ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
4,207	g/xa/f = 1/(xf)ga
-20,215	\dfrac{35 y + 30}{-y28 + 24 (-1)} = \dfrac{1}{-y7 + 6(-1)}(-7y + 6\left(-1\right)) \left(-5/4\right)
-20,461	\frac{s\left(-1\right)}{s + 10}\frac{1}{4}4 = \frac{(-4)s}{4*s + 40}
6,789	y_t \cdot y_l = y_l \cdot y_t
16,250	\|0(-1) + \frac1xx * x\| = \|x + 0*(-1)\|
-30,259	x^2 - x + 12\cdot (-1) = (3 + x)\cdot \left(x + 4\cdot (-1)\right)
13,375	p - \frac12*((-1) + p) = (p + 1)/2
16,618	a\times b\times 2 = a\times b + b\times a
10,645	(\frac{1}{2}e^{-ir} + \frac{e^{ir}}{2})^m = \cosh^m{ir} = \cos^m{r}
24,071	0 = U\cdot \sin(x)\cdot Z - \dfrac{g}{2}\cdot U^2\Longrightarrow U = Z\cdot \sin\left(x\right)\cdot 2/g
12,910	\left(z + 2(-1)\right)^2 + (z^2 - 1/2)^2 = z z - 4z + 4 + z^4 - z z + \tfrac{1}{4} = z^4 - 4*z + 17/4
-9,633	0.01\cdot \left(-80\right) = -\frac{80}{100} = -4/5
-20,482	\frac{1}{70 \times m + 28} \times (4 \times (-1) - m \times 10) = \frac{1}{m \times 10 + 4} \times (4 + 10 \times m) \times (-\dfrac{1}{7})
2,121	\sqrt{1 + \sqrt{y^3 + 2 + y + y^2}} = u \Rightarrow y^3 + 2 + y + y^2 = \left(u^2 + (-1)\right)^2
26,769	(X \cdot B)^3 = B \cdot X \cdot X \cdot B \cdot X \cdot B
-4,246	\frac{x^4 \times 72}{48 \times x \times x^2} = 72/48 \times \frac{1}{x^3} \times x^4
157	Var\left[x_X - x_Y\right] = Var\left[x_X\right] + Var\left[-x_Y\right] = Var\left[x_X\right] + Var\left[x_Y\right]
25,192	\|X \cdot D\| = \|X\| \cdot \|D\|
14,188	4*(4 + (-1))/3 = 6 < 7
21,575	1 + \frac13 \cdot (3000 \cdot (-1) + 3333) = 112
-19,619	9/74/3 = \frac179/(3*1/4)
18,922	x\Deltaz = \Deltaxz
-2,577	\sqrt{7}*2 + 4\sqrt{7} = \sqrt{7} \sqrt{4} + \sqrt{7} \sqrt{16}
-3,258	7^{\dfrac{1}{2}} \cdot (4 + (-1) + 2) = 5 \cdot 7^{1 / 2}
-1,583	\pi \frac{19}{12} = \frac{1}{6}5 \pi + 3/4 \pi
11,287	\sin{5\pi/2} = \sin{\pi/2} = 1
31,728	\cos{x} \cdot \cos{\alpha} - \sin{x} \cdot \sin{\alpha} = \cos(\alpha + x)
-7,660	\frac{1 - i7}{-3 - 4i} = \frac{1 - 7i}{-4i - 3}\frac{1}{-3 + i4}\left(4i - 3\right)
-28,113	\tan\left(x\right) \sec(x) = \frac{d}{dx} \sec(x)
5,747	y^6 + \left(-1\right) = (y^3 + 1) \cdot (y^3 + \left(-1\right)) = (y^3 + 1) \cdot (y + (-1)) \cdot (y^2 + y^1 + y^0)
9,725	\sqrt{2\times z^2 \times z} + 2\times \sqrt{2\times z^3} + \sqrt{2\times z \times z^2} = 4\times \sqrt{2\times z^3} = 4\times z\times \sqrt{2\times z}
-19,003	1/9 = \frac{1}{36 \pi}A_s*36 \pi = A_s
16,951	1 = \sqrt{-\sin(\pi) + 1}
28,196	\sin{u2} = \sin{u} \cos{u}2
27,007	4 + 4 k = 4 \left(k + 1\right)
5,621	\left(0 \lt B,C < \pi\Longrightarrow \pi \gt \|-B + C\|\right)\Longrightarrow C - B = 0
-4,605	-\frac{2}{4 + z} + \frac{1}{z + 5\cdot (-1)}\cdot 5 = \frac{30 + 3\cdot z}{20\cdot (-1) + z \cdot z - z}
20,332	\|b_n\|^{\tfrac1n} \leq \frac1p = \frac{1}{p} \Rightarrow p^{-n} \geq \|b_n\|
-20,430	\frac{21 (-1) - 7y}{12 + 4y} = -7/4 \frac{3 + y}{y + 3}
6,076	x^2 = 2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{\cdots2}}}}}} = 2x
4,433	1/3 = \frac{1/4}{1 - 1/4}\cdot 1
5,512	\frac{1}{(3 - 2^{1 / 2}) \cdot (3 - 2^{1 / 2})} = \dfrac{1}{11 - 6\cdot 2^{\frac{1}{2}}}
-25,022	4 - 64Y^2 + 1024Y^4 - Y^616384 + \dots = \frac{4}{Y^216 + 1}
15,280	\left(h \cdot a\right)^k = (a \cdot h)^k
8,580	\frac1{4N} + \frac1{N/3} + \frac1{4N/3} = \frac4N
-3,343	32^{1/2} + 18^{1/2} - 2^{1/2} = (162)^{1/2} + (92)^{1/2} - 2^{1/2}
14,905	\dfrac{1/b\cdot f}{h} = f/(b\cdot h)
7,162	60^3 = \left(2^2 \cdot 3 \cdot 5\right)^3 = 2^6 \cdot 3 \cdot 3^2 \cdot 5^3
35,727	(1 + 1)^2 = 4
21,955	A\cdot y + Z\cdot y = (Z + A)\cdot y
25,988	ab c = (a + b - ab) c = a + b - ab + c - (a + b - ab) c
9,210	h \cdot d_n - d_n = d_n \cdot (h + (-1))
25,544	\cos(x) = \frac{\sin\left(2x\right)}{2\sin(x)}
17,762	k! = k\times (k + (-1))! = k\times (k + (-1))\times \left(k + 2\times \left(-1\right)\right)!
8,829	2 \cdot (4 \cdot x^2 - 3 \cdot x + (-1)) = 2 \cdot (x + \left(-1\right)) \cdot (x + 1/4) = \dfrac{1}{2 \cdot \left(x + (-1)\right) \cdot (4 \cdot x + 1)}
26,729	(1 + s)\cdot \left((-1) + s\right) + 1 = s \cdot s
5,288	1 + 3 + \dotsm + 2x + (-1) + 2(x + 1) + (-1) = x^2 + 2*x + 1 = (x + 1)^2
14,174	0 = (G - I \cdot c) \cdot (G - I \cdot c) \Rightarrow -c \cdot c \cdot I + G \cdot c \cdot 2 = G^2
1,744	x^2 + x + 1 = (\frac12 + x)^2 + \frac14 \cdot 3
27,283	(-1)^z = e^{i \cdot z \cdot \pi} = \cos(z \cdot \pi) + i \cdot \sin\left(z \cdot \pi\right)
12,508	\binom{m + (-1)}{m + (-1) - m - x} = \binom{m + (-1)}{(-1) + x}
1,322	A^3 = A \times A \times A
47,982	7 = 4 \cdot 4 - 3^2
54,889	0.02 \approx -1/20.4 + \frac{1}{12.74}
19,578	\frac{1}{z \cdot z + 1} = -\frac{1}{-(z^2 + 2) + 1}
-2,998	9 \times 2^{1 / 2} = 2^{\frac{1}{2}} \times (4 + 5)
11,206	(x + 1) * (x + 1) = x^2 + 1 = x + 1 + 1 = x
-1,247	30/35 = \frac{30\cdot 1/5}{35\cdot 1/5} = 6/7
709	\frac{1}{\dfrac{1}{y}g - c/y} = \frac{1}{g - c}y
1,987	-1/x + 1 = \frac{1}{x}\cdot (x + \left(-1\right))
6,940	b \cdot c = 0 = c \cdot b
5,673	Y_{y_0} \Rightarrow Y_{y_0}
19,401	\dfrac{7}{3} \times z = z + z \times 4/3
-20,461	4/4 \frac{(-1) t}{10 + t} = \frac{(-1)4 t}{t4 + 40}
15,126	(b \cdot \dfrac{a}{b})^x = b \cdot a^x/b
-5,777	\dfrac{2}{(m + 5(-1))5} = \frac{2}{m5 + 25(-1)}
1,584	m = 316^2 - 3^617 = 316 316 - 3^43^217 = 316^2 - 3^4 (296^2 - m)
-10,540	\dfrac{4}{4} (-\dfrac{1}{4 x^3} 5) = -\frac{20}{x^3*16}
-24,276	6 \times 6 + 8 \times \dfrac{2}{1} = 6 \times 6 + 8 \times 2 = 36 + 8 \times 2 = 36 + 16 = 52
32,420	x + 2 + 2 \cdot \left(5x + 11\right) = 24 + 11 x
34,636	7 = 5^{\frac{1}{2}} \cdot 5^{1 / 2} + 2^{1 / 2} \cdot 2^{1 / 2}
23,824	\dfrac159 = 9/5
-20,411	\frac{x(-8)}{-72x + 72(-1)} = \frac88\dfrac{(-1)x}{-9x + 9*(-1)}
8,604	\frac{1}{x + 1} \cdot (2 \cdot \left(-1\right) + x \cdot x \cdot x + 4 \cdot x^2 + x) = 2 \cdot (-1) + x^2 + x \cdot 3
-15,839	56/10 = -8 \cdot \frac{1}{10} \cdot 2 + 9 \cdot \frac{8}{10}
13,004	(-z + x + y)^2 + (z - x + y)^2 + (z + x - y)^2 = 3\cdot x^2 + 3\cdot y \cdot y + 3\cdot z^2 - y\cdot x\cdot 2 - 2\cdot z\cdot x - y\cdot z\cdot 2
8,082	-(-y + x)^2 + (y + x)^2 = 4\times x\times y
-6,726	2/100 + 10^{-1} = 10/100 + \frac{2}{100}
6,608	2 + 4 + 6 + \cdots + 2n = n*(2n + 2)/2 = n^2 + n
19,868	S^k = \frac{1^k}{0^k} = \frac{1}{0} = S \Rightarrow S^k = S
4,351	\binom{6}{2} \binom{9}{3} \cdot 4 \cdot 4^2 \binom{6}{3} = 1612800
-11,596	0 + 5\cdot (-1) + i\cdot 4 = -5 + i\cdot 4
-20,452	-\dfrac{70}{56\cdot y + 49\cdot (-1)} = 7/7\cdot (-\frac{10}{8\cdot y + 7\cdot (-1)})
12,741	1 + 1/2 + 1/4 + \ldots + \frac{1}{2^k} = -\frac{1}{2^k} + 2
-8,065	\frac{1}{5 + 5i}(5i + 5) \dfrac{i\cdot 45 + 5}{-i\cdot 5 + 5} = \tfrac{1}{5 - i\cdot 5}(45 i + 5)
-7,004	4/8\cdot 5/9 = 5/18
30,248	\pi\cdot 5^2 = 25\cdot \pi
-14,594	84 = \dfrac{336}{4}
11,711	\mathbb{E}(-d) = -\mathbb{E}(d)
536	\left(-z + 1\right)^2 = ((-1) + z)^2

4,207

g/x*a/f = 1/(x*f)*g*a

-20,215

\dfrac{35 y + 30}{-y*28 + 24 (-1)} = \dfrac{1}{-y*7 + 6(-1)}(-7y + 6\left(-1\right)) \left(-5/4\right)

-20,461

\frac{s*\left(-1\right)}{s + 10}*\frac{1}{4}*4 = \frac{(-4)*s}{4*s + 40}

6,789

y_t \cdot y_l = y_l \cdot y_t

16,250

|0*(-1) + \frac1x*x * x| = |x + 0*(-1)|

-30,259

x^2 - x + 12\cdot (-1) = (3 + x)\cdot \left(x + 4\cdot (-1)\right)

13,375

p - \frac12*((-1) + p) = (p + 1)/2

16,618

a\times b\times 2 = a\times b + b\times a

10,645

(\frac{1}{2}e^{-ir} + \frac{e^{ir}}{2})^m = \cosh^m{ir} = \cos^m{r}

24,071

0 = U\cdot \sin(x)\cdot Z - \dfrac{g}{2}\cdot U^2\Longrightarrow U = Z\cdot \sin\left(x\right)\cdot 2/g

12,910

\left(z + 2*(-1)\right)^2 + (z^2 - 1/2)^2 = z * z - 4*z + 4 + z^4 - z * z + \tfrac{1}{4} = z^4 - 4*z + 17/4

-9,633

0.01\cdot \left(-80\right) = -\frac{80}{100} = -4/5

-20,482

\frac{1}{70 \times m + 28} \times (4 \times (-1) - m \times 10) = \frac{1}{m \times 10 + 4} \times (4 + 10 \times m) \times (-\dfrac{1}{7})

2,121

\sqrt{1 + \sqrt{y^3 + 2 + y + y^2}} = u \Rightarrow y^3 + 2 + y + y^2 = \left(u^2 + (-1)\right)^2

26,769

(X \cdot B)^3 = B \cdot X \cdot X \cdot B \cdot X \cdot B

-4,246

\frac{x^4 \times 72}{48 \times x \times x^2} = 72/48 \times \frac{1}{x^3} \times x^4

157

Var\left[x_X - x_Y\right] = Var\left[x_X\right] + Var\left[-x_Y\right] = Var\left[x_X\right] + Var\left[x_Y\right]

25,192

|X \cdot D| = |X| \cdot |D|

14,188

4*(4 + (-1))/3 = 6 < 7

21,575

1 + \frac13 \cdot (3000 \cdot (-1) + 3333) = 112

-19,619

9/7*4/3 = \frac17*9/(3*1/4)

18,922

x*\Delta*z = \Delta*x*z

-2,577

\sqrt{7}*2 + 4\sqrt{7} = \sqrt{7} \sqrt{4} + \sqrt{7} \sqrt{16}

-3,258

7^{\dfrac{1}{2}} \cdot (4 + (-1) + 2) = 5 \cdot 7^{1 / 2}

-1,583

\pi \frac{19}{12} = \frac{1}{6}5 \pi + 3/4 \pi

11,287

\sin{5\pi/2} = \sin{\pi/2} = 1

31,728

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

-7,660

\frac{1 - i*7}{-3 - 4*i} = \frac{1 - 7*i}{-4*i - 3}*\frac{1}{-3 + i*4}*\left(4*i - 3\right)

-28,113

\tan\left(x\right) \sec(x) = \frac{d}{dx} \sec(x)

5,747

y^6 + \left(-1\right) = (y^3 + 1) \cdot (y^3 + \left(-1\right)) = (y^3 + 1) \cdot (y + (-1)) \cdot (y^2 + y^1 + y^0)

9,725

\sqrt{2\times z^2 \times z} + 2\times \sqrt{2\times z^3} + \sqrt{2\times z \times z^2} = 4\times \sqrt{2\times z^3} = 4\times z\times \sqrt{2\times z}

-19,003

1/9 = \frac{1}{36 \pi}A_s*36 \pi = A_s

16,951

1 = \sqrt{-\sin(\pi) + 1}

28,196

\sin{u*2} = \sin{u} \cos{u}*2

27,007

4 + 4 k = 4 \left(k + 1\right)

5,621

\left(0 \lt B,C < \pi\Longrightarrow \pi \gt |-B + C|\right)\Longrightarrow C - B = 0

-4,605

-\frac{2}{4 + z} + \frac{1}{z + 5\cdot (-1)}\cdot 5 = \frac{30 + 3\cdot z}{20\cdot (-1) + z \cdot z - z}

20,332

|b_n|^{\tfrac1n} \leq \frac1p = \frac{1}{p} \Rightarrow p^{-n} \geq |b_n|

-20,430

\frac{21 (-1) - 7y}{12 + 4y} = -7/4 \frac{3 + y}{y + 3}

6,076

x^2 = 2*\sqrt{2*\sqrt{2*\sqrt{2*\sqrt{2*\sqrt{2*\sqrt{\cdots*2}}}}}} = 2*x

4,433

1/3 = \frac{1/4}{1 - 1/4}\cdot 1

5,512

\frac{1}{(3 - 2^{1 / 2}) \cdot (3 - 2^{1 / 2})} = \dfrac{1}{11 - 6\cdot 2^{\frac{1}{2}}}

-25,022

4 - 64*Y^2 + 1024*Y^4 - Y^6*16384 + \dots = \frac{4}{Y^2*16 + 1}

15,280

\left(h \cdot a\right)^k = (a \cdot h)^k

8,580

\frac1{4N} + \frac1{N/3} + \frac1{4N/3} = \frac4N

-3,343

32^{1/2} + 18^{1/2} - 2^{1/2} = (16*2)^{1/2} + (9*2)^{1/2} - 2^{1/2}

14,905

\dfrac{1/b\cdot f}{h} = f/(b\cdot h)

7,162

60^3 = \left(2^2 \cdot 3 \cdot 5\right)^3 = 2^6 \cdot 3 \cdot 3^2 \cdot 5^3

35,727

(1 + 1)^2 = 4

21,955

A\cdot y + Z\cdot y = (Z + A)\cdot y

25,988

ab c = (a + b - ab) c = a + b - ab + c - (a + b - ab) c

9,210

h \cdot d_n - d_n = d_n \cdot (h + (-1))

25,544

\cos(x) = \frac{\sin\left(2x\right)}{2\sin(x)}

17,762

k! = k\times (k + (-1))! = k\times (k + (-1))\times \left(k + 2\times \left(-1\right)\right)!

8,829

2 \cdot (4 \cdot x^2 - 3 \cdot x + (-1)) = 2 \cdot (x + \left(-1\right)) \cdot (x + 1/4) = \dfrac{1}{2 \cdot \left(x + (-1)\right) \cdot (4 \cdot x + 1)}

26,729

(1 + s)\cdot \left((-1) + s\right) + 1 = s \cdot s

5,288

1 + 3 + \dotsm + 2*x + (-1) + 2*(x + 1) + (-1) = x^2 + 2*x + 1 = (x + 1)^2

14,174

0 = (G - I \cdot c) \cdot (G - I \cdot c) \Rightarrow -c \cdot c \cdot I + G \cdot c \cdot 2 = G^2

1,744

x^2 + x + 1 = (\frac12 + x)^2 + \frac14 \cdot 3

27,283

(-1)^z = e^{i \cdot z \cdot \pi} = \cos(z \cdot \pi) + i \cdot \sin\left(z \cdot \pi\right)

12,508

\binom{m + (-1)}{m + (-1) - m - x} = \binom{m + (-1)}{(-1) + x}

1,322

A^3 = A \times A \times A

47,982

7 = 4 \cdot 4 - 3^2

54,889

0.02 \approx -1/20.4 + \frac{1}{12.74}

19,578

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

-2,998

9 \times 2^{1 / 2} = 2^{\frac{1}{2}} \times (4 + 5)

11,206

(x + 1) * (x + 1) = x^2 + 1 = x + 1 + 1 = x

-1,247

30/35 = \frac{30\cdot 1/5}{35\cdot 1/5} = 6/7

709

\frac{1}{\dfrac{1}{y}g - c/y} = \frac{1}{g - c}y

1,987

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

6,940

b \cdot c = 0 = c \cdot b

5,673

Y_{y_0} \Rightarrow Y_{y_0}

19,401

\dfrac{7}{3} \times z = z + z \times 4/3

-20,461

4/4 \frac{(-1) t}{10 + t} = \frac{(-1)*4 t}{t*4 + 40}

15,126

(b \cdot \dfrac{a}{b})^x = b \cdot a^x/b

-5,777

\dfrac{2}{(m + 5*(-1))*5} = \frac{2}{m*5 + 25*(-1)}

1,584

m = 316^2 - 3^6*17 = 316 * 316 - 3^4*3^2*17 = 316^2 - 3^4 (296^2 - m)

-10,540

\dfrac{4}{4} (-\dfrac{1}{4 x^3} 5) = -\frac{20}{x^3*16}

-24,276

6 \times 6 + 8 \times \dfrac{2}{1} = 6 \times 6 + 8 \times 2 = 36 + 8 \times 2 = 36 + 16 = 52

32,420

x + 2 + 2 \cdot \left(5x + 11\right) = 24 + 11 x

34,636

7 = 5^{\frac{1}{2}} \cdot 5^{1 / 2} + 2^{1 / 2} \cdot 2^{1 / 2}

23,824

\dfrac159 = 9/5

-20,411

\frac{x*(-8)}{-72*x + 72*(-1)} = \frac88*\dfrac{(-1)*x}{-9*x + 9*(-1)}

8,604

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

-15,839

56/10 = -8 \cdot \frac{1}{10} \cdot 2 + 9 \cdot \frac{8}{10}

13,004

(-z + x + y)^2 + (z - x + y)^2 + (z + x - y)^2 = 3\cdot x^2 + 3\cdot y \cdot y + 3\cdot z^2 - y\cdot x\cdot 2 - 2\cdot z\cdot x - y\cdot z\cdot 2

8,082

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

-6,726

2/100 + 10^{-1} = 10/100 + \frac{2}{100}

6,608

2 + 4 + 6 + \cdots + 2n = n*(2n + 2)/2 = n^2 + n

19,868

S^k = \frac{1^k}{0^k} = \frac{1}{0} = S \Rightarrow S^k = S

4,351

\binom{6}{2} \binom{9}{3} \cdot 4 \cdot 4^2 \binom{6}{3} = 1612800

-11,596

0 + 5\cdot (-1) + i\cdot 4 = -5 + i\cdot 4

-20,452

-\dfrac{70}{56\cdot y + 49\cdot (-1)} = 7/7\cdot (-\frac{10}{8\cdot y + 7\cdot (-1)})

12,741

1 + 1/2 + 1/4 + \ldots + \frac{1}{2^k} = -\frac{1}{2^k} + 2

-8,065

\frac{1}{5 + 5i}(5i + 5) \dfrac{i\cdot 45 + 5}{-i\cdot 5 + 5} = \tfrac{1}{5 - i\cdot 5}(45 i + 5)

-7,004

4/8\cdot 5/9 = 5/18

30,248

\pi\cdot 5^2 = 25\cdot \pi

-14,594

84 = \dfrac{336}{4}

11,711

\mathbb{E}(-d) = -\mathbb{E}(d)

536

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