ddrg/math_formulas · Datasets at Hugging Face

id int64 -30,985 55.9k	text stringlengths 5 437k
23,677	x^2 - 12x + 27 = 0 = \left(x + 3(-1)\right)(x + 9(-1))
-11,744	1/9 = (\dfrac{1}{3})^2
14,796	e^m = \sum_{l=0}^\infty \frac{m^l}{l!} \gt \frac{1}{l!}\cdot m^l
38,810	h = 15 \cdot h \cdot x = 1.2 \cdot h \cdot x
7,140	5/1 = 5*\frac16/(1/6)
52,504	\sum_{n=1}^{2^{l_2} + (-1)} \frac{1}{n^p} = \sum_{n=2^{l_1}}^{2^{l_1 + 1} + (-1)} \frac{1}{n^p} < \sum_{n=2^{l_1}}^{2^{l_1 + 1} + (-1)} \frac{1}{(2^{l_1})^p}
-4,268	\dfrac{b^4}{b \cdot b \cdot b} = \frac{b^4}{b\cdot b\cdot b}\cdot 1 = b
-3,297	\sqrt{7}\cdot \sqrt{9} + \sqrt{7}\cdot \sqrt{16} - \sqrt{7}\cdot \sqrt{25} = 3\cdot \sqrt{7} + 4\cdot \sqrt{7} - \sqrt{7}\cdot 5
21,811	2(g + \beta) = \beta + g + \beta + g
-2,889	\sqrt{11} \times (5 + 2 \times \left(-1\right)) = 3 \times \sqrt{11}
-28,758	\frac{1}{3} - \frac{4}{3 \cdot y + 6} = \frac{1}{3} - \frac{\frac{4}{3}}{y + 2} \cdot 1
34,579	(a\cdot d)^i = \left(a\cdot d\right)^i
-15,885	4/10 \times 5 - 8 \times \frac{1}{10} \times 6 = -28/10
12,072	e^{-x} = -6x \Rightarrow \frac{1}{e^x} = -6x
34,226	\frac{1}{1 + b}\times b = \frac{1 + b + (-1)}{1 + b} = 1 - \dfrac{1}{1 + b}
-22,291	z^2 - z*11 + 18 = (z + 2(-1)) (z + 9(-1))
-2,538	107^{1/2} = (4 + 1 + 5)7^{1/2}
-18,472	\dfrac{36}{21} = 12/7
-7,162	3/22 = \frac{1}{11} \cdot 3 \cdot \frac{1}{12} \cdot 6
-26,414	-12/5 (-2) = \dfrac15 24
-9,411	-2\cdot 2\cdot 3\cdot 7 + z\cdot 2\cdot 2\cdot 2\cdot 3\cdot 5 = z\cdot 120 + 84\cdot (-1)
-13,855	9 - 1 \cdot 2 + 8/8 = 9 - 1 \cdot 2 + 1 = 9 + 2 \cdot (-1) + 1 = 7 + 1 = 8
-4,511	-\frac{2}{y + 5 \cdot (-1)} + \frac{4}{y + (-1)} = \dfrac{1}{5 + y^2 - y \cdot 6} \cdot \left(y \cdot 2 + 18 \cdot \left(-1\right)\right)
-5,902	\frac{4}{(r + 8)\cdot (r + 4\cdot (-1))} = \frac{1}{32\cdot (-1) + r^2 + r\cdot 4}\cdot 4
18,974	\|z - \frac{1}{2}(z + y)\|^2 = \|\frac{z}{2} - y/2\|^2 = \|\frac{1}{2}(z + y) - y\|^2
10,215	\frac{\frac{1}{100}}{2} \cdot 20 = \frac{1}{5 \cdot 2} = \frac{1}{10}
16,933	\Omega^{-1}=\Omega^{-1/2}\Omega^{-1/2}
14,449	(-1) + 4\cdot x = 0 \Rightarrow \frac{1}{4} = x
11,807	f_2 \cdot f_2 + f_1 \cdot f_1 + f_2^2 \cdot f_1^2 = (f_2 - f_1)^2 + 2 \cdot f_2 \cdot f_1 + f_2^2 \cdot f_1^2 = 1 + 2 \cdot f_2 \cdot f_1 + f_2^2 \cdot f_1 \cdot f_1 = (1 + f_2 \cdot f_1)^2
38,826	{10 \choose 3}{7 \choose 2}{5 \choose 2}3! = 10!/(7!3!)\dfrac{1}{5!2!}7!5!/(3!2!)3! = \frac{10!}{3!2!2!}
8,605	1/(1/(g_2) g_1) = g_2/(g_1)
-5,112	10^{(-2)\cdot (-1) + 0}\cdot 0.93 = 0.93\cdot 10^2
33,148	-1 - e^{-2 \times i} = -(1 + e^{-2 \times i})
19,554	(y^2 + 4\cdot (-1))^{1 / 2} - y = \frac{y^2 + 4\cdot \left(-1\right) - y^2}{(y^2 + 4\cdot (-1))^{\tfrac{1}{2}} + y} = -\frac{4}{(y^2 + 4\cdot (-1))^{1 / 2} + y}
23,134	\cos(\pi + \pi \cdot z) = -\cos(\pi \cdot z)
14,451	\left(1 + x \cdot 2\right) \cdot (1 - x) = -x \cdot x \cdot 2 + 1 + x
23,397	\sin{k} = z\Longrightarrow \sin^2{k} = z^2
33,396	B\vartheta = B\vartheta
3,654	(a\cdot c) \cdot (a\cdot c) = c^2\cdot a^2
28,184	(\dfrac14)^4 \cdot 3/4 = 3/1024
-4,360	\frac{1}{t^4}\cdot t\cdot \dfrac{1}{48}\cdot 66 = \frac{66\cdot t}{48\cdot t^4}
13,128	\left(-1\right) + z^2 = (z + 1)*(\left(-1\right) + z)
-21,090	10/10\cdot \dfrac{3}{10} = 30/100
-6,344	\dfrac{1}{2 \cdot (x + 6) \cdot \left(8 + x\right)} \cdot 4 = \dfrac{2}{(6 + x) \cdot (x + 8)} \cdot \frac{2}{2}
19,810	m + 1 + a = 1 + a + m
6,234	\left((\dfrac{2}{n} + 1)^n\right)^3 = (2/n + 1)^{n\times 3}
36,142	u^2 + 3v^2 = (-v - u)^2 + 2v(-u - v) + \left(2v\right) * \left(2*v\right)
20,627	0 = x^4 - x * x*6 + 1 - 7.2^z\Longrightarrow x^2 = (6 \pm \sqrt{32 - 28.2^z})/2
-10,389	\frac{1}{240 \cdot (-1) + 60 \cdot s} \cdot 100 = \frac{5}{3 \cdot s + 12 \cdot (-1)} \cdot \dfrac{20}{20}
-10,560	-\frac{1}{60 (-1) + 15 z}30 = -\frac{10}{20 (-1) + 5z} \frac{1}{3}3
51,042	4\times 6\times 3 = 72
-20,535	\frac{9 + x}{x + 9}\left(-\frac{9}{10}\right) = \frac{1}{90 + 10x}(81(-1) - x*9)
1,199	x + 2 = 1 + 2 + 3 + \dotsm + x
10,224	(i + a)\times (j + b) = i\times j + i\times b + a\times j + b\times a
12,266	\frac{1}{2} \cdot (1 + \cos(2 \cdot p)) = \cos^2(p)
36,868	\frac{d}{dz} e^{e^z} = e^{e^z}\cdot e^z = e^{e^z + z}
-10,765	8 = -2 + 5a - 4 = 5a - 6
11,208	-1 = (\cos(\pi) + \sin(\pi)\cdot i)
-639	\left(e^{\frac76 \cdot π \cdot i}\right)^5 = e^{5 \cdot 7 \cdot i \cdot π/6}
-1,064	\frac{1/4\cdot 3}{\frac{1}{8}} = \dfrac{8}{1}\cdot \dfrac14\cdot 3
5,149	\frac{1}{z x} (x - z) = \dfrac{1}{z} - 1/x
27,348	\frac{1}{5!} = \frac{1}{120}
2,165	\frac{1}{S\cdot T} = 1/(T\cdot S)
-20,472	6/6\cdot \frac{1}{r\cdot 10}\cdot (r\cdot 9 + 9\cdot (-1)) = (r\cdot 54 + 54\cdot \left(-1\right))/\left(60\cdot r\right)
-4,998	39.510 10 = 39.5*10^{5 - 3}
33,749	\pi10^2 = \pi100
118	((-1) \cdot x)/3 = -\frac13 \cdot \frac{1}{3} \cdot x = -x/3
21,047	3/8 = -\frac18 \cdot 5 + 1
6,260	b \cdot d^k/b \cdot c \cdot \frac{d^x}{c} = d^x/c \cdot c \cdot b \cdot \frac{d^k}{b}
30,334	\sin{\varphi} = \cos(-\varphi + \pi/2)
-18,960	\frac19\cdot 2 = \frac{H_p}{81\cdot \pi}\cdot 81\cdot \pi = H_p
27,274	g \eta x = g \eta x
24,418	\tan(x)/\sec(x) = \dfrac{\sin\left(x\right)}{\frac{1}{\cos(x)}}1/\cos(x) = \sin(x)
3,982	x^2 - x \cdot z + z^2 = -3 \cdot z \cdot x + (z + x)^2
12,006	123\dots\left(2(-1) + k\right)((-1) + k)*k = k!
28,808	3(-y + 2z) = 6z - y3
3,949	\frac{1}{\left(z^2 + 1\right)^{1/2} - z} = (z^2 + 1)^{1/2} + z
30,387	301 = 350 + 49\cdot (-1)
14,261	f \cdot u + u \cdot x = u \cdot (x + f)
-6,993	\frac{4}{5} \cdot 2/3 = \frac{8}{15}
-27,688	4\sin(x) = d/dx \left(-4\cos(x)\right)
7,338	9\cdot r^2 = x^2\cdot 3\Longrightarrow x \cdot x = 3\cdot r^2
5,598	1278 (-1) + 8721 = 7443
-29,325	-3\cdot i - 1 + 10\cdot (-1) = -3\cdot i - 11
-9,189	60 - x24 = -2223x + 2235
10,120	3^{H/2} = 2^H + (-1) = 4^{\frac{H}{2}} + (-1)
12,494	2*x = 135 + 25 = 160 rightarrow 80 = x
36,297	\cos^2{r} - \sin^2{r} = \cos{r \cdot 2}
18,766	\left(a - b\right) \cdot \left(a^2 + b \cdot a + b^2\right) = -b \cdot b \cdot b + a^3
21,854	c \cdot c + a \cdot a + x^2 = \sqrt{a^2 + x^2 + c^2} \cdot \sqrt{a^2 + x^2 + c^2}
-20,891	\tfrac{l \cdot 72 + 63}{28 \cdot (-1) - l \cdot 32} = -9/4 \cdot \frac{-l \cdot 8 + 7 \cdot (-1)}{-8 \cdot l + 7 \cdot (-1)}
-7,672	\frac{19 \cdot i + 8}{4 - 3 \cdot i} = \tfrac{1}{4 + i \cdot 3} \cdot (i \cdot 3 + 4) \cdot \frac{1}{4 - i \cdot 3} \cdot (8 + 19 \cdot i)
21,230	(z3 + y + 5) (2z + y + 3(-1)) = 15 (-1) + 6z z + 5zy + y * y + z + y*2
16,657	\left(x - 35.7 = 4.1\cdot \left(-1.34\right) \Rightarrow x = 4.1\cdot (-1.34) + 35.7\right) \Rightarrow x = 30.206
-11,767	36/49 = \left(\frac{6}{7}\right)^2
11,165	(2 \cdot x + 3) \cdot x \leq 2 \Rightarrow 2 \cdot (-1) + 2 \cdot x^2 + 3 \cdot x \leq 0
929	x * x + b*x rightarrow \left(x + b/2\right)^2 - \left(b/2\right)^2
11,296	w^2 + 3 x^2 = (x - w)^2 + (x + w)^2 + (x - w) (x + w)
31,137	1 - \frac{1}{216}*125 = \dfrac{91}{216}
22,192	1/(b\cdot d) = \frac{1}{d\cdot b}

23,677

x^2 - 12*x + 27 = 0 = \left(x + 3*(-1)\right)*(x + 9*(-1))

-11,744

1/9 = (\dfrac{1}{3})^2

14,796

e^m = \sum_{l=0}^\infty \frac{m^l}{l!} \gt \frac{1}{l!}\cdot m^l

38,810

h = 15 \cdot h \cdot x = 1.2 \cdot h \cdot x

7,140

5/1 = 5*\frac16/(1/6)

52,504

\sum_{n=1}^{2^{l_2} + (-1)} \frac{1}{n^p} = \sum_{n=2^{l_1}}^{2^{l_1 + 1} + (-1)} \frac{1}{n^p} < \sum_{n=2^{l_1}}^{2^{l_1 + 1} + (-1)} \frac{1}{(2^{l_1})^p}

-4,268

\dfrac{b^4}{b \cdot b \cdot b} = \frac{b^4}{b\cdot b\cdot b}\cdot 1 = b

-3,297

\sqrt{7}\cdot \sqrt{9} + \sqrt{7}\cdot \sqrt{16} - \sqrt{7}\cdot \sqrt{25} = 3\cdot \sqrt{7} + 4\cdot \sqrt{7} - \sqrt{7}\cdot 5

21,811

2(g + \beta) = \beta + g + \beta + g

-2,889

\sqrt{11} \times (5 + 2 \times \left(-1\right)) = 3 \times \sqrt{11}

-28,758

\frac{1}{3} - \frac{4}{3 \cdot y + 6} = \frac{1}{3} - \frac{\frac{4}{3}}{y + 2} \cdot 1

34,579

(a\cdot d)^i = \left(a\cdot d\right)^i

-15,885

4/10 \times 5 - 8 \times \frac{1}{10} \times 6 = -28/10

12,072

e^{-x} = -6*x \Rightarrow \frac{1}{e^x} = -6*x

34,226

\frac{1}{1 + b}\times b = \frac{1 + b + (-1)}{1 + b} = 1 - \dfrac{1}{1 + b}

-22,291

z^2 - z*11 + 18 = (z + 2(-1)) (z + 9(-1))

-2,538

10*7^{1/2} = (4 + 1 + 5)*7^{1/2}

-18,472

\dfrac{36}{21} = 12/7

-7,162

3/22 = \frac{1}{11} \cdot 3 \cdot \frac{1}{12} \cdot 6

-26,414

-12/5 (-2) = \dfrac15 24

-9,411

-2\cdot 2\cdot 3\cdot 7 + z\cdot 2\cdot 2\cdot 2\cdot 3\cdot 5 = z\cdot 120 + 84\cdot (-1)

-13,855

9 - 1 \cdot 2 + 8/8 = 9 - 1 \cdot 2 + 1 = 9 + 2 \cdot (-1) + 1 = 7 + 1 = 8

-4,511

-\frac{2}{y + 5 \cdot (-1)} + \frac{4}{y + (-1)} = \dfrac{1}{5 + y^2 - y \cdot 6} \cdot \left(y \cdot 2 + 18 \cdot \left(-1\right)\right)

-5,902

\frac{4}{(r + 8)\cdot (r + 4\cdot (-1))} = \frac{1}{32\cdot (-1) + r^2 + r\cdot 4}\cdot 4

18,974

|z - \frac{1}{2}(z + y)|^2 = |\frac{z}{2} - y/2|^2 = |\frac{1}{2}(z + y) - y|^2

10,215

\frac{\frac{1}{100}}{2} \cdot 20 = \frac{1}{5 \cdot 2} = \frac{1}{10}

16,933

\Omega^{-1}=\Omega^{-1/2}\Omega^{-1/2}

14,449

(-1) + 4\cdot x = 0 \Rightarrow \frac{1}{4} = x

11,807

f_2 \cdot f_2 + f_1 \cdot f_1 + f_2^2 \cdot f_1^2 = (f_2 - f_1)^2 + 2 \cdot f_2 \cdot f_1 + f_2^2 \cdot f_1^2 = 1 + 2 \cdot f_2 \cdot f_1 + f_2^2 \cdot f_1 \cdot f_1 = (1 + f_2 \cdot f_1)^2

38,826

{10 \choose 3}*{7 \choose 2}*{5 \choose 2}*3! = 10!/(7!*3!)*\dfrac{1}{5!*2!}*7!*5!/(3!*2!)*3! = \frac{10!}{3!*2!*2!}

8,605

1/(1/(g_2) g_1) = g_2/(g_1)

-5,112

10^{(-2)\cdot (-1) + 0}\cdot 0.93 = 0.93\cdot 10^2

33,148

-1 - e^{-2 \times i} = -(1 + e^{-2 \times i})

19,554

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

23,134

\cos(\pi + \pi \cdot z) = -\cos(\pi \cdot z)

14,451

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

23,397

\sin{k} = z\Longrightarrow \sin^2{k} = z^2

33,396

B*\vartheta = B*\vartheta

3,654

(a\cdot c) \cdot (a\cdot c) = c^2\cdot a^2

28,184

(\dfrac14)^4 \cdot 3/4 = 3/1024

-4,360

\frac{1}{t^4}\cdot t\cdot \dfrac{1}{48}\cdot 66 = \frac{66\cdot t}{48\cdot t^4}

13,128

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

-21,090

10/10\cdot \dfrac{3}{10} = 30/100

-6,344

\dfrac{1}{2 \cdot (x + 6) \cdot \left(8 + x\right)} \cdot 4 = \dfrac{2}{(6 + x) \cdot (x + 8)} \cdot \frac{2}{2}

19,810

m + 1 + a = 1 + a + m

6,234

\left((\dfrac{2}{n} + 1)^n\right)^3 = (2/n + 1)^{n\times 3}

36,142

u^2 + 3*v^2 = (-v - u)^2 + 2*v*(-u - v) + \left(2*v\right) * \left(2*v\right)

20,627

0 = x^4 - x * x*6 + 1 - 7.2^z\Longrightarrow x^2 = (6 \pm \sqrt{32 - 28.2^z})/2

-10,389

\frac{1}{240 \cdot (-1) + 60 \cdot s} \cdot 100 = \frac{5}{3 \cdot s + 12 \cdot (-1)} \cdot \dfrac{20}{20}

-10,560

-\frac{1}{60 (-1) + 15 z}30 = -\frac{10}{20 (-1) + 5z} \frac{1}{3}3

51,042

4\times 6\times 3 = 72

-20,535

\frac{9 + x}{x + 9}*\left(-\frac{9}{10}\right) = \frac{1}{90 + 10*x}*(81*(-1) - x*9)

1,199

x + 2 = 1 + 2 + 3 + \dotsm + x

10,224

(i + a)\times (j + b) = i\times j + i\times b + a\times j + b\times a

12,266

\frac{1}{2} \cdot (1 + \cos(2 \cdot p)) = \cos^2(p)

36,868

\frac{d}{dz} e^{e^z} = e^{e^z}\cdot e^z = e^{e^z + z}

-10,765

8 = -2 + 5a - 4 = 5a - 6

11,208

-1 = (\cos(\pi) + \sin(\pi)\cdot i)

-639

\left(e^{\frac76 \cdot π \cdot i}\right)^5 = e^{5 \cdot 7 \cdot i \cdot π/6}

-1,064

\frac{1/4\cdot 3}{\frac{1}{8}} = \dfrac{8}{1}\cdot \dfrac14\cdot 3

5,149

\frac{1}{z x} (x - z) = \dfrac{1}{z} - 1/x

27,348

\frac{1}{5!} = \frac{1}{120}

2,165

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

-20,472

6/6\cdot \frac{1}{r\cdot 10}\cdot (r\cdot 9 + 9\cdot (-1)) = (r\cdot 54 + 54\cdot \left(-1\right))/\left(60\cdot r\right)

-4,998

39.5*10 * 10 = 39.5*10^{5 - 3}

33,749

\pi*10^2 = \pi*100

118

((-1) \cdot x)/3 = -\frac13 \cdot \frac{1}{3} \cdot x = -x/3

21,047

3/8 = -\frac18 \cdot 5 + 1

6,260

b \cdot d^k/b \cdot c \cdot \frac{d^x}{c} = d^x/c \cdot c \cdot b \cdot \frac{d^k}{b}

30,334

\sin{\varphi} = \cos(-\varphi + \pi/2)

-18,960

\frac19\cdot 2 = \frac{H_p}{81\cdot \pi}\cdot 81\cdot \pi = H_p

27,274

g \eta x = g \eta x

24,418

\tan(x)/\sec(x) = \dfrac{\sin\left(x\right)}{\frac{1}{\cos(x)}}1/\cos(x) = \sin(x)

3,982

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

12,006

1*2*3*\dots*\left(2*(-1) + k\right)*((-1) + k)*k = k!

28,808

3*(-y + 2*z) = 6*z - y*3

3,949

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

30,387

301 = 350 + 49\cdot (-1)

14,261

f \cdot u + u \cdot x = u \cdot (x + f)

-6,993

\frac{4}{5} \cdot 2/3 = \frac{8}{15}

-27,688

4*\sin(x) = d/dx \left(-4*\cos(x)\right)

7,338

9\cdot r^2 = x^2\cdot 3\Longrightarrow x \cdot x = 3\cdot r^2

5,598

1278 (-1) + 8721 = 7443

-29,325

-3\cdot i - 1 + 10\cdot (-1) = -3\cdot i - 11

-9,189

60 - x*24 = -2*2*2*3*x + 2*2*3*5

10,120

3^{H/2} = 2^H + (-1) = 4^{\frac{H}{2}} + (-1)

12,494

2*x = 135 + 25 = 160 rightarrow 80 = x

36,297

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

18,766

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

21,854

c \cdot c + a \cdot a + x^2 = \sqrt{a^2 + x^2 + c^2} \cdot \sqrt{a^2 + x^2 + c^2}

-20,891

\tfrac{l \cdot 72 + 63}{28 \cdot (-1) - l \cdot 32} = -9/4 \cdot \frac{-l \cdot 8 + 7 \cdot (-1)}{-8 \cdot l + 7 \cdot (-1)}

-7,672

\frac{19 \cdot i + 8}{4 - 3 \cdot i} = \tfrac{1}{4 + i \cdot 3} \cdot (i \cdot 3 + 4) \cdot \frac{1}{4 - i \cdot 3} \cdot (8 + 19 \cdot i)

21,230

(z*3 + y + 5) (2z + y + 3(-1)) = 15 (-1) + 6z * z + 5zy + y * y + z + y*2

16,657

\left(x - 35.7 = 4.1\cdot \left(-1.34\right) \Rightarrow x = 4.1\cdot (-1.34) + 35.7\right) \Rightarrow x = 30.206

-11,767

36/49 = \left(\frac{6}{7}\right)^2

11,165

(2 \cdot x + 3) \cdot x \leq 2 \Rightarrow 2 \cdot (-1) + 2 \cdot x^2 + 3 \cdot x \leq 0

929

x * x + b*x rightarrow \left(x + b/2\right)^2 - \left(b/2\right)^2

11,296

w^2 + 3 x^2 = (x - w)^2 + (x + w)^2 + (x - w) (x + w)

31,137

1 - \frac{1}{216}*125 = \dfrac{91}{216}

22,192

1/(b\cdot d) = \frac{1}{d\cdot b}