Pythagorean Theorem
Geometrya² + b² = c²
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Law of Sines
Trigonometrya/sin A = b/sin B = c/sin C
The ratio of a side of a triangle to the sine of its opposite angle is constant.
Law of Cosines
Trigonometryc² = a² + b² − 2ab·cos C
Generalization of the Pythagorean theorem to arbitrary triangles.
Binomial Theorem
Algebra(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ
Expands powers of a binomial into a sum of terms.
Quadratic Formula
Algebrax = (−b ± √(b² − 4ac)) / 2a
Solutions of the quadratic equation ax² + bx + c = 0.
Euler's Identity
Analysise^(iπ) + 1 = 0
Links five fundamental constants in one equation.
Euler's Formula
Analysise^(ix) = cos x + i·sin x
Bridges exponential and trigonometric functions via complex numbers.
Fundamental Thm. of Calculus
Calculus∫ₐᵇ f'(x) dx = f(b) − f(a)
Differentiation and integration are inverse operations.
Chain Rule
Calculus(f∘g)'(x) = f'(g(x))·g'(x)
Derivative of a composition of functions.
Product Rule
Calculus(fg)' = f'g + fg'
Derivative of a product of two functions.
Taylor Series
Analysisf(x) = Σ f⁽ⁿ⁾(a) (x−a)ⁿ / n!
Represents a smooth function as an infinite polynomial around a point.
De Morgan's Laws
Logic¬(A ∧ B) = ¬A ∨ ¬B
Rules relating negation, conjunction and disjunction.
Bayes' Theorem
ProbabilityP(A|B) = P(B|A)·P(A) / P(B)
Updates the probability of a hypothesis given new evidence.
Central Limit Theorem
StatisticsX̄ →ᵈ 𝒩(μ, σ²/n)
Means of large samples are approximately normally distributed.
Fermat's Little Theorem
Number Theoryaᵖ ≡ a (mod p)
For any integer a and prime p.
Cauchy–Schwarz Inequality
Linear Algebra|⟨u,v⟩| ≤ ‖u‖·‖v‖
Bounds the inner product of two vectors.
Triangle Inequality
Geometry‖u + v‖ ≤ ‖u‖ + ‖v‖
The length of any side of a triangle never exceeds the sum of the other two.
Heron's Formula
GeometryA = √(s(s−a)(s−b)(s−c))
Area of a triangle from its three sides, with s the semi-perimeter.
Euler's Polyhedron Formula
TopologyV − E + F = 2
For any convex polyhedron, vertices minus edges plus faces equals two.
Fermat's Last Theorem
Number Theoryaⁿ + bⁿ ≠ cⁿ (n>2)
No positive integers satisfy the equation for any integer exponent greater than two.
Euclid's Infinity of Primes
Number Theory|{p prime}| = ∞
There are infinitely many prime numbers.
Fundamental Thm. of Arithmetic
Number Theoryn = p₁^a₁ · p₂^a₂ · … · pₖ^aₖ
Every integer greater than one factors uniquely into primes.
Fundamental Thm. of Algebra
Algebradeg p = n ⇒ n complex roots
Every non-constant polynomial has as many complex roots as its degree.
L'Hôpital's Rule
Calculuslim f/g = lim f'/g'
Evaluates 0/0 or ∞/∞ limits by differentiating numerator and denominator.
Mean Value Theorem
Calculusf'(c) = (f(b) − f(a)) / (b − a)
A differentiable function attains its average slope at some interior point.
Integration by Parts
Calculus∫u dv = uv − ∫v du
Reverses the product rule to integrate products of functions.
Green's Theorem
Vector Calc∮ P dx + Q dy = ∬ (∂Q/∂x − ∂P/∂y) dA
Relates a line integral around a closed curve to a double integral over the enclosed area.
Stokes' Theorem
Vector Calc∮ F·dr = ∬ (∇×F)·dA
Generalizes Green's theorem to surfaces in three-dimensional space.
Divergence Theorem
Vector Calc∭ ∇·F dV = ∯ F·dA
Relates the flow of a field out of a volume to the divergence inside.
Fourier Series
Analysisf(x) = a₀ + Σ (aₙ cos nx + bₙ sin nx)
Represents any reasonable periodic function as a sum of sines and cosines.
Laplace Transform
Analysisℒ{f}(s) = ∫₀^∞ e^(−st) f(t) dt
Transforms differential equations into algebraic ones.
Cauchy Integral Formula
Complexf(a) = (1/2πi) ∮ f(z)/(z−a) dz
Values of a holomorphic function inside a contour follow from its values on the contour.
Residue Theorem
Complex∮ f dz = 2πi · Σ Res
Contour integrals of meromorphic functions equal 2πi times the sum of residues inside.
Gauss's Divergence Identity
Vector Calc∇·(∇×F) = 0
The divergence of any curl is always zero.
Jensen's Inequality
Probabilityf(E[X]) ≤ E[f(X)] (f convex)
Averages taken inside a convex function are bounded by the function of the average.
Chebyshev's Inequality
ProbabilityP(|X−μ| ≥ kσ) ≤ 1/k²
Bounds the probability that a random variable lies far from its mean.
Law of Large Numbers
StatisticsX̄ₙ → μ (n → ∞)
Sample averages converge to the true mean as the sample grows.
Euclidean Distance
Geometryd = √Σ(xᵢ − yᵢ)²
Straight-line distance between two points in n-dimensional space.
Circle Area & Circumference
GeometryA = πr², C = 2πr
Fundamental measurements of a circle from its radius.
Sphere Volume & Surface
GeometryV = 4/3·πr³, A = 4πr²
Volume and surface area of a sphere from its radius.
Golden Ratio
Number Theoryφ = (1 + √5)/2 ≈ 1.618
Ratio for which the whole is to the larger part as the larger is to the smaller.
Gauss–Bonnet Theorem
Topology∫_M K dA + ∮_∂M kg ds = 2π χ
Links the curvature of a surface to its topology through the Euler characteristic.
Zermelo–Fraenkel Axioms
LogicZF ⊂ ZFC
The standard foundation for modern set theory, with the axiom of choice giving ZFC.