Mathematics

Hexadecagon A hexadecagon is a sixteen-sided polygon. Covers definitions, regular-case measures and formulas, symmetry and constructibility, star polygons, historical notes and practical uses.

Hexadecimal (base-16 numeral system) Hexadecimal Hexadecimal (hex) is a base‑16 positional numeral system using digits 0–9 and A–F. It compactly represents binary data and is widely used in computing, color codes, addresses, and…

Hexagon: Geometry, Properties, and Uses Hexagon A hexagon is a six-sided polygon. This article describes its geometric properties, symmetry, types, natural occurrences (like honeycombs), tessellation, and common uses in design a…

Hexagram (six‑pointed star) Hexagram A hexagram is a six‑pointed star formed by two interlaced equilateral triangles. Used in geometry, art, religion and symbolism, it appears across many cultures with varied meanings…

Hexahedron A hexahedron is any polyhedron with six faces. This article explains definitions, varieties (convex and concave), typical shapes like the cube and dipyramid, topology, properties,…

Hilbert space: definition, structure, history and applications Hilbert space A Hilbert space is a complete inner-product vector space used throughout mathematics, physics and engineering; this article explains its structure, examples, history and major appl…

Hilbert's Grand Hotel paradox: how countable infinity defies finite intuition Hilbert's paradox of the Grand Hotel A clear explanation of Hilbert's Grand Hotel: its setup, examples, mathematical meaning for countable infinity, historical and pedagogical role, and distinctions from other infinit…

Hilbert's problems A guided overview of the 23 mathematical problems David Hilbert presented in 1900, their scope, notable resolutions and undecidability results, and their long-term influence on mat…

Hilbert's Tenth Problem: decidability of Diophantine equations Hilbert's tenth problem A complete overview of Hilbert's tenth problem: its statement, mathematical meaning, history of the undecidability proof, and its significance for number theory and computability.

Hillel Furstenberg: dynamics, probability and connections to number theory Hillel Furstenberg Overview of Hillel Furstenberg, his life, major contributions to ergodic theory and probability, applications to number theory and Lie groups, and his influence and honors.

Hindu–Arabic numeral system A positional decimal numeral system using ten symbols (including zero) that originated in South Asia and spread via the Islamic world to become the global standard for arithmetic a…

Histogram: definition, structure, history and practical use Histogram A histogram is a bar-based chart that summarizes the distribution of numerical data by grouping values into bins. This article explains its parts, history, uses, and differences fr…

History of Mathematics A concise, structured account of how mathematical ideas and practices developed across cultures, from ancient counting systems to modern abstract theory and applications.

Hole (opening or empty space in a solid object) Hole A hole is an opening, cavity, or void within or through a solid body or surface. It can be natural or manufactured and ranges from microscopic pores to large geological cavities.

Holland's schema theorem: principles, examples, and significance Holland's schema theorem An accessible overview of Holland's schema theorem for genetic algorithms: definition of schemata, the theorem's statement, historical context, uses, limitations and links to relat…

Holomorphic function A holomorphic function is a complex function that is complex-differentiable on an open set. This article explains the definition, key consequences, examples, history, and how it di…

Homology (mathematics): an introduction and overview Homology (mathematics) Homology assigns algebraic invariants—typically abelian groups or modules—to mathematical objects to detect and classify 'holes'. It underlies many tools in algebraic topology and…

Homotopy: continuous deformations and their role in topology Homotopy Homotopy describes continuous deformations between maps or shapes. It underpins algebraic topology, leading to invariants like homotopy groups, and distinguishes flexible equivalen…

Horn clause: definition, forms, role in logic programming and complexity Horn clause A Horn clause is a logical clause with at most one positive literal. This article explains its forms, first-order semantics, use in resolution and Prolog, model theory, and computa…

Hour (time unit) Hour An hour is a unit of time equal to 60 minutes or 3,600 seconds. Used in civil timekeeping and science, it originated from ancient sexagesimal divisions and is accepted for use with…

Hull: meanings, structures, mathematics, places and people Hull Hull — a term with multiple senses: external coverings of plants, bodies of ships and vehicles, mathematical constructions, place names, and notable people sharing the name Hull.

Hyperbola: definition, equations, properties, and applications Hyperbola A comprehensive overview of the hyperbola: geometric definition, standard equations, key features (foci, asymptotes), parameterizations, historical notes, and common applications.

Hypercube (n-dimensional cube) Hypercube An n-dimensional generalization of a square and cube: convex polytope with 2^n vertices, produced as a Cartesian product of intervals. Describes structure, counts of faces, history…

Hyperoperation — hierarchy of repeated binary operations Hyperoperation Hyperoperations form a sequence of increasingly powerful operations that extend addition, multiplication, exponentiation and beyond, with recursive definitions and extremely rapid…

Hypotenuse: Definition, Properties, History, and Applications Hypotenuse Hypotenuse: the side opposite the right angle in a right triangle; longest side. Includes definition, Pythagorean relation, history, examples (3-4-5), trigonometric use and notable…

Icosagon (20-sided polygon) Icosagon An icosagon is a polygon with 20 sides and 20 vertices. This article covers its defining properties, regular form, key formulas, constructibility, symmetry, and notable variations…

Icosahedron A polyhedron with twenty triangular faces; most familiar as the regular convex Platonic icosahedron. Covers geometry, history, variants, and common uses such as dice, architecture,…

Idempotence: definition, mathematical forms and uses in computing Idempotence Idempotence is the property that repeating an operation does not change the outcome after the first application. This article explains definitions, examples, history, and practical…

Identity (mathematics) An identity is an equality that holds for all values of its variables; this article explains definitions, common examples, notation, uses, and how identities differ from conditiona…

Identity element (neutral element) Identity element An identity (or neutral) element is an element in an algebraic structure that leaves other elements unchanged under a given operation — e.g., 0 for addition, 1 for multiplication.

If and only if (logical biconditional) If and only if Overview of the logical connective "if and only if" (iff): definition, semantics, notation, examples, relation to necessity and sufficiency, and common uses in logic and mathematic…

Ii — meanings, uses, and notable instances Ii Overview of Ii (or II): its use as a Roman numeral, letter sequence or digraph, place name in Finland, ISO language code, anatomical label, and cultural references such as an album…

Imaginary number An imaginary number is a real multiple of the imaginary unit i, where i^2 = −1. Imaginary and real numbers combine to form complex numbers, with wide use in algebra, geometry, engi…

Imaginary unit The imaginary unit i is a mathematical symbol whose square equals −1. It underpins complex numbers, complex arithmetic, Euler's formula, and many applications in analysis, physics…

Imperial units The Imperial system is a set of British-derived measurement units standardized by an 1824 Act; still used in some places today and distinct in key ways from U.S. customary units an…

Implication in logic: material conditional and logical consequence Implication (logic) A clear overview of implication in logic: material conditional, logical consequence, truth conditions, common inference rules, contrasts with natural language, and variants like in…

Improper rotation (rotoinversion) Improper rotation An improper rotation (rotoinversion) is a symmetry operation formed by a proper rotation followed by spatial inversion; it reverses orientation and appears in crystallography, mole…

Inch (unit of length) Inch The inch is a widely used unit of length in the Imperial and US customary systems. Defined exactly as 2.54 centimetres since 1959, it appears in many everyday and industrial measur…

Inclusive disjunction (logical OR) Inclusive disjunction Inclusive disjunction, commonly called OR, is the logical operation that is false only when all inputs are false; it appears in logic, mathematics, programming, and digital circuit…

Independence in Mathematical Logic Independence (mathematical logic) A clear exposition of independence for sentences and axioms: definitions, semantic and syntactic views, how independence is proved, key examples, and its role in foundations of mat…

Inductive reasoning Inductive reasoning is the process of drawing general conclusions from specific observations. It underpins empirical science, everyday inference, and many forms of hypothesis gener…

Inequality (mathematics): notation, properties and applications Inequality Overview of mathematical inequalities: symbols, basic rules for manipulation, common named inequalities, methods for solving, and their roles in analysis, optimization and probabil…

Inference: reasoning from premises to conclusions Inference The process of deriving conclusions from premises. Covers deductive, inductive and abductive forms, formal rules, statistical/Bayesian methods, and roles in science, law, and AI.

Inference (statistics) Statistical inference is the process of drawing conclusions about populations, models, or mechanisms from observed data, quantifying uncertainty and guiding decisions under uncerta…

Infinite monkey theorem A probability thought experiment that a random sequence generator will eventually produce any given finite text, often illustrated by monkeys typing Shakespeare; mathematically tru…

Infinitesimal A quantity smaller in magnitude than any positive real number but not zero; key to the intuition of calculus and formalized by limits, hyperreal numbers, and alternative foundation…

Infinity: mathematical concept, history, and uses Infinity An overview of infinity: its meanings in mathematics and philosophy, historical development, types (potential, actual, cardinal, ordinal), common examples, and how it is treated in…

Infix notation Infix notation writes operators between their operands (for example 2 + 2). It relies on parentheses, precedence, and associativity to show intended order and is common in mathemat…