ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
20,290	1.6^{(-1) + x} = 1.6 \cdot 1.6^{2 \cdot (-1) + x}
35,224	\sin{r \cdot 2} = \cos{r} \cdot \sin{r} \cdot 2
25,085	(a + b + c)^3 = a^3 + b^3 + c^3 + \left(c + a + b\right) \cdot (b \cdot a + b \cdot c + c \cdot a) rightarrow a^3 + b^3 + c^3 = 0
34,796	13570 = 430*\left(-1\right) + 14000
-6,442	\frac{5\cdot g}{g^2 - g\cdot 13 + 36} = \dfrac{5\cdot g}{(g + 4\cdot (-1))\cdot (g + 9\cdot \left(-1\right))}
-20,109	\frac{1}{6 \cdot (-1) + c} \cdot \left(6 \cdot \left(-1\right) + c\right) \cdot \frac37 = \frac{18 \cdot \left(-1\right) + c \cdot 3}{7 \cdot c + 42 \cdot (-1)}
-19,959	-\frac14*5 = -1.25
19,604	\left(x + i\right) \cdot w = i \cdot w + w \cdot x
22,032	-(\binom{s}{2} + s) + s^2 = (-s + s^2)/2
10,945	k\cdot b = k = b\cdot k
-23,019	\frac{3 \cdot 5}{10 \cdot 5} = \dfrac{15}{50}
23,671	1^2 \cdot 2 + 4^2 = 18
51,893	S = \sum_{k=1}^{10} \dfrac{1}{2^k}k = 2\sum_{k=1}^{10} \frac{1}{2^{k + 1}}k = 2\sum_{k=2}^{11} \frac{1}{2^k}(k + (-1)) = 2\sum_{k=1}^{10} \frac{1}{2^k}(k + (-1)) + \tfrac{10}{2^{10}} = 2S - \sum_{k=1}^{10} \dfrac{1}{2^{k + (-1)}} + \tfrac{10}{2^{10}}
33,563	\|w\| = \|y\| rightarrow w = y
39,097	(a - c)^2 = -(d - b)^2 rightarrow 0 = \left(a - c\right)^2 + \left(-b + d\right)^2
-19,016	11/12 = Y_p/(36\cdot \pi)\cdot 36\cdot \pi = Y_p
16,784	(Ix + I')^2 = Ix + I'
793	(z + 100 (-1))^2 + \left(y + 42 (-1)\right) * \left(y + 42 (-1)\right) = (z + 33 (-1))^2 + (y + 74 \left(-1\right))^2 = (z + 26)^2 + (y + 6(-1))^2
968	0.1944 = 60.3 0.3*0.6^2
-29,346	(3 \cdot y + 7) \cdot (3 \cdot y + 7 \cdot (-1)) = \left(3 \cdot y\right)^2 - 7^2 = 9 \cdot y^2 + 49 \cdot (-1)
-20,745	\dfrac19\cdot 9\cdot \frac{1}{-8}\cdot (-10\cdot x + 5) = \frac{1}{-72}\cdot (-x\cdot 90 + 45)
27,612	1 = -i\Longrightarrow 1^2 = (-i)^2 = -1
24,207	x_{2l + 1} = \dfrac{1}{1 + l*2} \implies 0 = \lim_{l \to \infty} x_{2l + 1}
43,214	1 + 1 + 1 + (-1) = 2
-20,964	\frac11\times 9\times \frac{1}{7 + k}\times (k + 7) = \frac{63 + 9\times k}{7 + k}
-4,612	\frac{1}{5(-1) + y \cdot y - y\cdot 4}(5y + 19 (-1)) = \dfrac{1}{5(-1) + y} + \frac{1}{1 + y}4
17,446	i^n = e^{i\cdot π\cdot n/2} = (-1)^{n/2}
37,577	{6 \choose 6} + {6 \choose 5} + {6 \choose 4} + {6 \choose 3} + 8\times (-1) = 34
35,513	3 = 2 + \cos{\pi24}
8,243	33 = 35 + 2 \left(-1\right)
-15,801	-\frac{6}{10} + 1 = \frac{1}{10}\cdot 4
12,307	2^22 + 3^2(-1) = 8 + 9*(-1) = -1
55,004	\lim_{n \to \infty} 2^n+1 = \lim_{n \to \infty} 2^n(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n\cdot \lim_{n\to \infty}(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n \cdot 1= \lim_{n \to \infty}2^n
-1,688	\dfrac{5}{12}\cdot \pi - \pi\cdot \frac{1}{4}\cdot 5 = -\tfrac{5}{6}\cdot \pi
6,862	\left(g + c\right) (g - c) = g^2 - c^2
-15,855	-\frac{2}{10}9 + 8/108 = 46/10
10,525	-\frac{1}{6}\pi + 2\pi = \frac{11}{6}\pi
-22,215	(n + 8)((-1) + n) = n^2 + 7n + 8*(-1)
13,053	x^{r + 1} = x^r\cdot x \gt x^r
22,676	\cos(y^3) + (-1) = 1 + \left(-1\right) - y^6/2 + \cdots = \frac{1}{2} \cdot ((-1) \cdot y^6) + \cdots
21,783	(1 + z^2 - z) (z^2 + z + 1) \left(z + 1\right) (1 - z) = -z^6 + 1
-18,336	\dfrac{k \cdot k - k \cdot 4 + 5 \cdot \left(-1\right)}{4 + k \cdot k + k \cdot 5} = \frac{1}{\left(k + 1\right) \cdot (k + 4)} \cdot (1 + k) \cdot (k + 5 \cdot (-1))
33,336	\frac{420 + t*41}{20 (-1) + t^2 - t} = \frac{625}{9 (t + 5 (-1))} - \frac{256}{9 (t + 4)}
36,471	x\times x + x\times y + y\times x + y\times y = x \times x + x\times y + y\times x + y^2 = x + x\times y + y\times x + y
14,848	16^2 + 2^2 = 8^2 + 14^2
-2,011	5/12 \times \pi - 23/12 \times \pi = -\pi \times 3/2
1,516	(z + 2)^{\frac{1}{2}} = \left(4 + z + 2\cdot (-1)\right)^{1/2} = 2\cdot (1 + (z + 2\cdot (-1))/4)^{1/2}
-20,611	\frac{-45 \cdot x + 27}{12 \cdot (-1) + x \cdot 20} = \frac{3 \cdot (-1) + 5 \cdot x}{3 \cdot (-1) + 5 \cdot x} \cdot (-\frac{1}{4} \cdot 9)
2,065	-\dfrac{1}{\left(1 - \frac1x\right)^2\cdot x^2} = -\frac{1}{x \cdot x\cdot \left(1 - \frac1x\right)^2} = -\frac{1}{(x + (-1)) \cdot (x + (-1))} = -\frac{1}{(x + (-1))^2}
16,761	\frac{\partial}{\partial z} (-r + z) = -r + \frac{dz}{dz}
6,397	\dfrac{1}{x_j} \cdot (I + J) = \frac{I}{x_j} + \frac{J}{x_j}
21,779	(1+\omega^2)^4=1+(\omega^4)^2=1+(\omega+1)^2=(1+\omega^2)+1
15,067	\sqrt{163} = \dfrac{12767}{\left(1 - \frac{1}{163 \cdot 10^6}3711\right)^{1/2}}\frac{1}{1000}
5,530	1 + x = \frac{1}{x + \left(-1\right)} \cdot ((-1) + x) \cdot (1 + x)
13,026	UL = LU
-10,415	2/2\left(-7/x\right) = -14/(2x)
-11,696	1/5 * 1/5 = 1/25
-22,036	\dfrac{4}{8}=\dfrac{1}{2}
9,084	E(U_i \cdot U_k) = \mathbb{Cov}(U_i,U_k) + E(U_i) \cdot E(U_k) = \mathbb{Cov}(U_i,U_k)
24,201	2 \times 2^n = 2^{n + 1}
7,766	\left(1 + 2\sqrt{3} = z \Rightarrow 12 = (z + \left(-1\right)) \cdot (z + \left(-1\right))\right) \Rightarrow z^2 - z \cdot 2 + 11 \left(-1\right) = 0
4,661	(1 + 2 + 3 + 4)/4 = \dfrac{1}{4} \cdot 10 = 5/2
14,686	( \|a\|, \|d\|) = 1 \implies 1 = \left(a, d\right)
-1,863	\frac{1}{12}\cdot 11\cdot \pi - \pi\cdot 0 = \pi\cdot 11/12
-20,119	\dfrac13\cdot 10\cdot \dfrac{1}{m + 2}\cdot (2 + m) = \frac{20 + m\cdot 10}{3\cdot m + 6}
4,830	b^2 \cdot k + \left(k + a\right) \cdot b + a = \left(b \cdot k + a\right) \cdot (b + 1)
7,075	317/(417) = 51/68
3,603	\sin\left(x\right) = \cos(-x + \frac{1}{2} \cdot π)
6,308	14 + 3 \cdot z^2 - 13 \cdot z = 0 \Rightarrow 0 = \left(7 \cdot (-1) + 3 \cdot z\right) \cdot (2 \cdot (-1) + z)
-9,358	-3\cdot 3 - 2\cdot 3\cdot 5\cdot t = 9\cdot (-1) - 30\cdot t
-20,466	-\frac149 \frac{10 + t}{t + 10} = \dfrac{1}{40 + 4t}(90 (-1) - t*9)
17,266	d \cdot b/f = b \cdot d/f
10,785	(n + 62)^2 - (34 + n) \cdot 113 = 2 + n^2 + n \cdot 11
9,460	\arccos(\cos(\pi - \arccos{t})) = \arccos{-t} \implies \pi - \arccos{t} = \arccos{-t}
9,658	(y + (-1)) (1 + y^6 + y^5 + y^4 + y^3 + y^2 + y) = y^7 + (-1)
20,220	x^2 - z z = (z + x) (x - z)
4,061	573 = 3\cdot 191
-28,971	14 + 12\cdot (-1) = 2
8,026	-\frac54 + (1/2 + q)^2 = q^2 + q + (-1)
20,062	\frac2x*1/2 = \frac{1}{x}
-4,183	\dfrac{x^4}{x^4} \cdot \tfrac{1}{8} \cdot 2 = \frac{2 \cdot x^4}{8 \cdot x^4} \cdot 1
7,578	13 + 2 \cdot 2 = 17
912	(\left(i\cdot 2 + 1\right)\cdot (-6))/(-2) = 3\cdot (2\cdot i + 1)
12,834	z^2 + (-1) = (z + (-1)) \cdot \left(1 + z\right)
-2,051	5/12\pi + \dfrac{5}{6}\pi = 5/4*\pi
28,534	p^n\cdot e^{x\cdot p} = \frac{1}{e^{-p\cdot x}}\cdot p^n
35,931	180 = 2^23^25
8,949	1 + \sqrt{-2} \cdot 2 = (-\sqrt{-2} + 1) \cdot \left(\sqrt{-2} - 1\right)
-20,019	8/8 \cdot \frac{1}{9 + 7 \cdot p} \cdot (10 \cdot \left(-1\right) + p) = \frac{1}{p \cdot 56 + 72} \cdot \left(p \cdot 8 + 80 \cdot (-1)\right)
24,628	(a + \nu) \cdot (\nu - a) = \nu \cdot \nu - a^2
-10,950	\frac{195}{13} = 15
2,357	640320^3 = \left(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29\right)^3
17,618	(x + 1)^{2 \cdot n} = (1 + x)^n \cdot \left(x + 1\right)^n
-7,605	\left(-14 + 5 i - 56 i + 20 \left(-1\right)\right)/17 = \frac{1}{17} \left(-34 - 51 i\right) = -2 - 3 i
-26,673	(x\cdot 2 + 5\cdot (-1))\cdot (1 + x\cdot 4) = x^2\cdot 8 - 18\cdot x + 5\cdot (-1)
-7,684	\frac{1}{25}(-60 + 30i - 80i + 40(-1)) = \frac{1}{25}(-100 - 50i) = -4 - 2*i
-20,558	\frac13 \cdot 3 \cdot \frac{1}{2 \cdot (-1) + j} \cdot \left(-3 \cdot j + 3 \cdot \left(-1\right)\right) = \frac{1}{6 \cdot (-1) + 3 \cdot j} \cdot (-j \cdot 9 + 9 \cdot \left(-1\right))
17,023	0 = (y^3 + (-1))*(y^6 + y^3 + 1) = y^9 + (-1)
2,440	\\|e\\|_1 = 0 \implies 0 = e
13,429	0 = x2^{1/2} + q5^{1/2} + r10^{1/2} + t = t + x2^{1/2} + (q + r2^{1/2})5^{1/2}

20,290

1.6^{(-1) + x} = 1.6 \cdot 1.6^{2 \cdot (-1) + x}

35,224

\sin{r \cdot 2} = \cos{r} \cdot \sin{r} \cdot 2

25,085

(a + b + c)^3 = a^3 + b^3 + c^3 + \left(c + a + b\right) \cdot (b \cdot a + b \cdot c + c \cdot a) rightarrow a^3 + b^3 + c^3 = 0

34,796

13570 = 430*\left(-1\right) + 14000

-6,442

\frac{5\cdot g}{g^2 - g\cdot 13 + 36} = \dfrac{5\cdot g}{(g + 4\cdot (-1))\cdot (g + 9\cdot \left(-1\right))}

-20,109

\frac{1}{6 \cdot (-1) + c} \cdot \left(6 \cdot \left(-1\right) + c\right) \cdot \frac37 = \frac{18 \cdot \left(-1\right) + c \cdot 3}{7 \cdot c + 42 \cdot (-1)}

-19,959

-\frac14*5 = -1.25

19,604

\left(x + i\right) \cdot w = i \cdot w + w \cdot x

22,032

-(\binom{s}{2} + s) + s^2 = (-s + s^2)/2

10,945

k\cdot b = k = b\cdot k

-23,019

\frac{3 \cdot 5}{10 \cdot 5} = \dfrac{15}{50}

23,671

1^2 \cdot 2 + 4^2 = 18

51,893

S = \sum_{k=1}^{10} \dfrac{1}{2^k}*k = 2*\sum_{k=1}^{10} \frac{1}{2^{k + 1}}*k = 2*\sum_{k=2}^{11} \frac{1}{2^k}*(k + (-1)) = 2*\sum_{k=1}^{10} \frac{1}{2^k}*(k + (-1)) + \tfrac{10}{2^{10}} = 2*S - \sum_{k=1}^{10} \dfrac{1}{2^{k + (-1)}} + \tfrac{10}{2^{10}}

33,563

|w| = |y| rightarrow w = y

39,097

(a - c)^2 = -(d - b)^2 rightarrow 0 = \left(a - c\right)^2 + \left(-b + d\right)^2

-19,016

11/12 = Y_p/(36\cdot \pi)\cdot 36\cdot \pi = Y_p

16,784

(Ix + I')^2 = Ix + I'

793

(z + 100 (-1))^2 + \left(y + 42 (-1)\right) * \left(y + 42 (-1)\right) = (z + 33 (-1))^2 + (y + 74 \left(-1\right))^2 = (z + 26)^2 + (y + 6(-1))^2

968

0.1944 = 6*0.3 * 0.3*0.6^2

-29,346

(3 \cdot y + 7) \cdot (3 \cdot y + 7 \cdot (-1)) = \left(3 \cdot y\right)^2 - 7^2 = 9 \cdot y^2 + 49 \cdot (-1)

-20,745

\dfrac19\cdot 9\cdot \frac{1}{-8}\cdot (-10\cdot x + 5) = \frac{1}{-72}\cdot (-x\cdot 90 + 45)

27,612

1 = -i\Longrightarrow 1^2 = (-i)^2 = -1

24,207

x_{2l + 1} = \dfrac{1}{1 + l*2} \implies 0 = \lim_{l \to \infty} x_{2l + 1}

43,214

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

-20,964

\frac11\times 9\times \frac{1}{7 + k}\times (k + 7) = \frac{63 + 9\times k}{7 + k}

-4,612

\frac{1}{5(-1) + y \cdot y - y\cdot 4}(5y + 19 (-1)) = \dfrac{1}{5(-1) + y} + \frac{1}{1 + y}4

17,446

i^n = e^{i\cdot π\cdot n/2} = (-1)^{n/2}

37,577

{6 \choose 6} + {6 \choose 5} + {6 \choose 4} + {6 \choose 3} + 8\times (-1) = 34

35,513

3 = 2 + \cos{\pi*2*4}

8,243

33 = 35 + 2 \left(-1\right)

-15,801

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

12,307

2^2*2 + 3^2*(-1) = 8 + 9*(-1) = -1

55,004

\lim_{n \to \infty} 2^n+1 = \lim_{n \to \infty} 2^n(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n\cdot \lim_{n\to \infty}(1+\frac{1}{2^n})=\lim_{n \to \infty} 2^n \cdot 1= \lim_{n \to \infty}2^n

-1,688

\dfrac{5}{12}\cdot \pi - \pi\cdot \frac{1}{4}\cdot 5 = -\tfrac{5}{6}\cdot \pi

6,862

\left(g + c\right) (g - c) = g^2 - c^2

-15,855

-\frac{2}{10}*9 + 8/10*8 = 46/10

10,525

-\frac{1}{6}\pi + 2\pi = \frac{11}{6}\pi

-22,215

(n + 8)*((-1) + n) = n^2 + 7*n + 8*(-1)

13,053

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

22,676

\cos(y^3) + (-1) = 1 + \left(-1\right) - y^6/2 + \cdots = \frac{1}{2} \cdot ((-1) \cdot y^6) + \cdots

21,783

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

-18,336

\dfrac{k \cdot k - k \cdot 4 + 5 \cdot \left(-1\right)}{4 + k \cdot k + k \cdot 5} = \frac{1}{\left(k + 1\right) \cdot (k + 4)} \cdot (1 + k) \cdot (k + 5 \cdot (-1))

33,336

\frac{420 + t*41}{20 (-1) + t^2 - t} = \frac{625}{9 (t + 5 (-1))} - \frac{256}{9 (t + 4)}

36,471

x\times x + x\times y + y\times x + y\times y = x \times x + x\times y + y\times x + y^2 = x + x\times y + y\times x + y

14,848

16^2 + 2^2 = 8^2 + 14^2

-2,011

5/12 \times \pi - 23/12 \times \pi = -\pi \times 3/2

1,516

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

-20,611

\frac{-45 \cdot x + 27}{12 \cdot (-1) + x \cdot 20} = \frac{3 \cdot (-1) + 5 \cdot x}{3 \cdot (-1) + 5 \cdot x} \cdot (-\frac{1}{4} \cdot 9)

2,065

-\dfrac{1}{\left(1 - \frac1x\right)^2\cdot x^2} = -\frac{1}{x \cdot x\cdot \left(1 - \frac1x\right)^2} = -\frac{1}{(x + (-1)) \cdot (x + (-1))} = -\frac{1}{(x + (-1))^2}

16,761

\frac{\partial}{\partial z} (-r + z) = -r + \frac{dz}{dz}

6,397

\dfrac{1}{x_j} \cdot (I + J) = \frac{I}{x_j} + \frac{J}{x_j}

21,779

(1+\omega^2)^4=1+(\omega^4)^2=1+(\omega+1)^2=(1+\omega^2)+1

15,067

\sqrt{163} = \dfrac{12767}{\left(1 - \frac{1}{163 \cdot 10^6}3711\right)^{1/2}}\frac{1}{1000}

5,530

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

13,026

U*L = L*U

-10,415

2/2*\left(-7/x\right) = -14/(2*x)

-11,696

1/5 * 1/5 = 1/25

-22,036

\dfrac{4}{8}=\dfrac{1}{2}

9,084

E(U_i \cdot U_k) = \mathbb{Cov}(U_i,U_k) + E(U_i) \cdot E(U_k) = \mathbb{Cov}(U_i,U_k)

24,201

2 \times 2^n = 2^{n + 1}

7,766

\left(1 + 2\sqrt{3} = z \Rightarrow 12 = (z + \left(-1\right)) \cdot (z + \left(-1\right))\right) \Rightarrow z^2 - z \cdot 2 + 11 \left(-1\right) = 0

4,661

(1 + 2 + 3 + 4)/4 = \dfrac{1}{4} \cdot 10 = 5/2

14,686

( |a|, |d|) = 1 \implies 1 = \left(a, d\right)

-1,863

\frac{1}{12}\cdot 11\cdot \pi - \pi\cdot 0 = \pi\cdot 11/12

-20,119

\dfrac13\cdot 10\cdot \dfrac{1}{m + 2}\cdot (2 + m) = \frac{20 + m\cdot 10}{3\cdot m + 6}

4,830

b^2 \cdot k + \left(k + a\right) \cdot b + a = \left(b \cdot k + a\right) \cdot (b + 1)

7,075

3*17/(4*17) = 51/68

3,603

\sin\left(x\right) = \cos(-x + \frac{1}{2} \cdot π)

6,308

14 + 3 \cdot z^2 - 13 \cdot z = 0 \Rightarrow 0 = \left(7 \cdot (-1) + 3 \cdot z\right) \cdot (2 \cdot (-1) + z)

-9,358

-3\cdot 3 - 2\cdot 3\cdot 5\cdot t = 9\cdot (-1) - 30\cdot t

-20,466

-\frac149 \frac{10 + t}{t + 10} = \dfrac{1}{40 + 4t}(90 (-1) - t*9)

17,266

d \cdot b/f = b \cdot d/f

10,785

(n + 62)^2 - (34 + n) \cdot 113 = 2 + n^2 + n \cdot 11

9,460

\arccos(\cos(\pi - \arccos{t})) = \arccos{-t} \implies \pi - \arccos{t} = \arccos{-t}

9,658

(y + (-1)) (1 + y^6 + y^5 + y^4 + y^3 + y^2 + y) = y^7 + (-1)

20,220

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

4,061

573 = 3\cdot 191

-28,971

14 + 12\cdot (-1) = 2

8,026

-\frac54 + (1/2 + q)^2 = q^2 + q + (-1)

20,062

\frac2x*1/2 = \frac{1}{x}

-4,183

\dfrac{x^4}{x^4} \cdot \tfrac{1}{8} \cdot 2 = \frac{2 \cdot x^4}{8 \cdot x^4} \cdot 1

7,578

13 + 2 \cdot 2 = 17

912

(\left(i\cdot 2 + 1\right)\cdot (-6))/(-2) = 3\cdot (2\cdot i + 1)

12,834

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

-2,051

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

28,534

p^n\cdot e^{x\cdot p} = \frac{1}{e^{-p\cdot x}}\cdot p^n

35,931

180 = 2^2*3^2*5

8,949

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

-20,019

8/8 \cdot \frac{1}{9 + 7 \cdot p} \cdot (10 \cdot \left(-1\right) + p) = \frac{1}{p \cdot 56 + 72} \cdot \left(p \cdot 8 + 80 \cdot (-1)\right)

24,628

(a + \nu) \cdot (\nu - a) = \nu \cdot \nu - a^2

-10,950

\frac{195}{13} = 15

2,357

640320^3 = \left(2^6 \cdot 3 \cdot 5 \cdot 23 \cdot 29\right)^3

17,618

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

-7,605

\left(-14 + 5 i - 56 i + 20 \left(-1\right)\right)/17 = \frac{1}{17} \left(-34 - 51 i\right) = -2 - 3 i

-26,673

(x\cdot 2 + 5\cdot (-1))\cdot (1 + x\cdot 4) = x^2\cdot 8 - 18\cdot x + 5\cdot (-1)

-7,684

\frac{1}{25}*(-60 + 30*i - 80*i + 40*(-1)) = \frac{1}{25}*(-100 - 50*i) = -4 - 2*i

-20,558

\frac13 \cdot 3 \cdot \frac{1}{2 \cdot (-1) + j} \cdot \left(-3 \cdot j + 3 \cdot \left(-1\right)\right) = \frac{1}{6 \cdot (-1) + 3 \cdot j} \cdot (-j \cdot 9 + 9 \cdot \left(-1\right))

17,023

0 = (y^3 + (-1))*(y^6 + y^3 + 1) = y^9 + (-1)

2,440

\|e\|_1 = 0 \implies 0 = e

13,429

0 = x*2^{1/2} + q*5^{1/2} + r*10^{1/2} + t = t + x*2^{1/2} + (q + r*2^{1/2})*5^{1/2}