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Formula & Equation Encyclopedia

Over 1255+ formulas organized by subject and topic — Math, Physics, Chemistry, Biology, Computer Science, Engineering, Economics, and more.

Search across all formulas or browse by subject and topic.

Algebra — Fundamentals

Quadratic Formula

x = (−b ± √(b² − 4ac)) / 2a

a, b, c = coefficients of ax² + bx + c = 0

Difference of Squares

a² − b² = (a + b)(a − b)

Perfect Square Trinomial

(a ± b)² = a² ± 2ab + b²

Sum of Cubes

a³ + b³ = (a + b)(a² − ab + b²)

Difference of Cubes

a³ − b³ = (a − b)(a² + ab + b²)

Binomial Theorem

(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ

Sum from k=0 to n

Logarithm Product Rule

log(xy) = log(x) + log(y)

Logarithm Quotient Rule

log(x/y) = log(x) − log(y)

Logarithm Power Rule

log(xⁿ) = n · log(x)

Change of Base

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

Exponential Identity

aˣ = eˣ·ˡⁿ⁽ᵃ⁾

Arithmetic Sequence

aₙ = a₁ + (n − 1)d

a₁ = first term, d = common difference

Arithmetic Series Sum

Sₙ = n/2 · (a₁ + aₙ)

Geometric Sequence

aₙ = a₁ · rⁿ⁻¹

r = common ratio

Geometric Series Sum (finite)

Sₙ = a₁(1 − rⁿ) / (1 − r)

Infinite Geometric Series

S = a₁ / (1 − r), |r| < 1

Algebra — Inequalities & Absolute Value

Absolute Value Definition

|x| = x if x ≥ 0; −x if x < 0

|ax + b| < c

−c < ax + b < c

|ax + b| > c

ax + b > c OR ax + b < −c

Triangle Inequality

|a + b| ≤ |a| + |b|

AM–GM Inequality

(a + b) / 2 ≥ √(ab), a,b ≥ 0

Geometry — Plane Shapes

Area of Triangle

A = ½bh

b = base, h = height

Area of Triangle (Heron's)

A = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2

Area of Rectangle

A = lw

Area of Parallelogram

A = bh

Area of Trapezoid

A = ½(b₁ + b₂)h

Area of Circle

A = πr²

Circumference of Circle

C = 2πr

Area of Sector

A = ½r²θ (θ in radians)

Arc Length

s = rθ

Area of Ellipse

A = πab

a, b = semi-axes

Area of Regular Polygon

A = ½ · perimeter · apothem

Interior Angle Sum (polygon)

S = (n − 2) · 180°

Pythagorean Theorem

a² + b² = c²

c = hypotenuse

Distance Formula

d = √((x₂−x₁)² + (y₂−y₁)²)

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Geometry — Solid Shapes

Volume of Cube

V = a³

Volume of Rectangular Prism

V = lwh

Surface Area of Rectangular Prism

SA = 2(lw + lh + wh)

Volume of Cylinder

V = πr²h

Surface Area of Cylinder

SA = 2πr² + 2πrh

Volume of Cone

V = ⅓πr²h

Slant Height of Cone

l = √(r² + h²)

Surface Area of Cone

SA = πr² + πrl

Volume of Sphere

V = (4/3)πr³

Surface Area of Sphere

SA = 4πr²

Volume of Pyramid

V = ⅓Bh

B = base area

Volume of Torus

V = 2π²Rr²

R = major radius, r = minor radius

Trigonometry

SOH-CAH-TOA

sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj

Pythagorean Identity

sin²θ + cos²θ = 1

Tangent Identity

tan θ = sin θ / cos θ

Secant Identity

sec²θ = 1 + tan²θ

Cosecant Identity

csc²θ = 1 + cot²θ

Sine Double Angle

sin(2θ) = 2 sin θ cos θ

Cosine Double Angle

cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

Tangent Double Angle

tan(2θ) = 2tanθ / (1 − tan²θ)

Sine Sum/Difference

sin(A ± B) = sinA cosB ± cosA sinB

Cosine Sum/Difference

cos(A ± B) = cosA cosB ∓ sinA sinB

Tangent Sum/Difference

tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)

Half Angle — sin

sin(θ/2) = ±√((1 − cosθ)/2)

Half Angle — cos

cos(θ/2) = ±√((1 + cosθ)/2)

Law of Sines

a/sinA = b/sinB = c/sinC

Law of Cosines

c² = a² + b² − 2ab cosC

Law of Tangents

(a−b)/(a+b) = tan(½(A−B)) / tan(½(A+B))

Product-to-Sum — sin·cos

sinA cosB = ½[sin(A+B) + sin(A−B)]

Product-to-Sum — cos·cos

cosA cosB = ½[cos(A−B) + cos(A+B)]

Product-to-Sum — sin·sin

sinA sinB = ½[cos(A−B) − cos(A+B)]

Euler's Formula

e^(iθ) = cosθ + i sinθ

Inverse Sine Range

arcsin: [−1,1] → [−π/2, π/2]

Inverse Cosine Range

arccos: [−1,1] → [0, π]

Inverse Tangent Range

arctan: ℝ → (−π/2, π/2)

Analytic Geometry & Conic Sections

Slope Formula

m = (y₂ − y₁) / (x₂ − x₁)

Slope-Intercept Form

y = mx + b

Point-Slope Form

y − y₁ = m(x − x₁)

Standard Form of Line

Ax + By + C = 0

Perpendicular Slope

m₂ = −1/m₁

Distance from Point to Line

d = |Ax₀ + By₀ + C| / √(A² + B²)

Circle (standard)

(x − h)² + (y − k)² = r²

Circle (general)

x² + y² + Dx + Ey + F = 0

Parabola (vertical)

y = a(x − h)² + k

Parabola (horizontal)

x = a(y − k)² + h

Focus of Parabola y=ax²

F = (0, 1/(4a))

Ellipse (standard)

x²/a² + y²/b² = 1

Hyperbola (standard)

x²/a² − y²/b² = 1

Eccentricity of Ellipse

e = c/a, c² = a² − b²

Eccentricity of Hyperbola

e = c/a, c² = a² + b²

Calculus — Limits & Derivatives

Limit Definition of Derivative

f′(x) = lim(h→0) [f(x+h)−f(x)] / h

Power Rule

d/dx[xⁿ] = nxⁿ⁻¹

Constant Rule

d/dx[c] = 0

Constant Multiple Rule

d/dx[cf(x)] = c·f′(x)

Sum Rule

d/dx[f+g] = f′ + g′

Product Rule

d/dx[fg] = f′g + fg′

Quotient Rule

d/dx[f/g] = (f′g − fg′) / g²

Chain Rule

d/dx[f(g(x))] = f′(g(x))·g′(x)

d/dx[eˣ]

eˣ

d/dx[aˣ]

aˣ ln(a)

d/dx[ln x]

1/x

d/dx[sin x]

cos x

d/dx[cos x]

−sin x

d/dx[tan x]

sec²x

d/dx[arcsin x]

1 / √(1 − x²)

d/dx[arctan x]

1 / (1 + x²)

L'Hôpital's Rule

lim f/g = lim f′/g′ (0/0 or ∞/∞)

Calculus — Integration

Power Rule (integral)

∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1

∫1/x dx

ln|x| + C

∫eˣ dx

eˣ + C

∫aˣ dx

aˣ / ln(a) + C

∫sin x dx

−cos x + C

∫cos x dx

sin x + C

∫sec²x dx

tan x + C

∫tan x dx

−ln|cos x| + C

∫1/√(1−x²) dx

arcsin x + C

∫1/(1+x²) dx

arctan x + C

Integration by Parts

∫u dv = uv − ∫v du

Fundamental Theorem of Calculus

∫[a→b] f(x)dx = F(b) − F(a)

Average Value of Function

f_avg = 1/(b−a) · ∫[a→b] f(x)dx

Arc Length

L = ∫[a→b] √(1 + [f′(x)]²) dx

Volume (disk method)

V = π∫[a→b] [f(x)]² dx

Volume (shell method)

V = 2π∫[a→b] x·f(x) dx

Trapezoidal Rule

∫≈ (Δx/2)[f(x₀)+2f(x₁)+…+2f(xₙ₋₁)+f(xₙ)]

Simpson's Rule

∫≈ (Δx/3)[f(x₀)+4f(x₁)+2f(x₂)+…+4f(xₙ₋₁)+f(xₙ)]

Calculus — Series & Sequences

Taylor Series

f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ

Maclaurin Series — eˣ

eˣ = 1 + x + x²/2! + x³/3! + …

Maclaurin Series — sin x

sin x = x − x³/3! + x⁵/5! − …

Maclaurin Series — cos x

cos x = 1 − x²/2! + x⁴/4! − …

Maclaurin Series — 1/(1−x)

1 + x + x² + x³ + … (|x|<1)

Ratio Test

lim|aₙ₊₁/aₙ| < 1 ⟹ converges

p-Series Test

Σ 1/nᵖ converges iff p > 1

Alternating Series Remainder

|Rₙ| ≤ aₙ₊₁

Linear Algebra

Dot Product

a·b = |a||b|cosθ = Σ aᵢbᵢ

Cross Product Magnitude

|a×b| = |a||b|sinθ

Determinant (2×2)

|A| = ad − bc

Determinant (3×3)

det(A) = a(ei−fh) − b(di−fg) + c(dh−eg)

Matrix Inverse (2×2)

A⁻¹ = 1/(ad−bc) · [[d,−b],[−c,a]]

Eigenvalue Equation

Av = λv → det(A − λI) = 0

Rank-Nullity Theorem

rank(A) + nullity(A) = n (columns)

Cauchy-Schwarz Inequality

|a·b| ≤ |a||b|

Projection of a onto b

proj_b(a) = (a·b/|b|²)·b

Gram-Schmidt (step 1)

e₁ = v₁/|v₁|

Statistics & Probability

Mean

μ = (Σxᵢ) / n

Variance (population)

σ² = Σ(xᵢ − μ)² / n

Standard Deviation

σ = √(σ²)

Sample Variance

s² = Σ(xᵢ − x̄)² / (n − 1)

Z-Score

z = (x − μ) / σ

Combination

C(n,k) = n! / (k!(n−k)!)

Permutation

P(n,k) = n! / (n−k)!

Binomial Probability

P(X=k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ

Poisson Probability

P(X=k) = (λᵏ e⁻λ) / k!

Normal PDF

f(x) = (1/σ√(2π)) e^(−(x−μ)²/(2σ²))

Bayes' Theorem

P(A|B) = P(B|A)·P(A) / P(B)

Expected Value

E[X] = Σ xᵢP(xᵢ)

Variance of X

Var(X) = E[X²] − (E[X])²

Correlation Coefficient

r = Σ[(xᵢ−x̄)(yᵢ−ȳ)] / (n·sₓ·sᵧ)

Linear Regression Slope

b = Σ[(x−x̄)(y−ȳ)] / Σ(x−x̄)²

Confidence Interval (mean)

x̄ ± z*(σ/√n)

Margin of Error

ME = z* · σ/√n

Chi-Square Statistic

χ² = Σ (O − E)² / E

Student's t-Statistic

t = (x̄ − μ) / (s/√n)

Chebyshev's Inequality

P(|X−μ| ≥ kσ) ≤ 1/k²

Number Theory & Discrete Math

Fundamental Theorem of Arithmetic

Every n > 1 = unique product of primes

Euclidean GCD

gcd(a,b) = gcd(b, a mod b)

Bezout Identity

There exist integers x,y: ax + by = gcd(a,b)

Euler Totient

phi(n) = n * prod(1 − 1/p) over primes p|n

Fermat's Little Theorem

a^p = a (mod p), p prime

Chinese Remainder Theorem

x = ai (mod ni) has unique solution mod prod(ni)

Stars and Bars

C(n+k-1, k-1) ways to distribute n items in k bins

Inclusion-Exclusion

|A union B| = |A| + |B| - |A intersect B|

Stirling Approximation

n! approx sqrt(2*pi*n) * (n/e)^n

Master Theorem

T(n) = aT(n/b)+f => T(n) = Theta(n^log_b(a)) if f = O(n^(log_b(a)-e))

Complex Numbers

Complex Number

z = a + bi, i² = −1

Modulus

|z| = √(a² + b²)

Argument (angle)

arg(z) = arctan(b/a)

Polar Form

z = r(cosθ + i sinθ) = re^(iθ)

Multiplication (polar)

z₁z₂ = r₁r₂ e^(i(θ₁+θ₂))

De Moivre's Theorem

(r e^(iθ))ⁿ = rⁿ e^(inθ)

n-th Roots of Unity

z_k = e^(2πik/n), k = 0,1,…,n−1

Complex Conjugate

z̄ = a − bi; z · z̄ = |z|²

Euler's Identity

e^(iπ) + 1 = 0

Real Part

Re(e^(iθ)) = cosθ

Imaginary Part

Im(e^(iθ)) = sinθ

Complex Division

z₁/z₂ = (z₁ · z̄₂) / |z₂|²

Differential Equations

1st Order Separable

dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx

1st Order Linear

dy/dx + P(x)y = Q(x); integrating factor μ = e^(∫P dx)

Integrating Factor Solution

y = (1/μ) ∫μ Q dx

2nd Order Homogeneous (const coeff)

ay'' + by' + cy = 0 → char. eq. ar² + br + c = 0

Char. Roots — two real

y = C₁e^(r₁x) + C₂e^(r₂x)

Char. Roots — repeated

y = (C₁ + C₂x)e^(rx)

Char. Roots — complex α±βi

y = e^(αx)(C₁cosβx + C₂sinβx)

Method of Undetermined Coefficients

Guess y_p based on form of Q(x)

Variation of Parameters

y_p = y₁∫(−y₂Q)/(W) dx + y₂∫(y₁Q)/(W) dx

W = Wronskian

Wronskian

W(y₁,y₂) = y₁y₂' − y₁'y₂

Laplace Transform — ODE method

L{y''} = s²Y − sy(0) − y'(0)

Convolution Theorem

L{f*g} = F(s)·G(s)

Euler's Method (numerical)

yₙ₊₁ = yₙ + hf(xₙ,yₙ)

Runge-Kutta 4th Order

yₙ₊₁ = yₙ + (k₁+2k₂+2k₃+k₄)/6

k₁=hf(xₙ,yₙ), k₂=hf(xₙ+h/2,yₙ+k₁/2), …

Logistic ODE Solution

y = K / (1 + ((K−y₀)/y₀)e^(−rt))

Heat Equation (PDE)

∂u/∂t = k ∂²u/∂x²

Wave Equation (PDE)

∂²u/∂t² = c² ∂²u/∂x²

Laplace Equation (PDE)

∇²u = ∂²u/∂x² + ∂²u/∂y² = 0

Vector Calculus

Gradient

∇f = (∂f/∂x)î + (∂f/∂y)ĵ + (∂f/∂z)k̂

Divergence

∇·F = ∂Fₓ/∂x + ∂Fᵧ/∂y + ∂F_z/∂z

Curl

∇×F = (∂F_z/∂y−∂Fᵧ/∂z)î − (∂F_z/∂x−∂Fₓ/∂z)ĵ + (∂Fᵧ/∂x−∂Fₓ/∂y)k̂

Laplacian

∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²

Green's Theorem

∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA

Stokes' Theorem

∮_C F·dr = ∬_S (∇×F)·dS

Divergence Theorem

∯_S F·dS = ∭_V (∇·F) dV

Line Integral of Scalar

∫_C f ds = ∫_a^b f(r(t)) |r'(t)| dt

Surface Area (parametric)

A = ∬_D |r_u × r_v| dA

Jacobian (2D)

J = ∂(x,y)/∂(u,v) = x_u y_v − x_v y_u

Change of Variables

∬f(x,y) dA = ∬f(x(u,v),y(u,v)) |J| du dv

Cylindrical Coordinates

x=r cosθ, y=r sinθ, z=z; dV = r dr dθ dz

Spherical Coordinates

x=ρ sinφ cosθ, y=ρ sinφ sinθ, z=ρ cosφ; dV = ρ² sinφ dρ dφ dθ

Combinatorics & Graph Theory

Handshaking Lemma

Σ deg(v) = 2|E|

Euler Formula (planar)

V − E + F = 2

Chromatic Polynomial

P(G,k) = number of proper k-colorings of G

Cayley's Formula (spanning trees)

T(K_n) = nⁿ⁻²

Turán Theorem (edges)

ex(n,K_{r+1}) = (1−1/r)n²/2 (approx)

Ramsey Number bound

R(s,t) ≤ C(s+t−2, s−1)

Catalan Number

C_n = C(2n,n)/(n+1)

Bell Number (set partitions)

B_{n+1} = Σ_{k=0}^{n} C(n,k) B_k

Inclusion-Exclusion (general)

|A₁∪…∪Aₙ| = Σ|Aᵢ| − Σ|Aᵢ∩Aⱼ| + …

Derangement Count

D_n = n! Σ_{k=0}^{n} (−1)ᵏ/k! ≈ n!/e

Burnside's Lemma

|orbits| = (1/|G|) Σ_{g∈G} |Fix(g)|

Pigeonhole Principle

n items in k < n bins → ≥ 1 bin has ≥ ⌈n/k⌉ items

Abstract Algebra

Lagrange's Theorem

|H| divides |G| for subgroup H of G

Order of Element

ord(g) = smallest n > 0 : gⁿ = e

Coset

aH = {ah : h ∈ H}

Index of Subgroup

[G:H] = |G|/|H|

First Isomorphism Theorem

G/ker(φ) ≅ Im(φ)

Cauchy's Theorem

If p | |G| and p prime → ∃ element of order p in G

Sylow's Theorem (count)

n_p ≡ 1 (mod p), n_p | |G|/p^k

Euler's Totient (group order)

Z_n^× has order φ(n)

Characteristic of a Ring

smallest n : n·1_R = 0_R

Ideal Quotient Ring

R/I (ring modulo ideal I)

About This Formula Encyclopedia

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