Math Calculators

Perform basic arithmetic

a+b, a-b, a×b, a÷b

X% of Y

X%×Y

Increase/decrease %

((V2-V1)/V1)×100

Add numbers

a+b+c

Subtract numbers

a-b-c

Multiply numbers

a×b×c

Divide numbers

a÷b

x^y

x^y

x²

x²

x³

x³

√x

√x

∛x

∛x

ⁿ√x

ⁿ√x

n!

n!=n×(n-1)×...

|x|

|x|

Round to precision

round(x,n)

Average of numbers

(a+b+...)/n

GCD(a,b)

GCD(a,b)

LCM(a,b)

LCM=ab/GCD

a mod b

a mod b

Is n prime?

n is prime

Break into primes

n=Πp_i^e_i

1+2+...+n

n(n+1)/2

1²+...+n²

n(n+1)(2n+1)/6

1³+...+n³

[n(n+1)/2]²

123→321

Reverse

123→1+2+3=6

Digit sum

→ a×10^b

a×10^b

Round to n sig figs

sig(x,n)

1/x

1/x

dr(n)

1+((n-1)mod9)

42→forty-two

English words

Solve ax+b=0

x=-b/a

Solve ax+b=cx+d

x=(d-b)/(a-c)

Solve 2 equations

a₁x+b₁y=c₁

ax²+bx+c=0

x=(-b±√Δ)/(2a)

Parabola vertex y=ax²+bx+c

V(-b/(2a),c-b²/(4a))

Depressed cubic

t=∛(-q/2+√Δ)+∛(-q/2-√Δ)

ax+b>0

ax+b>0

Evaluate P(x)

aₙxⁿ+...+a₀

Expand (a+b)²

a²+2ab+b²

Expand (a+b)³

a³+3a²b+3ab²+b³

nth term & sum

aₙ=a₁+(n-1)d

nth term & sum

aₙ=a₁rⁿ⁻¹

Sum if |r|<1

S=a₁/(1-r)

Decompose 1/((x+a)(x+b))

A/(x+a)+B/(x+b)

Polynomial ÷ (x-c)

Synthetic div

Solve a^x=b

x=ln(b)/ln(a)

Solve log_a(x)=b

x=a^b

Quadratic roots

Sum=-b/a,Prod=c/a

x²+bx

(x+b/2)²-(b/2)²

(a+bi)+(c+di)

Complex add

(a+bi)(c+di)

(ac-bd)+(ad+bc)i

|a+bi|

√(a²+b²)

r(cosθ+isinθ)

Polar form

A=½bh

A=½bh

Area from 3 sides

√(s(s-a)(s-b)(s-c))

Area & perimeter

A=lw,P=2(l+w)

All properties

A=s²,P=4s

Area

A=bh

Area

A=½(a+b)h

Area & side

A=½d₁d₂

A=½d₁d₂

A=½d₁d₂

Area & circumference

A=πr²,C=2πr

Area & arc length

A=½r²θ

Segment area

A=½r²(θ-sinθ)

Ring area

A=π(R²-r²)

Area

A=πab

Interior angles

Sum=(n-2)×180°

Area of regular polygon

A=½×n×s²×cot(π/n)

a²+b²=c²

a²+b²=c²

Distance between points

√((Δx)²+(Δy)²)

M=((x₁+x₂)/2,(y₁+y₂)/2)

Midpoint

Slope between points

m=(y₂-y₁)/(x₂-x₁)

Circumradius of triangle

R=abc/(4K)

Inradius of triangle

r=K/s

Triangle centroid

G=(x₁+x₂+x₃)/3

d=|Ax₀+By₀+C|/√(A²+B²)

Point-line distance

V=s³, SA=6s²

V=s³

V=lwh

V=lwh

V=⁴⁄₃πr³

V=⁴⁄₃πr³

V=πr²h

V=πr²h

V=⅓πr²h

V=⅓πr²h

V=⅓×base×h

V=⅓bh

Donut volume

V=2π²Rr²

Conical frustum volume

V=⅓πh(R²+Rr+r²)

Half sphere

V=⅔πr³

V=⁴⁄₃πabc

V=⁴⁄₃πabc

Cylinder+hemispheres

V=πr²h+⁴⁄₃πr³

Regular tetrahedron

V=√2/12×s³

Basic trig functions

sinθ,cosθ,tanθ

Inverse trig

sin⁻¹x

Reciprocal trig

csc=1/sin

a/sinA=b/sinB

a/sinA=b/sinB

c²=a²+b²-2ab·cosC

c²=a²+b²-2ab·cosC

Hyperbolic functions

sinhx

Convert angle units

180°=π rad

Verify trig identity

sin²θ+cos²θ=1

Double angle

sin2θ=2sinθcosθ

Half angle

sin(θ/2)=±√((1-cosθ)/2)

Sum of angles formulas

sin(A+B)=sinAcosB+cosAsinB

Derivative of power

d/dx(x^n)=nx^(n-1)

d/dx(ax³+bx²+cx+d)

3ax²+2bx+c

Derivative of sine

cos(x)

Derivative of e^x

e^x

(fg)'=f'g+fg'

(fg)'

(f/g)'=(f'g-fg')/g²

(f/g)'

Definite integral

x^(n+1)/(n+1)

Integral of sine

-cos(x)

Integral of cosine

sin(x)

Integral of e^x

e^x

Numerical integration

h/2[f(a)+2Σf+ f(b)]

Numerical integration

h/3[f(a)+4Σf+2Σf+f(b)]

Approximate e^x

Σx^n/n!

Approximate sin(x)

x-x³/3!+...

Find root

x_{n+1}=x_n-f/f'

Evaluate limit

lim_{x→a}f(x)

Solve y'=f(x,y)

y_{n+1}=y_n+h·f

Middle value

Median

Most frequent

Mode

Population std dev

σ=√(Σ(x-μ)²/n)

z=(x-μ)/σ

z=(x-μ)/σ

Line of best fit

y=mx+b

Correlation coefficient

r

nPr=n!/(n-r)!

nPr

nCr=n!/[r!(n-r)!]

nCr

P(k) = C(n,k)p^k(1-p)^(n-k)

Binomial

Normal density

(1/σ√2π)e^(-(x-μ)²/2σ²)

Normal cumulative

Φ(x)

P(k) = λ^k·e^(-λ)/k!

Poisson

P(k) = (1-p)^(k-1)×p

Geometric

P(A|B)=P(B|A)P(A)/P(B)

Bayes'

Σx·P(x)

E[X]

Efficient C(n,k)

C(n,k)

Uniform density f(x)=1/(b-a)

Uniform

nth Fibonacci

Fₙ=F_{n-1}+F_{n-2}

Generate sequence

F₀=0,F₁=1

φ=(1+√5)/2

φ

Cₙ=(2n)!/((n+1)!n!)

Catalan

Sum of divisors

σ(n)=Σd|n d

Number of divisors

τ(n)

Is n perfect?

σ(n)=2n

3n+1 problem

Collatz

m²-n², 2mn, m²+n²

Triple

φ(n)=nΠ(1-1/p)

φ(n)

Log any base

ln(x)/ln(b)

ln(x)

ln(x)

Base 10 log

log₁₀(x)

Base 2 log

log₂(x)

σ(x)=1/(1+e⁻ˣ)

σ(x)

max(0,x)

ReLU

ae^(-(x-b)²/(2c²))

Gaussian

P(t)=P₀e^(rt)

Growth

P(t)=P₀e^(-rt)

Decay

P(t)=K/(1+((K-P₀)/P₀)e^(-rt))

Logistic

T(t)=Tₐ+(T₀-Tₐ)e^(-kt)

Cooling

y=y₁+(x-x₁)(y₂-y₁)/(x₂-x₁)

Interpolate

L1 distance

|Δx|+|Δy|

L∞ distance

max|Δx|,|Δy|

Great-circle distance

d=2R·arcsin(√...)

Number→Roman

Numeral converter

Roman→Number

Parse

Approx n!

Stirling's

Γ(n)=(n-1)!

Γ(n)

B(t)=(1-t)²P₀+2(1-t)tP₁+t²P₂

Bezier

Solve a1x+b1y=c1, a2x+b2y=c2

Cramers rule

x = (-b +/- sqrt(b2-4ac))/(2a)

ax^2+bx+c=0

Find real root of ax3+bx2+cx+d=0

Bisection method

Evaluate a0+a1x+a2x2+a3x3

f(x)=sum(ak*x^k)

|x| = x if x>=0 else -x

|x|

y = y1+(x-x1)*(y2-y1)/(x2-x1)

y(x)

sqrt((x2-x1)^2+(y2-y1)^2)

Euclidean

((x1+x2)/2, (y1+y2)/2)

Midpoint formula

m = (y2-y1)/(x2-x1)

Slope formula

m_perp = -1/m

Negative reciprocal

Area and circumference of circle

A=pi*r^2, C=2*pi*r

|Ax0+By0+C|/sqrt(A^2+B^2)

Perpendicular distance

Decompose (Ax+B)/((x+c)(x+d))

Partial fractions

D = b^2-4ac

Number of real roots

(a1+b1, a2+b2)

v + w

A·B = a1*b1+a2*b2

Scalar product

AxB = a1*b2-a2*b1

Determinant

|v| = sqrt(x^2+y^2)

Length

u-hat = v/|v|

Normalization

theta = atan2(y,x)

Direction

A = 0.5*b*h

Base-height

A = sqrt(s(s-a)(s-b)(s-c))

Three sides

A = w*h, P = 2(w+h)

Area and perimeter

A = pi*r^2

Area of circle

A = (theta/360)*pi*r^2

Circular sector

V = (4/3)*pi*r^3

Volume of sphere

V = pi*r^2*h

Volume of cylinder

V = (1/3)*pi*r^2*h

Volume of cone

V = s^3

Volume of cube

V = l*w*h

Box volume

V = (1/3)*l*w*h

Square pyramid

A = b*h

Base-height

A = 0.5*(a+b)*h

Average bases

A = pi*a*b

Semi-axes

A = pi*(R^2-r^2)

Washer

A = 0.5*n*R^2*sin(2*pi/n)

Circumradius

V = 2*pi^2*R*r^2

Donut volume

d = s*sqrt(3)

Space diagonal

phi = (1+sqrt(5))/2

Divine proportion

sin(theta)

sin(theta)

cos(theta)

cos(theta)

tan(theta)

tan(theta)

csc(theta) = 1/sin(theta)

Reciprocal sine

sec(theta) = 1/cos(theta)

Reciprocal cosine

cot(theta) = 1/tan(theta)

Reciprocal tangent

theta = sin^-1(x)

Inverse sine

theta = cos^-1(x)

Inverse cosine

theta = tan^-1(x)

Inverse tangent

theta = atan2(y,x)

Two-argument arctan

sinh(x) = (e^x-e^-x)/2

sinh(x)

cosh(x) = (e^x+e^-x)/2

cosh(x)

tanh(x) = sinh(x)/cosh(x)

tanh(x)

a/sin(A)=b/sin(B)=c/sin(C)

Triangle law

c^2=a^2+b^2-2ab*cos(C)

Side from angle

rad = deg*pi/180

Angle conversion

deg = rad*180/pi

Angle conversion

sin(2theta) = 2sin(theta)cos(theta)

Double angle

cos(2theta) = cos^2-sin^2

Double angle

sin^2+cos^2=1

Identity check

x-bar = sum(x)/n

Arithmetic mean

Middle value of sorted data

Central value

Most frequent value

Frequency

max - min

Spread

sigma^2 = sum(x-x-bar)^2/n

Spread squared

sigma = sqrt(variance)

Spread

z = (x-mu)/sigma

Standard score

Q1, Q2, Q3

Five-number summary

cov = sum(x-x-bar)(y-y-bar)/n

Joint variation

r = cov/(sx*sy)

Pearson r

y = mx + b

Best fit line

n/(1/x1+1/x2+...)

Rate average

(x1*x2*...)^(1/n)

Multiplicative average

sqrt((x1^2+x2^2+...)/n)

Root mean square

g1 = (1/n)sum((x-x-bar)/sigma)^3

Distribution shape

g2 = (1/n)sum((x-x-bar)/sigma)^4-3

Tail shape

P-th percentile of data

Order statistic

IQR = Q3-Q1

Spread

MAD = (1/n)sum|x-x-bar|

Robust spread

P(|x-mu|>=k*sigma) <= 1/k^2

Probability bounds

% within k*sigma (68-95-99.7)

Normal distribution

SMA = sum(xi)/n

Time series

SE = sigma/sqrt(n)

Sampling error

ME = z*sigma/sqrt(n)

Confidence interval

an = a1+(n-1)d

Linear progression

an = a1*r^(n-1)

Exponential progression

S_inf = a/(1-r) for |r|<1

Convergent series

sum(1/k-1/(k+1)) = 1-1/(n+1)

Series

sum(1/k^p)

Convergence test

sum((-1)^(k+1)/k)

Leibniz series

n! = 1*2*3*...*n

Product sequence

n!! = n*(n-2)*(n-4)*...

Step 2 product

Is n in Fibonacci sequence?

Sequence membership

T_n = n(n+1)/2

Figurate number

S_n = n^2

Perfect squares

P_n = n(3n-1)/2

Figurate number

H_n = n(2n-1)

Figurate number

T(n)=T(n-1)+T(n-2)+T(n-3)

Third-order

(a+b)^n expansion

Pascal triangle

sum(f(k), k=a..b)

Series sum

prod(f(k), k=a..b)

Product series

A + B (2x2)

Element-wise

A x B (2x2)

Row-column

det = ad-bc

Square matrix

A^-1 = (1/det)*[d,-b;-c,a]

Matrix inverse

(a+bi)+(c+di)

Complex numbers

(a+bi)*(c+di)

Complex product

|a+bi| = sqrt(a^2+b^2)

Magnitude

x = r*cos, y = r*sin

Coordinate conversion

r = sqrt(x^2+y^2), theta = atan2(y,x)

Coordinate conversion

P(A or B)=P(A)+P(B)-P(A and B)

Addition rule

P(A|B) = P(A and B)/P(B)

Bayes rule

C(n,k) = n!/(k!(n-k)!)

N choose k

P(n,k) = n!/(n-k)!

N permute k

P(k) = C(n,k)*p^k*(1-p)^(n-k)

Binary outcome

E[X] = sum(x*P(x))

Weighted average

P(X=k) = C(n,k)*p^k*(1-p)^(n-k)

Discrete distribution

P(X=k) = (lambda^k*e^-lambda)/k!

Rare events

log10(x)

Common logarithm

ln(x)

Natural logarithm

log_b(x) = ln(x)/ln(b)

Arbitrary base

e^x

Natural exponential

x^y

Exponentiation

sqrt(x)

Radical

cbrt(x)

Radical

x^(1/n)

General radical

gcd(a,b) by Euclidean algorithm

Greatest common divisor

lcm(a,b) = |a*b|/gcd(a,b)

Least common multiple

Factor n into primes

Integer factorization

Convert number to roman

Roman numeral

Convert to scientific notation

Standard form

Count sig figs in number

Precision

a mod b

Remainder

Random number in range

RNG

Round to n decimal places

Rounding

Floor(x) = greatest int <= x

Floor function

Ceil(x) = smallest int >= x

Ceiling function

trunc(x) = integer part

Truncation

c = sqrt(a^2+b^2)

Pythagorean theorem

x(n+1)=r*x(n)*(1-x(n))

x(n+1) = r*x(n)*(1-x(n))

Approximate Feigenbaum delta

d1 = 4.6692

Check if c is in Mandelbrot set

z(n+1) = z(n)^2 + c

Generate Sierpinski triangle point

Chaos Game

Bifurcation diagram y value

x(n+1) = r*x(n)*(1-x(n))

dx/dt = sigma(y-x)

dx/dt = sigma*(y-x)

∫f(x)dx via Simpson rule

∫ f(x) dx

∫f(x)dx via trapezoid rule

∫ f(x) dx

f'(x) ≈ (f(x+h)-f(x-h))/(2h)

f'(x) = (f(x+h)-f(x-h))/(2h)

x(n+1)=x(n)-f(x)/f'(x)

x(n+1) = x(n) - f(x)/f'(x)

Find root by bisection

mid = (a+b)/2

sin(x) ≈ x - x³/3! + x⁵/5!

sin(x) = sum((-1)^n * x^(2n+1) / (2n+1)!)

e^x ≈ 1 + x + x²/2! + x³/3!

e^x = sum(x^n / n!)

∫f(x)dx left Riemann sum

sum(f(x_i) * h)

x(n+1) = x(n) - f(x(n))(x(n)-x(n-1))/(f(x(n))-f(x(n-1)))

x(n+1) = x(n) - f(n)*(x(n)-x(n-1))/(f(n)-f(n-1))

x = g(x)

x = g(x)

Interpolate points

y = sum(y_i * L_i(x))

C(n) = (2n)!/((n+1)!n!)

C(n) = (2n)! / ((n+1)!*n!)

F(n) = F(n-1) + F(n-2)

F(n) = F(n-1) + F(n-2)

L(n) = L(n-1) + L(n-2)

L(n) = L(n-1) + L(n-2)

Steps to reach 1

3n + 1

p + q = n (both prime)

p + q = n

φ(n) = count of coprimes

phi(n) = n * prod(1 - 1/p)

σ(n)

sigma(n) = sum(d)

μ(n)

mu(n)

B(n) via formula

B_n

Number of set partitions

B_n

a²+b²=c²

a^2 + b^2 = c^2

p(n) number of partitions

p(n)

ax+by = gcd(a,b)

a*x + b*y = gcd(a,b)

x ≡ a (mod m)

x = a_i (mod m_i)

Solve quadratic equations ax²+bx+c=0 using the quadratic formula.

x = (-b ± √(b²-4ac)) / 2a

Generate all prime numbers up to a given limit.

Sieve primes up to n

Generate Fibonacci sequence numbers.

F(n) = F(n-1) + F(n-2)

Calculate union, intersection, and difference of two sets.

A ∪ B, A ∩ B, A \ B

Calculate linear regression line (y = mx + b) from data points.

m = Σ(x-x̄)(y-ȳ) / Σ(x-x̄)²

Calculate binomial probability P(X = k) for n trials.

P(X=k) = C(n,k) × p^k × (1-p)^(n-k)

Estimate p-value from a z-score using normal approximation.

p ≈ 2 × (1 - Φ(|z|))

Calculate confidence interval for a population mean.

CI = x̄ ± z × σ/√n

Calculate permutations nPr = n!/(n-r)! and combinations nCr.

nPr = n!/(n-r)!, nCr = n!/(r!(n-r)!)

Update Beta posterior distribution with new data.

α´ = α + successes, β´ = β + failures

Calculate determinant of 2×2 matrices.

det = ad - bc

Approximate eˣ, sin(x), cos(x) using Taylor series up to n terms.

f(x) ≈ Σ f⁽ⁿ⁾(0) × xⁿ / n!