Mathematics

Probability space (measure-theoretic model of random experiments) Probability space A probability space is the measure-theoretic framework for modeling random experiments. It specifies a sample space, a collection of measurable events, and a probability measure th…

Probability theory: foundations, concepts, history, and applications Probability theory An accessible overview of probability theory: core concepts, historical development, common examples, and applications across science, engineering, and statistics.

Product (mathematics) In mathematics, a product is the result of multiplying numbers or other mathematical objects; this article explains definitions, properties, historical notes, examples and common d…

Professor's Cube (5×5×5 Rubik-style puzzle) Professor's Cube A 5×5×5 extension of the classic twisty cube, the Professor's Cube increases piece count and complexity; commonly solved by reduction into a 3×3 format and popular in speedcubing a…

Progression: ordered change in mathematics, music, and other fields Progression Progression denotes an ordered sequence or development of elements; used in math (arithmetic, geometric), music (chord sequences), and applied contexts such as age, color, and sema…

Projection (mathematics) Overview of mathematical projections: geometric shadows, linear idempotent operators, coordinate projections, topological and categorical projections, and common applications in sc…

Proof: meanings in logic, law, food, manufacture, and publishing Proof Overview of the many senses of proof — from mathematical demonstration and legal evidence to alcohol strength, bread proofing, proofreading, and proof objects in art and minting.

Proportionality: direct and inverse relationships in mathematics and applications Proportionality Proportionality describes multiplicative relationships between quantities: direct (x ∝ y) or inverse (xy = constant). Explains forms, constant of proportionality, graphs, examples,…

Proportions: definitions, properties, history and practical uses Proportions Overview of proportions: how ratios form equations, algebraic methods (cross-multiplication), geometric similarity, statistical proportions, history, and common examples and applic…

Propositional logic: basics, structure, history, and applications Propositional logic An overview of propositional logic (sentential calculus): its syntax and semantics, common connectives, proof methods, relation to other logics, and practical uses such as circuits…

Protractor — Instrument for Measuring Angles Protractor A protractor is a simple device for measuring and laying out angles. This article covers types, construction, use, accuracy, history, and common applications in education, drafting…

Psi (Greek letter Ψ, ψ) Psi (letter) Psi (Ψ, ψ) is the 23rd Greek letter, pronounced /ps/. It represents the 'ps' sound, has numeric value 700, and appears widely as a symbol in mathematics, physics, psychology and de…

Pure mathematics: theory, structure, and influence Pure mathematics Pure mathematics studies abstract mathematical structures and proofs for their own sake, developing theories that later often find unexpected applications across science, engineeri…

Puzzle (problem-solving activity and recreational challenge) Puzzle A puzzle is a problem designed to test ingenuity, knowledge, or pattern recognition. Puzzles appear in many forms—word, number, mechanical, and digital—and serve recreation, educat…

Pyramid (geometry): forms, properties, and formulas Pyramid (geometry) A pyramid is a polyhedron with a polygonal base and triangular lateral faces meeting at an apex. Covers definitions, parts, volume and area formulas, common types, frustums and pra…

Pythagorean theorem In Euclidean geometry, the Pythagorean theorem relates the lengths of the sides of a right triangle: the sum of the squares of the legs equals the square of the hypotenuse; foundat…

Pythagorean triple A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a^2 + b^2 = c^2. Covers definition, primitive triples, Euclid's formula, examples, history and common…

Quadratic equation: definition, solutions, graph, and applications Quadratic equation A quadratic equation is a second-degree polynomial ax²+bx+c=0. This article explains its form, graph (parabola), solution techniques, discriminant, historical notes, and common app…

Quadrilateral A quadrilateral is a four-sided polygon. This article describes its types, core properties, historical names, common theorems and practical uses in geometry and design.

Quadrivium: The Four Mathematical Arts in Medieval Liberal Education Quadrivium The Quadrivium grouped arithmetic, geometry, music and astronomy as the mathematical stage of the medieval liberal arts, preparing students for advanced study and informing later s…

Quantity: concepts of amount, measurement, and magnitude Quantity Quantity denotes how much of something exists, either countable or measurable. It includes mathematical cardinality, physical quantities with units, discrete vs continuous measures…

Quart (unit of volume) Quart A quart is a unit of volume equal to a quarter of a gallon or two pints. Its exact size differs between the US (liquid and dry) and the British imperial systems and is commonly use…

Quartic equation (fourth-degree polynomial) Quartic equation A quartic equation is a polynomial equation of degree four. This article explains its general form, key properties, historical solution methods, applications, and notable distincti…

Queen's Gambit (chess opening) Queen's Gambit A classical chess opening beginning 1.d4 d5 2.c4 in which White contests Black's central pawn. Includes Accepted, Declined and Slav systems and leads to rich strategic play and man…

Quicksort: divide-and-conquer comparison sorting algorithm Quicksort Quicksort is a fast, in-place comparison sorting method that uses divide-and-conquer partitioning around a pivot. Widely used in practice, it is efficient on average but has worst-…

Quota: Limits, Allocations, and Policy Uses Quota A quota is a numerical limit, target or allocation used to control quantities of goods, resources, people or rights across trade, environment, immigration, employment, computing an…

Quotient (division result, integer quotient, and algebraic quotients) Quotient The quotient is the result of dividing one quantity by another; it appears in arithmetic as exact or integer division and in algebra and topology as a construction of equivalence c…

Radian (unit of angular measure) Radian A radian is the standard angular unit defined by an angle whose arc length equals the circle's radius. It is dimensionless in SI and is preferred in calculus, physics, and engineer…

Radical: meanings in mathematics, science, language, medicine and politics Radical “Radical” denotes roots, origins, or extremes across fields — from mathematical symbols and chemical species to linguistic roots, political movements, and medical procedures.

Radius (geometry): definition, formulas, history and applications Radius The radius is the straight-line distance from a circle or sphere's center to its boundary. This article explains definitions, common formulas, history, and practical uses with exam…

Radix point (radix character) Radix point Symbol that separates the integer part from the fractional part of a number in a given base; commonly a decimal point or comma, with variants for binary, octal, hexadecimal and loc…

Ramanujan prime A Ramanujan prime is a special prime related to the prime-counting function and a strengthening of Bertrand's postulate, introduced from Ramanujan's work and studied in analytic nu…

Ramsey theory Branch of combinatorics that studies when complete disorder is impossible and a particular kind of structure must appear; includes Ramsey's theorem, Ramsey numbers, and many applic…

Random (concept) Random Random refers to lack of predictable order or pattern. Covers mathematical definitions, physical and algorithmic sources of randomness, common uses in sampling and cryptography, an…

Random mating (panmixia) in population genetics Random mating Random mating, or panmixia, is an idealized mating pattern where mates pair without regard to genotype or phenotype; it underpins Hardy–Weinberg expectations and contrasts with ass…

Random Sample: Definition, Methods, Uses, and Limitations Random sample A random sample is a subset of a population selected so each member has a known chance of inclusion; key for unbiased inference. Covers types, practical issues, history, and common…

Random variable: definition, types, properties, history and applications Random variable An accessible encyclopedia article on random variables: formal meaning, discrete/continuous types, key functions (PMF/PDF/CDF), basic statistics, brief history, and common applicat…

Random walk A random walk is a mathematical model of successive random steps. It underlies diffusion, stochastic processes, and many applications in physics, finance, biology, and computer sci…

Range (mathematics) The mathematical notion of range appears in several areas: as the image of a function, a simple statistical dispersion measure, and the column space of a linear map. Definitions, e…

Rate: meanings and uses in mathematics, science, finance, and society Rate Overview of 'rate' as a ratio or frequency across mathematics, physics, chemistry, computing, finance, demography, taxation, and historical naval usage, with notes on interpretatio…

Rate (mathematics): measures of change and ratios between quantities Rate (mathematics) A rate is a ratio comparing two related quantities, often expressing change per unit (commonly per unit time). Covers definitions, types, calculation, history, examples and distinc…

Ratio: definition, forms, simplification, and common applications Ratio A ratio compares quantities by showing their relative sizes. This article explains notation, simplification, differences from rates, historical context, and practical uses in recip…

Rational function A rational function is a quotient of two polynomials. This article explains definition, singularities, asymptotic behavior, common examples, uses in calculus and engineering, and d…

Rational number A rational number is any quantity expressible as the quotient of two integers (denominator nonzero). It includes integers, finite decimals and repeating decimals, and is dense with…

Rationalization (mathematics) Method for removing radicals or irrational expressions from a fraction's denominator by multiplying by a suitable expression (often the same radical or a conjugate).

Real number A real number is any value on the continuous number line, including rational and irrational numbers; it underpins calculus, measurement and real analysis and is denoted by the symb…

Reciprocal (multiplicative inverse) — definition, properties and examples Reciprocal The reciprocal (multiplicative inverse) of a nonzero number is 1 divided by that number. This article explains definitions, arithmetic rules, examples, and generalizations in algeb…

Rectangle: definition, properties, uses, and related concepts Rectangle A rectangle is a four‑sided polygon with four right angles. This article explains its defining properties, formulas for area and perimeter, historical notes, common uses, and how i…