ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
4,937	b/c + a/c = \frac1c\cdot (b + a)
16,327	a_t + b_t = a_t + b_t
20,836	-5 = 4*(-1) - 1
-15,574	\frac{z^4}{\left(z^2\cdot t^2\right)^3} = \frac{z^4}{t^6\cdot z^6}
15,807	-(x \cdot x^2 + 1) + (x + 1)^3 = 3 \cdot (x + x^2)
33,998	213 13 = 338
18,805	1 - \cos{4 \cdot y} = (4 \cdot y)^2/2! + ... = 8 \cdot y^2 + ...
15,351	\cos(x + y) = \cos{x} \cdot \cos{y} - \sin{y} \cdot \sin{x}
1,779	a^{1/2} \cdot a^{1/2} \cdot a^{1/2} = a^{3/2}
9,553	4 + \eta = v \Rightarrow \eta = v + 4 (-1)
-1,600	-\frac{5}{12}\cdot \pi + 2\cdot \pi = 19/12\cdot \pi
-22,899	\dfrac{56}{8 \cdot 10} \cdot 1 = 56/80
-23,942	7 + \frac{48}{8} = 7 + 6 = 13
3,410	(A + E)^2 = (A + E) (A + E) = A A + A E + E A + E^2 \neq A^2 + E^2
11,918	S \cdot T = (S^{1/2} \cdot T^{\frac{1}{2}})^2
20,036	(2 + \sqrt{3})\left(2 - \sqrt{3}\right) = 2 2 - \sqrt{3} * \sqrt{3} = 1
30,811	\frac{1/3}{2} \cdot 2 = \dfrac13
43,381	286 = 12 \cdot 3 + 50 \cdot 5
-4,580	\frac{-2 \cdot x + 8}{15 + x \cdot x - 8 \cdot x} = -\dfrac{1}{5 \cdot (-1) + x} - \frac{1}{x + 3 \cdot \left(-1\right)}
-11,501	-30 - i20 = -5 + 25 (-1) - i20
983	Z = Q - Y \Rightarrow Q = Z + Y
27,209	\cos\left(\sin^{-1}{-y}\right) = \cos(-\sin^{-1}{y}) = \cos\left(\sin^{-1}{y}\right)
-486	e^{19 \frac{i\pi}{6}11} = (e^{\frac{i\pi}{6}11})^{19}
27,098	\cos(\pi/2) = \cos(\frac32 \cdot \pi) = 0
1,050	z \cdot 21/20 = 4 \cdot z/5 + z/4
-5,464	\frac{l\cdot 3}{25\cdot \left(-1\right) + l^2} = \frac{l\cdot 3}{(5 + l)\cdot (5\cdot (-1) + l)}
15,859	\cot(-B) = z \Rightarrow \operatorname{arccot}(z) = -B
13,534	\frac12\sin{y \cdot 2} = \cos{y} \sin{y}
-90	-13 + 2\cdot (-1) = -15
8,008	\frac{3 + n}{(-1) + n} = 1 + \frac{4}{n + (-1)}
12,180	0 = 315 \cdot (-1) - 3 \cdot b \cdot b + 66 \cdot b rightarrow b^2 - 22 \cdot b + 105 = (b + 15 \cdot (-1)) \cdot (b + 7 \cdot (-1)) = 0
8,186	80/10 \cdot 5 = 8 \cdot 5 = 40
5,211	(j + 1)/j = 1 + \tfrac1j
25,363	\sqrt{X\cdot 4}/2 = \sqrt{X}
32,158	i! = \dfrac{(i!)!}{(i! + (-1))!}
42,230	2 + 67 + 22(-1) + 6\left(-1\right) = 41
22,153	\tfrac{1}{(1 - y)^3} = (1 + y + y^2 + \dots)^3
-5,367	10^40.24 = 10^{(-5) (-1) - 1}0.24
20,536	\frac{1}{2} + \dfrac{1}{4} + 1/8 + \frac{1}{12} + 1/24 = 1
-2,701	16^{1/2}3^{1/2} - 3^{1/2}9^{1/2} = 3^{1/2}4 - 33^{1/2}
-18,506	5c + 10 = 6\left(2c + 2\right) = 12c + 12
-12,496	3 = \frac{19.5}{6.5}
10,540	e^c\cdot e^d = e^{c + d}
-3,409	11^{1/2} \cdot 2 = 11^{1/2} \cdot (5 + 4 \cdot \left(-1\right) + 1)
5,289	\sigma(BA) \backslash 0 = \sigma\left(BA\right) \backslash 0
6,821	-\frac{9}{25} + 16/25 + \dfrac{12}{25} = 19/25
11,599	3 \cdot 5 + 2 \cdot 2 + 2 \cdot 4 + 2 \cdot 6 = -5 \cdot 3 + (2 + 5) \cdot 2 + 2 \cdot (6 + 5) + (4 + 5) \cdot 2
48,656	108\cdot (-1) + 120 = 12
7,621	h + z \cdot (1 + k) = y rightarrow \frac{-h + y}{1 + k} = z
26,780	\left(1 + x^4 + x^3 + x^2 + x\right)\cdot (\left(-1\right) + x) = x^5 + (-1)
14,137	1 + t^4 + t^2 = (t \cdot t + 1)^2 - t^2
40,161	3 \cdot 3 \cdot 4 = 36
-20,869	\frac{1}{z \cdot (-60)} \cdot (100 \cdot \left(-1\right) + 10 \cdot z) = \frac{1}{\left(-6\right) \cdot z} \cdot (z + 10 \cdot (-1)) \cdot 10/10
-20,817	\frac{1}{-24} \times (-n \times 9 + 21 \times (-1)) = (-n \times 3 + 7 \times (-1))/(-8) \times \dfrac33
-4,052	\frac{m}{40m}100 = 100/40*\dfrac{m}{m}
1,527	2\cdot k = m \implies (-1) + m\cdot 3 = (-1) + k\cdot 6
-23,571	12/25 = \frac{3}{5} \cdot \frac45
-4,939	7.4 \cdot 10^0 = 7.4 \cdot 10^{1 - 1}
10,916	1989/867 = \dfrac{51 \cdot 39}{51 \cdot 17} = 39/17
8,397	1 = \tfrac12*(3 + (-1))
8,418	363^{1/2} = 723^{1/2}/2
-9,158	49\cdot (-1) - 7\cdot k = -7\cdot 7 - 7\cdot k
-29,935	\frac{d}{dx} (-x^3) = -\frac{d}{dx} x \cdot x^2 = -3 \cdot x^2 = -3 \cdot x \cdot x
26,118	(\left(-1\right) + 1)*20 + 250 = 250
27,569	h \cdot b \cdot d = h \cdot b \cdot d = \frac{1}{b \cdot d} \cdot h
35,082	\frac{1}{1 - x^2} = (\frac{1}{1 - x} + \dfrac{1}{x + 1})/2
-5,081	10^{11}48.0 = 48.010^{6 + 5}
31,515	-\dfrac{u}{u * u + 1} + u = \frac{1}{u * u + 1}u u * u
213	1 + 2*u = -u^2 + (u + 1)^2
1,862	(-1) + m \leq -r + x \Rightarrow 1 + x \geq m + r
-22,378	(n + 10)\cdot (n + 3\cdot (-1)) = n^2 + 7\cdot n + 30\cdot (-1)
27,729	a^2 \cdot a + b^3 = \left(a^2 - ab + b^2\right) (a + b)
-20,422	\frac{x \cdot 70 + 14 \cdot (-1)}{14 + x \cdot 35} = \frac{1}{7} \cdot 7 \cdot \dfrac{1}{2 + 5 \cdot x} \cdot (2 \cdot (-1) + 10 \cdot x)
-4,146	8m^2 = 8m^2
27,861	1/a = \tfrac1a
14,592	1/2\cdot 1/2\cdot \dfrac{1}{2} = 1/8
19,931	3^{2x + 2} + (-1) = 3^{2x}3^2 + (-1) = 93^{2x} + (-1) = 83^{2x} + 3^{2x} + (-1)
40,937	26^6 + 140552 (-1) = 308775224
20,600	m^2 + 2\cdot m + 1 = (m + 1)^2
9,015	\tfrac{0}{(0 + 2 \cdot \left(-1\right)) \cdot (0 + 1)} = \frac{0}{\left(-2\right)} = 0/2 = 0
10,550	2^k\cdot k + 2^{1 + k} = 2^{1 + k}\cdot (k + 1) - k\cdot 2^k
-8,387	\left(-6\right)*(-2) = 12
35,995	-2\times x \times x + 2\times (x^2 + x)^2 - 2\times x^4 = 4\times x^3
-26,673	8z^2 - z18 + 5(-1) = \left(4z + 1\right)(5(-1) + 2*z)
26,832	2^{15} + \left(-1\right) = \frac{1}{((-1) + 2 \cdot 2 \cdot 2) \cdot (\left(-1\right) + 2^5)} \cdot (2^{15} + (-1)) \cdot ((-1) + 2^5) \cdot (\left(-1\right) + 2^3)
2,540	AA^U = A^U A
28,190	x^2 - a^2 = \left(x - a\right) \cdot \left(a + x\right)
4,867	19 + s^4 - 20 \cdot s^2 = (s^2 + 19 \cdot \left(-1\right)) \cdot \left(s \cdot s + (-1)\right)
31,392	z^4 + 2*z + 1 = z^4 + 1 = (z + 1)^4
29,521	4\cdot a = \left(a + 1 + \rho\right)^2 = a^2 + 2\cdot \left(1 + \rho\right)\cdot a + (1 + \rho)^2
44,378	x = \dfrac{x}{1}
19,041	{l + (-1) + x - l + 1 + (-1) \choose l + (-1)} = {(-1) + x \choose l + (-1)}
1,351	(D - x) \cdot B \cdot D = B \cdot D \cdot \left(-x + D\right)
-20,396	-\frac{1}{4} \cdot \frac{1}{4 \cdot (-1) - r \cdot 5} \cdot \left(4 \cdot \left(-1\right) - r \cdot 5\right) = \dfrac{5 \cdot r + 4}{-r \cdot 20 + 16 \cdot (-1)}
-12,617	141*\left(-1\right) + 172 = 31
-20,235	\frac{x \cdot 90 + 50}{-20 \cdot x + 40 \cdot \left(-1\right)} = \frac{1}{10} \cdot 10 \cdot \frac{5 + x \cdot 9}{4 \cdot (-1) - x \cdot 2}
3,200	0 = 3\cdot \left(-1\right) + z^2\cdot 3\Longrightarrow 1 = z^2
1,817	\frac{2^4}{3^4} = 16/81
-24,251	710 + 7\frac22 = 7*10 + 7 = 70 + 7 = 70 + 7 = 77
-27,626	-1 + 3 \cdot \left(-1\right) + 9 + 5 \cdot \left(-1\right) = -4 + 9 + 5 \cdot (-1) = 5 + 5 \cdot (-1) = 0

4,937

b/c + a/c = \frac1c\cdot (b + a)

16,327

a_t + b_t = a_t + b_t

20,836

-5 = 4*(-1) - 1

-15,574

\frac{z^4}{\left(z^2\cdot t^2\right)^3} = \frac{z^4}{t^6\cdot z^6}

15,807

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

33,998

2*13 * 13 = 338

18,805

1 - \cos{4 \cdot y} = (4 \cdot y)^2/2! + ... = 8 \cdot y^2 + ...

15,351

\cos(x + y) = \cos{x} \cdot \cos{y} - \sin{y} \cdot \sin{x}

1,779

a^{1/2} \cdot a^{1/2} \cdot a^{1/2} = a^{3/2}

9,553

4 + \eta = v \Rightarrow \eta = v + 4 (-1)

-1,600

-\frac{5}{12}\cdot \pi + 2\cdot \pi = 19/12\cdot \pi

-22,899

\dfrac{56}{8 \cdot 10} \cdot 1 = 56/80

-23,942

7 + \frac{48}{8} = 7 + 6 = 13

3,410

(A + E)^2 = (A + E) (A + E) = A A + A E + E A + E^2 \neq A^2 + E^2

11,918

S \cdot T = (S^{1/2} \cdot T^{\frac{1}{2}})^2

20,036

(2 + \sqrt{3})*\left(2 - \sqrt{3}\right) = 2 * 2 - \sqrt{3} * \sqrt{3} = 1

30,811

\frac{1/3}{2} \cdot 2 = \dfrac13

43,381

286 = 12 \cdot 3 + 50 \cdot 5

-4,580

\frac{-2 \cdot x + 8}{15 + x \cdot x - 8 \cdot x} = -\dfrac{1}{5 \cdot (-1) + x} - \frac{1}{x + 3 \cdot \left(-1\right)}

-11,501

-30 - i*20 = -5 + 25 (-1) - i*20

983

Z = Q - Y \Rightarrow Q = Z + Y

27,209

\cos\left(\sin^{-1}{-y}\right) = \cos(-\sin^{-1}{y}) = \cos\left(\sin^{-1}{y}\right)

-486

e^{19 \frac{i\pi}{6}11} = (e^{\frac{i\pi}{6}11})^{19}

27,098

\cos(\pi/2) = \cos(\frac32 \cdot \pi) = 0

1,050

z \cdot 21/20 = 4 \cdot z/5 + z/4

-5,464

\frac{l\cdot 3}{25\cdot \left(-1\right) + l^2} = \frac{l\cdot 3}{(5 + l)\cdot (5\cdot (-1) + l)}

15,859

\cot(-B) = z \Rightarrow \operatorname{arccot}(z) = -B

13,534

\frac12\sin{y \cdot 2} = \cos{y} \sin{y}

-90

-13 + 2\cdot (-1) = -15

8,008

\frac{3 + n}{(-1) + n} = 1 + \frac{4}{n + (-1)}

12,180

0 = 315 \cdot (-1) - 3 \cdot b \cdot b + 66 \cdot b rightarrow b^2 - 22 \cdot b + 105 = (b + 15 \cdot (-1)) \cdot (b + 7 \cdot (-1)) = 0

8,186

80/10 \cdot 5 = 8 \cdot 5 = 40

5,211

(j + 1)/j = 1 + \tfrac1j

25,363

\sqrt{X\cdot 4}/2 = \sqrt{X}

32,158

i! = \dfrac{(i!)!}{(i! + (-1))!}

42,230

2 + 67 + 22*(-1) + 6*\left(-1\right) = 41

22,153

\tfrac{1}{(1 - y)^3} = (1 + y + y^2 + \dots)^3

-5,367

10^4*0.24 = 10^{(-5) (-1) - 1}*0.24

20,536

\frac{1}{2} + \dfrac{1}{4} + 1/8 + \frac{1}{12} + 1/24 = 1

-2,701

16^{1/2}*3^{1/2} - 3^{1/2}*9^{1/2} = 3^{1/2}*4 - 3*3^{1/2}

-18,506

5*c + 10 = 6*\left(2*c + 2\right) = 12*c + 12

-12,496

3 = \frac{19.5}{6.5}

10,540

e^c\cdot e^d = e^{c + d}

-3,409

11^{1/2} \cdot 2 = 11^{1/2} \cdot (5 + 4 \cdot \left(-1\right) + 1)

5,289

\sigma(BA) \backslash 0 = \sigma\left(BA\right) \backslash 0

6,821

-\frac{9}{25} + 16/25 + \dfrac{12}{25} = 19/25

11,599

3 \cdot 5 + 2 \cdot 2 + 2 \cdot 4 + 2 \cdot 6 = -5 \cdot 3 + (2 + 5) \cdot 2 + 2 \cdot (6 + 5) + (4 + 5) \cdot 2

48,656

108\cdot (-1) + 120 = 12

7,621

h + z \cdot (1 + k) = y rightarrow \frac{-h + y}{1 + k} = z

26,780

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

14,137

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

40,161

3 \cdot 3 \cdot 4 = 36

-20,869

\frac{1}{z \cdot (-60)} \cdot (100 \cdot \left(-1\right) + 10 \cdot z) = \frac{1}{\left(-6\right) \cdot z} \cdot (z + 10 \cdot (-1)) \cdot 10/10

-20,817

\frac{1}{-24} \times (-n \times 9 + 21 \times (-1)) = (-n \times 3 + 7 \times (-1))/(-8) \times \dfrac33

-4,052

\frac{m}{40*m}*100 = 100/40*\dfrac{m}{m}

1,527

2\cdot k = m \implies (-1) + m\cdot 3 = (-1) + k\cdot 6

-23,571

12/25 = \frac{3}{5} \cdot \frac45

-4,939

7.4 \cdot 10^0 = 7.4 \cdot 10^{1 - 1}

10,916

1989/867 = \dfrac{51 \cdot 39}{51 \cdot 17} = 39/17

8,397

1 = \tfrac12*(3 + (-1))

8,418

36*3^{1/2} = 72*3^{1/2}/2

-9,158

49\cdot (-1) - 7\cdot k = -7\cdot 7 - 7\cdot k

-29,935

\frac{d}{dx} (-x^3) = -\frac{d}{dx} x \cdot x^2 = -3 \cdot x^2 = -3 \cdot x \cdot x

26,118

(\left(-1\right) + 1)*20 + 250 = 250

27,569

h \cdot b \cdot d = h \cdot b \cdot d = \frac{1}{b \cdot d} \cdot h

35,082

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

-5,081

10^{11}*48.0 = 48.0*10^{6 + 5}

31,515

-\dfrac{u}{u * u + 1} + u = \frac{1}{u * u + 1}*u * u * u

213

1 + 2*u = -u^2 + (u + 1)^2

1,862

(-1) + m \leq -r + x \Rightarrow 1 + x \geq m + r

-22,378

(n + 10)\cdot (n + 3\cdot (-1)) = n^2 + 7\cdot n + 30\cdot (-1)

27,729

a^2 \cdot a + b^3 = \left(a^2 - ab + b^2\right) (a + b)

-20,422

\frac{x \cdot 70 + 14 \cdot (-1)}{14 + x \cdot 35} = \frac{1}{7} \cdot 7 \cdot \dfrac{1}{2 + 5 \cdot x} \cdot (2 \cdot (-1) + 10 \cdot x)

-4,146

8m^2 = 8m^2

27,861

1/a = \tfrac1a

14,592

1/2\cdot 1/2\cdot \dfrac{1}{2} = 1/8

19,931

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

40,937

26^6 + 140552 (-1) = 308775224

20,600

m^2 + 2\cdot m + 1 = (m + 1)^2

9,015

\tfrac{0}{(0 + 2 \cdot \left(-1\right)) \cdot (0 + 1)} = \frac{0}{\left(-2\right)} = 0/2 = 0

10,550

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

-8,387

\left(-6\right)*(-2) = 12

35,995

-2\times x \times x + 2\times (x^2 + x)^2 - 2\times x^4 = 4\times x^3

-26,673

8*z^2 - z*18 + 5*(-1) = \left(4*z + 1\right)*(5*(-1) + 2*z)

26,832

2^{15} + \left(-1\right) = \frac{1}{((-1) + 2 \cdot 2 \cdot 2) \cdot (\left(-1\right) + 2^5)} \cdot (2^{15} + (-1)) \cdot ((-1) + 2^5) \cdot (\left(-1\right) + 2^3)

2,540

AA^U = A^U A

28,190

x^2 - a^2 = \left(x - a\right) \cdot \left(a + x\right)

4,867

19 + s^4 - 20 \cdot s^2 = (s^2 + 19 \cdot \left(-1\right)) \cdot \left(s \cdot s + (-1)\right)

31,392

z^4 + 2*z + 1 = z^4 + 1 = (z + 1)^4

29,521

4\cdot a = \left(a + 1 + \rho\right)^2 = a^2 + 2\cdot \left(1 + \rho\right)\cdot a + (1 + \rho)^2

44,378

x = \dfrac{x}{1}

19,041

{l + (-1) + x - l + 1 + (-1) \choose l + (-1)} = {(-1) + x \choose l + (-1)}

1,351

(D - x) \cdot B \cdot D = B \cdot D \cdot \left(-x + D\right)

-20,396

-\frac{1}{4} \cdot \frac{1}{4 \cdot (-1) - r \cdot 5} \cdot \left(4 \cdot \left(-1\right) - r \cdot 5\right) = \dfrac{5 \cdot r + 4}{-r \cdot 20 + 16 \cdot (-1)}

-12,617

141*\left(-1\right) + 172 = 31

-20,235

\frac{x \cdot 90 + 50}{-20 \cdot x + 40 \cdot \left(-1\right)} = \frac{1}{10} \cdot 10 \cdot \frac{5 + x \cdot 9}{4 \cdot (-1) - x \cdot 2}

3,200

0 = 3\cdot \left(-1\right) + z^2\cdot 3\Longrightarrow 1 = z^2

1,817

\frac{2^4}{3^4} = 16/81

-24,251

7*10 + 7*\frac22 = 7*10 + 7 = 70 + 7 = 70 + 7 = 77

-27,626

-1 + 3 \cdot \left(-1\right) + 9 + 5 \cdot \left(-1\right) = -4 + 9 + 5 \cdot (-1) = 5 + 5 \cdot (-1) = 0