Formula Reference

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Showing 117 formulas

Quadratic Formula

Class 10algebra
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Roots of a quadratic equation ax^2 + bx + c = 0

#roots#equations#quadratic

Difference of Squares

Class 10algebra
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Factoring difference of two squares

#factoring#polynomials

Sum of Cubes

Class 10algebra
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring sum of two cubes

#factoring#cubes

Difference of Cubes

Class 10algebra
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring difference of two cubes

#factoring#cubes

Binomial Theorem

Class 12algebra
(x+y)n=∑k=0n(nk)xn−kyk(x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Expansion of a binomial power

#expansion#series#binomial

Arithmetic Progression (nth Term)

Class 10algebra
an=a1+(n−1)da_n = a_1 + (n-1)d

nth term of an AP with common difference d

#sequences#ap

Arithmetic Progression (Sum)

Class 10algebra
Sn=n2(2a1+(n−1)d)=n2(a1+an)S_n = \frac{n}{2}(2a_1 + (n-1)d) = \frac{n}{2}(a_1 + a_n)

Sum of first n terms of an AP

#series#ap

Geometric Progression (nth Term)

Class 12algebra
an=a1rn−1a_n = a_1 r^{n-1}

nth term of a GP with common ratio r

#sequences#gp

Geometric Progression (Sum)

Class 12algebra
Sn=a1(1−rn)1−r(r≠1)S_n = \frac{a_1(1-r^n)}{1-r} \quad (r \neq 1)

Sum of first n terms of a GP

#series#gp

Infinite Geometric Series

Class 12algebra
S∞=a11−r(∣r∣<1)S_\infty = \frac{a_1}{1-r} \quad (|r| < 1)

Sum of infinite GP when ratio is strictly between -1 and 1

#series#infinite

Vieta's Formulas (Quadratic)

Class 10algebra
x1+x2=−ba,x1x2=cax_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Sum and product of roots of ax^2 + bx + c = 0

#roots#vieta#polynomials

Completing the Square

Class 10algebra
ax2+bx+c=a(x+b2a)2+(c−b24a)ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

Rewriting quadratic into vertex form

#quadratic#factoring

Pythagorean Identity

Class 10trigonometry
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Fundamental trigonometric identity relating sine and cosine

#identities#pythagorean

Secant & Tangent Identity

Class 10trigonometry
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta

Pythagorean identity for tangent and secant

#identities#pythagorean

Cosecant & Cotangent Identity

Class 10trigonometry
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Pythagorean identity for cotangent and cosecant

#identities#pythagorean

Sine Addition & Subtraction

Class 12trigonometry
sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Sine of sum or difference of two angles

#compound#addition

Cosine Addition & Subtraction

Class 12trigonometry
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Cosine of sum or difference of two angles

#compound#addition

Tangent Addition & Subtraction

Class 12trigonometry
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Tangent of sum or difference of two angles

#compound#addition

Double Angle (Sine)

Class 12trigonometry
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta

Sine of double angle

#double-angle

Double Angle (Cosine)

Class 12trigonometry
cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Cosine of double angle formulas

#double-angle

Half Angle Formulas

Class 12trigonometry
sin⁡(θ2)=±1−cos⁡θ2,cos⁡(θ2)=±1+cos⁡θ2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}, \quad \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}

Sine and cosine of half angle

#half-angle

Law of Sines

Class 10trigonometry
asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Relates sides of a triangle to the sines of its angles

Interactive Triangle Solver (SAS):
Side c (opposite to ∠C):—
#triangles#sines

Law of Cosines

Class 10trigonometry
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Relates sides of a triangle to cosine of an angle

Interactive Triangle Solver (SAS):
Side c (opposite to ∠C):—
#triangles#cosines

Power Rule (Derivative)

Class 12calculus
ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}

Derivative of a power function

#derivative#power

Product Rule

Class 12calculus
ddx[u⋅v]=u′v+uv′\frac{d}{dx}[u \cdot v] = u'v + uv'

Derivative of product of two functions

#derivative#product

Quotient Rule

Class 12calculus
ddx[uv]=u′v−uv′v2\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}

Derivative of quotient of two functions

#derivative#quotient

Chain Rule

Class 12calculus
ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Derivative of composite functions

#derivative#chain

Derivative of Exponential

Class 12calculus
ddx[ex]=ex,ddx[ax]=axln⁡a\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln a

Derivative of natural and base-a exponential functions

#derivative#exponential

Derivative of Logarithm

Class 12calculus
ddx[ln⁡x]=1x,ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

Derivative of natural and base-a logarithms

#derivative#logarithm

Standard Trig Derivatives

Class 12calculus
ddx[sin⁡x]=cos⁡x,ddx[cos⁡x]=−sin⁡x,ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin x, \quad \frac{d}{dx}[\tan x] = \sec^2 x

Derivatives of basic trigonometric functions

#derivative#trigonometry

Power Rule (Integral)

Class 12calculus
∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Indefinite integral of power function

#integral#power

Reciprocal Integral

Class 12calculus
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C

Integral of 1/x yielding natural logarithm

#integral#logarithm

Integration by Parts

Higher Edcalculus
∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Standard integration technique for products

#integral#parts

Fundamental Theorem of Calculus

Higher Edcalculus
∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)

Evaluating definite integral from antiderivative F(x)

#integral#theorem

L'Hôpital's Rule

Higher Edcalculus
lim⁡x→cf(x)g(x)=lim⁡x→cf′(x)g′(x)\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

Evaluating indeterminate forms (0/0 or ∞/∞)

#limit#lhopital

Product Rule for Logs

Class 10logarithms
log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)

Logarithm of a product equals sum of logarithms

#product#logarithm

Quotient Rule for Logs

Class 10logarithms
log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)

Logarithm of a quotient equals difference of logarithms

#quotient#logarithm

Power Rule for Logs

Class 10logarithms
log⁡b(xk)=klog⁡b(x)\log_b(x^k) = k \log_b(x)

Exponent comes out as a multiplier

#power#exponent#logarithm

Change of Base Formula

Class 12logarithms
log⁡b(x)=log⁡k(x)log⁡k(b)=ln⁡xln⁡b=log⁡10xlog⁡10b\log_b(x) = \frac{\log_k(x)}{\log_k(b)} = \frac{\ln x}{\ln b} = \frac{\log_{10} x}{\log_{10} b}

Converting logarithm to a different base k

#base#conversion

Identity & Zero Properties

Class 10logarithms
log⁡b(b)=1,log⁡b(1)=0\log_b(b) = 1, \quad \log_b(1) = 0

Basic identity properties of logarithms

#identity#logarithm

Inverse Properties

Class 12logarithms
blog⁡bx=x,log⁡b(bx)=xb^{\log_b x} = x, \quad \log_b(b^x) = x

Logarithms and exponentials are inverse functions

#inverse#exponential

Natural Log & Euler Constant

Higher Edlogarithms
ln⁡(x)=log⁡e(x),e=lim⁡n→∞(1+1n)n≈2.71828\ln(x) = \log_e(x), \quad e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

Definition of natural logarithm and base e

#natural#euler

Exponential Laws

Class 10logarithms
am⋅an=am+n,aman=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}

Fundamental laws of exponent arithmetic

#exponential#powers

Permutations Formula

Class 10probability
P(n,r)=nPr=n!(n−r)!P(n, r) = {}^n P_r = \frac{n!}{(n-r)!}

Number of ordered arrangements of r elements from n

#combinatorics#permutations

Combinations Formula

Class 10probability
C(n,r)=nCr=(nr)=n!r!(n−r)!C(n, r) = {}^n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r elements from n

#combinatorics#combinations

Addition Rule of Probability

Class 10probability
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability of union of two events

#probability#union

Multiplication Rule

Class 12probability
P(A∩B)=P(A)⋅P(B∣A)P(A \cap B) = P(A) \cdot P(B|A)

Probability of simultaneous occurrence of two events

#probability#intersection

Conditional Probability

Class 12probability
P(A∣B)=P(A∩B)P(B)(P(B)>0)P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given event B has occurred

#probability#conditional

Bayes' Theorem

Higher Edprobability
P(A∣B)=P(B∣A)⋅P(A)P(B)=P(B∣A)P(A)∑P(B∣Ai)P(Ai)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{P(B|A) P(A)}{\sum P(B|A_i) P(A_i)}

Calculates posterior probability based on prior knowledge

#probability#bayes

Binomial Distribution PMF

Class 12probability
P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent Bernoulli trials

#distribution#binomial

Poisson Distribution PMF

Higher Edprobability
P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval with average rate λ

#distribution#poisson

Expected Value (Discrete)

Class 12probability
E[X]=∑ixi⋅P(X=xi)E[X] = \sum_{i} x_i \cdot P(X = x_i)

Weighted average of all possible values of discrete random variable

#statistics#mean#expectation

Variance & Standard Deviation

Class 12probability
Var(X)=E[X2]−(E[X])2,σ=Var(X)\text{Var}(X) = E[X^2] - (E[X])^2, \quad \sigma = \sqrt{\text{Var}(X)}

Measure of dispersion of a random variable

#statistics#variance

Standard Normal Distribution (PDF)

Higher Edprobability
f(z)=12πe−z2/2,z=x−μσf(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}, \quad z = \frac{x - \mu}{\sigma}

Probability density function of standard normal variable

#distribution#normal#z-score

Distance Formula (2D)

Class 10geometry
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Distance between two points (x1, y1) and (x2, y2)

Interactive Solver:
Distance:—
Midpoint:—
Slope (m):—
#coordinate#distance

Midpoint Formula

Class 10geometry
M=(x1+x22,y1+y22)M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)

Midpoint of a line segment

#coordinate#midpoint

Slope of a Line

Class 10geometry
m=y2−y1x2−x1=tan⁡θm = \frac{y_2-y_1}{x_2-x_1} = \tan\theta

Slope of line passing through two points

#coordinate#line#slope

Slope-Intercept Form

Class 10geometry
y=mx+cy = mx + c

Equation of a straight line with slope m and y-intercept c

#coordinate#line

Point-Slope Form

Class 10geometry
y−y1=m(x−x1)y - y_1 = m(x - x_1)

Equation of line passing through (x1, y1) with slope m

#coordinate#line

Standard Circle Equation

Class 12geometry
(x−h)2+(y−k)2=r2(x-h)^2 + (y-k)^2 = r^2

Circle with center (h, k) and radius r

#conic#circle

General Parabola Equation

Class 12geometry
y=a(x−h)2+kor(y−k)2=4a(x−h)y = a(x-h)^2 + k \quad \text{or} \quad (y-k)^2 = 4a(x-h)

Parabola in vertex form

#conic#parabola

Standard Ellipse Equation

Class 12geometry
(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Ellipse centered at (h, k) with semi-major/minor axes a, b

#conic#ellipse

Standard Hyperbola Equation

Class 12geometry
(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Horizontal hyperbola centered at (h, k)

#conic#hyperbola

Section Formula (Internal)

Class 10geometry
P=(mx2+nx1m+n,my2+ny1m+n)P = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right)

Coordinates of point dividing line segment in ratio m:n

Interactive Solver:
Point P:—
#coordinate#section

Perpendicular Distance from Point to Line

Class 12geometry
d=∣Ax1+By1+C∣A2+B2d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}

Distance from point (x1, y1) to line Ax + By + C = 0

#coordinate#distance#line

Vector Dot Product

Class 12vectors
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=axbx+ayby+azbz\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_x b_x + a_y b_y + a_z b_z

Scalar product of two vectors

#vectors#dot-product

Vector Cross Product

Class 12vectors
a⃗×b⃗=∣a⃗∣∣b⃗∣sin⁡θ n^=∣i^j^k^axayazbxbybz∣\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix}

Vector product yielding orthogonal vector

#vectors#cross-product

Vector Magnitude

Class 12vectors
∣a⃗∣=ax2+ay2+az2|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}

Euclidean norm or length of a vector

#vectors#magnitude

Unit Vector

Class 12vectors
a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of length 1 pointing in same direction

#vectors#unit-vector

Scalar Projection of A onto B

Class 12vectors
projb⃗(a⃗)=a⃗⋅b⃗∣b⃗∣\text{proj}_{\vec{b}}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of vector a in direction of vector b

#vectors#projection

Matrix Determinant (2x2)

Class 12vectors
det⁡[abcd]=ad−bc\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

Determinant of a 2 by 2 matrix

#matrices#determinant

Matrix Inverse (2x2)

Class 12vectors
[abcd]−1=1ad−bc[d−b−ca]\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Inverse of an invertible 2 by 2 matrix

#matrices#inverse

Characteristic Equation (Eigenvalues)

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

Equation for finding eigenvalues λ of matrix A

#matrices#eigenvalues

Euler's Formula

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

Links complex exponential with trig functions

#euler#polar

De Moivre's Theorem

Higher Edcomplex
(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))

Power of a complex number in polar form

#power#polar

Modulus of Complex Number

Class 12complex
∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

Absolute value of z = a + bi

#magnitude#modulus

Complex Conjugate

Class 12complex
zˉ=a−bi\bar{z} = a - bi

Conjugate of z = a + bi

#conjugate

Polar Form

Class 12complex
z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta)

Polar representation of a complex number

#polar#argument

Argument of Complex Number

Class 12complex
arg⁡(z)=tan⁡−1(ba)\arg(z) = \tan^{-1}\left(\frac{b}{a}\right)

Angle of z = a + bi in the complex plane

#argument#angle

Complex Multiplication

Class 12complex
(a+bi)(c+di)=(ac−bd)+(ad+bc)i(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Product of two complex numbers

#multiplication#arithmetic

Complex Division

Class 12complex
a+bic+di=(ac+bd)+(bc−ad)ic2+d2\frac{a+bi}{c+di} = \frac{(ac+bd)+(bc-ad)i}{c^2+d^2}

Quotient of two complex numbers

#division#arithmetic

Roots of Unity

Higher Edcomplex
zk=ei⋅2πk/n,k=0,1,…,n−1z_k = e^{i \cdot 2\pi k/n}, \quad k=0,1,\ldots,n-1

The n-th roots of unity

#roots#unity

Exponential Form

Higher Edcomplex
z=reiθz = re^{i\theta}

Exponential representation using Euler formula

#exponential#polar

Circle Area

Class 10mensuration
A=πr2A = \pi r^2

Area of a circle with radius r

#2d#area#circle

Circle Circumference

Class 10mensuration
C=2πrC = 2\pi r

Perimeter of a circle

#2d#perimeter#circle

Triangle Area

Class 10mensuration
A=12bhA = \frac{1}{2}bh

Area of a triangle with base b and height h

#2d#area#triangle

Heron's Formula

Class 10mensuration
A=s(s−a)(s−b)(s−c)A = \sqrt{s(s-a)(s-b)(s-c)}

Triangle area from side lengths, s = (a+b+c)/2

#2d#area#triangle

Rectangle Area

Class 10mensuration
A=l×wA = l \times w

Area of a rectangle

#2d#area#rectangle

Trapezium Area

Class 10mensuration
A=12(a+b)hA = \frac{1}{2}(a+b)h

Area of a trapezium with parallel sides a, b

#2d#area#trapezium

Sphere Volume

Class 10mensuration
V=43πr3V = \frac{4}{3}\pi r^3

Volume of a sphere

#3d#volume#sphere

Sphere Surface Area

Class 10mensuration
S=4πr2S = 4\pi r^2

Surface area of a sphere

#3d#surface#sphere

Cylinder Volume

Class 10mensuration
V=πr2hV = \pi r^2 h

Volume of a cylinder

#3d#volume#cylinder

Cylinder Surface Area

Class 10mensuration
S=2πr(r+h)S = 2\pi r(r+h)

Total surface area of a cylinder

#3d#surface#cylinder

Cone Volume

Class 10mensuration
V=13πr2hV = \frac{1}{3}\pi r^2 h

Volume of a cone

#3d#volume#cone

Cone Curved Surface Area

Class 10mensuration
S=πrlS = \pi r l

Lateral surface area of a cone with slant height l

#3d#surface#cone

Cuboid Volume

Class 10mensuration
V=l×w×hV = l \times w \times h

Volume of a cuboid

#3d#volume#cuboid

Hemisphere Volume

Class 10mensuration
V=23πr3V = \frac{2}{3}\pi r^3

Volume of a hemisphere

#3d#volume#hemisphere

GCD-LCM Relationship

Class 10numberTheory
gcd⁡(a,b)×lcm(a,b)=∣ab∣\gcd(a,b) \times \text{lcm}(a,b) = |ab|

Product of GCD and LCM equals product of numbers

#gcd#lcm

Fermat's Little Theorem

Higher EdnumberTheory
ap−1≡1(modp)a^{p-1} \equiv 1 \pmod{p}

For prime p and integer a not divisible by p

#modular#prime#fermat

Euler's Totient Theorem

Higher EdnumberTheory
aϕ(n)≡1(modn)a^{\phi(n)} \equiv 1 \pmod{n}

Generalization of Fermat's Little Theorem

#modular#euler#totient

Wilson's Theorem

Higher EdnumberTheory
(p−1)!≡−1(modp)(p-1)! \equiv -1 \pmod{p}

A number p is prime iff this holds

#prime#factorial#wilson

Euclidean Algorithm

Class 10numberTheory
gcd⁡(a,b)=gcd⁡(b,a mod b)\gcd(a,b) = \gcd(b, a \bmod b)

Recursive algorithm for finding GCD

#gcd#algorithm#euclidean

Divisibility Rule for 3

Class 10numberTheory
n≡0(mod3)  ⟺  digit sum≡0(mod3)n \equiv 0 \pmod{3} \iff \text{digit sum} \equiv 0 \pmod{3}

A number is divisible by 3 if its digit sum is

#divisibility#rules

Modular Addition

Class 12numberTheory
(a+b) mod n=((a mod n)+(b mod n)) mod n(a + b) \bmod n = ((a \bmod n) + (b \bmod n)) \bmod n

Modular arithmetic addition property

#modular#addition

Modular Multiplication

Class 12numberTheory
(a⋅b) mod n=((a mod n)⋅(b mod n)) mod n(a \cdot b) \bmod n = ((a \bmod n) \cdot (b \bmod n)) \bmod n

Modular arithmetic multiplication property

#modular#multiplication

Chinese Remainder Theorem

Higher EdnumberTheory
x≡ai(modni) has unique solution mod N=∏nix \equiv a_i \pmod{n_i} \text{ has unique solution mod } N = \prod n_i

System of congruences with coprime moduli

#crt#congruence

Prime Counting (Approx)

Higher EdnumberTheory
π(x)∼xln⁡x\pi(x) \sim \frac{x}{\ln x}

Approximate number of primes up to x

#prime#counting#approximation

De Morgan's Law (Union)

Class 12setsAndFunctions
(A∪B)c=Ac∩Bc(A \cup B)^c = A^c \cap B^c

Complement of a union is intersection of complements

#sets#complement#de morgan

De Morgan's Law (Intersection)

Class 12setsAndFunctions
(A∩B)c=Ac∪Bc(A \cap B)^c = A^c \cup B^c

Complement of an intersection is union of complements

#sets#complement#de morgan

Inclusion-Exclusion (2 Sets)

Class 10setsAndFunctions
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|

Cardinality of union of two sets

#sets#cardinality#counting

Inclusion-Exclusion (3 Sets)

Class 12setsAndFunctions
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Cardinality of union of three sets

#sets#cardinality#counting

Function Composition

Class 12setsAndFunctions
(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

Applying function f to result of function g

#functions#composition

Inverse Function

Class 12setsAndFunctions
f(f−1(x))=x=f−1(f(x))f(f^{-1}(x)) = x = f^{-1}(f(x))

Property of an inverse function

#functions#inverse

Cartesian Product

Class 12setsAndFunctions
A×B={(a,b)∣a∈A,b∈B}A \times B = \{(a,b) \mid a \in A, b \in B\}

Set of all ordered pairs from A and B

#sets#product#cartesian

Power Set Cardinality

Class 12setsAndFunctions
∣P(A)∣=2∣A∣|\mathcal{P}(A)| = 2^{|A|}

Number of subsets of a set with n elements

#sets#power set#cardinality

Set Difference

Class 10setsAndFunctions
A∖B=A∩BcA \setminus B = A \cap B^c

Elements in A but not in B

#sets#difference

Symmetric Difference

Class 12setsAndFunctions
A△B=(A∖B)∪(B∖A)A \triangle B = (A \setminus B) \cup (B \setminus A)

Elements in exactly one of the two sets

#sets#symmetric#difference