ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
-15,907	-7\cdot 4/10 + 10\cdot \tfrac{6}{10} = \frac{1}{10}\cdot 32
-4,456	(z + (-1)) \cdot (z + 3) = z^2 + z \cdot 2 + 3 \cdot (-1)
63	\|B^B\| = 1 \gt \|B\|
940	n + n + (-1) + n + 2\cdot (-1) = 3\cdot (-1) + 3\cdot n
11,747	\sin^4(z) + \cos^4(z) = (\sin^2(z) + \cos^2\left(z\right)) \cdot (\sin^2(z) + \cos^2\left(z\right)) - 2\sin^2(z) \cos^2\left(z\right) = 1 - \sin^{22}(z)/2
27,686	\left\|{D^2}\right\| = \left\|{D}\right\| \times \left\|{D}\right\| = 0 rightarrow \left\|{D}\right\| = 0
27,367	(\sqrt{c\cdot b})^2 = (\sqrt{c}\cdot \sqrt{b})^2
6,649	x \cdot x - x\cdot z\cdot 6 + 9\cdot z^2 = 16 \implies 16 = \left(-3\cdot z + x\right) \cdot \left(-3\cdot z + x\right)
14,161	5\cdot \left(-1\right) + x + 11 - x^2\cdot 2 + 3\cdot x = 6 - 2\cdot x^2 + 4\cdot x
-182	8\cdot 7\cdot 6\cdot 5 = \dfrac{8!}{\left(4\cdot (-1) + 8\right)!}
-5,149	\frac{48.0}{10^5} = \frac{1}{10^5}\cdot 48
-20,874	\dfrac{1}{t \cdot 5 + 30} \cdot (-t \cdot 6 + 36 \cdot (-1)) = -6/5 \cdot \frac{6 + t}{t + 6}
921	15/51 = \frac{1}{\left(-1\right) + 52}*((-1) + 16)
24,979	70 = 5 + 20 + 10 + 5 + 5\cdot 6
24,779	1 - \dfrac{1}{F^2}\cdot a = \frac{\partial}{\partial F} \left(F + \dfrac{a}{F}\right)
-19,496	8/3\cdot \frac27 = \frac{\dfrac13\cdot 8}{\dfrac{1}{2}\cdot 7}
-2,178	-\frac{2}{15} + \frac{9}{15} = \tfrac{7}{15}
28,103	\sin(z) = (e^{iz} - e^{-iz})/\left(2i\right)\cos(z) = \left(e^{iz} + e^{-iz}\right)/2
3,205	\dfrac{1}{\left(1 - V\right) V} = 1/V + \frac{1}{-V + 1}
38,781	0 + 2 \cdot 34 = 68
331	417 17 = 24^2 + 2 * 2 + 24^2
17,359	Z^{2k + 1} = Z^{2k}Z = \left(Z^2\right)^kZ = (\frac165)^kZ
13,458	{82 \choose 2} - {3 \choose 1} \cdot {31 \choose 2} = 1926
228	(b \cdot a/a)^n = \dfrac{b^n}{a} \cdot a
-27,624	-4 + 5\cdot \left(-1\right) + 4 + 5\cdot (-1) = -4 + 4 + 5\cdot (-1) + 5\cdot (-1) = 0 + 10\cdot (-1) = -10
19,386	z = x\cdot 3 + 1\Longrightarrow x = \frac13\cdot \left((-1) + z\right)
7,525	\sin((-x)^2) = \sin\left(x^2\right)
20,421	z^3 - y^3 = (z - y)\cdot (z \cdot z + y\cdot z + y^2)
7,560	\sin(\frac{4}{3} \cdot π) = \sin(-\tfrac{1}{3} \cdot π)
-5,234	10^40.98 = 0.9810^{(-5) (-1) - 1}
-9,772	-0.45 = -\frac{1}{10}4.5 = -9/20
14,456	7 \cdot 7 + 1^2 = 2\cdot 5^2
-464	13/12 \pi = -\pi*6 + \dfrac{1}{12}85 \pi
25,521	(x + 9\cdot (-1))\cdot \left(x + 9\right) = x^2 + 81\cdot (-1)
29,621	3 = \tfrac{1}{2}\cdot (3\cdot (-1) + 9)
26,467	77 = 23 \cdot \left(-1\right) + 10^2
26,481	x^2 - x\cdot 8 + 16 = (x + 4\cdot (-1)) \cdot (x + 4\cdot (-1))
8,690	0 = -11\frac{1}{5}(-b_2 + b_12) + b_3 - b_17 \implies 0 = b_35 - 57b_1 + 11*b_2
-16,820	{3} = ({3} \times 5t) + ({3} \times -7) = (15t) + (-21) = 15t - 21
-15,458	\dfrac{{x^{6}r^{-10}}}{{x^{10}r^{-20}}} = \dfrac{{x^{6}}}{{x^{10}}} \cdot \dfrac{{r^{-10}}}{{r^{-20}}} = x^{{6} - {10}} \cdot r^{{-10} - {(-20)}} = x^{-4}r^{10}
3,316	ab = a^1b^1
16,273	\frac{1}{c^2} \cdot (1 - c \cdot 2) = y \cdot 2 + y^2 \Rightarrow \sqrt{-2 \cdot c + 1}/c = \sqrt{y^2 + y \cdot 2}
29,612	(4 + 4 + 4)\cdot c = 6\cdot 2 \implies c = 1
4,085	\frac{1}{ux} = \frac{1}{xu}
-20,378	\frac{1}{j\cdot \left(-6\right)}\cdot (-j\cdot 5 + 2\cdot (-1))\cdot 6/6 = \dfrac{1}{j\cdot (-36)}\cdot \left(12\cdot (-1) - j\cdot 30\right)
31,804	g \cdot f \cdot h = (h + i) \cdot 10^2 + f \cdot 10 + g - i = h \cdot 10^2 + f \cdot 10 + g + i \cdot 10^2 - i = h \cdot 10^2 + f \cdot 10 + g + 99 \cdot i
22,430	-f + g = -f + g
12,146	\dfrac{75}{100} \cdot (1 - \dfrac{1}{100} \cdot 90) = \frac{1}{100} \cdot 75 \cdot 10/100 = 3/40
7,601	\sec(v - w) = \frac{1}{\cos(v - w)} = \frac{1}{\cos(v) \cdot \cos(w) + \sin(v) \cdot \sin(w)}
49,075	(2 - 2^{1/2})^{1/2}(2 + 2^{1/2})^{1/2} = (2^2 + 2(-1))^{1/2} = 2^{1/2}
8,424	x\cdot x^{l + k} = x^{k + l + 1}
19,224	2 \cdot z^2 - z \cdot 4 = 2 \cdot (-1) + 2 \cdot ((-1) + z)^2
24,261	a + c \coloneqq c + a
13,717	z^h - z^c = 1 - z^c - 1 - z^h
-18,558	5r + 5 = 3\left(2r + 7\left(-1\right)\right) = 6r + 21(-1)
6,206	3 = -1/(\left(-1\right) \frac13)
-1,200	((-1)\cdot 6\cdot 1/5)/(\dfrac17 (-4)) = -\frac{7}{4} (-6/5)
43,556	2\times 3 = 3\times 2
14,987	\left(-1 = 1 rightarrow 1 \cdot 1 = (-1) \cdot (-1)\right) rightarrow 1 = 1
-16,351	(411)^{\frac{1}{2}}3 = 3*44^{\frac{1}{2}}
1,146	0 = -x + 1 \implies 1 = x
30,493	z_2 = \frac{\partial}{\partial t} z_1 = \frac{\partial}{\partial z_2} z_1 \cdot \frac{\partial}{\partial t} z_2
16,062	10^{2t0.05} = 10^{t*0.1}
4,421	(-1) + q^3 = (q^2 + q + 1) \cdot ((-1) + q)
-6,976	198 = 3116
362	1 + 3 + 5 + 7\cdot \cdots\cdot k = k^2
-17,080	5 = 5 \times 3 \times n + 5 \times (-3) = 15 \times n - 15 = 15 \times n + 15 \times (-1)
35,051	2^2 \cdot 2/32 = 2^{3 + 5 \cdot (-1)} = 1/4
18,674	A^\complement \cap F^\complement = A^\complement - F
14,581	\tfrac25\cdot 4/7 = \tfrac{8}{35}
9,664	\frac{1}{9(-1) + y^2}(-y3 + y^2) = 1 - \frac{y3 + 9(-1)}{9\left(-1\right) + y^2}
3,182	3 + 2(3k + \left(-1\right) + 2k^2 - k) = 4k^2 + 4k + 1 = (2k + 1)^2
1,310	\dfrac{1}{x + (-1)}(x^2 + (-1)) = \frac{1}{x + (-1)}(x + 1)*(x + (-1)) = x + 1
6,966	C - F\cdot C\cdot F = C\cdot F^2 - F\cdot C\cdot F = (C, F)
-26,169	13 - 0.75*12 + \frac{1}{2}8 = 13 + 9(-1) + 4 = 8
-3,924	\frac{r^4}{r^5}\tfrac{6}{22} = \frac{6r^4}{22r^5}1
5,532	\int \sin\left(z\cdot 2\right)\,dz = \int 1\cdot 2\cos(z) \sin\left(z\right)\,dz
21,050	1/\left(XX'\right) = \frac{1}{X'X}
26,837	\frac{1}{3 + y}\left(3 - y\right) = \frac{1}{6}(3 - y)*\dfrac{3 - y}{y + 3} + (3 - y)/6
36,908	C - 0.6667\cdot C = (1 - 0.6667)\cdot C = 0.3333\cdot C
172	(S + 2 (-1)) \left(1 + S\right) = S^2 - S + 2 (-1)
3,139	(y^{20} + y^{21} + y^{22} + \dotsm + y^{50})^3 = (y^{20})^3 \times (1 + y + y^2 + \dotsm + y^{30})^3
20,357	\dfrac{1}{\left(-1\right) + 3}(3^{k + 1} + (-1)) = 2 \cdot 3^k \implies 0 = -3^k + (-1)
26,115	(z + 2(-1)) \left(z + 3(-1)\right) = z^2 - z \cdot 5 + 6
2,898	(e^y + 1)^{1/y} = (e^y)^{\dfrac1y} (1 + e^{-y})^{1/y} = e*(1 + e^{-y})^{\frac1y}
19,393	(-\frac{1}{2} + 5) \cdot (-\dfrac{1}{5} + 3) \cdot (-\tfrac13 + 2) = 21
31,349	x^2 + 3 \cdot x + 10 \cdot (-1) = \left(x + 2 \cdot (-1)\right)^2 + 7 \cdot x + 14 \cdot (-1) = (x + 2 \cdot (-1))^2 + 7 \cdot x + 14 \cdot (-1) = (x + 2 \cdot (-1))^2 + 7 \cdot \left(x + 2 \cdot (-1)\right)
-7,393	\tfrac{6}{7}\cdot 4/9 = 8/21
4,418	22500 - 450 z + z^2 \cdot 4.25 = (-1.5 z + 150) \cdot (-1.5 z + 150) + z^2 \cdot 2
-20,659	-\frac45 \cdot \tfrac{6 \cdot k}{6 \cdot k} = \frac{(-24) \cdot k}{30 \cdot k}
14,132	a \cdot \eta \cdot T_l = T_l \cdot \eta \cdot a
16,031	\dfrac{1}{2}\cdot ((-1) + 7) = 3
-6,437	\frac{4}{y^2 + y5 + 50\left(-1\right)} = \tfrac{4}{\left(y + 5(-1)\right)\left(y + 10\right)}
-2,473	\sqrt{25} \times \sqrt{3} - \sqrt{3} \times \sqrt{16} = 5 \times \sqrt{3} - \sqrt{3} \times 4
6,043	(-a' + h)\cdot (3 + h + a') = -(a'^2 + 3\cdot a' + 1) + h^2 + h\cdot 3 + 1
2,465	h^x = 1 + \ln(h) x + \tfrac{\ln(h)^2 x^2}{2}1 + \frac{1}{6}x^3 \ln(h)^3 \ldots
-3,764	\frac{11 d^2}{9} = \tfrac{11}{9} d^2
38,503	4\times\dfrac{3!}{2!} = 12
29,553	x!^2 (1 + x) = (1 + x)! x!
-9,973	0.01*(-4) = -4/100 = -0.04

-15,907

-7\cdot 4/10 + 10\cdot \tfrac{6}{10} = \frac{1}{10}\cdot 32

-4,456

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

63

|B^B| = 1 \gt |B|

940

n + n + (-1) + n + 2\cdot (-1) = 3\cdot (-1) + 3\cdot n

11,747

\sin^4(z) + \cos^4(z) = (\sin^2(z) + \cos^2\left(z\right)) \cdot (\sin^2(z) + \cos^2\left(z\right)) - 2\sin^2(z) \cos^2\left(z\right) = 1 - \sin^{22}(z)/2

27,686

\left|{D^2}\right| = \left|{D}\right| \times \left|{D}\right| = 0 rightarrow \left|{D}\right| = 0

27,367

(\sqrt{c\cdot b})^2 = (\sqrt{c}\cdot \sqrt{b})^2

6,649

x \cdot x - x\cdot z\cdot 6 + 9\cdot z^2 = 16 \implies 16 = \left(-3\cdot z + x\right) \cdot \left(-3\cdot z + x\right)

14,161

5\cdot \left(-1\right) + x + 11 - x^2\cdot 2 + 3\cdot x = 6 - 2\cdot x^2 + 4\cdot x

-182

8\cdot 7\cdot 6\cdot 5 = \dfrac{8!}{\left(4\cdot (-1) + 8\right)!}

-5,149

\frac{48.0}{10^5} = \frac{1}{10^5}\cdot 48

-20,874

\dfrac{1}{t \cdot 5 + 30} \cdot (-t \cdot 6 + 36 \cdot (-1)) = -6/5 \cdot \frac{6 + t}{t + 6}

921

15/51 = \frac{1}{\left(-1\right) + 52}*((-1) + 16)

24,979

70 = 5 + 20 + 10 + 5 + 5\cdot 6

24,779

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

-19,496

8/3\cdot \frac27 = \frac{\dfrac13\cdot 8}{\dfrac{1}{2}\cdot 7}

-2,178

-\frac{2}{15} + \frac{9}{15} = \tfrac{7}{15}

28,103

\sin(z) = (e^{i*z} - e^{-i*z})/\left(2*i\right)*\cos(z) = \left(e^{i*z} + e^{-i*z}\right)/2

3,205

\dfrac{1}{\left(1 - V\right) V} = 1/V + \frac{1}{-V + 1}

38,781

0 + 2 \cdot 34 = 68

331

4*17 * 17 = 24^2 + 2 * 2 + 24^2

17,359

Z^{2*k + 1} = Z^{2*k}*Z = \left(Z^2\right)^k*Z = (\frac16*5)^k*Z

13,458

{82 \choose 2} - {3 \choose 1} \cdot {31 \choose 2} = 1926

228

(b \cdot a/a)^n = \dfrac{b^n}{a} \cdot a

-27,624

-4 + 5\cdot \left(-1\right) + 4 + 5\cdot (-1) = -4 + 4 + 5\cdot (-1) + 5\cdot (-1) = 0 + 10\cdot (-1) = -10

19,386

z = x\cdot 3 + 1\Longrightarrow x = \frac13\cdot \left((-1) + z\right)

7,525

\sin((-x)^2) = \sin\left(x^2\right)

20,421

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

7,560

\sin(\frac{4}{3} \cdot π) = \sin(-\tfrac{1}{3} \cdot π)

-5,234

10^4*0.98 = 0.98*10^{(-5) (-1) - 1}

-9,772

-0.45 = -\frac{1}{10}4.5 = -9/20

14,456

7 \cdot 7 + 1^2 = 2\cdot 5^2

-464

13/12 \pi = -\pi*6 + \dfrac{1}{12}85 \pi

25,521

(x + 9\cdot (-1))\cdot \left(x + 9\right) = x^2 + 81\cdot (-1)

29,621

3 = \tfrac{1}{2}\cdot (3\cdot (-1) + 9)

26,467

77 = 23 \cdot \left(-1\right) + 10^2

26,481

x^2 - x\cdot 8 + 16 = (x + 4\cdot (-1)) \cdot (x + 4\cdot (-1))

8,690

0 = -11*\frac{1}{5}*(-b_2 + b_1*2) + b_3 - b_1*7 \implies 0 = b_3*5 - 57*b_1 + 11*b_2

-16,820

{3} = ({3} \times 5t) + ({3} \times -7) = (15t) + (-21) = 15t - 21

-15,458

\dfrac{{x^{6}r^{-10}}}{{x^{10}r^{-20}}} = \dfrac{{x^{6}}}{{x^{10}}} \cdot \dfrac{{r^{-10}}}{{r^{-20}}} = x^{{6} - {10}} \cdot r^{{-10} - {(-20)}} = x^{-4}r^{10}

3,316

a*b = a^1*b^1

16,273

\frac{1}{c^2} \cdot (1 - c \cdot 2) = y \cdot 2 + y^2 \Rightarrow \sqrt{-2 \cdot c + 1}/c = \sqrt{y^2 + y \cdot 2}

29,612

(4 + 4 + 4)\cdot c = 6\cdot 2 \implies c = 1

4,085

\frac{1}{u*x} = \frac{1}{x*u}

-20,378

\frac{1}{j\cdot \left(-6\right)}\cdot (-j\cdot 5 + 2\cdot (-1))\cdot 6/6 = \dfrac{1}{j\cdot (-36)}\cdot \left(12\cdot (-1) - j\cdot 30\right)

31,804

g \cdot f \cdot h = (h + i) \cdot 10^2 + f \cdot 10 + g - i = h \cdot 10^2 + f \cdot 10 + g + i \cdot 10^2 - i = h \cdot 10^2 + f \cdot 10 + g + 99 \cdot i

22,430

-f + g = -f + g

12,146

\dfrac{75}{100} \cdot (1 - \dfrac{1}{100} \cdot 90) = \frac{1}{100} \cdot 75 \cdot 10/100 = 3/40

7,601

\sec(v - w) = \frac{1}{\cos(v - w)} = \frac{1}{\cos(v) \cdot \cos(w) + \sin(v) \cdot \sin(w)}

49,075

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

8,424

x\cdot x^{l + k} = x^{k + l + 1}

19,224

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

24,261

a + c \coloneqq c + a

13,717

z^h - z^c = 1 - z^c - 1 - z^h

-18,558

5*r + 5 = 3*\left(2*r + 7*\left(-1\right)\right) = 6*r + 21*(-1)

6,206

3 = -1/(\left(-1\right) \frac13)

-1,200

((-1)\cdot 6\cdot 1/5)/(\dfrac17 (-4)) = -\frac{7}{4} (-6/5)

43,556

2\times 3 = 3\times 2

14,987

\left(-1 = 1 rightarrow 1 \cdot 1 = (-1) \cdot (-1)\right) rightarrow 1 = 1

-16,351

(4*11)^{\frac{1}{2}}*3 = 3*44^{\frac{1}{2}}

1,146

0 = -x + 1 \implies 1 = x

30,493

z_2 = \frac{\partial}{\partial t} z_1 = \frac{\partial}{\partial z_2} z_1 \cdot \frac{\partial}{\partial t} z_2

16,062

10^{2*t*0.05} = 10^{t*0.1}

4,421

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

-6,976

198 = 3*11*6

362

1 + 3 + 5 + 7\cdot \cdots\cdot k = k^2

-17,080

5 = 5 \times 3 \times n + 5 \times (-3) = 15 \times n - 15 = 15 \times n + 15 \times (-1)

35,051

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

18,674

A^\complement \cap F^\complement = A^\complement - F

14,581

\tfrac25\cdot 4/7 = \tfrac{8}{35}

9,664

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

3,182

3 + 2(3k + \left(-1\right) + 2k^2 - k) = 4k^2 + 4k + 1 = (2k + 1)^2

1,310

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

6,966

C - F\cdot C\cdot F = C\cdot F^2 - F\cdot C\cdot F = (C, F)

-26,169

13 - 0.75*12 + \frac{1}{2}8 = 13 + 9(-1) + 4 = 8

-3,924

\frac{r^4}{r^5}*\tfrac{6}{22} = \frac{6*r^4}{22*r^5}*1

5,532

\int \sin\left(z\cdot 2\right)\,dz = \int 1\cdot 2\cos(z) \sin\left(z\right)\,dz

21,050

1/\left(X*X'\right) = \frac{1}{X'*X}

26,837

\frac{1}{3 + y}*\left(3 - y\right) = \frac{1}{6}*(3 - y)*\dfrac{3 - y}{y + 3} + (3 - y)/6

36,908

C - 0.6667\cdot C = (1 - 0.6667)\cdot C = 0.3333\cdot C

172

(S + 2 (-1)) \left(1 + S\right) = S^2 - S + 2 (-1)

3,139

(y^{20} + y^{21} + y^{22} + \dotsm + y^{50})^3 = (y^{20})^3 \times (1 + y + y^2 + \dotsm + y^{30})^3

20,357

\dfrac{1}{\left(-1\right) + 3}(3^{k + 1} + (-1)) = 2 \cdot 3^k \implies 0 = -3^k + (-1)

26,115

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

2,898

(e^y + 1)^{1/y} = (e^y)^{\dfrac1y} (1 + e^{-y})^{1/y} = e*(1 + e^{-y})^{\frac1y}

19,393

(-\frac{1}{2} + 5) \cdot (-\dfrac{1}{5} + 3) \cdot (-\tfrac13 + 2) = 21

31,349

x^2 + 3 \cdot x + 10 \cdot (-1) = \left(x + 2 \cdot (-1)\right)^2 + 7 \cdot x + 14 \cdot (-1) = (x + 2 \cdot (-1))^2 + 7 \cdot x + 14 \cdot (-1) = (x + 2 \cdot (-1))^2 + 7 \cdot \left(x + 2 \cdot (-1)\right)

-7,393

\tfrac{6}{7}\cdot 4/9 = 8/21

4,418

22500 - 450 z + z^2 \cdot 4.25 = (-1.5 z + 150) \cdot (-1.5 z + 150) + z^2 \cdot 2

-20,659

-\frac45 \cdot \tfrac{6 \cdot k}{6 \cdot k} = \frac{(-24) \cdot k}{30 \cdot k}

14,132

a \cdot \eta \cdot T_l = T_l \cdot \eta \cdot a

16,031

\dfrac{1}{2}\cdot ((-1) + 7) = 3

-6,437

\frac{4}{y^2 + y*5 + 50*\left(-1\right)} = \tfrac{4}{\left(y + 5*(-1)\right)*\left(y + 10\right)}

-2,473

\sqrt{25} \times \sqrt{3} - \sqrt{3} \times \sqrt{16} = 5 \times \sqrt{3} - \sqrt{3} \times 4

6,043

(-a' + h)\cdot (3 + h + a') = -(a'^2 + 3\cdot a' + 1) + h^2 + h\cdot 3 + 1

2,465

h^x = 1 + \ln(h) x + \tfrac{\ln(h)^2 x^2}{2}1 + \frac{1}{6}x^3 \ln(h)^3 \ldots

-3,764

\frac{11 d^2}{9} = \tfrac{11}{9} d^2

38,503

4\times\dfrac{3!}{2!} = 12

29,553

x!^2 (1 + x) = (1 + x)! x!

-9,973

0.01*(-4) = -4/100 = -0.04