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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
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Arithmetic Sequences & Series

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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

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Square (side s)

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

Rectangle (length l, width w)

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

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)

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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)
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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)

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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

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

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

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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

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)

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

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

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
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))
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

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‖

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

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

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.

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))
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

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.