ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
17,700	(-g + z)^2 = (z - g) (z - g)
4,450	(-1) + 6\cdot n^2 + n = \left(1 + n\cdot 2\right)\cdot (n\cdot 3 + (-1))
22,967	l + 1 = -(l + (-1)) + l\cdot 2
30,133	\frac{\binom{68}{8}}{\binom{80}{20}} = 51/24391353490
-2,662	(5 + 3\cdot (-1))\cdot 7^{\frac{1}{2}} = 2\cdot 7^{1 / 2}
13,844	n\cdot 2 + 1 = 11 + 5\cdot \left(-1\right) + 3\cdot (-1) + n\cdot 2 + 2\cdot (-1)
37,784	2l + (-1) + 2 = 2l + 1 = 2*(l + 1) + (-1)
-15,809	\frac{1}{10} \cdot 28 = -5 \cdot \frac{1}{10} \cdot 4 + 8 \cdot \frac{6}{10}
13,522	z_1\cdot z_2 = 0\Longrightarrow 0 = z_1\text{ or }z_2 = 0
11,919	\left(\left(2 (-1) + z*2 = z/2 \Rightarrow z = 4 (-1) + 4 z\right) \Rightarrow 4 = 3 z\right) \Rightarrow 4/3 = z
-4,931	10^5\cdot 29.4 = 29.4\cdot 10^{0 + 5}
-25,510	\frac{d}{dp} (\dfrac{4}{2 + p}) = -\frac{1}{(2 + p)^2}\cdot 4
10,038	\frac{1}{1!\cdot 2^1}(-1)^1 = -1/2
29,597	(x + 1) \times (x + 1) = x^2 + 2\times x + 1 = x \times x + 1
-536	\frac12 \cdot 3 \cdot \pi = -16 \cdot \pi + 35/2 \cdot \pi
-2,147	-5/12 \pi = 7/6 \pi - 19/12 \pi
12,256	\frac{\partial}{\partial y} e^{i \cdot l \cdot y} = e^{y \cdot l \cdot i} \cdot l \cdot i
13,892	(-1) + m^4 = ((-1) + m)(1 + m^2)(m + 1)
-19,420	\dfrac32\cdot 6/1 = \frac{1}{\frac16}\cdot \frac{3}{2}
13,576	V^{(-1) + i} = \frac{V^i}{V}
36,555	0 + \frac13 + \frac12 = 5/6
-19,517	\frac65\cdot \frac19\cdot 2 = 6\cdot 2/\left(5\cdot 9\right) = \dfrac{12}{45}
-6,191	\frac{p + 5\cdot (-1)}{p + 5\cdot (-1)}\cdot \frac{1}{2\cdot (p + 5\cdot (-1))}\cdot 2 = \frac{2\cdot (p + 5\cdot (-1))}{(5\cdot \left(-1\right) + p)\cdot \left(p + 5\cdot (-1)\right)\cdot 2}
-4,646	(x + 5) \cdot (4 + x) = x^2 + 9 \cdot x + 20
-22,198	((-1) + a) \left(a + 3(-1)\right) = a^2 - 4a + 3
12,991	2^2 + 4 \cdot 4 \cdot 2 = 4 \cdot 3^2
2,936	\frac12g g - \frac12*a^2 = (g^2 - a^2)/2
17,546	49/100 = \frac{7}{10}\frac{1}{10}7
16,298	\left( 5\cdot x + 5\cdot z, 5\cdot x + 5\cdot z\right) + \left( 2\cdot x - 2\cdot z, 2\cdot z - 2\cdot x\right) = ( 7\cdot x + 3\cdot z, 7\cdot z + 3\cdot x) = \left( 7\cdot x + 3\cdot z, 3\cdot x + 7\cdot z\right)
10,114	\mathbb{Var}\left[X + Y\right] = \mathbb{Var}\left[X\right] + \mathbb{Var}\left[Y\right] + 2\mathbb{Cov}\left[X, Y\right] = \mathbb{Var}\left[X\right] + \mathbb{Var}\left[Y\right]
14,875	{100 + 10 (-1) + 21 (-1) + 3 \choose 3} = {72 \choose 3}
-16,803	-3z = -3z3z + -3z8 = -9z^2 - 24z = -9z^2 - 24z
-22,324	(y + 8\left(-1\right)) (y + 9) = 72 (-1) + y^2 + y
4,363	1 + \cos(y) = 1 + 2\cos^2(\frac{y}{2}) + (-1) = 2\cos^2(y/2)
-24,448	\frac{1}{8 + 7}\cdot 135 = 135/15 = 135/15 = 9
-6,691	\frac{50}{100} + \frac{1}{100} = \dfrac{1}{100} + 5/10
13,096	((-1) + x^4) \cdot (x^4 + 1) = x^8 + (-1)
12,013	(g + b)\cdot (g + b) = g + g\cdot b + b\cdot g + b = g + b
15,644	x + 3 - 10 + x^2 - 7x = -x^2 + 8x + 7*(-1)
25,998	2 + (z + (-1))^4 + 3\cdot (z + (-1))^2 = 5 + (z + \left(-1\right))^4 + z^2\cdot 3 - z\cdot 6
28,270	1575 = \frac{1}{4 \cdot 4!^2} \cdot 10!
16,819	5\cdot 23 = (-5) (-23)
16,443	B + E = E + B
32,120	4^2 + 3^2*27 = 259
-5,325	1.2 \cdot 10^2 = 10^{2 + 0 \cdot (-1)} \cdot 1.2
-24,352	\frac{3}{1 + 2} = \frac{3}{3} = \frac{3}{3} = 1
350	3 + (3!)!/\left(3!\cdot 3!\right) = 23
45,200	1 \times 2 + 6 = 8 = 176 + 168 \times (-1)
1,450	2014 \cdot \pi/12 - 84 \cdot 2 \cdot \pi = (\pi \cdot (-1))/6
-27,592	4\cdot \frac19/4 = 1/9
5,795	n\cdot 2^{n + (-1)} = (\sum_{i=0}^n {n \choose i})\cdot i = (\sum_{i=1}^n {n \choose i})\cdot i
29,040	\frac{1 - z^{q + 1}}{-z + 1} = 1 + z + z z + z^3 + \dotsm + z^q
14,773	3/10\cdot 7/8\cdot \frac29 = 7/120
4,544	x_1mSh + Smx_2 = S(hx_1m + x_2*m)
36,733	\frac{1}{l^x}\cdot l^k = l^{k - x}
35,605	(b + b)b + b\left(b + b\right) = (b + b)*(b + b)
27,095	\left\|{A \cdot B \cdot C + x}\right\| = \left\|{B \cdot C \cdot A + x}\right\|
34,487	X^5 = I \Rightarrow 0 = X^5 - I
40,492	e^{iz} = e^{iz}
15,161	\frac{x^3 + (-1)}{\left(-1\right) + x} = x \cdot x + x + 1
32,479	0 = e^{j \cdot 2} + 1 + j \cdot 2 \Rightarrow e^{j \cdot 2} = (-1) - j \cdot 2
4,680	(-1) + \left(z + 1\right) \cdot \left(z + 1\right) = (-1) + z^2 + z \cdot 2 + 1
38,258	1 - \frac{1}{(m + 1)!} + \frac{m + 1}{\left(m + 2\right)!} = 1 - \dfrac{m + 2}{\left(m + 2\right)!} + \tfrac{1}{(m + 2)!}*(m + 1) = 1 - \frac{1}{(m + 2)!}
-23,263	-\dfrac25 + 1 = 3/5
9,042	9 = -4 \times p \Rightarrow p = -\frac{9}{4}
14,090	\sin(2 u) = \sin(u) \cos\left(u\right)*2
36,667	3^m - x_{m + \left(-1\right)} = 3^m - (3^m + \left(-1\right))/2 = (3^m + 1)/2 = x_{m + (-1)} + 1
16,253	w*(-x) = -wx
2,638	\sqrt{99^2 + 16\cdot n} = \sqrt{9801 + 16\cdot n}
4,564	r < s \implies s - r > 0
15,523	x^2 + x^2 - 2\cdot x\cdot y + y \cdot y - x \cdot x - y\cdot x = x^2 + y^2 - y\cdot x
-30,897	3\cdot d + 8\cdot (-1) = 8\cdot (-1) + 3\cdot d
905	b^2 + 0 \cdot b^1 + 0 \cdot b^0 = 4 \Rightarrow 2 = b
-10,791	28 = \frac{1}{6} 168
1,876	\frac{x}{5} + \frac{1}{4} = \frac{1}{20}(4x + 5)
32,313	\binom{n}{1} + (-1) = n + \left(-1\right)
18,920	\left(e/2\right)^m = e^m \cdot (\frac{1}{2})^m
-2,051	5/6\pi + \frac{5}{12}\pi = \frac{5}{4}*\pi
1,994	4(5y)^2 = y^2\cdot 100
-182	\frac{1}{(4 (-1) + 8)!} 8! = 876*5
24,875	z = 3\Longrightarrow z \in ]3,4]
21,105	\|y\| \lt 2 \Rightarrow 1 > \|y/2\|
-25,836	y^24 + 2y + 1 = \dfrac{1}{y + 4(-1)}(4y^3 - 14 y^2 - y7 + 4(-1))
-26,373	-32 = (-2)\cdot (-2)\cdot (-2)\cdot \left(-2\right)\cdot (-2)
18,030	3 \cdot (x + 1) = ((1 + x) \cdot 3 + 2.5) \cdot x\Longrightarrow x = \frac{1}{3} \cdot 2
19,216	\mathbb{E}\left(\cos\left(y + z\right)\right) = -\mathbb{E}\left(\sin{y} \times \sin{z}\right) + \mathbb{E}\left(\cos{z} \times \cos{y}\right)
-596	92/3\pi - 30\pi = \pi\frac132
18,271	h\cdot (x - B) = -B\cdot h + x\cdot h
9,299	u = u - P(u) + P(u)
-25,828	x^22 - x + 3 + \frac{2}{6(-1) + x} = \dfrac{1}{x + 6(-1)}\left(16(-1) + 2x^3 - 13x^2 + x9\right)
20,128	\left(a^4 + 3 \cdot a^3 - 3 \cdot a + 1 = a^4 + 3 \cdot a + 2 \cdot (-1) \implies a^3 - 2 \cdot a + 1 = 0\right) \implies (a + (-1)) \cdot (a^2 + a + (-1)) = 0
1,590	g \cdot d \cdot Y \Rightarrow Y \cdot g \cdot d
22,425	(10\cdot \left(-1\right)\cdot \dfrac{1}{6})/3 = -5/9
-16,500	\sqrt{4 \cdot 3} \cdot 3 = \sqrt{12} \cdot 3
-1,113	-35/35 = ((-35)1/35)/(35\frac{1}{35}) = -1
13,956	y_0a_1 + y_1 = y_1 + a_1y_0
10,459	x^2 + x = (x+1/2)^2 - 1/4
10,628	y + 1 = y^2 + y - y^2 + (-1)
8,382	49 = 3 + 3\cdot 11 + 13
20,466	\dfrac{1}{\frac{1}{1/(1/25)}} = 5^{-2\cdot (-(-1)\cdot (-1))} = 5 \cdot 5 = 25

17,700

(-g + z)^2 = (z - g) (z - g)

4,450

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

22,967

l + 1 = -(l + (-1)) + l\cdot 2

30,133

\frac{\binom{68}{8}}{\binom{80}{20}} = 51/24391353490

-2,662

(5 + 3\cdot (-1))\cdot 7^{\frac{1}{2}} = 2\cdot 7^{1 / 2}

13,844

n\cdot 2 + 1 = 11 + 5\cdot \left(-1\right) + 3\cdot (-1) + n\cdot 2 + 2\cdot (-1)

37,784

2*l + (-1) + 2 = 2*l + 1 = 2*(l + 1) + (-1)

-15,809

\frac{1}{10} \cdot 28 = -5 \cdot \frac{1}{10} \cdot 4 + 8 \cdot \frac{6}{10}

13,522

z_1\cdot z_2 = 0\Longrightarrow 0 = z_1\text{ or }z_2 = 0

11,919

\left(\left(2 (-1) + z*2 = z/2 \Rightarrow z = 4 (-1) + 4 z\right) \Rightarrow 4 = 3 z\right) \Rightarrow 4/3 = z

-4,931

10^5\cdot 29.4 = 29.4\cdot 10^{0 + 5}

-25,510

\frac{d}{dp} (\dfrac{4}{2 + p}) = -\frac{1}{(2 + p)^2}\cdot 4

10,038

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

29,597

(x + 1) \times (x + 1) = x^2 + 2\times x + 1 = x \times x + 1

-536

\frac12 \cdot 3 \cdot \pi = -16 \cdot \pi + 35/2 \cdot \pi

-2,147

-5/12 \pi = 7/6 \pi - 19/12 \pi

12,256

\frac{\partial}{\partial y} e^{i \cdot l \cdot y} = e^{y \cdot l \cdot i} \cdot l \cdot i

13,892

(-1) + m^4 = ((-1) + m)*(1 + m^2)*(m + 1)

-19,420

\dfrac32\cdot 6/1 = \frac{1}{\frac16}\cdot \frac{3}{2}

13,576

V^{(-1) + i} = \frac{V^i}{V}

36,555

0 + \frac13 + \frac12 = 5/6

-19,517

\frac65\cdot \frac19\cdot 2 = 6\cdot 2/\left(5\cdot 9\right) = \dfrac{12}{45}

-6,191

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

-4,646

(x + 5) \cdot (4 + x) = x^2 + 9 \cdot x + 20

-22,198

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

12,991

2^2 + 4 \cdot 4 \cdot 2 = 4 \cdot 3^2

2,936

\frac12*g * g - \frac12*a^2 = (g^2 - a^2)/2

17,546

49/100 = \frac{7}{10}*\frac{1}{10}*7

16,298

\left( 5\cdot x + 5\cdot z, 5\cdot x + 5\cdot z\right) + \left( 2\cdot x - 2\cdot z, 2\cdot z - 2\cdot x\right) = ( 7\cdot x + 3\cdot z, 7\cdot z + 3\cdot x) = \left( 7\cdot x + 3\cdot z, 3\cdot x + 7\cdot z\right)

10,114

\mathbb{Var}\left[X + Y\right] = \mathbb{Var}\left[X\right] + \mathbb{Var}\left[Y\right] + 2\mathbb{Cov}\left[X, Y\right] = \mathbb{Var}\left[X\right] + \mathbb{Var}\left[Y\right]

14,875

{100 + 10 (-1) + 21 (-1) + 3 \choose 3} = {72 \choose 3}

-16,803

-3*z = -3*z*3*z + -3*z*8 = -9*z^2 - 24*z = -9*z^2 - 24*z

-22,324

(y + 8\left(-1\right)) (y + 9) = 72 (-1) + y^2 + y

4,363

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

-24,448

\frac{1}{8 + 7}\cdot 135 = 135/15 = 135/15 = 9

-6,691

\frac{50}{100} + \frac{1}{100} = \dfrac{1}{100} + 5/10

13,096

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

12,013

(g + b)\cdot (g + b) = g + g\cdot b + b\cdot g + b = g + b

15,644

x + 3 - 10 + x^2 - 7*x = -x^2 + 8*x + 7*(-1)

25,998

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

28,270

1575 = \frac{1}{4 \cdot 4!^2} \cdot 10!

16,819

5\cdot 23 = (-5) (-23)

16,443

B + E = E + B

32,120

4^2 + 3^2*27 = 259

-5,325

1.2 \cdot 10^2 = 10^{2 + 0 \cdot (-1)} \cdot 1.2

-24,352

\frac{3}{1 + 2} = \frac{3}{3} = \frac{3}{3} = 1

350

3 + (3!)!/\left(3!\cdot 3!\right) = 23

45,200

1 \times 2 + 6 = 8 = 176 + 168 \times (-1)

1,450

2014 \cdot \pi/12 - 84 \cdot 2 \cdot \pi = (\pi \cdot (-1))/6

-27,592

4\cdot \frac19/4 = 1/9

5,795

n\cdot 2^{n + (-1)} = (\sum_{i=0}^n {n \choose i})\cdot i = (\sum_{i=1}^n {n \choose i})\cdot i

29,040

\frac{1 - z^{q + 1}}{-z + 1} = 1 + z + z z + z^3 + \dotsm + z^q

14,773

3/10\cdot 7/8\cdot \frac29 = 7/120

4,544

x_1*m*S*h + S*m*x_2 = S*(h*x_1*m + x_2*m)

36,733

\frac{1}{l^x}\cdot l^k = l^{k - x}

35,605

(b + b)*b + b*\left(b + b\right) = (b + b)*(b + b)

27,095

\left|{A \cdot B \cdot C + x}\right| = \left|{B \cdot C \cdot A + x}\right|

34,487

X^5 = I \Rightarrow 0 = X^5 - I

40,492

e^{iz} = e^{iz}

15,161

\frac{x^3 + (-1)}{\left(-1\right) + x} = x \cdot x + x + 1

32,479

0 = e^{j \cdot 2} + 1 + j \cdot 2 \Rightarrow e^{j \cdot 2} = (-1) - j \cdot 2

4,680

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

38,258

1 - \frac{1}{(m + 1)!} + \frac{m + 1}{\left(m + 2\right)!} = 1 - \dfrac{m + 2}{\left(m + 2\right)!} + \tfrac{1}{(m + 2)!}*(m + 1) = 1 - \frac{1}{(m + 2)!}

-23,263

-\dfrac25 + 1 = 3/5

9,042

9 = -4 \times p \Rightarrow p = -\frac{9}{4}

14,090

\sin(2 u) = \sin(u) \cos\left(u\right)*2

36,667

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

16,253

w*(-x) = -wx

2,638

\sqrt{99^2 + 16\cdot n} = \sqrt{9801 + 16\cdot n}

4,564

r < s \implies s - r > 0

15,523

x^2 + x^2 - 2\cdot x\cdot y + y \cdot y - x \cdot x - y\cdot x = x^2 + y^2 - y\cdot x

-30,897

3\cdot d + 8\cdot (-1) = 8\cdot (-1) + 3\cdot d

905

b^2 + 0 \cdot b^1 + 0 \cdot b^0 = 4 \Rightarrow 2 = b

-10,791

28 = \frac{1}{6} 168

1,876

\frac{x}{5} + \frac{1}{4} = \frac{1}{20}*(4*x + 5)

32,313

\binom{n}{1} + (-1) = n + \left(-1\right)

18,920

\left(e/2\right)^m = e^m \cdot (\frac{1}{2})^m

-2,051

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

1,994

4(5y)^2 = y^2\cdot 100

-182

\frac{1}{(4 (-1) + 8)!} 8! = 8*7*6*5

24,875

z = 3\Longrightarrow z \in ]3,4]

21,105

|y| \lt 2 \Rightarrow 1 > |y/2|

-25,836

y^2*4 + 2y + 1 = \dfrac{1}{y + 4(-1)}(4y^3 - 14 y^2 - y*7 + 4(-1))

-26,373

-32 = (-2)\cdot (-2)\cdot (-2)\cdot \left(-2\right)\cdot (-2)

18,030

3 \cdot (x + 1) = ((1 + x) \cdot 3 + 2.5) \cdot x\Longrightarrow x = \frac{1}{3} \cdot 2

19,216

\mathbb{E}\left(\cos\left(y + z\right)\right) = -\mathbb{E}\left(\sin{y} \times \sin{z}\right) + \mathbb{E}\left(\cos{z} \times \cos{y}\right)

-596

92/3*\pi - 30*\pi = \pi*\frac13*2

18,271

h\cdot (x - B) = -B\cdot h + x\cdot h

9,299

u = u - P(u) + P(u)

-25,828

x^2*2 - x + 3 + \frac{2}{6*(-1) + x} = \dfrac{1}{x + 6*(-1)}*\left(16*(-1) + 2*x^3 - 13*x^2 + x*9\right)

20,128

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

1,590

g \cdot d \cdot Y \Rightarrow Y \cdot g \cdot d

22,425

(10\cdot \left(-1\right)\cdot \dfrac{1}{6})/3 = -5/9

-16,500

\sqrt{4 \cdot 3} \cdot 3 = \sqrt{12} \cdot 3

-1,113

-35/35 = ((-35)*1/35)/(35*\frac{1}{35}) = -1

13,956

y_0*a_1 + y_1 = y_1 + a_1*y_0

10,459

x^2 + x = (x+1/2)^2 - 1/4

10,628

y + 1 = y^2 + y - y^2 + (-1)

8,382

49 = 3 + 3\cdot 11 + 13

20,466

\dfrac{1}{\frac{1}{1/(1/25)}} = 5^{-2\cdot (-(-1)\cdot (-1))} = 5 \cdot 5 = 25