Mathematics

Epigraph (mathematics) The epigraph of a function is the set of points lying on or above its graph. Central in convex analysis and optimization, it characterizes convexity, lower semicontinuity, and supp…

Epsilon (Greek letter ε) Epsilon Epsilon, the fifth letter of the Greek alphabet (Ε, ε), denotes the short “e” sound, has numeric value 5, and appears widely in mathematics, science, typography and derived alphabe…

Equality (mathematics): definition, properties, notation and examples Equality (mathematics) Mathematical equality is the fundamental relation stating two expressions denote the same object or value. This article explains formal properties, notation, variants, applications…

Equals sign (=): meaning, history, and uses Equals sign The equals sign (=) is the standard mathematical symbol indicating equality. This article explains its form, history, mathematical role, and how its use differs in logic and comput…

Equation (mathematics): definition, types, history and uses Equation An equation states that two mathematical expressions are equal. This article covers definitions, common types, brief history, methods for solving, and typical applications across m…

Equation solving: methods, types, history and applications Equation solving The theory and practice of finding values or functions that make an equation true, covering basic techniques, classes of equations, existence and uniqueness, numerical methods, and…

Equilateral triangle A triangle with all three sides equal and all internal angles 60°. Key properties, metric formulas, constructions, history, uses, and distinctions from other triangles.

Equivalence relation (mathematics) Equivalence relation A binary relation on a set that is reflexive, symmetric and transitive. Equivalence relations partition sets into classes, underlie quotient constructions, and appear across algebr…

Erdős number A measure of collaborative distance in mathematical publishing named for Paul Erdős, indicating how many coauthorship steps separate an author from Erdős.

Ernst Eduard Kummer Ernst Kummer German 19th-century mathematician known for contributions to number theory, ideal numbers, Kummer theory, and work on hypergeometric functions and algebraic surfaces.

Statistical errors and residuals: definitions, properties and applications Errors and residuals in statistics An accessible overview of statistical errors and residuals: what they are, how they differ, common types and properties, example illustrations, and their role in modelling and diag…

Estimation Estimation is the practice of producing useful approximate values from incomplete data. This article explains types, common methods, history, uses, errors, and how it differs from…

Estimator (statistics): definition, properties, methods and examples Estimator An estimator is a rule or formula that produces an estimate of a population quantity from sample data. This article covers definitions, common properties, main methods, examples, a…

Eta (Greek letter) Eta Eta (Η, η) is a letter of the Greek alphabet. Historically a consonant, it later became a long e vowel and today represents /i/. It has numeric value 8 and is used in science and m…

Euclid's Elements: Structure, Influence, and Legacy Euclid's Elements A comprehensive overview of Euclid's Elements: its organization into thirteen books, logical method, historical context in Alexandria, mathematical content, and lasting influence o…

Euclid's lemma Euclid's lemma states that when a prime divides a product of integers, it divides at least one factor. It underpins the fundamental theorem of arithmetic and many basic divisibilit…

Euclidean geometry: foundations, development, and significance Euclidean geometry Overview of Euclidean geometry: its axiomatic foundations, main concepts, historical origin with Euclid, key results, contrasts with non‑Euclidean systems, and applications across…

Euclidean space — n-dimensional Cartesian geometry and its properties Euclidean space Mathematical setting for classical geometry: the n-dimensional Cartesian space with points, vectors, distance, and inner product. Origins in Euclid and Cartesian coordinates; used…

Euler characteristic: a topological invariant of shapes and spaces Euler characteristic A concise overview of the Euler characteristic, its combinatorial and algebraic definitions, typical values for common surfaces, historical background, and key applications.

Euler–Mascheroni constant A mathematical constant denoted γ that links harmonic numbers and the natural logarithm; appears in analysis, special functions and number theory with several integral and series r…

Euler–Poisson–Darboux equation A classical linear partial differential equation linked to radial reductions of the wave equation; used in mathematical physics and analysis to construct fundamental solutions and…

Euler's formula Euler's formula e^{ix} = cos x + i sin x links exponentials and trigonometry in the complex plane; it underpins polar representation, rotations, Fourier analysis, and Euler's ident…

Euler's homogeneous function theorem A multivariable calculus identity: for a differentiable function f homogeneous of degree k, the sum of inputs times partial derivatives equals k·f. Important for scaling, geometry…

Euler's identity (e^{iπ} + 1 = 0) Euler's identity Euler's identity is the elegant equation e^{iπ} + 1 = 0, linking the constants 0, 1, e, i and π. It arises from Euler's formula and is celebrated for its mathematical unity and app…

Euler's theorem Overview of the mathematical results commonly called Euler's theorem, including the number-theoretic congruence, the polyhedral formula, the homogeneous function identity, and the…

Euler's totient function (phi): definition, properties, and applications Euler's totient function Euler's totient function phi(n) counts integers ≤ n that are coprime to n. This article explains its definition, formulas, algebraic role, history, computational aspects, examples,…

Euler's totient theorem (Fermat–Euler theorem) Euler's totient theorem A fundamental result in number theory: if a and n are coprime then a^{φ(n)} ≡ 1 (mod n). Explains the statement, proof idea, examples, history and applications such as RSA.

Eulerian path (graph theory) Eulerian path A path in a graph that traverses every edge exactly once. Key criteria, historical origin, directed vs undirected cases, algorithms, applications, and differences from Hamiltonian…

Even number An even number is an integer divisible by two. This article explains definition, key properties, common tests, examples, uses in computing and mathematics, and brief historical not…

Exabyte (EB) — unit of digital information Exabyte An exabyte (EB) is an SI unit of digital information equal to 10^18 bytes. It describes enormous volumes of data used in cloud services, scientific computing, and global internet t…

Exclusive disjunction (XOR) Exclusive disjunction Exclusive disjunction, or XOR, is a logical operation true when exactly one input is true. It appears in Boolean algebra, digital circuits, cryptography, and parity checks.

Existential quantifier: notation, semantics, and logical rules Existence quantifier Overview of the existential quantifier (∃): its notation, formal semantics and domain dependence, scope and binding, negation equivalences, and proof rules like instantiation and g…

Expected value (mathematical expectation) Expected value The expected value is the long-run average or mean of a random variable. This article defines discrete and continuous cases, lists key properties, examples, history, applications,…

Exponent: notation, rules, history, and applications Exponent An exponent indicates repeated multiplication and powers. This article explains notation, fundamental laws, extensions to negative and fractional exponents, historical notes, and c…

Exponential distribution Continuous probability distribution on nonnegative reals with density f(x)=λ e^{-λx}. Models interarrival times in Poisson processes; memoryless property; mean 1/λ and variance 1/λ…

Exponential function The exponential function exp(x)=e^x is a fundamental mathematical function with base e. It equals its own derivative, extends to complex numbers, and appears across calculus, diffe…

Exponentiation: definition, rules, extensions and applications Exponentiation A clear introduction to exponentiation: notation, integer, fractional and real exponents, algebraic laws, extensions to complex numbers and matrices, and common applications.

Exponentiation by Squaring (Square-and-Multiply, Binary Exponentiation) Exponentiation by squaring A fast method to compute integer powers using repeated squaring and binary expansion of the exponent; widely used for modular exponentiation and efficient powering in algorithms.

Factor A multifaceted term used across disciplines to mean a contributing element, a divisor in mathematics, a biological agent, an economic input, a financial service, or a computing con…

False (untrue) False False describes a statement, proposition, or belief that does not correspond to facts or reality. It is a central concept in everyday language, philosophy, logic, mathematics and c…

Fathom: definition, history, conversions and nautical use Fathom The fathom is a traditional unit of length used mainly at sea for measuring depth. This article explains its size, origin, historical uses, conversions and how it relates to modern…

February 30: history, calendar rules, and notable exceptions February 30 February 30 does not exist in standard Julian or Gregorian calendars. This article explains why February is shorter, describes the rare historical exception in Sweden, and notes cu…

Fermat number An integer of the form F_n = 2^{2^n} + 1. Fermat numbers have special arithmetic properties, a short recorded history, and links to constructible polygons and prime research.

Fermat's Last Theorem: statement, history, and the modern proof Fermat's Last Theorem A readable account of Fermat's Last Theorem: its simple statement, the centuries-long search for a proof, key ideas and contributors, and the role of modern number theory in its re…

Fermat's Little Theorem A fundamental result in elementary number theory describing congruences for powers of integers modulo a prime; widely used in primality testing and cryptography.

Fermat's primality test A probabilistic primality check based on Fermat’s little theorem; simple and fast but vulnerable to certain composite numbers (pseudoprimes and Carmichael numbers).

Fibonacci number An entry describing the Fibonacci numbers: definition, basic properties, history, examples in nature and mathematics, and common applications.

Field (mathematics): definition, properties, and examples Field (mathematics) A field is an algebraic system where addition, subtraction, multiplication and division by nonzero elements are defined. Overview of axioms, examples, history, and applications.