left(X^2 + 5X + 7\right) \left(X + 4\right) + (-1)

-1,372

-\frac{1}{72} \cdot 18 = \frac{1}{72 \cdot 1/18} \cdot ((-18) \cdot \dfrac{1}{18}) = -\frac{1}{4}

-29,514

\frac{1}{(4\cdot (-1) + 5)!}\cdot 5! = 120

7,608

t^2\cdot p^2 = (p\cdot t)^2

9,883

1 = 1^{1/2} = ((-1)^2)^{1/2} = (\left(-1\right)^{1/2})^2 = i^2 = -1

22,817

4 + 4 + 3\times \left(-1\right) = 5

9,294

1 + q^4 - 2\cdot q \cdot q = (1 - q \cdot q) \cdot (1 - q \cdot q)

16,785

-(c + b) + d = d - b - c

15,985

2^l \cdot (2^{m - l} + (-1)) = -2^l + 2^m

-7,107

\frac{5}{66} = 5/11 \cdot \frac{1}{12} \cdot 2

-10,019

96\% = 96/100 = 0.96

5,869

-B\cdot C + C\cdot A = (-B + A)\cdot C

-20,021

\frac{1}{4*r + 4*(-1)}*(-9*r + 9) = \frac{(-1) + r}{(-1) + r}*(-9/4)

2,299

\frac{\sqrt{4 - 4*z + z^2}}{z + 2*(-1)} = \frac{1}{z + 2*(-1)}*\sqrt{(z + 2*\left(-1\right))^2} = \frac{|z + 2*\left(-1\right)|}{z + 2*(-1)}

-10,513

-\frac{24}{15 + r \cdot 6} = 3/3 \cdot (-\dfrac{1}{2 \cdot r + 5} \cdot 8)

-6,342

\frac{1}{(z + 6\cdot (-1))\cdot \left(z + 7\cdot \left(-1\right)\right)}\cdot 4 = \tfrac{4}{z^2 - z\cdot 13 + 42}

-18,979

\frac{2}{9} = \dfrac{A_s}{36\times \pi}\times 36\times \pi = A_s

-19,998

\frac91\cdot \frac{-9\cdot l + 3}{-l\cdot 9 + 3} = \frac{1}{3 - l\cdot 9}\cdot (-l\cdot 81 + 27)

-23,275

24\% = -0.01\cdot 76 + 100\%

-9,127

-20 \cdot z + 20 \cdot z^2 = z \cdot z \cdot 2 \cdot 2 \cdot 5 - z \cdot 2 \cdot 2 \cdot 5

19,260

\frac{84}{49 \cdot (-1) + 1} = -7/4

13,315

\frac{1}{4 \times 3} = 1/12

8,663

\frac{1}{2^{10}}*116 = \frac{1}{256}*29 \approx 0.11328125

-22,339

a^2 - 4a - 5 = (a - 5)(a + 1)

-7,744

\left(4 + i\cdot 6\right)/\left(-2\right) = \frac{6}{-2}\cdot i + \frac{4}{-2}

-18,646

-\frac{1}{11}\cdot 28 = -28/11

-20,026

\frac{1}{48 \cdot (-1) + 48 \cdot p} \cdot (56 \cdot p + 56 \cdot (-1)) = \dfrac{7}{6} \cdot \tfrac{8 \cdot p + 8 \cdot (-1)}{8 \cdot \left(-1\right) + p \cdot 8}

19,713

\sin\left(0\right) \cdot \cos(0) = 0

7,488

\sum_{n=1}^\infty \dfrac{n}{n \cdot n} = \sum_{n=1}^\infty \frac{1}{n}

15,127

exp(j + z) = exp(j) exp(z)

-2,514

(1 + 4) \cdot \sqrt{10} = \sqrt{10} \cdot 5

-15,213

\frac{1}{a^4 \cdot k^4} \cdot a^{15} = \frac{1}{k^4 \cdot a^4} \cdot a^5 \cdot (a^5)^2

34,277

b_2\cdot b_1\cdot b_3 = b_1\cdot b_3\cdot b_2

3,997

-Y^2 + A^2 = (A - Y) \cdot (A + Y)

-19,392

\frac{4}{3} \cdot \frac56 = \frac{1/3 \cdot 4}{6 \cdot \frac{1}{5}}

-2,882

-2\cdot \sqrt{6} + 3\cdot \sqrt{6} = -\sqrt{6}\cdot \sqrt{4} + \sqrt{6}\cdot \sqrt{9}

29,125

\cos^2{\psi_1} + \cos^2{\psi_2} + \cos^2{\psi_3} = 3/2 = \sin^2{\psi_1} + \sin^2{\psi_2} + \sin^2{\psi_3}

3,574

\left|{F \cdot A + x}\right| = \left|{x + A \cdot F}\right|

-19,276

\frac{9}{5} 1/2 = 1/\left(5*2/9\right)

-16,003

-6/10*6 + 9*\frac{1}{10}*4 = 0

19,817

\left(-1\right)^{b\cdot d} = (-1)^{d\cdot b}

32,229

\frac{1}{2} (\cos(y\cdot 2) + 1) = \cos^2(y)

-25,012

\frac{1}{25}\cdot 33 = 1.32

1,030

36 \cdot (-1) + z^2 = (6 + z) \cdot (z + 6 \cdot (-1))

-2,414

\sqrt{6} \cdot (2 + 3) = \sqrt{6} \cdot 5

-10,250

-\dfrac{1}{100} \times 43 = -0.43

15,880

x_e*x_d = x_d*x_e

7,729

8 = \frac{4!}{1^1\cdot 1!\cdot 3^1\cdot 1!}

21,028

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

8,897

n \lt 0 \Rightarrow |n| - n = |n|\times 2

16,816

1=n/n=1/n + ... + 1/n = (1/n)(1+...+1)\to 0

4,042

\sqrt{(1/u)^2 + 1}\cdot u = \sqrt{((\frac{1}{u})^2 + 1)\cdot u^2} = \sqrt{1 + u^2}

-5,460

\dfrac{1}{l^2\cdot 2 + 98\cdot \left(-1\right)}\cdot (l\cdot 6 + 42\cdot \left(-1\right) - l + 7\cdot (-1) + 4\cdot (-1)) = \dfrac{53\cdot (-1) + 5\cdot l}{2\cdot l^2 + 98\cdot (-1)}

17,876

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

10,287

1 = \frac{c}{c} \Rightarrow c = 1/(\frac{1}{c})

27,986

2 = \dfrac{1}{3 + 2*(-1)}*2

-3,318

3^{1 / 2} + 9^{\frac{1}{2}} \cdot 3^{\dfrac{1}{2}} = 3^{\dfrac{1}{2}} + 3^{1 / 2} \cdot 3

13,737

9y^2 + 36 (-1) = 3y \cdot y - 6^2 = (3y + 6) (3y + 6\left(-1\right)) = 3(y + 2) (3y + 6(-1)) = 9(y + 2) (y + 2(-1))

19,039

\sin{x} = \sin{2 \cdot x} = \dots = \sin{100 \cdot x}

21,807

d_2^2\cdot d_1\cdot d_2^2 = \frac{1}{d_2\cdot d_1\cdot d_2} = d_1\cdot d_2\cdot d_1

6,859

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

-9,252

-3\cdot 2\cdot 2 - t\cdot 2\cdot 2\cdot 5 = -20\cdot t + 12\cdot (-1)

12,460

z^2*3 + 6*z + 1 = 2*(-1) + 3*(z + 1)^2

8,373

x + 2 = k\cdot x\cdot 2 + 1\Longrightarrow (2\cdot k + (-1))\cdot x = 1

44,148

\frac{\mathrm{d}x}{\mathrm{d}t} = \frac{\mathrm{d}x^1 + 1}{\mathrm{d}x^1 + 1}

-16,546

\sqrt{8} \cdot 9 = 9 \cdot \sqrt{4 \cdot 2}

17,263

-z^3 + x^3 = (x - z)\cdot (x^2 + x\cdot z + z^2)

22,723

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

27,481

1 + r^3 - r^2 + r^2 - r + r + (-1) = r^3

11,431

\vartheta x = \vartheta x

13,597

\dfrac{48}{{8 \choose 3}}1 = 6/7

4,033

(2^4)^2 + (2.2^{4.2})^6 + 2^{12} = (2^4)^2 + (2.2^{4.2})^6 + (2^6)^2 = (2^4 + 2^6)^2

30,106

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

25,750

k = \frac{1}{2}\cdot (k\cdot 2 + 0)

28,682

\lim_{x \to 0^+} \sin(\frac1x) = \lim_{x \to \infty} \sin\left(x\right)

-7,792

(-10 + i*20)/5 = i*20/5 - \frac{10}{5}

7,845

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

16,064

x/s = \frac{x}{1} \cdot 1/s

12,054

-\cos(A + B) + \cos(A - B) = \sin(A)\cdot \sin(B)\cdot 2

25,116

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

25,460

3^m - 3^{m + (-1)} = 3^{m + (-1)}*\left(3 + (-1)\right) = 2*3^{m + (-1)}

25,350

-7/4 = 1/4 - 2

37,530

(\sqrt{21} \cdot \sqrt{17})^2 = 21 \cdot 17 = (-1) \cdot 6 = 5

18,429

5 \cdot k + 5 = (1 + k) \cdot 5

-12,319

2^2 \cdot 3 = 12

463

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

2,923

0 = 3K - 1 rightarrow K = \dfrac{1}{3}

31,226

1 - (1 - p)\cdot (1 - c) = 1 - 1 - p - c + p\cdot c = p + c - c\cdot p

-27,746

\frac{\text{d}}{\text{d}x} (4 \cdot \tan{x}) = 4 \cdot \frac{\text{d}}{\text{d}x} \tan{x} = 4 \cdot \sec^2{x}

7,039

\cos(\pi/2 - a) = \sin a

5,449

s\cdot v_1 + s\cdot v_2 = (v_1 + v_2)\cdot s

-2,054

\frac{\pi}{4} + 7/12 \pi = 5/6 \pi

16,138

x^2/x^3 = 1/x

19,288

1 + 2 + 3 + ... + 21 = \dfrac{21\cdot 22}{2} = 231

-8,061

\frac{-2 + i \cdot 5}{-2 + 5 \cdot i} \cdot \frac{26 + 7 \cdot i}{-2 - 5 \cdot i} = \frac{26 + i \cdot 7}{-2 - 5 \cdot i}

-8,302

-4 = -\dfrac{1}{3} \cdot 12

-30,576

70 (-1) + 40 x = (7(-1) + 4x)*10

22,793

xs + \left(-1\right) - x + s + 2(-1) = xs - x - s + 1 = (x + (-1)) \left(s + (-1)\right)

-23,129

-3/8 = \dfrac{3}{4} \cdot (-1/2)