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Math Formulas

Understanding key math formulas is essential for solving complex problems in calculus, algebra, trigonometry, geometry, and statistics. Our comprehensive math formula sheet brings together the most frequently searched concepts such as derivative formulas, integral formulas, limit formulas, and complete trigonometry identities that students rely on for fast problem-solving. High-value topics like the derivative calculator, integral calculator, statistics formulas and limit calculator often demand precise formulas including the power rule, chain rule, L'Hôpital's rule, Taylor series, and Maclaurin expansions.

For students preparing for competitive exams, our curated list includes essential geometry formulas, covering distance, midpoint, slope, circle equation, Heron's formula, and volume formulas for 3D shapes. We also feature high search topics such as matrix calculator, determinant rules, Cramer's rule, permutations, combinations, mean, median, standard deviation, and probability formulas—making it easy to simplify data analysis and statistics questions.

Whether you're working on engineering assignments, competitive test preparation, or advanced university math, these formulas provide the foundation you need to solve problems quickly and accurately. This formula library is designed to be lightweight, exam-centric, and optimized for students, teachers, and professionals searching for reliable math formula calculators and step-by-step solutions.

Total Formulas: 294

Showing 294 of 294 formulas

Power Rule

calculus
ddx(xn)=nxn−1\frac{d}{dx}(x^n)=n x^{n-1}

Constant Rule

calculus
ddx(c)=0\frac{d}{dx}(c)=0

Constant Multiple

calculus
ddx(cf)=cf′\frac{d}{dx}(c f)=c f'

Sum Rule

calculus
ddx(f+g)=f′+g′\frac{d}{dx}(f+g)=f'+g'

Product Rule

calculus
ddx(fg)=fg′+gf′\frac{d}{dx}(f g)=f g' + g f'

Quotient Rule

calculus
ddx(fg)=gf′−fg′g2\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g f' - f g'}{g^2}

Chain Rule

calculus
ddxf(g(x))=f′(g(x))g′(x)\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Derivative of e^x

calculus
ddxex=ex\frac{d}{dx}e^x=e^x

Derivative of a^x

calculus
ddxax=axln⁡a\frac{d}{dx}a^x=a^x\ln a

Derivative of ln x

calculus
ddxln⁡x=1x\frac{d}{dx}\ln x=\frac{1}{x}

Sum of many functions

calculus
ddx(∑ifi(x))=∑ifi′(x)\frac{d}{dx}\left(\sum_i f_i(x)\right)=\sum_i f_i'(x)

Derivative of ln|x|

calculus
ddxln⁡∣x∣=1x\frac{d}{dx}\ln|x|=\frac{1}{x}

Derivative of x^x

calculus
ddx(xx)=xx(ln⁡x+1)\frac{d}{dx}(x^x)=x^x(\ln x + 1)

Derivative of inverse function

calculus
(f−1)′(y)=1f′(x)(f^{-1})'(y)=\frac{1}{f'(x)}

Derivative of sec

calculus
ddxsec⁡x=sec⁡xtan⁡x\frac{d}{dx}\sec x = \sec x\tan x

Derivative of csc

calculus
ddxcsc⁡x=−csc⁡xcot⁡x\frac{d}{dx}\csc x = -\csc x\cot x

Derivative of arccot

calculus
ddxcot⁡−1x=−11+x2\frac{d}{dx}\cot^{-1}x=-\frac{1}{1+x^2}

Derivative of arcsec

calculus
ddxsec⁡−1x=1∣x∣x2−1\frac{d}{dx}\sec^{-1}x=\frac{1}{|x|\sqrt{x^2-1}}

Derivative of arccsc

calculus
ddxcsc⁡−1x=−1∣x∣x2−1\frac{d}{dx}\csc^{-1}x=-\frac{1}{|x|\sqrt{x^2-1}}

Derivative of sinh

calculus
ddxsinh⁡x=cosh⁡x\frac{d}{dx}\sinh x=\cosh x

Derivative of cosh

calculus
ddxcosh⁡x=sinh⁡x\frac{d}{dx}\cosh x=\sinh x

Derivative of tanh

calculus
ddxtanh⁡x=sech⁡2x\frac{d}{dx}\tanh x=\operatorname{sech}^2 x

Integral power rule

calculus
∫xn dx=xn+1n+1+C (n≠−1)\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\ (n\neq -1)

Integral of 1/x

calculus
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx=\ln|x|+C

Integral of sin

calculus
∫sin⁡x dx=−cos⁡x+C\int \sin x\,dx=-\cos x + C

Integral of cos

calculus
∫cos⁡x dx=sin⁡x+C\int \cos x\,dx=\sin x + C

Integral of sec^2

calculus
∫sec⁡2x dx=tan⁡x+C\int \sec^2 x\,dx=\tan x + C

Integral tan

calculus
∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int\tan x\,dx=-\ln|\cos x|+C

Integral cot

calculus
∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int\cot x\,dx=\ln|\sin x|+C

Integral sec

calculus
∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C\int\sec x\,dx=\ln|\sec x+\tan x|+C

Integral csc

calculus
∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+C\int\csc x\,dx=\ln|\csc x - \cot x|+C

Integral e^{ax}

calculus
∫eaxdx=1aeax+C\int e^{ax}dx=\frac{1}{a}e^{ax}+C

Repeated integration by parts

calculus
∫udv=uv−∫vdu\int u dv = uv - \int v du

Weierstrass substitution

calculus
t=tan⁡x2⇒dx=2dt1+t2t=\tan\frac{x}{2}\Rightarrow dx=\frac{2dt}{1+t^2}

Improper integral definition

calculus
∫a∞f(x) dx=lim⁡t→∞∫atf(x) dx\int_a^\infty f(x)\,dx=\lim_{t\to\infty}\int_a^t f(x)\,dx

Parametric dy/dx

calculus
dydx=dydtdxdt\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}

Second derivative (parametric)

calculus
d2ydx2=ddt(dydx)dxdt\frac{d^2y}{dx^2}=\frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}

e^x series

calculus
ex=∑n=0∞xnn!e^x=\sum_{n=0}^\infty\frac{x^n}{n!}

sin x series

calculus
sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!\sin x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{(2n+1)!}

cos x series

calculus
cos⁡x=∑n=0∞(−1)nx2n(2n)!\cos x=\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}

L'Hôpital 0/0

calculus
lim⁡fg=lim⁡f′g′ if 00\lim\frac{f}{g}=\lim\frac{f'}{g'}\text{ if }\frac{0}{0}

L'Hôpital ∞/∞

calculus
lim⁡fg=lim⁡f′g′ if ∞∞\lim\frac{f}{g}=\lim\frac{f'}{g'}\text{ if }\frac{\infty}{\infty}

Partial derivative

calculus
∂f∂x\frac{\partial f}{\partial x}

Gradient

calculus
∇f=(∂f∂x,∂f∂y,∂f∂z)\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)

Divergence

calculus
∇⋅F⃗\nabla\cdot \vec{F}

Curl

calculus
∇×F⃗\nabla\times \vec{F}

Jacobian

calculus
J=det⁡[∂(x1,…,xn)∂(u1,…,un)]J=\det\left[\frac{\partial(x_1,\dots,x_n)}{\partial(u_1,\dots,u_n)}\right]

Green's Theorem

calculus
∮C(L dx+M dy)=∬D(∂M∂x−∂L∂y)dA\oint_C (L\,dx+M\,dy)=\iint_D\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dA

Stokes Theorem

calculus
∫S(∇×F)⋅dS=∮∂SF⋅dr\int_S (\nabla\times F)\cdot dS=\oint_{\partial S}F\cdot dr

Gauss Divergence Theorem

calculus
∭V(∇⋅F)dV=∬∂VF⋅n dS\iiint_V(\nabla\cdot F)dV=\iint_{\partial V}F\cdot n\,dS

Pythagorean identity

trigonometry
sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1

Tangent/Secant identity

trigonometry
1+tan⁡2x=sec⁡2x1+\tan^2 x = \sec^2 x

Cot/Cosecant identity

trigonometry
1+cot⁡2x=csc⁡2x1+\cot^2 x = \csc^2 x

Reciprocal of sin

trigonometry
csc⁡x=1sin⁡x\csc x = \frac{1}{\sin x}

Reciprocal of cos

trigonometry
sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}

Reciprocal of tan

trigonometry
cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x}

Tan quotient

trigonometry
tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

Cot quotient

trigonometry
cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}

Sine sum

trigonometry
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)=\sin A \cos B \pm \cos A \sin B

Cosine sum

trigonometry
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B)=\cos A \cos B \mp \sin A \sin B

tan(A+B)

trigonometry
tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}

tan(A−B)

trigonometry
tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡Atan⁡B\tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}

cot(A+B)

trigonometry
cot⁡(A+B)=cot⁡Acot⁡B−1cot⁡A+cot⁡B\cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}

Cofunction sin

trigonometry
sin⁡(π2−x)=cos⁡x\sin\left(\frac{\pi}{2}-x\right)=\cos x

Cofunction cos

trigonometry
cos⁡(π2−x)=sin⁡x\cos\left(\frac{\pi}{2}-x\right)=\sin x

Cofunction tan

trigonometry
tan⁡(π2−x)=cot⁡x\tan\left(\frac{\pi}{2}-x\right)=\cot x

Cos even

trigonometry
cos⁡(−x)=cos⁡x\cos(-x)=\cos x

Sin odd

trigonometry
sin⁡(−x)=−sin⁡x\sin(-x)=-\sin x

Tan odd

trigonometry
tan⁡(−x)=−tan⁡x\tan(-x)=-\tan x

Double-angle sin

trigonometry
sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2\sin A \cos A

Double-angle cos

trigonometry
cos⁡2A=cos⁡2A−sin⁡2A\cos 2A = \cos^2 A - \sin^2 A

tan 2A

trigonometry
tan⁡2A=2tan⁡A1−tan⁡2A\tan 2A = \frac{2\tan A}{1-\tan^2 A}

cot 2A

trigonometry
cot⁡2A=cot⁡2A−12cot⁡A\cot 2A = \frac{\cot^2 A-1}{2\cot A}

sin 2A (alt)

trigonometry
sin⁡2A=2tan⁡A1+tan⁡2A\sin 2A = \frac{2\tan A}{1+\tan^2 A}

sin 3A

trigonometry
sin⁡3A=3sin⁡A−4sin⁡3A\sin 3A = 3\sin A - 4\sin^3 A

cos 3A

trigonometry
cos⁡3A=4cos⁡3A−3cos⁡A\cos 3A = 4\cos^3 A - 3\cos A

tan 3A

trigonometry
tan⁡3A=3tan⁡A−tan⁡3A1−3tan⁡2A\tan 3A = \frac{3\tan A - \tan^3 A}{1-3\tan^2 A}

Half-angle (sin^2)

trigonometry
sin⁡2x=1−cos⁡2x2\sin^2 x = \frac{1-\cos 2x}{2}

Half-angle (cos^2)

trigonometry
cos⁡2x=1+cos⁡2x2\cos^2 x = \frac{1+\cos 2x}{2}

sin A sin B

trigonometry
sin⁡Asin⁡B=12[cos⁡(A−B)−cos⁡(A+B)]\sin A \sin B = \frac{1}{2}[\cos(A-B)-\cos(A+B)]

cos A cos B

trigonometry
cos⁡Acos⁡B=12[cos⁡(A−B)+cos⁡(A+B)]\cos A \cos B = \frac{1}{2}[\cos(A-B)+\cos(A+B)]

sin A cos B

trigonometry
sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin(A+B)+\sin(A-B)]

sin A + sin B

trigonometry
sin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}

sin A − sin B

trigonometry
sin⁡A−sin⁡B=2cos⁡A+B2sin⁡A−B2\sin A-\sin B=2\cos\frac{A+B}{2}\sin\frac{A-B}{2}

cos A + cos B

trigonometry
cos⁡A+cos⁡B=2cos⁡A+B2cos⁡A−B2\cos A+\cos B=2\cos\frac{A+B}{2}\cos\frac{A-B}{2}

cos A − cos B

trigonometry
cos⁡A−cos⁡B=−2sin⁡A+B2sin⁡A−B2\cos A-\cos B=-2\sin\frac{A+B}{2}\sin\frac{A-B}{2}

sin(arcsin x)

trigonometry
sin⁡(sin⁡−1x)=x\sin(\sin^{-1}x)=x

cos(arccos x)

trigonometry
cos⁡(cos⁡−1x)=x\cos(\cos^{-1}x)=x

tan(arctan x)

trigonometry
tan⁡(tan⁡−1x)=x\tan(\tan^{-1}x)=x

arcsin relation

trigonometry
sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}

Law of Sines

trigonometry
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Law of Cosines

trigonometry
c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C

Law of Tangents

trigonometry
a−ba+b=tan⁡A−B2tan⁡A+B2\frac{a-b}{a+b}=\frac{\tan\frac{A-B}{2}}{\tan\frac{A+B}{2}}

sinh definition

trigonometry
sinh⁡x=ex−e−x2\sinh x=\frac{e^x - e^{-x}}{2}

cosh definition

trigonometry
cosh⁡x=ex+e−x2\cosh x=\frac{e^x + e^{-x}}{2}

tanh definition

trigonometry
tanh⁡x=sinh⁡xcosh⁡x\tanh x=\frac{\sinh x}{\cosh x}

cosh²-sinh² identity

trigonometry
cosh⁡2x−sinh⁡2x=1\cosh^2 x - \sinh^2 x = 1

Radian-degree conversion

trigonometry
180∘=π radians180^\circ = \pi\ \text{radians}

Degrees to radians

trigonometry
θrad=θdeg⋅π180\theta_{\text{rad}}=\theta_{\text{deg}}\cdot\frac{\pi}{180}

Radians to degrees

trigonometry
θdeg=θrad⋅180π\theta_{\text{deg}}=\theta_{\text{rad}}\cdot\frac{180}{\pi}

(a+b)^2

algebra
(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2

algebra
(a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2

(a+b)^3

algebra
(a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

(a-b)^3

algebra
(a−b)3=a3−3a2b+3ab2−b3(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

a³ - b³

algebra
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)

a³ + b³

algebra
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)

Rational root theorem

algebra
Possible roots=±factors of constantfactors of leading\text{Possible roots} = \pm \frac{\text{factors of constant}}{\text{factors of leading}}

Difference of squares

algebra
a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)

Completing the square

algebra
a(x−h)2+ka(x-h)^2 + k

Sum of polynomial

algebra
P(a)+P(b)P(a)+P(b)

Remainder theorem

algebra
P(a)=remainder of P(x)÷(x−a)P(a)=\text{remainder of }P(x)\div(x-a)

Factor theorem

algebra
(x−a) divides P(x)  ⟺  P(a)=0(x-a)\text{ divides }P(x)\iff P(a)=0

Quadratic formula

algebra
x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Quadratic vertex x-coordinate

algebra
x=−b2ax=-\frac{b}{2a}

Sum of arithmetic series

algebra
Sn=n2(2a+(n−1)d)S_n=\frac{n}{2}(2a+(n-1)d)

Sum of geometric series

algebra
Sn=a1−rn1−rS_n=a\frac{1-r^n}{1-r}

Infinite geometric series

algebra
S∞=a1−r, ∣r∣<1S_\infty=\frac{a}{1-r},\ |r|<1

nth harmonic number

algebra
Hn=∑k=1n1kH_n=\sum_{k=1}^n \frac{1}{k}

a^m * a^n

algebra
aman=am+na^m a^n = a^{m+n}

a^m / a^n

algebra
aman=am−n\frac{a^m}{a^n}=a^{m-n}

(a^m)^n

algebra
(am)n=amn(a^m)^n = a^{mn}

nth root

algebra
an=a1/n\sqrt[n]{a}=a^{1/n}

Function composition

algebra
(f∘g)(x)=f(g(x))(f\circ g)(x)=f(g(x))

Inverse function

algebra
f(f−1(x))=xf(f^{-1}(x))=x

|ab|

algebra
∣ab∣=∣a∣∣b∣|ab| = |a||b|

|a/b|

algebra
∣ab∣=∣a∣∣b∣\left|\frac{a}{b}\right|=\frac{|a|}{|b|}

Triangle inequality

algebra
∣a+b∣≤∣a∣+∣b∣|a+b|\le |a|+|b|

AM-GM Inequality

algebra
a+b2≥ab\frac{a+b}{2}\ge\sqrt{ab}

Cauchy-Schwarz

algebra
(a12+⋯+an2)(b12+⋯+bn2)≥(a1b1+⋯+anbn)2(a_1^2+\dots+a_n^2)(b_1^2+\dots+b_n^2)\ge(a_1b_1+\dots+a_n b_n)^2

Jensen's Inequality

algebra
f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)f(\lambda x + (1-\lambda)y)\le\lambda f(x)+(1-\lambda)f(y)

Cramer's rule (2x2)

algebra
x=∣ba2db2∣∣aa2cc2∣x=\frac{\begin{vmatrix}b & a_2 \\ d & b_2\end{vmatrix}}{\begin{vmatrix}a & a_2 \\ c & c_2\end{vmatrix}}

Linear system general

algebra
Ax⃗=b⃗A\vec{x}=\vec{b}

Euler formula

algebra
eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta + i\sin\theta

cis notation

algebra
cis⁡θ=cos⁡θ+isin⁡θ\operatorname{cis}\theta=\cos\theta+i\sin\theta

Multiply complex numbers

algebra
(a+ib)(c+id)=(ac−bd)+i(ad+bc)(a+ib)(c+id)=(ac-bd)+i(ad+bc)

Symmetric sum

algebra
σ1=a+b+c\sigma_1=a+b+c

General binomial

algebra
(x+y)n=∑k=0n(nk)xn−kyk(x+y)^n=\sum_{k=0}^n{n\choose k}x^{n-k}y^k

Multinomial theorem

algebra
(x1+x2+⋯+xm)n=∑k1+⋯+km=nn!k1!…km!∏ixiki(x_1+x_2+\dots+x_m)^n=\sum_{k_1+\dots+k_m=n}\frac{n!}{k_1!\dots k_m!}\prod_i x_i^{k_i}

Binomial theorem

algebra
(a+b)n=∑k=0n(nk)an−kbk(a+b)^n=\sum_{k=0}^n {n \choose k} a^{n-k} b^k

Arithmetic sequence

algebra
an=a1+(n−1)da_n=a_1+(n-1)d

Geometric sequence

algebra
an=a1r n−1a_n=a_1 r^{\,n-1}

Infinite GP sum

algebra
S=a1−r(∣r∣<1)S=\frac{a}{1-r}\quad(|r|<1)

Log product

logarithms
log⁡(xy)=log⁡x+log⁡y\log(xy)=\log x + \log y

Log quotient

logarithms
log⁡(xy)=log⁡x−log⁡y\log\left(\frac{x}{y}\right)=\log x - \log y

Log power

logarithms
log⁡(xn)=nlog⁡x\log(x^n)=n\log x

Change of base

logarithms
log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

Definition of divisibility

number theory
a∣b  ⟺  ∃k∈Z such that b=aka\mid b \iff \exists k\in\mathbb{Z}\text{ such that }b=ak

Divisibility by zero

number theory
a∣0 for all integers aa\mid 0\ \text{for all integers }a

Divisibility rule for 2

number theory
If last digit is even\text{If last digit is even}

Divisibility rule for 3

number theory
If sum of digits divisible by 3\text{If sum of digits divisible by 3}

Divisibility rule for 5

number theory
Last digit is 0 or 5\text{Last digit is 0 or 5}

Divisibility rule for 9

number theory
Sum of digits divisible by 9\text{Sum of digits divisible by 9}

Divisibility rule for 11

number theory
Alternating sum of digits divisible by 11\text{Alternating sum of digits divisible by 11}

Greatest common divisor (GCD)

number theory
gcd⁡(a,b)=max⁡{d∣d∣a and d∣b}\gcd(a,b)=\max\{d\mid d\mid a\ \text{and}\ d\mid b\}

Least common multiple (LCM)

number theory
lcm⁡(a,b)=∣ab∣gcd⁡(a,b)\operatorname{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)}

Euclidean Algorithm

number theory
gcd⁡(a,b)=gcd⁡(b,a mod b)\gcd(a,b)=\gcd(b,a\bmod b)

Linear Diophantine equation

number theory
ax+by=c has solution iff gcd⁡(a,b)∣cax+by=c\ \text{has solution iff }\gcd(a,b)\mid c

General linear equation solutions

number theory
x=x0+bgcd⁡(a,b)t, y=y0−agcd⁡(a,b)tx=x_0+\frac{b}{\gcd(a,b)}t,\ y=y_0-\frac{a}{\gcd(a,b)}t

Prime number definition

number theory
p is prime iff p>1 and p∣ab⇒p∣a or p∣bp\text{ is prime iff }p>1\text{ and }p\mid ab\Rightarrow p\mid a\text{ or }p\mid b

Prime counting function

number theory
π(n)≈nln⁡n\pi(n)\approx\frac{n}{\ln n}

Wilson's Theorem

number theory
(p−1)!≡−1(modp) for prime p(p-1)!\equiv -1\pmod p\ \text{for prime }p

Fundamental theorem of arithmetic

number theory
n=∏piki unique prime factorizationn=\prod p_i^{k_i}\ \text{unique prime factorization}

Modular equivalence

number theory
a≡b(modn)  ⟺  n∣(a−b)a\equiv b\pmod{n}\iff n\mid(a-b)

Mod addition

number theory
(a+b) mod n(a+b)\bmod n

Mod subtraction

number theory
(a−b) mod n(a-b)\bmod n

Mod multiplication

number theory
(ab) mod n(ab)\bmod n

Mod exponentiation

number theory
ak mod na^k\bmod n

Modular inverse

number theory
a−1≡x(modn)  ⟺  ax≡1(modn)a^{-1}\equiv x\pmod{n}\iff ax\equiv 1\pmod{n}

Fermat's Little Theorem

number theory
ap−1≡1(modp) (p prime)a^{p-1}\equiv 1\pmod{p}\ (p\ \text{prime})

Euler Totient function

number theory
ϕ(n)=n∏p∣n(1−1p)\phi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right)

Euler's Theorem

number theory
aϕ(n)≡1(modn) if gcd⁡(a,n)=1a^{\phi(n)}\equiv 1\pmod{n}\text{ if }\gcd(a,n)=1

ax ≡ b (mod n)

number theory
ax≡b(modn) solvable iff gcd⁡(a,n)∣bax\equiv b\pmod{n}\text{ solvable iff }\gcd(a,n)\mid b

Chinese Remainder Theorem

number theory
x≡a1(modn1), x≡a2(modn2)⇒∃! x(modn1n2) when gcd⁡(n1,n2)=1x\equiv a_1\pmod{n_1},\ x\equiv a_2\pmod{n_2}\Rightarrow\exists!\ x\pmod{n_1n_2}\text{ when }\gcd(n_1,n_2)=1

Divisor function τ(n)

number theory
τ(n)=∏(ki+1)\tau(n)=\prod (k_i+1)

Sum-of-divisors σ(n)

number theory
σ(n)=∏piki+1−1pi−1\sigma(n)=\prod\frac{p_i^{k_i+1}-1}{p_i-1}

Sum of first n numbers

number theory
∑k=1nk=n(n+1)2\sum_{k=1}^n k = \frac{n(n+1)}{2}

Sum of squares

number theory
∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}

Sum of cubes

number theory
∑k=1nk3=(n(n+1)2)2\sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2

Pell's Equation

number theory
x2−Dy2=1x^2 - Dy^2 = 1

Legendre symbol

number theory
(ap)≡ap−12(modp)\left(\frac{a}{p}\right)\equiv a^{\frac{p-1}{2}}\pmod{p}

Mobius function μ(n)

number theory
μ(n)=(−1)k if n is product of k distinct primes\mu(n)=(-1)^k\text{ if n is product of k distinct primes}

Riemann zeta (discrete)

number theory
ζ(s)=∑n=1∞1ns\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}

Bézout Identity

number theory
gcd⁡(a,b)=ax+by for some integers x,y\gcd(a,b)=ax+by\ \text{for some integers }x,y

Triangle area

geometry
A=12bhA=\frac{1}{2}bh

Heron's formula

geometry
A=s(s−a)(s−b)(s−c), s=a+b+c2A=\sqrt{s(s-a)(s-b)(s-c)},\ s=\frac{a+b+c}{2}

Area using sine

geometry
A=12absin⁡CA=\frac{1}{2}ab\sin C

Triangle inradius

geometry
r=Asr=\frac{A}{s}

Triangle circumradius

geometry
R=abc4AR=\frac{abc}{4A}

Centroid of triangle

geometry
G=(x1+x2+x33,y1+y2+y33)G=\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)

Orthocenter condition

geometry
Intersection of altitudes\text{Intersection of altitudes}

Midpoint theorem

geometry
Line joining midpoints =12 third side\text{Line joining midpoints }=\frac{1}{2}\text{ third side}

Square area

geometry
A=s2A=s^2

Rectangle area

geometry
A=lwA=lw

Parallelogram area

geometry
A=bhA=bh

Parallelogram using sine

geometry
A=absin⁡θA=ab\sin\theta

Trapezoid area

geometry
A=12(a+b)hA=\frac{1}{2}(a+b)h

Regular polygon area

geometry
A=12nR2sin⁡2πnA=\frac{1}{2}nR^2\sin\frac{2\pi}{n}

Side of regular n-gon

geometry
s=2Rsin⁡πns=2R\sin\frac{\pi}{n}

Circle area

geometry
A=πr2A=\pi r^2

Circle circumference

geometry
C=2πrC=2\pi r

Sector area

geometry
A=θ2ππr2=12r2θA=\frac{\theta}{2\pi}\pi r^2=\frac{1}{2}r^2\theta

Arc length

geometry
L=rθL=r\theta

Chord length

geometry
c=2rsin⁡θ2c=2r\sin\frac{\theta}{2}

Circle segment area

geometry
A=r2cos⁡−1r−hr−(r−h)2rh−h2A=r^2\cos^{-1}\frac{r-h}{r}-(r-h)\sqrt{2rh-h^2}

Power of a point

geometry
PA⋅PB=PC⋅PDPA\cdot PB = PC\cdot PD

Slope

geometry
m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}

Two-point form

geometry
y−y1=y2−y1x2−x1(x−x1)y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)

Point-slope form

geometry
y−y1=m(x−x1)y-y_1=m(x-x_1)

Slope-intercept form

geometry
y=mx+by=mx+b

Intercept form

geometry
xa+yb=1\frac{x}{a}+\frac{y}{b}=1

Normal form

geometry
xcos⁡α+ysin⁡α=px\cos\alpha+y\sin\alpha=p

Distance from point to line

geometry
d=∣Ax+By+C∣A2+B2d=\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}

Parabola vertex form

geometry
y=a(x−h)2+ky=a(x-h)^2+k

General ellipse

geometry
x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

General hyperbola

geometry
x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

Ellipse eccentricity

geometry
e=1−b2a2e=\sqrt{1-\frac{b^2}{a^2}}

Hyperbola eccentricity

geometry
e=1+b2a2e=\sqrt{1+\frac{b^2}{a^2}}

Rotation matrix

geometry
(x′y′)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)(xy)\begin{pmatrix}x'\\y'\end{pmatrix}=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}

Reflection in x-axis

geometry
(x,y)→(x,−y)(x,y)\to(x,-y)

Reflection in y-axis

geometry
(x,y)→(−x,y)(x,y)\to(-x,y)

Reflection in origin

geometry
(x,y)→(−x,−y)(x,y)\to(-x,-y)

Translation

geometry
(x,y)→(x+a,y+b)(x,y)\to(x+a,y+b)

Dilation

geometry
(x,y)→(kx,ky)(x,y)\to(kx,ky)

Distance in 3D

geometry
d=(x2−x1)2+(y2−y1)2+(z2−z1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Plane equation

geometry
Ax+By+Cz+D=0Ax+By+Cz+D=0

Distance from point to plane

geometry
d=∣Ax+By+Cz+D∣A2+B2+C2d=\frac{|Ax+By+Cz+D|}{\sqrt{A^2+B^2+C^2}}

Direction cosines

geometry
cos⁡α=aa2+b2+c2, cos⁡β=ba2+b2+c2, cos⁡γ=ca2+b2+c2\cos\alpha=\frac{a}{\sqrt{a^2+b^2+c^2}},\ \cos\beta=\frac{b}{\sqrt{a^2+b^2+c^2}},\ \cos\gamma=\frac{c}{\sqrt{a^2+b^2+c^2}}

Cube volume

geometry
V=a3V=a^3

Cube surface area

geometry
A=6a2A=6a^2

Rectangular prism volume

geometry
V=lwhV=lwh

Rectangular prism surface area

geometry
A=2(lw+wh+lh)A=2(lw+wh+lh)

Sphere volume

geometry
V=43πr3V=\frac{4}{3}\pi r^3

Sphere surface area

geometry
A=4πr2A=4\pi r^2

Cylinder volume

geometry
V=πr2hV=\pi r^2 h

Cylinder surface area

geometry
A=2πr(r+h)A=2\pi r(r+h)

Cone volume

geometry
V=13πr2hV=\frac{1}{3}\pi r^2h

Cone surface area

geometry
A=πr(r+r2+h2)A=\pi r(r+\sqrt{r^2+h^2})

Frustum volume

geometry
V=13πh(r12+r1r2+r22)V=\frac{1}{3}\pi h(r_1^2+r_1 r_2+r_2^2)

Pyramid volume

geometry
V=13BhV=\frac{1}{3}Bh

Triangular prism volume

geometry
V=12bhLV=\frac{1}{2}b h L

Distance between points

geometry
d=(x2−x1)2+(y2−y1)2d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

Midpoint

geometry
M=(x1+x22,y1+y22)M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)

Equation of circle

geometry
(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

Parabola (standard)

geometry
y2=4axy^2=4ax

Ellipse (standard)

geometry
x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

Hyperbola (standard)

geometry
x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

Rectangular form

complex numbers
z=x+iyz=x+iy

Modulus & argument

complex numbers
∣z∣=x2+y2, arg⁡z=tan⁡−1yx|z|=\sqrt{x^2+y^2},\ \arg z=\tan^{-1}\frac{y}{x}

De Moivre

complex numbers
(r(cos⁡θ+isin⁡θ))n=rn(cos⁡nθ+isin⁡nθ)(r(\cos\theta+i\sin\theta))^n = r^n(\cos n\theta + i\sin n\theta)

Dot product

vectors
a⋅b=axbx+ayby+azbz\mathbf{a}\cdot\mathbf{b}=a_x b_x + a_y b_y + a_z b_z

Cross product magnitude

vectors
∣a×b∣=∣a∣∣b∣sin⁡θ|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta

Projection of a on b

vectors
proj⁡ba=(a⋅b∣b∣2)b\operatorname{proj}_{\mathbf{b}}\mathbf{a}=\left(\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}\right)\mathbf{b}

Determinant 2x2

matrices
det⁡(abcd)=ad−bc\det\begin{pmatrix}a & b \\ c & d\end{pmatrix} = ad - bc

Determinant 3x3

matrices
det⁡(abcdefghi)=a(ei−fh)−b(di−fg)+c(dh−eg)\det\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)

Determinant n×n

matrices
det⁡(A)=∑j=1n(−1)1+ja1jdet⁡(M1j)\det(A) = \sum_{j=1}^{n} (-1)^{1+j} a_{1j} \det(M_{1j})

Inverse 2x2

matrices
(abcd)−1=1ad−bc(d−b−ca)\begin{pmatrix}a & b \\ c & d\end{pmatrix}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Matrix multiplication

matrices
(AB)ij=∑kAikBkj(AB)_{ij} = \sum_{k} A_{ik} B_{kj}

Transpose

matrices
(AT)ij=Aji(A^T)_{ij} = A_{ji}

Rank-Nullity Theorem

matrices
rank(A)+nullity(A)=n\text{rank}(A) + \text{nullity}(A) = n

Eigenvalue equation

matrices
Av⃗=λv⃗A \vec{v} = \lambda \vec{v}

Characteristic equation

matrices
det⁡(A−λI)=0\det(A - \lambda I) = 0

Inner product

matrices
⟨u⃗,v⃗⟩=∑iuivi\langle \vec{u}, \vec{v} \rangle = \sum_i u_i v_i

Linear transformation

matrices
T(x⃗)=Ax⃗T(\vec{x}) = A \vec{x}

Mean

statistics
xˉ=1n∑i=1nxi\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i

Variance (population)

statistics
σ2=1n∑i=1n(xi−xˉ)2\sigma^2=\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2

Variance (sample)

statistics
s2=1n−1∑i=1n(xi−xˉ)2s^2=\frac{1}{n-1}\sum_{i=1}^n (x_i-\bar{x})^2

Standard deviation (population)

statistics
σ=σ2\sigma=\sqrt{\sigma^2}

Standard deviation (sample)

statistics
s=s2s=\sqrt{s^2}

Median

statistics
Median=middle value of ordered data\text{Median} = \text{middle value of ordered data}

Mode

statistics
Mode=value that occurs most frequently\text{Mode} = \text{value that occurs most frequently}

Range

statistics
Range=xmax−xmin\text{Range} = x_\text{max} - x_\text{min}

Classical probability

statistics
P(A)=favorable outcomestotal outcomesP(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}

Conditional probability

statistics
P(A∣B)=P(A∩B)P(B)P(A|B)=\frac{P(A \cap B)}{P(B)}

Addition rule of probability

statistics
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Z-score

statistics
z=x−xˉσz = \frac{x-\bar{x}}{\sigma}

Covariance

statistics
Cov(X,Y)=1n∑i=1n(xi−xˉ)(yi−yˉ)\text{Cov}(X,Y) = \frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})

Correlation coefficient

statistics
r=Cov(X,Y)σXσYr = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}

Limit sinx/x

limits series
lim⁡x→0sin⁡xx=1\lim_{x\to 0}\frac{\sin x}{x} = 1

Exponential limit

limits series
lim⁡x→∞(1+kx)x=ek\lim_{x\to\infty}\left(1+\frac{k}{x}\right)^x = e^{k}

(1-cos)/x^2

limits series
lim⁡x→01−cos⁡xx2=12\lim_{x\to 0}\frac{1-\cos x}{x^2} = \frac{1}{2}

Geometric series

limits series
∑k=0∞ark=a1−r(∣r∣<1)\sum_{k=0}^\infty ar^k = \frac{a}{1-r}\quad(|r|<1)

p-series

limits series
∑n=1∞1np converges iff p>1\sum_{n=1}^\infty \frac{1}{n^p}\ \text{converges iff } p>1

1st order linear ODE (integrating factor)

differential equations
dydx+P(x)y=Q(x)  ⟹  yμ=∫Qμ dx, μ=e∫Pdx\frac{dy}{dx} + P(x)y = Q(x)\implies y\mu = \int Q\mu\,dx,\ \mu=e^{\int Pdx}

Separable ODE

differential equations
dydx=g(x)h(y)  ⟹  ∫1h(y)dy=∫g(x)dx\frac{dy}{dx}=g(x)h(y)\implies \int \frac{1}{h(y)}dy=\int g(x)dx

Trapezoidal rule (composite)

numerical methods
∫abf(x) dx≈h2(y0+yn+2∑k=1n−1yk)\int_a^b f(x)\,dx \approx \frac{h}{2}\Big(y_0+y_n+2\sum_{k=1}^{n-1}y_k\Big)

Simpson's rule

numerical methods
∫abf(x) dx≈h3(y0+yn+4∑oddyk+2∑evenyk)\int_a^b f(x)\,dx \approx \frac{h}{3}\Big(y_0+y_n+4\sum_{odd} y_k + 2\sum_{even} y_k\Big)

Euler's method

numerical methods
yn+1=yn+hf(xn,yn)y_{n+1} = y_n + h f(x_n, y_n)

Runge-Kutta 4th order (RK4)

numerical methods
yn+1=yn+h6(k1+2k2+2k3+k4),with k1=f(xn,yn),k2=f(xn+h2,yn+h2k1),k3=f(xn+h2,yn+h2k2),k4=f(xn+h,yn+hk3)y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4), \text{with } k_1=f(x_n,y_n), k_2=f(x_n+\frac{h}{2},y_n+\frac{h}{2}k_1), k_3=f(x_n+\frac{h}{2},y_n+\frac{h}{2}k_2), k_4=f(x_n+h,y_n+hk_3)

Forward difference derivative

numerical methods
f′(xi)≈f(xi+1)−f(xi)hf'(x_i) \approx \frac{f(x_{i+1}) - f(x_i)}{h}

Backward difference derivative

numerical methods
f′(xi)≈f(xi)−f(xi−1)hf'(x_i) \approx \frac{f(x_i) - f(x_{i-1})}{h}

Central difference derivative

numerical methods
f′(xi)≈f(xi+1)−f(xi−1)2hf'(x_i) \approx \frac{f(x_{i+1}) - f(x_{i-1})}{2h}

Newton-Raphson method

numerical methods
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Secant method

numerical methods
xn+1=xn−f(xn)xn−xn−1f(xn)−f(xn−1)x_{n+1} = x_n - f(x_n)\frac{x_n-x_{n-1}}{f(x_n)-f(x_{n-1})}

Bisection method

numerical methods
xmid=a+b2,choose subinterval where f(a)f(xmid)<0x_{mid} = \frac{a+b}{2}, \text{choose subinterval where } f(a)f(x_{mid}) < 0
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