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About

Math Formula Reference

Your comprehensive collection of essential mathematics formulas. Bookmark this page for quick access during homework or exam prep.

Algebra Formulas

Quadratic Formula

x = (-b ± √(b² − 4ac)) / (2a)

Solves ax² + bx + c = 0. The discriminant Δ = b² − 4ac determines root type.

Factoring Identities

a² − b² = (a + b)(a − b)

a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²

a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

Binomial Theorem

(a + b)ⁿ = Σ (from k=0 to n) C(n,k) · aⁿ⁻ᵏ · bᵏ

Where C(n,k) = n! / (k!(n−k)!) is the binomial coefficient.

(a + b)² = a² + 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

Arithmetic Sequences & Series

nth term: aₙ = a₁ + (n − 1)d
Sum of n terms: Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n − 1)d)

Geometric Sequences & Series

nth term: aₙ = a₁ · rⁿ⁻¹
Sum of n terms: Sₙ = a₁(1 − rⁿ) / (1 − r), r ≠ 1
Infinite sum (|r| < 1): S∞ = a₁ / (1 − r)

Exponent Laws

aᵐ · aⁿ = aᵐ⁺ⁿ aᵐ / aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ (ab)ⁿ = aⁿbⁿ
a⁰ = 1 (a ≠ 0) a⁻ⁿ = 1/aⁿ
a^(m/n) = ⁿ√(aᵐ)

Logarithm Laws

log_b(xy) = log_b(x) + log_b(y)
log_b(x/y) = log_b(x) − log_b(y)
log_b(xⁿ) = n · log_b(x)
log_b(1) = 0 log_b(b) = 1
log_b(x) = ln(x) / ln(b) (change of base)

If log_b(x) = y, then bʸ = x. Logarithms and exponentials are inverse functions.

Geometry Formulas

2D Shapes — Area & Perimeter

Square (side s)

Area = s²
Perimeter = 4s
Diagonal = s√2

Rectangle (length l, width w)

Area = lw
Perimeter = 2(l + w)
Diagonal = √(l² + w²)

Triangle (base b, height h)

Area = ½bh
Area = ½ab sin(C) (two sides and included angle)
Area = √(s(s−a)(s−b)(s−c)) (Heron's formula, s = (a+b+c)/2)

Circle (radius r)

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Area = πr²
Circumference = 2πr
Arc length = rθ (θ in radians)
Sector area = ½r²θ

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Trapezoid (parallel sides a, b; height h)

Area = ½(a + b)h

Parallelogram (base b, height h)

Area = bh
Perimeter = 2(a + b)

Ellipse (semi-axes a, b)

Area = πab
Circumference ≈ π(3(a + b) − √((3a + b)(a + 3b))) (Ramanujan approx.)

3D Solids — Volume & Surface Area

Cube (side s)

Volume = s³
Surface Area = 6s²

Rectangular Prism (l × w × h)

Volume = lwh
Surface Area = 2(lw + lh + wh)

Sphere (radius r)

Volume = (4/3)πr³
Surface Area = 4πr²

Cylinder (radius r, height h)

Volume = πr²h
Surface Area = 2πr² + 2πrh = 2πr(r + h)

Cone (radius r, height h, slant height l)

Volume = (1/3)πr²h
Surface Area = πr² + πrl
Slant height: l = √(r² + h²)

Pyramid (base area B, height h)

Volume = (1/3)Bh

Coordinate Geometry

Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope: m = (y₂ − y₁) / (x₂ − x₁)

Slope-intercept form: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Standard form: Ax + By = C

Circle equation: (x − h)² + (y − k)² = r²
Center: (h, k), Radius: r

Trigonometry Formulas

Basic Ratios (Right Triangle)

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

csc(θ) = 1/sin(θ) sec(θ) = 1/cos(θ) cot(θ) = 1/tan(θ)

Unit Circle — Key Values

θ sin(θ) cos(θ) tan(θ)
0° 0 1 0
30° 1/2 √3/2 √3/3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
90° 1 0 undefined

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Pythagorean Identities

sin²(θ) + cos²(θ) = 1
1 + tan²(θ) = sec²(θ)
1 + cot²(θ) = csc²(θ)

Sum & Difference Formulas

sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))

Double Angle Formulas

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ) − 1 = 1 − 2sin²(θ)
tan(2θ) = 2tan(θ) / (1 − tan²(θ))

Half Angle Formulas

sin(θ/2) = ±√((1 − cos(θ)) / 2)
cos(θ/2) = ±√((1 + cos(θ)) / 2)
tan(θ/2) = sin(θ) / (1 + cos(θ)) = (1 − cos(θ)) / sin(θ)

Law of Sines

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a/sin(A) = b/sin(B) = c/sin(C) = 2R

Where R is the circumradius of the triangle.

Law of Cosines

c² = a² + b² − 2ab·cos(C)

Law of Tangents

(a − b) / (a + b) = tan((A − B)/2) / tan((A + B)/2)

The ± sign in half angle formulas depends on the quadrant of θ/2. Always check which quadrant the half angle falls in.

Calculus Formulas

Limits

lim (x→0) sin(x)/x = 1
lim (x→0) (1 − cos(x))/x = 0
lim (x→∞) (1 + 1/x)ˣ = e
lim (x→0) (eˣ − 1)/x = 1
lim (x→0) ln(1 + x)/x = 1

L'Hôpital's Rule

If lim f(x)/g(x) is 0/0 or ∞/∞, then:
lim f(x)/g(x) = lim f'(x)/g'(x)

Derivative Rules

Constant: d/dx [c] = 0
Power: d/dx [xⁿ] = nxⁿ⁻¹
Constant mult.: d/dx [cf(x)] = cf'(x)
Sum/Diff: d/dx [f ± g] = f' ± g'
Product: d/dx [fg] = f'g + fg'
Quotient: d/dx [f/g] = (f'g − fg') / g²
Chain: d/dx [f(g(x))] = f'(g(x)) · g'(x)

Common Derivatives

d/dx [eˣ] = eˣ d/dx [aˣ] = aˣ ln(a)
d/dx [ln(x)] = 1/x d/dx [log_a(x)] = 1/(x ln(a))
d/dx [sin(x)] = cos(x) d/dx [cos(x)] = −sin(x)
d/dx [tan(x)] = sec²(x) d/dx [cot(x)] = −csc²(x)
d/dx [sec(x)] = sec(x)tan(x) d/dx [csc(x)] = −csc(x)cot(x)
d/dx [arcsin(x)] = 1/√(1−x²) d/dx [arccos(x)] = −1/√(1−x²)
d/dx [arctan(x)] = 1/(1+x²)

Integral Rules

∫ cf(x) dx = c ∫ f(x) dx
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
∫ u dv = uv − ∫ v du (integration by parts)

Common Integrals

∫ 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
∫ sec²(x) dx = tan(x) + C
∫ csc²(x) dx = −cot(x) + C
∫ sec(x)tan(x) dx = sec(x) + C
∫ csc(x)cot(x) dx = −csc(x) + C
∫ 1/(1+x²) dx = arctan(x) + C
∫ 1/√(1−x²) dx = arcsin(x) + C

Fundamental Theorem of Calculus

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Part 1: d/dx [∫ₐˣ f(t) dt] = f(x)
Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)

Taylor / Maclaurin Series

Taylor series about x = a:
f(x) = Σ (n=0 to ∞) f⁽ⁿ⁾(a)/n! · (x − a)ⁿ

Maclaurin series (a = 0):
eˣ = 1 + x + x²/2! + x³/3! + ...
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + ...
cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + ...
ln(1+x) = x − x²/2 + x³/3 − x⁴/4 + ..., |x| ≤ 1
1/(1−x) = 1 + x + x² + x³ + ..., |x| < 1

Don't forget the + C (constant of integration) for indefinite integrals! This is one of the most common mistakes on exams.

Statistics Formulas

Measures of Central Tendency

Mean (average): x̄ = (Σ xᵢ) / n
Median: middle value when data is ordered
Mode: most frequently occurring value

Measures of Spread

Range = max − min
Variance (population): σ² = Σ(xᵢ − μ)² / N
Variance (sample): s² = Σ(xᵢ − x̄)² / (n − 1)
Standard deviation: σ = √(σ²), s = √(s²)

Z-Score

z = (x − μ) / σ

Measures how many standard deviations a value is from the mean.

Probability Rules

0 ≤ P(A) ≤ 1
P(A') = 1 − P(A) (complement)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (addition rule)
P(A ∩ B) = P(A) · P(B|A) (multiplication rule)
P(A ∩ B) = P(A) · P(B) (if A and B are independent)

Conditional Probability

P(A|B) = P(A ∩ B) / P(B)

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

Bayes' Theorem

P(A|B) = P(B|A) · P(A) / P(B)

Permutations & Combinations

Permutations: P(n,r) = n! / (n − r)!
Combinations: C(n,r) = n! / (r!(n − r)!)

Discrete Distributions

Binomial Distribution

P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ
Mean: μ = np
Variance: σ² = np(1−p)

Poisson Distribution

P(X = k) = (λᵏ · e⁻ˡ) / k!
Mean: μ = λ
Variance: σ² = λ

Continuous Distributions

Normal Distribution

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f(x) = (1 / (σ√(2π))) · e^(−(x−μ)² / (2σ²))
68-95-99.7 Rule:
68% of data within μ ± 1σ
95% of data within μ ± 2σ
99.7% of data within μ ± 3σ

Linear Regression

ŷ = b₀ + b₁x
Slope: b₁ = (nΣxᵢyᵢ − ΣxᵢΣyᵢ) / (nΣxᵢ² − (Σxᵢ)²)
Intercept: b₀ = ȳ − b₁x̄

Correlation Coefficient

r = (nΣxᵢyᵢ − ΣxᵢΣyᵢ) / √((nΣxᵢ² − (Σxᵢ)²)(nΣyᵢ² − (Σyᵢ)²))

r ranges from −1 (perfect negative) to +1 (perfect positive), with 0 indicating no linear correlation.

Use n − 1 (Bessel's correction) in the denominator for sample variance and standard deviation. Use N for population parameters.

Linear Algebra Formulas

Vector Operations

Addition: (a₁, a₂) + (b₁, b₂) = (a₁+b₁, a₂+b₂)
Scalar multiplication: c(a₁, a₂) = (ca₁, ca₂)
Magnitude: ‖v‖ = √(v₁² + v₂² + ... + vₙ²)
Unit vector: û = v / ‖v‖

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Dot Product

a · b = a₁b₁ + a₂b₂ + ... + aₙbₙ
a · b = ‖a‖ ‖b‖ cos(θ)

If a · b = 0, the vectors are orthogonal (perpendicular).

Cross Product (3D)

a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
‖a × b‖ = ‖a‖ ‖b‖ sin(θ)

The result is a vector perpendicular to both a and b.

Matrix Operations

Addition: (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ (same dimensions)
Scalar mult.: (cA)ᵢⱼ = cAᵢⱼ
Multiplication: (AB)ᵢⱼ = Σₖ AᵢₖBₖⱼ (A is m×n, B is n×p → AB is m×p)
Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ

Matrix multiplication is NOT commutative: AB ≠ BA in general. Always check dimensions before multiplying.

Determinants

2×2 Determinant

det [a b] = ad − bc
[c d]

3×3 Determinant (cofactor expansion along first row)

det [a b c]
[d e f] = a(ei − fh) − b(di − fg) + c(dh − eg)
[g h i]

Matrix Inverse

2×2 Inverse

A = [a b] A⁻¹ = (1/det(A)) · [ d −b]
[c d] [−c a]

A⁻¹ exists only if det(A) ≠ 0 (the matrix is non-singular).

Properties

AA⁻¹ = A⁻¹A = I (identity matrix)
(AB)⁻¹ = B⁻¹A⁻¹
(Aᵀ)⁻¹ = (A⁻¹)ᵀ

Eigenvalues & Eigenvectors

Av = λv

Where λ is an eigenvalue and v is the corresponding eigenvector.

Characteristic equation: det(A − λI) = 0

Solve for λ to find eigenvalues, then solve (A − λI)v = 0 for eigenvectors.

Key Properties

Σ λᵢ = trace(A) = Σ Aᵢᵢ
Π λᵢ = det(A)

Number Theory Formulas

Divisibility & GCD

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Division Algorithm: a = bq + r, where 0 ≤ r < b
GCD via Euclidean Algorithm: gcd(a, b) = gcd(b, a mod b)
Bézout's Identity: gcd(a, b) = ax + by for some integers x, y
LCM formula: lcm(a, b) = |a · b| / gcd(a, b)

Modular Arithmetic

a ≡ b (mod n) means n | (a − b)
(a + b) mod n = ((a mod n) + (b mod n)) mod n
(a · b) mod n = ((a mod n) · (b mod n)) mod n
aᵏ mod n = ((a mod n)ᵏ) mod n

Euler's Totient Function

φ(n) = n · ∏(p|n) (1 − 1/p)
φ(pᵏ) = pᵏ − pᵏ⁻¹ = pᵏ(1 − 1/p)
φ(mn) = φ(m)φ(n) when gcd(m,n) = 1

Fermat's & Euler's Theorems

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Fermat's Little Theorem: aᵖ ≡ a (mod p) if p is prime
Equivalently: aᵖ⁻¹ ≡ 1 (mod p) if gcd(a,p) = 1
Euler's Theorem: a^φ(n) ≡ 1 (mod n) if gcd(a,n) = 1

Chinese Remainder Theorem

If gcd(mᵢ, mⱼ) = 1 for all i ≠ j, then the system:
x ≡ a₁ (mod m₁)
x ≡ a₂ (mod m₂)
...has a unique solution modulo M = m₁ · m₂ · ... · mₖ

Prime Number Formulas

Fundamental Theorem of Arithmetic: every n > 1 has a unique prime factorization
Number of divisors: τ(n) = ∏(eᵢ + 1) for n = p₁^e₁ · p₂^e₂ · ...
Sum of divisors: σ(n) = ∏((pᵢ^(eᵢ+1) − 1) / (pᵢ − 1))
Prime Counting Function: π(x) ≈ x / ln(x) (Prime Number Theorem)

Wilson's Theorem

(p − 1)! ≡ −1 (mod p) if and only if p is prime

Number theory formulas form the security backbone of RSA encryption. The RSA algorithm relies on Euler's theorem and the difficulty of factoring large numbers.

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Differential Equations Formulas

Separable Equations

dy/dx = f(x)g(y) → ∫ dy/g(y) = ∫ f(x) dx + C

First-Order Linear (Integrating Factor)

dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^(∫ P(x) dx)
Solution: y = (1/μ) ∫ μQ dx + C/μ

Second-Order Linear (Constant Coefficients)

ay″ + by′ + cy = 0
Characteristic equation: ar² + br + c = 0

Two real roots r₁ ≠ r₂: y = C₁e^(r₁x) + C₂e^(r₂x)
Repeated root r: y = (C₁ + C₂x)e^(rx)
Complex roots α ± βi: y = e^(αx)(C₁cos(βx) + C₂sin(βx))

Method of Undetermined Coefficients

For ay″ + by′ + cy = g(x):
If g(x) = polynomial → try yₚ = polynomial of same degree
If g(x) = eᵏˣ → try yₚ = Aeᵏˣ
If g(x) = sin(kx) or cos(kx) → try yₚ = A·cos(kx) + B·sin(kx)

Multiply by x if the trial solution overlaps with the homogeneous solution.

Variation of Parameters

For y″ + P(x)y′ + Q(x)y = g(x):
yₚ = −y₁ ∫ (y₂g)/W dx + y₂ ∫ (y₁g)/W dx
where W = y₁y₂′ − y₂y₁′ (Wronskian)

Laplace Transform Pairs

ℒ{1} = 1/s
ℒ{t} = 1/s²
ℒ{tⁿ} = n!/s^(n+1)
ℒ{eᵃᵗ} = 1/(s − a)
ℒ{sin(bt)} = b/(s² + b²)
ℒ{cos(bt)} = s/(s² + b²)
ℒ{eᵃᵗf(t)} = F(s − a) (first shift)
ℒ{f′(t)} = sF(s) − f(0)
ℒ{f″(t)} = s²F(s) − sf(0) − f′(0)

Famous Differential Equations

Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ
Logistic growth: dy/dt = ky(1 − y/L) → y = L/(1 + Ce⁻ᵏᵗ)
Simple harmonic oscillator: y″ + ω²y = 0 → y = A·cos(ωt) + B·sin(ωt)
Damped oscillator: y″ + 2ζωy′ + ω²y = 0
Heat equation: ∂u/∂t = k · ∂²u/∂x²
Wave equation: ∂²u/∂t² = c² · ∂²u/∂x²
Laplace equation: ∂²u/∂x² + ∂²u/∂y² = 0

Mathematical Constants

π (pi) = 3.14159265358979...
e (Euler's number) = 2.71828182845904...
φ (golden ratio) = (1 + √5)/2 = 1.61803398874989...
√2 = 1.41421356237310...
√3 = 1.73205080756888...
γ (Euler–Mascheroni constant) = 0.57721566490153...
ln(2) = 0.69314718055995...
ln(10) = 2.30258509299405...

Important Identities

e^(iπ) + 1 = 0 (Euler's identity)
e^(iθ) = cos(θ) + i·sin(θ) (Euler's formula)
Γ(n) = (n − 1)! for positive integers
Γ(1/2) = √π
ζ(2) = π²/6 (Basel problem)
∫₋∞^∞ e^(−x²) dx = √π (Gaussian integral)

This formula sheet covers the most commonly needed formulas across mathematics. For topic-specific explanations and worked examples, visit the individual topic pages linked in the navigation.