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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)
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
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)
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

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)
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
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

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)

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
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

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

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

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
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)!)
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

Discrete Distributions

Binomial Distribution

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
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

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‖

Dot Product

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
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)

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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.
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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)

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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.