ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
32,114	b \cdot a - a - b = -b + a \cdot (\left(-1\right) + b)
46,250	8 \frac{7!}{x! (7 - x!)} = \frac{1}{x! (8 + (-1) - x)!} 8! = \frac{8!}{x! \left(8 - x + 1\right)!}
8,710	1 + m^5 \cdot 6 + 15 \cdot m^4 + 20 \cdot m^3 + 15 \cdot m^2 + m \cdot 6 = -m^6 + (1 + m)^6
11,096	\|gNg^{-1}\|=\|N\|\implies gNg^{-1}=N
1,458	1 + n = 2^n - \left((-1) + n\right)\cdot 2^{2\cdot \left(-1\right) + n} + 2^{4\cdot (-1) + n}\cdot \frac{1}{2!}\cdot (n + 3\cdot (-1))\cdot \left(n + 2\cdot (-1)\right) - \dots
16,514	1 * 1 (1 + 1)^2/4 = \frac{4}{4} = 2
-7,503	\frac{1}{2} \cdot 15 = \frac16 \cdot 45
25,690	\mathbb{E}(h\cdot k) = \mathbb{E}(h)\cdot \mathbb{E}(k)
17,530	\frac{3}{2} \dfrac{1/2}{2}3 = \frac{3}{2}*\frac{3}{4} = \frac{1}{8}9
-1,869	-5/6\pi = -\pi11/12 + \pi/12
-9,272	-55x^2 = -511xx
27,274	\xi\cdot x\cdot b = x\cdot \xi\cdot b
23,289	208=256-48
29,766	\frac{\partial}{\partial f} \left(u \cdot x\right) = \frac{\partial}{\partial f} (u \cdot x)
16,187	-z + z*2 = z
-6,341	\dfrac{1}{30(-1) + z^2 - 7z}4 = \frac{4}{\left(z + 10(-1)\right)*\left(3 + z\right)}
-26,170	\frac64 + \frac{1}{4} \cdot 14 = 1.5 + 3.5 = 5
30,260	z\cdot x + i\cdot y = x\cdot z + i\cdot y + y\cdot z
-22,289	p^2 + 4\cdot p + 12\cdot (-1) = (2\cdot (-1) + p)\cdot (6 + p)
11,497	(d\cdot f)^{1 - n} = d\cdot f\cdot (d\cdot f)^{-n} = d\cdot f\cdot f^{-n}\cdot d^{-n} = d\cdot f^{1 - n}\cdot d^{-n}
5,075	\left(-1\right) + z/2 = \dfrac{100}{2} - z \implies z = 34
-22,307	C^2 + C3 + 2 = (1 + C)(C + 2)
19,729	-\cos\left(x\right) = \cos(x - \pi)
1,421	\dfrac{1}{(-1) \cdot x} = \frac{1}{(-1) \cdot x} + \tfrac{0}{x} = \frac{1}{(-1) \cdot x} + \frac1x \cdot (1 - 1) = \frac{1}{(-1) \cdot x} + 1/x - \frac{1}{x}
14,890	0 = \sin(x)\Longrightarrow 0 = x
26,074	5 \cdot 1109 + 4999 \left(-1\right) = 546
19,685	4k + k^2 - 2k + 1 = 1 + k^2 + k*2
22,911	2^{1 + n}\cdot (1 + n) = \left(n + 1\right)\cdot 2\cdot 2^n
-6,137	\frac{1}{3(x + 9)}4 = \frac{4}{27 + 3*x}
11,771	-1/(\sqrt{3}) = \tan(5\cdot \pi/6)
43,245	80 \cdot 11 + 1 = 881
19,626	\|1/x - \frac{1}{3}\| = \frac{1}{3 \cdot x} \cdot \|x + 3 \cdot (-1)\| \lt \dfrac16 \cdot \|x + 3 \cdot (-1)\|
1,431	-\frac{1}{n + 1} + \tfrac{1}{(-1) + n} = \dfrac{2}{(-1) + n^2}
34,889	3\left(-1\right) + 36 = 33
8,223	\cos^\sin{z}{z} = (\cos^2{z})^{\tfrac{1}{2} \cdot \sin{z}} = \left(1 - \sin^2{z}\right)^{\sin{z/2}}
980	\left(\frac1a\cdot a\cdot x\right)^2 = \dfrac{x}{a}\cdot \frac{a\cdot x}{a}\cdot a
-29,004	-35.5 = \dfrac{1}{2}(-27 - 44)
2,764	1 = 15x + 4\gamma\Longrightarrow x = -1,\gamma = 4
29,454	\tan\left(z + 2 \cdot \pi\right) = \tan(z)
34,249	\sin{5/2} - \left(\frac{5}{5 + 1}\right)^{1/2} = -0.314 \dots < 0
31,933	(-1) + 2 + 2 + 2 + (-1) + (-1) = 3
5,703	\bar{z}\cdot (T^2 - \lambda^2) = \bar{z}\cdot (T + \lambda)\cdot (-\lambda + T)
-9,145	t237t = 42*t^2
-18,420	\frac{x + x^2}{x^2 - 6x + 7(-1)} = \frac{x*\left(x + 1\right)}{(x + 1) (x + 7(-1))}
6,250	\mathbb{E}(D_2D_1) = \mathbb{E}(D_2)\mathbb{E}(D_1)
25,243	π = π\times 2/2
-12,152	\dfrac15 = \frac{s}{8 \cdot \pi} \cdot 8 \cdot \pi = s
6,590	\frac{1}{2} (-1 + 9) = 4
16,595	\sin(x + π*2) = \sin{x}
30,199	B = X^2 \Rightarrow B^{\dfrac{1}{2}} = X
8,802	a \cdot p \cdot x + d \cdot x \cdot q = x \cdot (d \cdot q + a \cdot p)
-25,073	\sec^2{4x}\tan{4x}8 = d/dx \sec^2{4*x}
41,584	4\cdot 17 + 3\cdot (-1) - 2\cdot 5 = 68 + 3\cdot \left(-1\right) + 10\cdot (-1) = 55
30,446	1 = x \cdot e^{x \cdot 6} \Rightarrow x \approx 0.2387
11,938	n \cdot 4 + 4 = 4 \cdot (1 + n)
-20,803	\dfrac11 10*5/5 = \frac{1}{5} 50
-16,424	28^{1 / 2}8 = (47)^{1 / 2}*8
-679	e^{12i\pi/12} = (e^{\pi*i/12})^{12}
11,325	1/\left(\dfrac56\cdot 6\right) = \dfrac15
51,535	\cos{x} = \cos(\dfrac12\times x + \frac12\times x) = \cos^2{\frac{x}{2}} - \sin^2{\frac{x}{2}} = 1 - 2\times \sin^2{\frac{x}{2}}
23,483	r \cdot r^r = r^{r + 1}
-1,817	\tfrac{\pi}{12} + \pi = \pi \frac{13}{12}
-18,617	5x + 5 = 7 \cdot (3x + 9) = 21 x + 63
27,892	a\cdot z + d = \dfrac{1}{z} rightarrow z^2\cdot a + d\cdot z + (-1) = 0
-20,648	-\frac{48}{24 - t \cdot 60} = 6/6 \cdot (-\frac{1}{4 - 10 \cdot t} \cdot 8)
-549	(e^{\frac{1}{12}\cdot 19\cdot π\cdot i})^6 = e^{6\cdot π\cdot i\cdot 19/12}
7,151	(1 + x + \dots + x^5)^8 = (\frac{1 - x^6}{1 - x})^8 = \frac{1}{\left(1 - x\right)^8}\cdot (1 - x^6)^8
3,021	\sin\left(x + z\right) = \cos(z)\cdot \sin(x) + \sin(z)\cdot \cos(x)
25,371	\pi20 = \pi8 + 8\pi + \pi4
35,409	\binom{F}{-k + F} = \binom{F}{k}
-4,464	x^2 - x + 20(-1) = (x + 5\left(-1\right))*(x + 4)
-20,080	-\frac{54}{54 \cdot (-1) + 6 \cdot l} = -\frac{1}{l + 9 \cdot (-1)} \cdot 9 \cdot \frac{6}{6}
27,160	1/52 - \dfrac{1}{52} \cdot 1/51 = \frac{1}{52} \cdot 50/51
2,880	x \cdot 2 + (-1) = (-\dfrac12 + x) \cdot 2
17,473	2^x + (-1) + 2^x = 2 \cdot 2^x + (-1) = 2^{x + 1} + \left(-1\right)
-22,347	(p + 1) \cdot (p + 10 \cdot (-1)) = 10 \cdot \left(-1\right) + p^2 - p \cdot 9
-20,969	\dfrac{1}{z9 + 18 (-1)} (90 \left(-1\right) + 9 z) = \frac{10 (-1) + z}{2 (-1) + z}9/9
25,094	b < -2 \Rightarrow 2 \lt -b
53,102	\frac{\text{d}y}{\text{d}x} = (y + x) \cdot (y + x) \Rightarrow 1 + \frac{\text{d}y}{\text{d}x} = \frac{\partial}{\partial x} (y + x) = 1 + (y + x)^2
-23,404	\frac{4\cdot 1/5}{6} = \frac{2}{15}
9,156	xf = x x * xf^3 = (fx) * \left(fx\right)^2 = fx
-2,772	48^{1 / 2} + 12^{1 / 2} + 3^{1 / 2} = (43)^{1 / 2} + 3^{1 / 2} + (163)^{\dfrac{1}{2}}
-7,956	\frac{i + 8}{-2 + i} = \dfrac{1}{i - 2}\cdot (i + 8)\cdot \dfrac{1}{-2 - i}\cdot \left(-2 - i\right)
-10,475	-\frac{21}{2} = -\frac12 \cdot 21
14,658	kr_0\pi2/\left(2\pir_1\right) = kr_0/r_1
27,177	\dfrac{1}{(2 \cdot l + 2) \cdot (1 + 2 \cdot l)} = \frac{1}{(2 + l \cdot 2)!} \cdot (2 \cdot l)!
-493	\dfrac{\pi}{2} = \frac{17}{2}\pi - 8\pi
18,741	\frac{1}{5 + V^2} \times (20 + V \times V \times 2) = 2 + \frac{10}{V^2 + 5}
6,103	px + pz = p \cdot (z + x)
35,414	5 - \frac17 = \frac{1}{7}\times ((-1) + 35)
-30,803	4\times x^2 + 24 = 4\times (x^2 + 6)
8,636	b \cdot C \cdot \frac{l}{b \cdot C} \cdot C = l \cdot C = b \cdot \frac{l}{b} \cdot C
42,433	\\|g + 0*(-1)\\| = \\|g\\|
7,402	A^{2n} = 0 \Rightarrow 0 = A^n
-15,700	\frac{\dfrac1a \frac{1}{z^4}}{a^3 z^3} = \dfrac{1/a \frac{1}{z^4}}{z^3 a^3}1
18,857	\frac{50}{17 \cdot 16} = \frac{1}{16} \cdot 2 + \tfrac{1}{17}
-1,378	\frac45 \cdot \frac13 \cdot 2 = \dfrac{4 \cdot 1/5}{3 \cdot \dfrac{1}{2}}
-9,351	-y \cdot y\cdot 121 = -y\cdot 11\cdot 11 y
31,549	z^3 + z^2 - 10z + 8 = (4 + z)(z + \left(-1\right))(z + 2(-1))
20,314	9 \cdot x^4 \cdot 2 \cdot x + 6 \cdot x^5 + x \cdot x \cdot 5 \cdot 4 \cdot x^3 = 44 \cdot x^5

32,114

b \cdot a - a - b = -b + a \cdot (\left(-1\right) + b)

46,250

8 \frac{7!}{x! (7 - x!)} = \frac{1}{x! (8 + (-1) - x)!} 8! = \frac{8!}{x! \left(8 - x + 1\right)!}

8,710

1 + m^5 \cdot 6 + 15 \cdot m^4 + 20 \cdot m^3 + 15 \cdot m^2 + m \cdot 6 = -m^6 + (1 + m)^6

11,096

|gNg^{-1}|=|N|\implies gNg^{-1}=N

1,458

1 + n = 2^n - \left((-1) + n\right)\cdot 2^{2\cdot \left(-1\right) + n} + 2^{4\cdot (-1) + n}\cdot \frac{1}{2!}\cdot (n + 3\cdot (-1))\cdot \left(n + 2\cdot (-1)\right) - \dots

16,514

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

-7,503

\frac{1}{2} \cdot 15 = \frac16 \cdot 45

25,690

\mathbb{E}(h\cdot k) = \mathbb{E}(h)\cdot \mathbb{E}(k)

17,530

\frac{3}{2} \dfrac{1/2}{2}3 = \frac{3}{2}*\frac{3}{4} = \frac{1}{8}9

-1,869

-5/6*\pi = -\pi*11/12 + \pi/12

-9,272

-55*x^2 = -5*11*x*x

27,274

\xi\cdot x\cdot b = x\cdot \xi\cdot b

23,289

208=256-48

29,766

\frac{\partial}{\partial f} \left(u \cdot x\right) = \frac{\partial}{\partial f} (u \cdot x)

16,187

-z + z*2 = z

-6,341

\dfrac{1}{30*(-1) + z^2 - 7*z}*4 = \frac{4}{\left(z + 10*(-1)\right)*\left(3 + z\right)}

-26,170

\frac64 + \frac{1}{4} \cdot 14 = 1.5 + 3.5 = 5

30,260

z\cdot x + i\cdot y = x\cdot z + i\cdot y + y\cdot z

-22,289

p^2 + 4\cdot p + 12\cdot (-1) = (2\cdot (-1) + p)\cdot (6 + p)

11,497

(d\cdot f)^{1 - n} = d\cdot f\cdot (d\cdot f)^{-n} = d\cdot f\cdot f^{-n}\cdot d^{-n} = d\cdot f^{1 - n}\cdot d^{-n}

5,075

\left(-1\right) + z/2 = \dfrac{100}{2} - z \implies z = 34

-22,307

C^2 + C*3 + 2 = (1 + C)*(C + 2)

19,729

-\cos\left(x\right) = \cos(x - \pi)

1,421

\dfrac{1}{(-1) \cdot x} = \frac{1}{(-1) \cdot x} + \tfrac{0}{x} = \frac{1}{(-1) \cdot x} + \frac1x \cdot (1 - 1) = \frac{1}{(-1) \cdot x} + 1/x - \frac{1}{x}

14,890

0 = \sin(x)\Longrightarrow 0 = x

26,074

5 \cdot 1109 + 4999 \left(-1\right) = 546

19,685

4*k + k^2 - 2*k + 1 = 1 + k^2 + k*2

22,911

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

-6,137

\frac{1}{3*(x + 9)}*4 = \frac{4}{27 + 3*x}

11,771

-1/(\sqrt{3}) = \tan(5\cdot \pi/6)

43,245

80 \cdot 11 + 1 = 881

19,626

|1/x - \frac{1}{3}| = \frac{1}{3 \cdot x} \cdot |x + 3 \cdot (-1)| \lt \dfrac16 \cdot |x + 3 \cdot (-1)|

1,431

-\frac{1}{n + 1} + \tfrac{1}{(-1) + n} = \dfrac{2}{(-1) + n^2}

34,889

3\left(-1\right) + 36 = 33

8,223

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

980

\left(\frac1a\cdot a\cdot x\right)^2 = \dfrac{x}{a}\cdot \frac{a\cdot x}{a}\cdot a

-29,004

-35.5 = \dfrac{1}{2}(-27 - 44)

2,764

1 = 15*x + 4*\gamma\Longrightarrow x = -1,\gamma = 4

29,454

\tan\left(z + 2 \cdot \pi\right) = \tan(z)

34,249

\sin{5/2} - \left(\frac{5}{5 + 1}\right)^{1/2} = -0.314 \dots < 0

31,933

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

5,703

\bar{z}\cdot (T^2 - \lambda^2) = \bar{z}\cdot (T + \lambda)\cdot (-\lambda + T)

-9,145

t*2*3*7*t = 42*t^2

-18,420

\frac{x + x^2}{x^2 - 6x + 7(-1)} = \frac{x*\left(x + 1\right)}{(x + 1) (x + 7(-1))}

6,250

\mathbb{E}(D_2*D_1) = \mathbb{E}(D_2)*\mathbb{E}(D_1)

25,243

π = π\times 2/2

-12,152

\dfrac15 = \frac{s}{8 \cdot \pi} \cdot 8 \cdot \pi = s

6,590

\frac{1}{2} (-1 + 9) = 4

16,595

\sin(x + π*2) = \sin{x}

30,199

B = X^2 \Rightarrow B^{\dfrac{1}{2}} = X

8,802

a \cdot p \cdot x + d \cdot x \cdot q = x \cdot (d \cdot q + a \cdot p)

-25,073

\sec^2{4*x}*\tan{4*x}*8 = d/dx \sec^2{4*x}

41,584

4\cdot 17 + 3\cdot (-1) - 2\cdot 5 = 68 + 3\cdot \left(-1\right) + 10\cdot (-1) = 55

30,446

1 = x \cdot e^{x \cdot 6} \Rightarrow x \approx 0.2387

11,938

n \cdot 4 + 4 = 4 \cdot (1 + n)

-20,803

\dfrac11 10*5/5 = \frac{1}{5} 50

-16,424

28^{1 / 2}*8 = (4*7)^{1 / 2}*8

-679

e^{12*i*\pi/12} = (e^{\pi*i/12})^{12}

11,325

1/\left(\dfrac56\cdot 6\right) = \dfrac15

51,535

\cos{x} = \cos(\dfrac12\times x + \frac12\times x) = \cos^2{\frac{x}{2}} - \sin^2{\frac{x}{2}} = 1 - 2\times \sin^2{\frac{x}{2}}

23,483

r \cdot r^r = r^{r + 1}

-1,817

\tfrac{\pi}{12} + \pi = \pi \frac{13}{12}

-18,617

5x + 5 = 7 \cdot (3x + 9) = 21 x + 63

27,892

a\cdot z + d = \dfrac{1}{z} rightarrow z^2\cdot a + d\cdot z + (-1) = 0

-20,648

-\frac{48}{24 - t \cdot 60} = 6/6 \cdot (-\frac{1}{4 - 10 \cdot t} \cdot 8)

-549

(e^{\frac{1}{12}\cdot 19\cdot π\cdot i})^6 = e^{6\cdot π\cdot i\cdot 19/12}

7,151

(1 + x + \dots + x^5)^8 = (\frac{1 - x^6}{1 - x})^8 = \frac{1}{\left(1 - x\right)^8}\cdot (1 - x^6)^8

3,021

\sin\left(x + z\right) = \cos(z)\cdot \sin(x) + \sin(z)\cdot \cos(x)

25,371

\pi*20 = \pi*8 + 8*\pi + \pi*4

35,409

\binom{F}{-k + F} = \binom{F}{k}

-4,464

x^2 - x + 20*(-1) = (x + 5*\left(-1\right))*(x + 4)

-20,080

-\frac{54}{54 \cdot (-1) + 6 \cdot l} = -\frac{1}{l + 9 \cdot (-1)} \cdot 9 \cdot \frac{6}{6}

27,160

1/52 - \dfrac{1}{52} \cdot 1/51 = \frac{1}{52} \cdot 50/51

2,880

x \cdot 2 + (-1) = (-\dfrac12 + x) \cdot 2

17,473

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

-22,347

(p + 1) \cdot (p + 10 \cdot (-1)) = 10 \cdot \left(-1\right) + p^2 - p \cdot 9

-20,969

\dfrac{1}{z*9 + 18 (-1)} (90 \left(-1\right) + 9 z) = \frac{10 (-1) + z}{2 (-1) + z}*9/9

25,094

b < -2 \Rightarrow 2 \lt -b

53,102

\frac{\text{d}y}{\text{d}x} = (y + x) \cdot (y + x) \Rightarrow 1 + \frac{\text{d}y}{\text{d}x} = \frac{\partial}{\partial x} (y + x) = 1 + (y + x)^2

-23,404

\frac{4\cdot 1/5}{6} = \frac{2}{15}

9,156

x*f = x * x * x*f^3 = (f*x) * \left(f*x\right)^2 = f*x

-2,772

48^{1 / 2} + 12^{1 / 2} + 3^{1 / 2} = (4*3)^{1 / 2} + 3^{1 / 2} + (16*3)^{\dfrac{1}{2}}

-7,956

\frac{i + 8}{-2 + i} = \dfrac{1}{i - 2}\cdot (i + 8)\cdot \dfrac{1}{-2 - i}\cdot \left(-2 - i\right)

-10,475

-\frac{21}{2} = -\frac12 \cdot 21

14,658

k*r_0*\pi*2/\left(2*\pi*r_1\right) = k*r_0/r_1

27,177

\dfrac{1}{(2 \cdot l + 2) \cdot (1 + 2 \cdot l)} = \frac{1}{(2 + l \cdot 2)!} \cdot (2 \cdot l)!

-493

\dfrac{\pi}{2} = \frac{17}{2}*\pi - 8*\pi

18,741

\frac{1}{5 + V^2} \times (20 + V \times V \times 2) = 2 + \frac{10}{V^2 + 5}

6,103

px + pz = p \cdot (z + x)

35,414

5 - \frac17 = \frac{1}{7}\times ((-1) + 35)

-30,803

4\times x^2 + 24 = 4\times (x^2 + 6)

8,636

b \cdot C \cdot \frac{l}{b \cdot C} \cdot C = l \cdot C = b \cdot \frac{l}{b} \cdot C

42,433

\|g + 0*(-1)\| = \|g\|

7,402

A^{2n} = 0 \Rightarrow 0 = A^n

-15,700

\frac{\dfrac1a \frac{1}{z^4}}{a^3 z^3} = \dfrac{1/a \frac{1}{z^4}}{z^3 a^3}1

18,857

\frac{50}{17 \cdot 16} = \frac{1}{16} \cdot 2 + \tfrac{1}{17}

-1,378

\frac45 \cdot \frac13 \cdot 2 = \dfrac{4 \cdot 1/5}{3 \cdot \dfrac{1}{2}}

-9,351

-y \cdot y\cdot 121 = -y\cdot 11\cdot 11 y

31,549

z^3 + z^2 - 10*z + 8 = (4 + z)*(z + \left(-1\right))*(z + 2*(-1))

20,314

9 \cdot x^4 \cdot 2 \cdot x + 6 \cdot x^5 + x \cdot x \cdot 5 \cdot 4 \cdot x^3 = 44 \cdot x^5