id
int64 -30,985
55.9k
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stringlengths 5
437k
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|---|---|
28,585 |
\mathbb{E}\left((A - \mu)^2\right) = (\mathbb{E}\left(A\right) - \mu)^2 + \mathbb{Var}\left(A\right)
|
22,216 |
-0.618 = \frac12 (1 - \sqrt{5}) \approx -0.618
|
9,789 |
(1 + x)\cdot (x^2 + 1)\cdot (1 + x^4)\cdot \dotsm\cdot (x^{2^n} + 1) = \frac{1}{-x + 1}\cdot (-x^{2^{n + 1}} + 1)
|
21,927 |
l^2 = b_{l + 1} - b_l = \frac{1}{l + 1 - l} \cdot (b_{l + 1} - b_l)
|
16,267 |
2 (-1) + x = -(2 - x)
|
17,192 |
x\beta n = nx \beta
|
-20,436 |
\frac{1}{x + 10 \left(-1\right)} \left(x + 10 (-1)\right)/9 = \frac{1}{90 \left(-1\right) + x*9}(x + 10 \left(-1\right))
|
29,374 |
(1 + n)^2 = 1 + n^2 + 2*n
|
32,367 |
0 \cdot z = (0 + 0) \cdot z = 0 \cdot z + 0 \cdot z
|
32,027 |
y r x = r y x
|
16,943 |
2/10\cdot \frac{3}{10} = 6/100
|
-5,933 |
\frac{5}{j \cdot 5 + 45 \cdot \left(-1\right)} = \frac{5}{5 \cdot (j + 9 \cdot (-1))}
|
12,677 |
y^2 + 8\cdot y + 14 - q = 0 rightarrow y = (-8 \pm (10 + 4\cdot q)^{1/2})/2
|
11,040 |
-h^t + f^t = \left(-h + f\right) \cdot \left(f^{t + (-1)} + f^{2 \cdot (-1) + t} \cdot h + h^2 \cdot f^{3 \cdot (-1) + t} + \dots + f \cdot h^{t + 2 \cdot \left(-1\right)} + h^{t + \left(-1\right)}\right)
|
37,633 |
12 = \frac{18}{3} \cdot (5 + 4 \cdot \left(-1\right) + 1)
|
-26,575 |
\left(8 + x\right) (-x + 8) = 8 \cdot 8 - x^2
|
-2,690 |
7^{1/2} + 25^{1/2} \cdot 7^{1/2} = 7^{1/2} \cdot 5 + 7^{1/2}
|
32,987 |
\tfrac{45}{216} = \frac{1}{24}\cdot 5
|
20,697 |
\operatorname{acos}(-q) = \operatorname{acos}(\cos(\pi - \operatorname{acos}(q)))\Longrightarrow -\operatorname{acos}(q) + \pi = \operatorname{acos}(-q)
|
-10,262 |
-\frac{20}{15\cdot z + 5} = -\frac{1}{1 + z\cdot 3}\cdot 4\cdot 5/5
|
3,446 |
\dfrac52 \cdot \dfrac{1}{9} = 5/18
|
25,714 |
0 - z^3\cdot 2 + 9\cdot z^2 - z\cdot 12 + 5 = -2\cdot z^2 \cdot z + 9\cdot z^2 - z\cdot 12 + 5
|
28,282 |
\frac{\text{d}z}{\text{d}y} = \frac{1}{2 \cdot z \cdot y - y^2} \cdot (z^2 - 2 \cdot z \cdot y) = \frac{1}{2 \cdot y/z - (\frac{y}{z})^2} \cdot \left(1 - 2 \cdot y/z\right)
|
30,843 |
48 = 2! \cdot 2! \cdot 2! \cdot 3!
|
36,090 |
\dfrac{1}{8} = (\dfrac12)^3
|
31,164 |
22 = 8 \cdot (-1) + 30
|
-1,773 |
\frac{1}{3}\pi = -\pi \frac135 + 2\pi
|
7,249 |
17^5\cdot 13^4\cdot 3 \cdot 3\cdot 7 \cdot 7 = 17^5 (3\cdot 7\cdot 13^2)^2
|
-26,542 |
(10 - 3 \cdot x) \cdot \left(10 + 3 \cdot x\right) = -9 \cdot x^2 + 100
|
-15,771 |
-\frac{1}{10}\cdot 44 = -8\cdot 8/10 + 10\cdot \frac{1}{10}\cdot 2
|
-30,160 |
\frac{\mathrm{d}}{\mathrm{d}z} z^9 = 9\cdot z^{9 + (-1)} = 9\cdot z^8
|
1,409 |
\frac{1}{\sqrt{n} + \sqrt{n + 1}} = -\sqrt{n} + \sqrt{1 + n}
|
-4,733 |
\frac{1}{x^2 + 3*x + 2}*(5*(-1) - x*2) = -\frac{1}{x + 1}*3 + \frac{1}{x + 2}
|
4,806 |
t\cdot y^i = t\cdot y^i
|
32,515 |
9 = -8\cdot 2 + 25
|
11,192 |
6/11 \cdot 7/12 = \frac{1}{132} \cdot 42
|
-29,571 |
(7 + 5 \cdot z^2 + z)/z = 5 \cdot z^2/z + z/z + 7/z
|
-4,182 |
\frac{48\cdot z^5}{24\cdot z} = \frac{48}{24}\cdot z^5/z
|
12,880 |
1 + 6\xi + 1 = \xi*6 + 2
|
24,108 |
(11 + 3) \cdot (11 + 7 \cdot (-1)) = 56
|
-5,182 |
4.92\cdot 10 = \frac{49.2}{10}\cdot 1 = 4.92\cdot 10^0
|
15,482 |
k + k + k = 3\cdot k
|
6,015 |
y + y^2 = -\frac{1}{4} + (y + \dfrac{1}{2})^2
|
-713 |
e^{5 \cdot i \cdot \pi/3 \cdot 15} = (e^{\frac53 \cdot i \cdot \pi})^{15}
|
1,192 |
\sin(x) \sin(\alpha) + \cos(\alpha) \cos(x) = \cos(-x + \alpha)
|
3,978 |
\tfrac{2}{4 + 7 + 2} = \dfrac{2}{13}
|
21,617 |
-8 + x\cdot 4 + 2\cdot h = (2\cdot h^2 - 8\cdot h + 4\cdot x\cdot h)/h
|
18,674 |
E^c - D = D^c \cap E^c
|
12,180 |
-b^2\cdot 3 + 66\cdot b + 315\cdot (-1) = 0 \implies b^2 - 22\cdot b + 105 = \left(b + 15\cdot (-1)\right)\cdot (b + 7\cdot (-1)) = 0
|
-22,993 |
21/27 = \frac{7 \cdot 3}{3 \cdot 9}
|
3,006 |
-u^3 + x \cdot x \cdot x = \left(-u + x\right) (u \cdot u + x^2 + xu)
|
22,683 |
-b^2 + a^2 - ab + ba = \left(a + b\right) (-b + a)
|
-6,147 |
\dfrac{1}{2*i + 18*(-1)}*5 = \frac{5}{(i + 9*(-1))*2}
|
-9,926 |
0.01 (-28) = -28/100 = -\frac{1}{25}7
|
-13,128 |
-22.5/\left(-0.5\right) = 45
|
3,415 |
64 S^4 r r + 4 = (r^2 S^4\cdot 16 + 1)\cdot 4
|
17,337 |
b = db = bd
|
20,805 |
π/6 = \arccos(\sqrt{3}/2)
|
33,820 |
\frac{3 - x}{(x + 3*(-1))^2} = -\frac{1}{(x + 3*(-1))^2}*(x + 3*\left(-1\right)) = -\frac{1}{x + 3*(-1)}
|
34,616 |
1 + \cos{x} = 1 + \cos{2 x/2} = 2 \cos^2{x/2}
|
28,720 |
2^{1/2} \approx 1 + \frac12 - 1/8 + \dfrac{1}{16} = 23/16 \approx 1.4375
|
21,342 |
\sin(y\cdot 2) = \cos(y) \sin\left(y\right)\cdot 2
|
26,899 |
\cos\left(x \cdot 2\right) = -2 \cdot \sin^2(x) + 1 \implies \sin(x) = \sqrt{\frac12 \cdot (1 - \cos\left(2 \cdot x\right))}
|
-29,374 |
(x - g) \cdot (x + g) = -g \cdot g + x^2
|
11,972 |
545140134^2*12^2 = (12^3 + 640320^3)*163
|
-20,336 |
2/2 \cdot \tfrac{1}{-x \cdot 7 + 5 \cdot (-1)} \cdot 2 = \frac{1}{10 \cdot \left(-1\right) - 14 \cdot x} \cdot 4
|
1,922 |
7 = 6 \cdot (-1) + 18 + 5 \cdot \left(-1\right)
|
34,072 |
512^2 = 64^2 64
|
14,850 |
(1 + 2\cdot (-1))^2 = (-1) \cdot (-1) = 1 \leq 2\cdot \left(1 + (-2)^2\right) = 2\cdot \left(1 + 4\right) = 10
|
4,912 |
-x_2 + x_1 = x_0 - x_1 \implies x_2\cdot ... = -x_1\cdot 2 + x_0
|
33,256 |
73 = 145 + 72 \times (-1)
|
1,632 |
\sin(\alpha) \cdot \cos(J) - \cos(\alpha) \cdot \sin\left(J\right) = \sin\left(-J + \alpha\right)
|
717 |
1/z=\bar{z}=z^n\implies z^{n+1}=1
|
-9,413 |
2*2*7*x = 28*x
|
3,974 |
x := x_1 x_2 \cdots x_n := x_1 x_2 \cdots x_n
|
47,991 |
2 + 8 (-1) = -6 = 3 (-2)
|
14,747 |
\dfrac{117.7}{1 + 0.07} \frac{1}{1 + 0.1} = 100
|
1,961 |
\frac{1}{-y + 1} = \frac{1 + 0\cdot (-1)}{-y + 1}
|
9,038 |
\frac{1}{x + 2\left(-1\right)}\left(4 + 2x^3 - 10 x\right) = 2(-1) + 2x^2 + 4x
|
-20,472 |
\frac{q\cdot 9 + 9(-1)}{q\cdot 10}\cdot 6/6 = \frac{1}{60 q}\left(54 q + 54 (-1)\right)
|
9,484 |
5 + \left(1 + n\right)^2 = n \cdot n + 2 \cdot n + 6
|
48,701 |
x^2 + x + 1 = (x^2 + x + \tfrac 1 4) + \frac 3 4 = \left(x + \frac 1 2 \right)^2 + \frac 3 4
|
41,398 |
\cos{\frac{\pi}{4}} = \sin{\frac{1}{4}\cdot \pi} = 1
|
-12,190 |
1/2 = \frac{q}{16 \pi} \cdot 16 \pi = q
|
2,804 |
t^3-3t^2+t+1=(t-1)(t^2-2t-1)
|
-7,145 |
3/13\cdot \frac{5}{12} = 5/52
|
14,115 |
\frac{100}{6} = \frac{1}{x}\cdot 40 rightarrow 2.4 = x
|
-11,621 |
i\cdot 4 + 0 + 20 (-1) = i\cdot 4 - 20
|
6,649 |
16 = y^2 \cdot 9 + z^2 - 6yz \Rightarrow (z - 3y) \cdot (z - 3y) = 16
|
15,999 |
z^4 - z^2 + 1 = (1 + z * z - \sqrt{3}*z)*(z * z + z*\sqrt{3} + 1)
|
29,059 |
cb = b = bc
|
-29,802 |
d/dx (5 + x^2*2 - x*6) = 6\left(-1\right) + x*4
|
-9,675 |
\dfrac{2}{2} = 1
|
16,160 |
c = a/b \Rightarrow a = b \times c = 0 \times c = 0
|
15,823 |
\sin{\frac{1}{5}\cdot \pi/2} = \sin{\pi/10}
|
12,958 |
d + x + c = x + c + d = c + d + x
|
-27,702 |
\frac{\text{d}}{\text{d}y} \cos(y) = -\sin\left(y\right)
|
15,520 |
\frac16(1 + 2 + 3 + 4 + 5 + 6) = \frac{6}{2*6}7 = \tfrac{7}{2}
|
10,616 |
x - b = (\sqrt{x} - \sqrt{b})\cdot (\sqrt{x} + \sqrt{b}) \geq (\sqrt{x} - \sqrt{b}) \cdot (\sqrt{x} - \sqrt{b})
|
-22,102 |
\dfrac{1}{20}30 = \dfrac123
|
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