Science

Mathematics

Branches, theorems, constants, and famous problems of mathematics.

A study reference, not a substitute for primary sources. Updated 2026-06-02.

Number Systems and Types

Natural numbers (ℕ) — positive integers 1, 2, 3, … (some definitions include 0); closed under addition and multiplication.
Integers (ℤ) — natural numbers plus zero and their negatives; closed under addition, subtraction, and multiplication.
Rational numbers (ℚ) — numbers expressible as p/q with integers p, q (q ≠ 0); decimals terminate or repeat.
Irrational numbers — real numbers not expressible as a ratio of integers; decimals are non-terminating and non-repeating. Examples: √2, π, e.
Real numbers (ℝ) — all rationals and irrationals; form a complete ordered field and correspond to points on the number line.
Complex numbers (ℂ) — numbers of the form a + bi where i = √(−1); extend ℝ to the complex plane.
Algebraic vs transcendental — algebraic numbers are roots of polynomials with integer coefficients (includes all rationals, √2); transcendental numbers (e.g., π and e) are not. π and e were both proved transcendental in the 19th century (Lindemann 1882 for π; Hermite 1873 for e).
Cardinal and ordinal numbers — cardinals measure set size; ordinals encode order type. Cantor showed infinite cardinals themselves form a hierarchy (ℵ₀ < ℵ₁ < …).

Arithmetic and Algebra Fundamentals

Fundamental theorem of arithmetic — every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors; proves the existence and uniqueness of prime factorization.
Rational root theorem — if a polynomial aₙxⁿ + … + a₀ with integer coefficients has a rational root p/q (in lowest terms), then p divides a₀ and q divides aₙ; used to find candidate rational roots before factoring.
Commutative, associative, distributive laws — commutativity: a + b = b + a; associativity: (a + b) + c = a + (b + c); distributivity: a(b + c) = ab + ac.
Order of operations (PEMDAS/BODMAS) — Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
Absolute value — x = x if x ≥ 0, −x if x < 0; measures distance from zero.
Quadratic formula — for ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a; the discriminant b² − 4ac determines whether roots are real and distinct, real and repeated, or complex.
Binomial theorem — (a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ; coefficients given by Pascal’s triangle.
Fundamental theorem of algebra — every non-constant polynomial with complex coefficients has at least one complex root; equivalently, a degree-n polynomial has exactly n roots in ℂ (counting multiplicity). Proved rigorously by Gauss (1799).
Logarithm laws — log(ab) = log a + log b; log(aⁿ) = n log a; change of base: log_b(x) = ln x / ln b.
Modular arithmetic — a ≡ b (mod n) means n divides a − b; forms the basis of clock arithmetic and much of cryptography.
Pascal’s triangle — triangular array where each entry is the sum of the two above it; row n gives the binomial coefficients C(n,0), C(n,1), …, C(n,n); also encodes Fibonacci numbers (diagonal sums) and Sierpinski triangle (odd entries).

Geometry

Euclidean Foundations

Euclid’s Elements — ~300 BCE; 13 books systematizing geometry from five postulates. The fifth postulate (parallel postulate) is the one that distinguishes Euclidean from non-Euclidean geometry.
Congruence criteria — triangles are congruent by SSS, SAS, ASA, AAS; not by SSA (ambiguous case).
Similar triangles — corresponding angles equal, sides proportional; AA is sufficient for similarity.
Angle sum of a triangle — 180° in Euclidean geometry; less than 180° in hyperbolic, more than 180° on a sphere.
Thales’ theorem — an angle inscribed in a semicircle is a right angle.
Area formulas — triangle: ½bh or Heron’s formula √(s(s−a)(s−b)(s−c)); circle: πr²; sector: ½r²θ.

Polygons and Polyhedra

Quadrilateral — a polygon with four sides and four vertices; the sum of interior angles is always 360°; types include parallelogram (two pairs of parallel sides; opposite sides equal), rectangle (parallelogram with right angles), rhombus (parallelogram with equal sides), square (rectangle and rhombus), trapezoid (exactly one pair of parallel sides in US usage; called trapezium in UK), and kite (two pairs of adjacent equal sides); any quadrilateral can be divided into two triangles, giving the 360° angle sum; a cyclic quadrilateral has all four vertices on a circle, and opposite angles sum to 180°.
Perimeter — the total length of the boundary of a two-dimensional figure; for a polygon, it is the sum of all side lengths; for a circle, the perimeter is the circumference C = 2πr = πd; for a rectangle, P = 2(l + w); for an equilateral triangle with side s, P = 3s; distinct from area (which measures the enclosed region, not the boundary); in optimization, the isoperimetric inequality states that among all plane figures with a given perimeter, the circle has the maximum enclosed area (equivalently, among shapes of a given area, the circle has the minimum perimeter): 4πA ≤ L², where A is area and L is perimeter, with equality only for circles.
Regular polygons — equal sides and angles; interior angle sum = (n − 2) × 180°; each interior angle = (n − 2) × 180° / n.
Platonic solids — exactly 5: tetrahedron (4 faces), cube/hexahedron (6), octahedron (8), dodecahedron (12), icosahedron (20). All faces are congruent regular polygons with the same number meeting at each vertex.
Euler’s polyhedron formula — V − E + F = 2 for any convex polyhedron (V vertices, E edges, F faces).
Archimedean solids — 13 semi-regular convex polyhedra with two or more types of regular polygon faces.

Topology

Topological space — a set X together with a collection of “open” subsets satisfying: the empty set and X are open; arbitrary unions of open sets are open; finite intersections of open sets are open; generalizes metric spaces and underpins all of modern topology.
Compact space — a topological space where every open cover has a finite subcover; in ℝⁿ, equivalently, closed and bounded (Heine-Borel theorem); compactness guarantees continuity theorems and extreme value results.
Simply connected — a space is simply connected if it is path-connected and every loop can be continuously contracted to a point (trivial fundamental group); the sphere S² is simply connected; the torus is not.
Möbius strip — a surface with only one side and one boundary edge, formed by giving a rectangular strip a half-twist and joining the ends; the canonical example of a non-orientable surface.
Klein bottle — a closed non-orientable surface with no interior or exterior; cannot be embedded in 3D space without self-intersection; a 4D object; named for Felix Klein (1882).
Euler characteristic — topological invariant χ = V − E + F for polyhedra (equals 2 for convex polyhedra); for surfaces χ = 2 − 2g where g is the number of handles (genus); central to the classification of surfaces.
Homeomorphism — a bijective continuous map with a continuous inverse; two spaces are topologically equivalent if homeomorphic (“rubber-sheet geometry”); e.g., a donut and a coffee cup are homeomorphic.
Brouwer fixed-point theorem — every continuous function from a closed ball to itself has at least one fixed point; e.g., any stirred cup of coffee has at least one unmoved point; proved by L. E. J. Brouwer (~1911).
Hairy ball theorem — there is no everywhere-nonzero continuous tangent vector field on an even-dimensional sphere; colloquially: you cannot comb a hairy ball flat without a cowlick; a corollary is that there is always a point on Earth with zero horizontal wind.
Ham-sandwich theorem — for n measurable “ingredients” in ℝⁿ, there exists a single hyperplane that simultaneously bisects all n of them; proved using the Borsuk-Ulam theorem.

Projective and Differential Geometry

Projective geometry — geometry of properties invariant under projection; parallel lines meet at a “point at infinity”; cross-ratio is the fundamental projective invariant; underpins perspective drawing and homogeneous coordinates in computer graphics.
Riemannian geometry — generalizes Euclidean geometry to curved smooth manifolds equipped with a metric tensor; Bernhard Riemann introduced the framework in his 1854 Habilitation lecture; the mathematical language of Einstein’s general relativity.
Gauss-Bonnet theorem — for a closed surface, ∫∫ K dA = 2π χ where K is the Gaussian curvature and χ is the Euler characteristic; connects local geometry (curvature) to global topology.

Non-Euclidean and Analytic Geometry

Hyperbolic geometry — Lobachevsky and Bolyai (independently, 1820s–1830s) developed a consistent geometry where parallel postulate fails and triangles have angle sums less than 180°.
Spherical geometry — great circles are “straight lines”; no parallel lines exist; triangles have angle sums greater than 180°.
Cartesian coordinates — Descartes’ analytic geometry (17th century) assigned numerical coordinates to geometric points, bridging algebra and geometry.
Distance formula — d = √((x₂ − x₁)² + (y₂ − y₁)²) in the Euclidean plane.
Conic sections — curves from slicing a double cone: circle, ellipse, parabola, hyperbola. Standard forms expressible as quadratic equations in x and y.
Triangle centers — the centroid (intersection of medians; center of mass) is always inside the triangle; the orthocenter (intersection of altitudes) can be outside for obtuse triangles; the incenter (intersection of angle bisectors) is equidistant from all three sides; the circumcenter (intersection of perpendicular bisectors) is equidistant from all three vertices. All four are collinear on the Euler line (except in equilateral triangles).
Catalan numbers — sequence 1, 1, 2, 5, 14, 42, …; Cₙ = C(2n,n)/(n+1); count structurally equivalent combinatorial objects including valid parenthesizations, binary trees, and non-crossing polygon triangulations.
Polar coordinates — alternative 2D coordinate system using distance r from the origin and angle θ from the positive x-axis; x = r cos θ, y = r sin θ; simplifies equations of circles, spirals, and roses.

Vectors in ℝⁿ — quantities with magnitude and direction; dot product u·v =

cos θ gives the projection; cross product u×v (in ℝ³) gives a perpendicular vector of magnitude

sin θ.

Trigonometry

Six functions — sin, cos, tan, csc (1/sin), sec (1/cos), cot (1/tan); defined for right triangles via opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent; extended to all angles via the unit circle.
Pythagorean identities — sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ.
Angle addition formulas — sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.
Law of sines — a/sin A = b/sin B = c/sin C = 2R (R = circumradius).
Law of cosines — c² = a² + b² − 2ab cos C; generalizes the Pythagorean theorem.
Radian measure — one radian is the angle subtending an arc equal to the radius; 2π radians = 360°.
Key values — sin 30° = ½; sin 45° = √2/2; sin 60° = √3/2; cos 0° = 1; tan 45° = 1.
Euler’s formula — e^(iθ) = cos θ + i sin θ; at θ = π gives Euler’s identity: e^(iπ) + 1 = 0.
De Moivre’s theorem — (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ); enables computation of powers and roots of complex numbers and derivation of multiple-angle identities.
Hyperbolic functions — sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x; satisfy cosh²x − sinh²x = 1; appear in solutions to wave and heat equations and in catenary curves.

Calculus

Origins and Fundamentals

Founders — Newton (fluxions, 1660s) and Leibniz (notation d/dx and ∫, 1670s–1680s) developed calculus independently; a priority dispute consumed much of their later careers. Leibniz’s notation became standard.
Limits — formal foundation of calculus (Cauchy, Weierstrass, 19th century); lim(x→a) f(x) describes the value f approaches but need not equal f(a).
Continuity — f is continuous at a if the limit equals the function value; differentiability implies continuity (not vice versa).

Differential Calculus

Rolle’s theorem — if f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) where f′(c) = 0; a special case of (and the basis for proving) the mean value theorem.
Intermediate value theorem — if f is continuous on [a,b] and k is any value between f(a) and f(b), there exists c in (a,b) with f(c) = k; implies, e.g., that a continuous function changing sign must have a zero.
Taylor’s theorem — a smooth function can be approximated near a point by a polynomial (the Taylor polynomial) with an explicit remainder term; underpins numerical analysis, asymptotic methods, and series expansions.
Derivative — instantaneous rate of change; f′(x) = lim(h→0) [f(x+h) − f(x)] / h.
Basic rules — power rule: d/dx xⁿ = nxⁿ⁻¹; product rule: (fg)′ = f′g + fg′; quotient rule: (f/g)′ = (f′g − fg′)/g²; chain rule: d/dx f(g(x)) = f′(g(x))g′(x).
Common derivatives — d/dx sin x = cos x; d/dx eˣ = eˣ; d/dx ln x = 1/x.
Critical points and optimization — f′(c) = 0 or undefined at local extrema; second derivative test: f″(c) > 0 → local min; < 0 → local max.
Mean value theorem — if f is continuous on [a,b] and differentiable on (a,b), there exists c where f′(c) = (f(b) − f(a)) / (b − a).

Integral Calculus

Antiderivative / indefinite integral — F is an antiderivative of f if F′ = f; written ∫ f(x) dx = F(x) + C.
Definite integral — the signed area under f from a to b; defined as the limit of Riemann sums.
Fundamental theorem of calculus — Part 1: d/dx ∫(a to x) f(t) dt = f(x); Part 2: ∫(a to b) f(x) dx = F(b) − F(a). Unifies differentiation and integration.
Integration techniques — substitution (u-substitution), integration by parts (∫ u dv = uv − ∫ v du), partial fractions, trigonometric substitution.
Improper integrals — integrals over infinite intervals or with singularities; convergent if the limit exists.

Multivariable and Beyond

Partial derivatives — ∂f/∂x holds other variables constant; gradient ∇f points in the direction of steepest ascent.
Multiple integrals — double and triple integrals over regions; Fubini’s theorem permits iterating the integrals.
Taylor / Maclaurin series — f(x) = Σ f⁽ⁿ⁾(a)/n! (x − a)ⁿ; Maclaurin is the special case a = 0. Classic examples: eˣ = Σ xⁿ/n!; sin x = x − x³/3! + …; cos x = 1 − x²/2! + …
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) (provided the latter limit exists); resolves indeterminate forms.
Lagrange multipliers — method for optimizing f(x,y,…) subject to constraint g(x,y,…) = 0; set ∇f = λ∇g and solve simultaneously for variables and multiplier λ.
Calculus of variations — optimizes functionals (functions of functions) rather than ordinary functions; the Euler-Lagrange equation gives necessary conditions; classic problems include the brachistochrone (curve of fastest descent, solved by Bernoulli and Newton) and geodesics.
Green’s theorem — relates a line integral around a closed planar curve C to a double integral over the enclosed region D: ∮_C (P dx + Q dy) = ∬_D (∂Q/∂x − ∂P/∂y) dA.
Stokes’ theorem — generalizes Green’s theorem to surfaces in 3D: ∬S (∇ × F) · dS = ∮∂S F · dr; flux of the curl through a surface equals the circulation around its boundary.
Divergence theorem (Gauss’s theorem) — ∭V (∇ · F) dV = ∯∂V F · dS; total divergence in a volume equals net flux through the enclosing surface.
Implicit function theorem — given conditions on a system of equations F(x, y) = 0, the theorem guarantees the existence of a locally defined function y = f(x) and gives its derivative; foundational in manifold theory and economics.
Inverse function theorem — if a differentiable map has an invertible Jacobian at a point, it is locally a diffeomorphism near that point; the derivative of the inverse is the inverse of the Jacobian.
Jacobian — the matrix of all first-order partial derivatives of a vector-valued function; its determinant measures local volume scaling and appears in change-of-variables for multiple integrals.

Linear Algebra

Cauchy-Schwarz inequality —

⟨u, v⟩

² ≤ ⟨u,u⟩ · ⟨v,v⟩ for inner product spaces; in ℝⁿ: (Σ aᵢbᵢ)² ≤ (Σ aᵢ²)(Σ bᵢ²); ubiquitous in analysis, probability, and linear algebra.

Triangle inequality — for a normed space: ‖u + v‖ ≤ ‖u‖ + ‖v‖; generalizes the geometric fact that one side of a triangle is shorter than the sum of the other two; underpins metric space theory.
Matrix — rectangular array of numbers; addition and scalar multiplication are element-wise; matrix multiplication is row-by-column (not commutative in general).
Determinant — scalar det(A) encoding whether a matrix is invertible (det ≠ 0) and the signed volume scaling factor of its linear transformation; det(AB) = det(A) det(B).
Eigenvalue / eigenvector — scalar λ and nonzero vector v such that Av = λv; eigenvalues are roots of the characteristic polynomial det(A − λI) = 0; diagonalization decomposes A = PDP⁻¹ when n independent eigenvectors exist.
Gaussian elimination — systematic row operations to reduce a matrix to row echelon form; solves linear systems and computes rank and inverse.
Vector space — set closed under addition and scalar multiplication satisfying eight axioms (e.g., ℝⁿ, polynomials, function spaces); a basis is a minimal spanning set; dimension is the number of basis vectors.
Cayley-Hamilton theorem — every square matrix satisfies its own characteristic polynomial; e.g., if p(λ) = det(A − λI), then p(A) = 0; allows matrix powers and inverses to be expressed as lower-degree polynomials.
Spectral theorem — a real symmetric (or complex Hermitian) matrix is diagonalizable by an orthogonal (unitary) matrix; its eigenvalues are real; fundamental to principal component analysis, quantum mechanics, and differential equations.
Linear transformation — function T: V → W preserving addition and scalar multiplication; representable as matrix multiplication once bases are chosen.
One-to-one function (injective function) — a function f: A → B in which distinct inputs map to distinct outputs: f(x₁) = f(x₂) ⟹ x₁ = x₂; equivalently, no two elements of the domain share the same image; the horizontal line test detects whether a function’s graph is one-to-one; a function that is both injective (one-to-one) and surjective (onto) is bijective, meaning it has a well-defined inverse function; important in cryptography (injective encryption functions) and formal logic; contrasted with a many-to-one function (like f(x) = x²) which is not injective.
Rank-nullity theorem — for a linear map T: V → W, dim(V) = rank(T) + nullity(T), where rank is the dimension of the image and nullity is the dimension of the kernel; a fundamental dimension-counting result.
Singular value decomposition (SVD) — any m×n matrix A can be written as UΣVᵀ where U and V are orthogonal and Σ is diagonal with non-negative entries (singular values); basis of principal component analysis, data compression, and pseudoinverse computation.
Gram-Schmidt process — algorithm that converts any basis of an inner product space into an orthonormal basis; foundation of QR decomposition.
Positive definite matrix — a symmetric matrix A is positive definite if xᵀAx > 0 for all nonzero x; equivalently, all eigenvalues are positive; appears in optimization (Hessian criteria), statistics (covariance matrices), and physics.

Differential Equations

Ordinary differential equation (ODE) — equation relating a function of one variable to its derivatives; order = highest derivative present; e.g., y′ = ky has solution y = Ce^(kt).
Separable equation — ODE of the form dy/dx = f(x)g(y); solved by separating variables and integrating both sides: ∫ dy/g(y) = ∫ f(x) dx.
Laplace transform — integral transform ℒ{f(t)} = ∫₀^∞ e^(−st) f(t) dt; converts ODEs into algebraic equations in s; inverse transform recovers the solution.
Fourier series — decomposes a periodic function into sums of sines and cosines: f(x) = a₀/2 + Σ (aₙ cos nx + bₙ sin nx); coefficients given by orthogonality integrals.
Fourier transform — extends Fourier series to non-periodic functions: F̂(ξ) = ∫ f(x) e^(−2πiξx) dx; fundamental to signal processing, PDEs, and quantum mechanics.
Heat equation — ∂u/∂t = α ∇²u; models diffusion of heat (or other quantities) over time; one of the canonical parabolic PDEs; solution via Fourier series or the heat kernel.
Wave equation — ∂²u/∂t² = c² ∇²u; models propagation of waves (sound, light, strings); one of the canonical hyperbolic PDEs.
Laplace equation — ∇²u = 0; solutions are harmonic functions; models steady-state heat, gravitational potential, and electrostatics; one of the canonical elliptic PDEs.
Sturm-Liouville theory — framework for second-order linear ODEs with boundary conditions yielding eigenfunctions that form orthogonal bases; unifies the theory of Fourier series, Legendre polynomials, and Bessel functions.

Abstract Algebra

Group — set G with a binary operation satisfying closure, associativity, identity, and invertibility; e.g., (ℤ, +), permutations of a set. An abelian (commutative) group has ab = ba for all elements.

Lagrange’s theorem (group theory) — the order of any subgroup H of a finite group G divides the order of G; equivalently

= the number of cosets; key consequence: the order of any element divides

Ring — set with two operations (addition and multiplication) where addition forms an abelian group and multiplication is associative and distributes over addition; e.g., (ℤ, +, ×).
Field — a commutative ring where every nonzero element has a multiplicative inverse; e.g., ℚ, ℝ, ℂ, and finite fields 𝔽_p for prime p.
Homomorphism / isomorphism — a homomorphism is a structure-preserving map between algebraic objects (e.g., groups, rings); an isomorphism is a bijective homomorphism with a homomorphic inverse; isomorphic structures are algebraically identical.
Normal subgroup / quotient group — a subgroup N of G is normal if gNg⁻¹ = N for all g ∈ G; the quotient group G/N consists of the cosets of N; the first isomorphism theorem states G/ker(φ) ≅ im(φ) for any homomorphism φ.
Sylow theorems — for a finite group G of order pⁿm (p prime, p ∤ m), Sylow’s theorems guarantee the existence, count, and conjugacy of subgroups of order pⁿ; central tools in the classification of groups.
Classification of finite simple groups — the complete list of finite simple groups consists of cyclic groups of prime order, alternating groups Aₙ (n ≥ 5), 16 families of groups of Lie type, and 26 sporadic groups (including the Monster group, the largest, of order ~8 × 10⁵³); the proof spans tens of thousands of pages across hundreds of papers, completed ~1981–2004.
Noetherian ring — a ring in which every ascending chain of ideals eventually stabilizes (ascending chain condition); named for Emmy Noether; polynomial rings over fields are Noetherian (Hilbert basis theorem).
Principal ideal domain (PID) — an integral domain in which every ideal is generated by a single element; examples include ℤ and polynomial rings over fields; unique factorization holds in any PID.

p-adic numbers — for a prime p, the p-adic numbers ℚ_p complete ℚ with respect to the p-adic absolute value (

pⁿ a/b

_p = p⁻ⁿ for p ∤ a, b); form a field used in number theory and the proof of Fermat’s Last Theorem.

Galois theory — connects field extensions to group theory; Galois (1830s) proved that a polynomial equation is solvable by radicals if and only if its Galois group is a solvable group; explains why no general formula exists for degree ≥ 5.

Graph Theory

Seven Bridges of Königsberg — Euler’s 1736 problem asking whether one could cross all seven bridges of the city of Königsberg exactly once and return to the start; Euler proved it impossible by showing some vertices had odd degree, founding graph theory.
Eulerian path / circuit — a path (circuit) that traverses every edge exactly once; an Eulerian circuit exists if and only if every vertex has even degree; an Eulerian path exists iff exactly 0 or 2 vertices have odd degree.
Hamiltonian path / circuit — visits every vertex exactly once; unlike the Eulerian case, no simple degree condition characterizes existence; the related Hamiltonian cycle problem is NP-complete.
Planar graphs — graphs that can be drawn in the plane without edge crossings; Euler’s formula V − E + F = 2 applies; Kuratowski’s theorem: a graph is planar iff it contains no subdivision of K₅ or K₃,₃.
Graph coloring — assigning colors to vertices so adjacent vertices differ; the chromatic number χ(G) is the minimum number needed; the four color theorem states χ ≤ 4 for planar graphs.
Trees — connected acyclic graphs; a tree on n vertices has exactly n − 1 edges; spanning trees connect all vertices of a graph with no cycles; Cayley’s formula: the number of labeled trees on n vertices is nⁿ⁻².
Bipartite graph — a graph whose vertices can be 2-colored so no edge joins same-color vertices; König’s theorem: in a bipartite graph the maximum matching size equals the minimum vertex cover size.
Ramsey theory — studies conditions guaranteeing structure in large enough combinatorial objects; Ramsey’s theorem states that for any r and s there exists N such that any 2-coloring of K_N contains a monochromatic K_r or K_s; the smallest such N is a Ramsey number (most are unknown).
Network flow / max-flow min-cut theorem — in a directed network, the maximum flow from source to sink equals the minimum capacity of a cut separating them; proved by Ford and Fulkerson (1956).
Chromatic polynomial — counts the number of proper colorings of a graph G with k colors as a polynomial in k; evaluating at specific k gives the chromatic number when the polynomial first becomes positive.

Numerical Analysis

Wilson’s theorem — an integer p > 1 is prime if and only if (p − 1)! ≡ −1 (mod p); provides a theoretical primality characterization but is computationally impractical for large numbers.
Real analysis — rigorous study of the real number line, limits, continuity, differentiability, and Riemann/Lebesgue integration; key figures: Cauchy (epsilon-delta definitions), Weierstrass (pathological functions), Riemann (integration), Lebesgue (measure theory).
Lebesgue measure and integration — Henri Lebesgue (1902) extended Riemann integration to a far broader class of functions by measuring sets rather than partitioning intervals; the Lebesgue integral handles pointwise limits and is foundational to functional analysis and probability.
Measure theory — the axiomatic study of “size” for sets; a measure μ assigns non-negative numbers to measurable sets satisfying countable additivity; underpins probability theory (probability measures) and integration.
Metric space — a set X with a distance function d(x,y) satisfying non-negativity, symmetry, and the triangle inequality; convergence, continuity, and completeness generalize naturally to metric spaces.
Banach space — a complete normed vector space (every Cauchy sequence converges); e.g., Lᵖ spaces; the Hahn-Banach theorem extends linear functionals and is fundamental in functional analysis.
Hilbert space — a complete inner product space; the infinite-dimensional analogue of Euclidean space; central to quantum mechanics (state vectors are elements of a Hilbert space) and Fourier analysis; named for David Hilbert.
Complex analysis — study of functions of a complex variable; holomorphic functions obey the Cauchy-Riemann equations; Cauchy’s integral theorem states that the integral of an analytic function over a closed contour is zero; residue theorem enables evaluation of many real integrals.
Riemann mapping theorem — any simply connected proper open subset of ℂ is conformally equivalent (biholomorphic) to the open unit disk; a cornerstone of complex analysis.
Knot theory — studies mathematical knots (closed curves in 3D space) up to continuous deformation; key invariants include the Alexander polynomial, Jones polynomial (Vaughan Jones, 1984, Fields Medal 1990), and knot genus; the unknot is the trivial knot; the trefoil is the simplest non-trivial knot.
Category theory — abstract framework unifying mathematical structures via objects and morphisms (arrows); concepts include functors (structure-preserving maps between categories), natural transformations, and adjunctions; developed by Eilenberg and Mac Lane (1945); increasingly central to theoretical computer science and algebraic topology.
Algebraic topology — uses algebraic invariants (homotopy groups, homology groups, cohomology rings) to distinguish topological spaces; the fundamental group π₁ captures loops up to deformation; Henri Poincaré originated both homology and the conjecture bearing his name.
Combinatorics — counting and arrangement of discrete structures; key tools include permutations (n!), combinations (C(n,k) = n!/k!(n−k)!), the inclusion-exclusion principle, the pigeonhole principle, and generating functions.
Generating functions — formal power series Σ aₙ xⁿ encode a sequence (aₙ); multiplication, differentiation, and partial fractions extract combinatorial identities; ordinary and exponential generating functions are the two main types.
Stirling numbers — two related families: Stirling numbers of the first kind count permutations by cycle structure; those of the second kind count partitions of a set of n elements into k non-empty subsets; appear in combinatorics and analysis of algorithms.
Pigeonhole principle — if n+1 objects are placed into n containers, at least one container holds more than one object; despite simplicity, yields non-trivial results (e.g., two people in any city of > 365,000 share a birthday).

Inclusion-exclusion principle —

A ∪ B ∪ C

−

A∩B

−

A∩C

−

B∩C

A∩B∩C

; generalizes to n sets; used in counting and probability.

Newton-Raphson method — iterative root-finding: xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ); converges quadratically near a simple root; fails if f′(xₙ) = 0.
Runge-Kutta methods — family of numerical ODE solvers; the classical 4th-order method (RK4) achieves high accuracy by evaluating slopes at multiple intermediate points per step; widely used default in scientific computing.
Big-O notation — O(f(n)) describes an upper bound on the asymptotic growth rate of a function; standard hierarchy: O(1) < O(log n) < O(n) < O(n log n) < O(n²) < O(2ⁿ); central to algorithm analysis.
Turing machine — Alan Turing’s (1936) abstract model of computation consisting of an infinite tape, a read/write head, and a finite state table; defines the class of “computable” functions; the halting problem (determining whether an arbitrary Turing machine halts) is undecidable.
Traveling Salesperson Problem (TSP) — given a list of cities and the distances between them, find the shortest possible route that visits each city exactly once and returns to the origin; one of the most famous NP-hard optimization problems; the decision version (is there a route shorter than length k?) is NP-complete; exact solutions require exponential time in the worst case; notable approaches include the Held-Karp dynamic programming algorithm (O(2ⁿn²)), branch-and-bound methods, and the Concorde solver; practical approximation algorithms include the Christofides-Serdyuk algorithm (guaranteed within 3/2 of optimal for metric TSP) and local search heuristics (2-opt, 3-opt); has applications in logistics, circuit board drilling, DNA sequencing, and telescope scheduling; a canonical example used to motivate the P vs NP question.
NP-completeness — a problem is NP-complete if it is in NP and every NP problem reduces to it in polynomial time; Cook (1971) and Levin proved SAT (Boolean satisfiability) is NP-complete; thousands of problems (e.g., traveling salesman, graph coloring) are NP-complete; solving any one in polynomial time would imply P = NP.

Information Theory and Cryptography

Feigenbaum constants — two universal constants (δ ≈ 4.6692 and α ≈ 2.5029) discovered by Mitchell Feigenbaum (1975); describe the rate at which period-doubling bifurcations converge to chaos in a wide class of nonlinear maps; their universality was among the first deep results of chaos theory.
Strange attractor — the long-term behavior of a chaotic dynamical system; a fractal set in phase space to which trajectories converge but on which nearby trajectories diverge exponentially; examples include the Lorenz and Rössler attractors.
Lyapunov exponent — quantifies the rate of separation of infinitesimally close trajectories in a dynamical system; positive Lyapunov exponent is the hallmark of chaos.
Sierpinski triangle — a fractal formed by recursively removing the central triangle from an equilateral triangle; has fractal dimension log 3 / log 2 ≈ 1.585; also appears in Pascal’s triangle mod 2.
Cantor set — formed by iteratively removing the middle third of intervals starting from [0,1]; uncountable but of Lebesgue measure zero; a prototypical fractal; totally disconnected and perfect.
Mandelbrot set — the set of complex numbers c for which the iteration z → z² + c does not diverge; named for Benoît Mandelbrot; the iconic fractal with infinitely complex boundary; self-similar at all scales.
Julia sets — for a fixed complex parameter c, the Julia set is the boundary of the set of starting points z₀ for which the iteration z → z² + c remains bounded; connected when c is in the Mandelbrot set, a Cantor dust otherwise.
Fractal — a geometric object exhibiting self-similarity at multiple scales and typically a non-integer (fractal) dimension; examples include the Mandelbrot set, the Koch snowflake, and the Cantor set. The term was coined by Mandelbrot (1975).
Shannon entropy — H(X) = −Σ p(x) log₂ p(x); measures average information (in bits) of a random variable; maximized by a uniform distribution; introduced by Claude Shannon (1948).
RSA encryption — public-key cryptosystem (Rivest, Shamir, Adleman, 1977); security relies on the computational difficulty of factoring large semiprime n = pq; public key (n, e), private key d with ed ≡ 1 (mod φ(n)).
Chaos theory / Lorenz attractor — small differences in initial conditions of certain deterministic systems yield wildly divergent trajectories (“butterfly effect”); the Lorenz attractor (Edward Lorenz, 1963) is the canonical strange attractor arising from simplified fluid convection equations, exhibiting fractal structure.

Important Constants

Constant	Symbol	Approximate value	Significance
Pi	π	3.14159…	ratio of a circle’s circumference to its diameter; transcendental
Euler’s number	e	2.71828…	base of natural logarithm; transcendental; limit of (1 + 1/n)ⁿ as n → ∞
Golden ratio	φ	1.61803…	(1 + √5)/2; satisfies φ² = φ + 1; appears in Fibonacci sequence ratios
Imaginary unit	i	√(−1)	basis of complex numbers; i² = −1
Euler–Mascheroni constant	γ	0.57721…	limit of (Σ 1/k − ln n) as n → ∞; whether it is irrational is unknown
√2	—	1.41421…	first number proved irrational (attributed to the Pythagoreans)

Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, 21, …; each term is the sum of the two preceding. The ratio of consecutive Fibonacci numbers converges to φ.
Apéry’s constant — ζ(3) = Σ 1/n³ ≈ 1.20206…; proved irrational by Roger Apéry (1979) in a result that shocked the mathematical community; whether ζ(3)/π³ is irrational remains open.
Transcendence of π — proved by Ferdinand von Lindemann (1882) using Hermite’s method; immediately settled the ancient problem of squaring the circle: it is impossible with compass and straightedge.
verify: Hilbert’s seventh problem — confirm the problem asks whether aᵇ is transcendental for algebraic a ≠ 0,1 and irrational algebraic b; confirm it was solved affirmatively by Gelfond and Schneider (1934) independently (Gelfond-Schneider theorem).

Set Theory and Logic

Cantor’s set theory — Georg Cantor (1874 onward) formalized infinite sets and showed that not all infinities are equal in size.

Cardinality —

ℕ

= ℵ₀ (countably infinite); Cantor’s diagonal argument proves

ℝ

ℕ

; the real numbers are uncountably infinite.

Cantor’s theorem — for any set S, the power set P(S) has strictly greater cardinality; there is no largest infinity.

Continuum hypothesis — the assertion that there is no infinite cardinality between ℵ₀ and

ℝ

; shown to be independent of ZFC set theory (Gödel 1940, Cohen 1963).

Zermelo-Fraenkel axioms (ZFC) — standard axiomatic foundation for set theory, including the axiom of choice (C); the axiom of choice is equivalent to the well-ordering theorem and Zorn’s lemma.
Russell’s paradox — naive set theory is inconsistent: the set of all sets that do not contain themselves leads to a contradiction; motivated formal axiomatization (Bertrand Russell, 1901).
Propositional logic — connectives AND (∧), OR (∨), NOT (¬), implication (→), biconditional (↔); truth tables determine validity.
First-order logic — adds quantifiers ∀ (for all) and ∃ (there exists) to propositional logic.
Gödel’s completeness theorem — (1929, Gödel’s dissertation) every valid first-order sentence (true in all models) has a formal proof; distinct from and opposite to the incompleteness theorems, which address specific axiom systems for arithmetic.
Gödel’s incompleteness theorems — (1931) First theorem: any consistent formal system strong enough to express basic arithmetic contains true statements that cannot be proved within the system. Second theorem: such a system cannot prove its own consistency.

Number Theory

Prime numbers — integers greater than 1 with no positive divisors other than 1 and themselves; 2 is the only even prime.
Infinitely many primes — proved by Euclid via a reductio ad absurdum argument.
Sieve of Eratosthenes — ancient algorithm for finding all primes up to a given limit.
Prime number theorem — the number of primes up to n is asymptotically n / ln n; independently proved by Hadamard and de la Vallée Poussin (1896).
Euclid’s algorithm — iterative application of division to find the greatest common divisor (GCD) of two integers.
Fermat’s little theorem — if p is prime and p does not divide a, then aᵖ⁻¹ ≡ 1 (mod p).
Goldbach’s conjecture — every even integer greater than 2 is the sum of two primes (1742; unproved as of 2026).
Twin prime conjecture — there are infinitely many pairs of primes differing by 2 (e.g., 11 and 13); unproved.
Mersenne primes — primes of the form 2ⁿ − 1; requires n to be prime (though not all such n yield primes). Used in searches for the largest known primes.
Perfect numbers — equal the sum of their proper divisors (e.g., 6 = 1 + 2 + 3); all known perfect numbers are even and linked to Mersenne primes.
Euler’s totient function φ(n) — counts integers from 1 to n that are coprime to n; φ(p) = p − 1 for prime p; appears in Euler’s theorem aᵠ⁽ⁿ⁾ ≡ 1 (mod n) for gcd(a,n) = 1, which underlies RSA.
Chinese Remainder Theorem — a system of simultaneous congruences x ≡ aᵢ (mod mᵢ) has a unique solution mod M = m₁m₂…mₖ when the moduli are pairwise coprime; used in cryptography and computer arithmetic.
Elliptic curves — smooth projective curves of the form y² = x³ + ax + b (with non-zero discriminant); form an abelian group under a geometric “chord and tangent” addition law; central to modern number theory, the proof of Fermat’s Last Theorem (Wiles used the modularity theorem for elliptic curves), and elliptic-curve cryptography.
Modularity theorem — every elliptic curve over ℚ is associated with a modular form; conjectured by Taniyama and Shimura (1955–1957), extended by Weil; Wiles proved it for semistable curves (1995), enabling the proof of Fermat’s Last Theorem; fully proved by Breuil, Conrad, Diamond, and Taylor (2001).
L-functions — complex analytic functions generalizing the Riemann zeta function; the Birch and Swinnerton-Dyer conjecture connects the rank of an elliptic curve’s group of rational points to the order of vanishing of its L-function at s = 1.
Quadratic reciprocity — Gauss’s “theorema aureum”: for distinct odd primes p and q, whether p is a quadratic residue mod q is reciprocally related to whether q is a quadratic residue mod p; Gauss gave the first rigorous proof and subsequently found five more.
Diophantine equations — polynomial equations sought in integer (or rational) solutions; named for Diophantus of Alexandria (~3rd century CE); examples include Pythagorean triples and Pell’s equation x² − ny² = 1.
Waring’s problem — asks, for each k, the minimum number g(k) such that every positive integer is a sum of at most g(k) perfect k-th powers; g(2) = 4 (Lagrange’s four-square theorem, 1770); Hilbert proved g(k) is finite for all k (1909).
Analytic number theory — uses tools from analysis (especially complex analysis and Fourier analysis) to study integers; major results include the prime number theorem and Dirichlet’s theorem on primes in arithmetic progressions.
Dirichlet’s theorem on primes in arithmetic progressions — for coprime a and d, there are infinitely many primes of the form a + nd; proved by Dirichlet (1837) using L-functions, inaugurating analytic number theory.
Euler’s theorem — for gcd(a, n) = 1, aᵠ⁽ⁿ⁾ ≡ 1 (mod n) where φ(n) is Euler’s totient; generalizes Fermat’s little theorem (which is the case n = prime).
Euler’s formula (polyhedra) — V − E + F = 2; already in the Geometry section as “Euler’s polyhedron formula.”
Riemann zeta function — ζ(s) = Σ n⁻ˢ for Re(s) > 1; analytically continued to all s ≠ 1; the Riemann hypothesis conjectures all non-trivial zeros have Re(s) = ½; the distribution of primes is intimately tied to the zeros of ζ.
Collatz conjecture — start with any positive integer n; if even divide by 2, if odd multiply by 3 and add 1; the conjecture states every starting value eventually reaches 1; verified for all n up to ~2 × 10²⁰ but unproved; also called the 3n+1 problem.
Hilbert’s problems — 23 unsolved problems posed by David Hilbert at the 1900 International Congress of Mathematicians; shaped the agenda of 20th-century mathematics; include questions on Cantor’s continuum hypothesis (#1), the consistency of arithmetic (#2), and the Riemann hypothesis (#8); most have been resolved, but #8 remains open.
Fermat primes — primes of the form Fₙ = 2^(2ⁿ) + 1; only five are known: F₀=3, F₁=5, F₂=17, F₃=257, F₄=65537; Gauss proved a regular n-gon is constructible with compass and straightedge iff n’s odd prime factors are distinct Fermat primes.

Famous Theorems and Problems

Pythagorean theorem — a² + b² = c² for a right triangle; known to the Babylonians earlier; hundreds of proofs exist, including ones by Euclid, Garfield (U.S. President), and Bhaskara.
Fermat’s Last Theorem — xⁿ + yⁿ = zⁿ has no positive integer solutions for n > 2; conjectured by Pierre de Fermat (~1637, wrote it in a book margin); proved by Andrew Wiles (1995) using elliptic curves and modular forms.
Four color theorem — any map on a plane can be colored with at most four colors so that no two adjacent regions share a color; proved by Appel and Haken (1976) with computer assistance (the first major theorem requiring a computer proof).
Poincaré conjecture — every simply connected, closed 3-manifold is homeomorphic to the 3-sphere; proved by Grigori Perelman (2002–2003) using Ricci flow with surgery. The only Millennium Prize Problem solved; Perelman declined both the prize and the Fields Medal.
Banach-Tarski paradox — using the axiom of choice, a ball in 3D space can be decomposed into finitely many pieces and reassembled into two balls identical to the original; relies on the existence of non-measurable sets; has no physical interpretation because real matter is not infinitely divisible.
P vs NP problem — the central open question of theoretical computer science: can every problem whose solution can be quickly verified also be quickly solved? (see Millennium Problems).

Millennium Prize Problems

Seven problems named by the Clay Mathematics Institute in 2000; each carries a $1 million prize. As of 2026, only the Poincaré conjecture has been solved.

Problem	Status	Notes
Poincaré conjecture	Solved (Perelman, 2003)	Prize declined
Riemann hypothesis	Unsolved	All non-trivial zeros of ζ(s) lie on the line Re(s) = ½
P vs NP	Unsolved	Does P = NP?
Navier-Stokes equations	Unsolved	Existence and smoothness of solutions in 3D
Hodge conjecture	Unsolved	Algebraic cycles on complex projective manifolds
Yang-Mills existence and mass gap	Unsolved	Rigorous quantum field theory foundation
Birch and Swinnerton-Dyer conjecture	Unsolved	Relates elliptic-curve solutions to L-functions

Probability and Statistics

Sample space / event — set of all possible outcomes; an event is any subset.
Probability axioms (Kolmogorov, 1933) — P(Ω) = 1; P(A) ≥ 0; P(A ∪ B) = P(A) + P(B) for disjoint A, B.
Conditional probability — P(A B) = P(A ∩ B) / P(B).
Bayes’ theorem — P(A B) = P(B A) P(A) / P(B); foundation of Bayesian inference.
Independence — A and B are independent if P(A ∩ B) = P(A) P(B).
Expected value — E[X] = Σ x P(X = x) for discrete X; ∫ x f(x) dx for continuous X.
Variance and standard deviation — Var(X) = E[(X − μ)²]; σ = √Var(X).
Normal distribution — bell-shaped curve; characterized by mean μ and standard deviation σ; the empirical rule: ~68% within 1σ, ~95% within 2σ, ~99.7% within 3σ.
Central limit theorem — the sum (or mean) of a large number of independent, identically distributed random variables is approximately normally distributed, regardless of the original distribution.
Law of large numbers — sample average converges to the population mean as sample size grows.

Notable Mathematicians

Euclid (~300 BCE) — Elements organized geometry from axioms; also treated number theory and the infinitude of primes.
Archimedes (~287–212 BCE) — approximated π; calculated areas and volumes using methods resembling integration; lever and buoyancy principles.
Brahmagupta (598–668 CE) — Indian mathematician; formalized rules for arithmetic with zero and negative numbers.
Al-Khwarizmi (~780–850) — Kitāb al-mukhtaṣar fī ḥisāb al-jabr gave “algebra” its name; systematized solution of linear and quadratic equations.
René Descartes (1596–1650) — Cartesian coordinate system bridging algebra and geometry.
Pierre de Fermat (1607–1665) — contributions to number theory, probability (with Pascal), and the last theorem bearing his name.
Blaise Pascal (1623–1662) — Pascal’s triangle, probability theory (with Fermat), and early mechanical calculators.
Isaac Newton (1643–1727) — co-developer of calculus; binomial theorem; laws of motion and gravitation.
Gottfried Wilhelm Leibniz (1646–1716) — co-developer of calculus; standard derivative and integral notation; binary number system.
Leonhard Euler (1707–1783) — prodigious contributor to analysis, graph theory, number theory; introduced notation f(x), e, i, π, Σ; Euler’s formula, identity, polyhedron formula, and countless theorems; solved the Basel problem (Σ 1/n² = π²/6); the most prolific mathematician in history by volume of output.
Carl Friedrich Gauss (1777–1855) — “Prince of Mathematics”; fundamental theorem of algebra, Gaussian distribution, number theory (Disquisitiones Arithmeticae), non-Euclidean geometry (unpublished), least squares.
Évariste Galois (1811–1832) — founded group theory and Galois theory, explaining solvability of polynomial equations; died in a duel at 20.
Georg Cantor (1845–1918) — set theory, transfinite numbers, diagonal argument, continuum hypothesis.
Henri Poincaré (1854–1912) — topology, celestial mechanics, special relativity foundations, the conjecture bearing his name.
David Hilbert (1862–1943) — formalist program; 23 problems presented in 1900 shaped 20th-century mathematics; Hilbert spaces.
Emmy Noether (1882–1935) — abstract algebra (ring and module theory); Noether’s theorem (symmetry ↔ conservation laws in physics). Often called the most important woman in the history of mathematics.
Srinivasa Ramanujan (1887–1920) — self-taught Indian mathematician; extraordinary results in infinite series, number theory, and partitions; collaborated with G. H. Hardy; taxicab numbers.
Alan Turing (1912–1954) — Turing machines (theoretical basis of computation), computability, the halting problem, Enigma codebreaking.
Kurt Gödel (1906–1978) — incompleteness theorems; consistency of the axiom of choice and continuum hypothesis with ZF.
Andrew Wiles (b. 1953) — proved Fermat’s Last Theorem (1995); awarded the Abel Prize (2016).
Ada Lovelace (1815–1852) — wrote the first published algorithm intended for execution on Babbage’s Analytical Engine (notes on Menabrea’s paper, 1843); often cited as the first computer programmer.
Niels Henrik Abel (1802–1829) — proved at age 22 that no general algebraic formula exists for roots of degree-5 (and higher) polynomials; foundational work on elliptic functions; died of tuberculosis at 26; the Abel Prize is named in his honor.
Joseph-Louis Lagrange (1736–1813) — reformulated Newtonian mechanics as Lagrangian mechanics; Lagrange multipliers; contributions to number theory (Lagrange’s four-square theorem), group theory, and celestial mechanics.
Augustin-Louis Cauchy (1789–1857) — rigorous foundations of calculus via epsilon-delta limits; Cauchy sequences; Cauchy’s integral theorem and residue theorem in complex analysis; prolific author of ~800 papers.
Bernhard Riemann (1826–1866) — Riemannian geometry (curved spaces, essential for general relativity); Riemann integral; Riemann zeta function and the hypothesis bearing his name; Riemann surfaces.
Fibonacci (Leonardo of Pisa) (~1170–1250) — introduced Hindu-Arabic numerals to medieval Europe via Liber Abaci (1202); the Fibonacci sequence appears in a problem about rabbit population growth in that book.
G. H. Hardy (1877–1947) — rigorous British analyst; discovered Ramanujan via correspondence; coauthored landmark papers on the Hardy-Weinberg principle (genetics) and Hardy-Ramanujan asymptotic formula for integer partitions; A Mathematician’s Apology (1940).
Paul Erdős (1913–1996) — Hungarian mathematician who co-authored papers with ~500 collaborators, giving rise to the concept of the Erdős number (degree of separation from Erdős in the coauthorship graph); prolific in combinatorics, graph theory, and number theory.
Alexander Grothendieck (1928–2014) — revolutionized algebraic geometry by introducing schemes, sheaves, and topos theory; Fields Medal (1966); foundational to modern number theory and arithmetic geometry; lived as a recluse later in life.
John Nash (1928–2015) — game theory (Nash equilibrium: a strategy profile where no player can unilaterally improve their payoff); Nobel Prize in Economics (1994); subject of the film A Beautiful Mind; also made contributions to differential geometry.
Grigori Perelman (b. 1966) — proved the Poincaré conjecture (2002–2003) using Ricci flow with surgery; declined the Fields Medal and the $1 million Millennium Prize; largely withdrawn from mathematics since.
Terence Tao (b. 1975) — Fields Medal (2006); work on arithmetic progressions in primes (Green-Tao theorem with Ben Green, 2004), compressed sensing, and partial differential equations; regarded as among the most versatile mathematicians alive.
Maryam Mirzakhani (1977–2017) — first woman to win the Fields Medal (2014); work on the dynamics and geometry of Riemann surfaces; died of cancer at 40.
Sophie Germain (1776–1831) — proved Fermat’s Last Theorem for all primes p where 2p+1 is also prime (Germain primes); work on elasticity theory; overcame gender barriers by corresponding with Gauss under the pseudonym “Monsieur LeBlanc.”
Karl Weierstrass (1815–1897) — “father of modern analysis”; gave rigorous epsilon-delta foundations to calculus; constructed the Weierstrass function, everywhere continuous but nowhere differentiable, overturning intuitions about smoothness.
John von Neumann (1903–1957) — game theory, von Neumann algebras, quantum mechanics formalization, von Neumann architecture (stored-program computers), cellular automata (Game of Life concept ancestor); also contributed to set theory and functional analysis.
Richard Feynman (path integrals) — physicist who developed the path integral formulation of quantum mechanics; the Feynman diagram is a calculational tool in quantum field theory; though primarily a physicist, his work on the Feynman-Kac formula and integration methods bridges mathematics and physics.
verify: Fields Medal age limit — confirm the Fields Medal is awarded to mathematicians under 40 at the time of the award, and is given every four years at the International Congress of Mathematicians to 2–4 recipients.
Blaise Pascal / probability — Pascal’s wager; correspondence with Fermat (1654) founding probability theory; Pascal’s triangle; built one of the first mechanical calculators (Pascaline, 1642).
al-Khwarizmi (extended) — the word “algorithm” derives from the Latinization of his name (Algoritmi); “algebra” from the title of his book Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr waʾl-muqābala.
Cryptographic hash functions — one-way functions mapping arbitrary inputs to fixed-length outputs (e.g., SHA-256 produces 256-bit hashes); collision-resistance and preimage-resistance are the security properties; underpin digital signatures, blockchains, and certificate verification.
Diffie-Hellman key exchange — (1976) allows two parties to establish a shared secret over a public channel using the computational difficulty of the discrete logarithm problem; the first practical public-key protocol.
Elliptic-curve cryptography (ECC) — public-key cryptography based on the discrete logarithm problem on elliptic curves; achieves equivalent security to RSA with much shorter key lengths; widely used in TLS, Bitcoin, and mobile devices.
verify: Cauchy-Riemann equations — the standard statement is ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x for f = u + iv; a complex function satisfying these (plus differentiability) is holomorphic.
verify: Green-Tao theorem (2004) — confirm Ben Green and Terence Tao proved that the primes contain arbitrarily long arithmetic progressions.