Mathematics

Recursion: concept, structure, history, and applications Recursion Recursion is a method of defining objects in terms of themselves. This article explains its form, examples in mathematics and computing, historical notes, uses, and common distinct…

Reductio ad absurdum: method, history, and uses Reductio ad absurdum Reductio ad absurdum is an indirect proof technique showing a claim true because denying it yields contradiction or absurd results; used in logic, mathematics, and philosophy since…

Reed–Solomon error correction: principles, properties, and applications Reed–Solomon error correction Reed–Solomon codes are block-based, non-binary forward error correction codes that protect data against symbol and burst errors by encoding messages as polynomials over finite fiel…

Reflex angle A reflex angle is an angle greater than 180° and less than 360° (π to 2π radians). It is the larger of the two angles determined by two rays and appears in geometry, graphics, and…

Reflexive relation A reflexive relation on a set is one that relates every element to itself. Covers definition, examples and non-examples, connections to equivalence relations and orders, and basic…

Regular expression (regex) Regular expression A concise, practical overview of regular expressions: what they are, core syntax, history, common uses, and important distinctions between theory and real-world implementations.

Regular polygon A polygon that is both equilateral and equiangular: definitions, properties, constructions, examples, and distinctions including star (nonconvex) regular polygons.

Regular polytope A regular polytope is a highly symmetric generalization of regular polygons to any dimension; this article explains definition, properties, classification by dimension, history, ex…

Relation (mathematics): definition, properties, and applications Relation (mathematics) A mathematical relation is a subset of a Cartesian product. This article explains n-ary and binary relations, key properties, operations, examples, historical context, and uses in…

Relational model (database design) Relational model A conceptual framework for organizing data into relations (tables) using tuples and predicates; introduced by E. F. Codd and realized widely via SQL-based relational database syste…

Remainder (integer division) Remainder The remainder is the integer left after dividing two integers. Article explains the division algorithm, sign conventions, relation to modular arithmetic, examples, computation, and…

Repeating decimal (recurring decimal) Repeating decimal A repeating (recurring) decimal is a decimal expansion in which a digit or group of digits repeats infinitely. Common for rational numbers; convertible to fractions and classified…

Rhind Mathematical Papyrus: Egypt’s Principal Mathematical Manuscript Rhind Mathematical Papyrus Overview of the Rhind Mathematical Papyrus: its content, physical form, dating, authorship, discovery, and importance for the history of mathematics in ancient Egypt.

Rho (Greek letter) Rho Rho (Ρ, ρ) is the 17th letter of the Greek alphabet, representing the r sound. It has numeric value 100 and appears widely in science, mathematics and alphabets derived from Greek.

Rhombus A rhombus is an equilateral quadrilateral — a parallelogram with four equal sides. This entry summarizes its defining properties, formulas, historical name origin, common uses, and…

Riemann hypothesis Conjecture that all nontrivial zeros of the Riemann zeta function have real part 1/2; a central open problem in number theory with deep consequences for the distribution of prime n…

Riemann sum — approximation of definite integrals Riemann sum A Riemann sum approximates a definite integral by summing products of function values and interval widths; it underlies the Riemann integral and many numerical integration methods.

Riemann zeta function A complex analytic function central to number theory and the study of prime distribution; foundation of the Riemann hypothesis and links to physics and statistics.

Right angle (90°) — definition, notation, tests and uses Right angle A right angle measures exactly 90 degrees; lines, segments or planes that meet this way are perpendicular or orthogonal. Covers notation, measurement, construction, examples and ma…

Right triangle A triangle with one 90° angle; defined by a hypotenuse and two legs. Key tools include the Pythagorean theorem and trigonometric ratios for solving sides and angles.

Right-hand rule (cross product and spatial orientation) Right-hand rule A geometric convention used to determine direction of vector cross products and to define right-handed coordinate systems; common in physics, engineering and 3D mathematics.

Rijndael (AES) key schedule Rijndael key schedule How the Rijndael key schedule (AES) expands a secret key into per-round round keys: operations, differences for 128/192/256-bit keys, structure of round keys, and security/implemen…

Ring (mathematics) An algebraic structure with two operations (addition and multiplication) satisfying distributive laws; includes integers, polynomials, matrices, and many constructions used across…

Ring theory (abstract algebra) Ring theory Ring theory studies algebraic structures with two binary operations—addition and multiplication—generalizing integers and providing a framework for number theory, geometry, and alg…

RMS — common meanings, history, and notable uses RMS RMS is an initialism with several widely used meanings, including the maritime prefix Royal Mail Ship, the mathematical root-mean-square, learned societies, and other technical or…

Robert F. Coleman American number theorist and UC Berkeley professor (1954–2014), known for pioneering p-adic methods in arithmetic geometry, including Coleman integration, overconvergent modular fo…

Roman numerals: system, history, and modern uses Roman numerals Overview of Roman numerals, their symbols and rules, historical development, and where they remain in use today, with examples and notable features.

Root of a Function (Zero of a Function) Root of a function A root (zero) of a function is an input where the function value equals zero. Covers definition, real and complex roots, multiplicity, basic theorems, common analytic and numerical…

ROT13 — simple substitution cipher and its uses ROT13 ROT13 is a simple letter-substitution cipher that shifts Latin alphabet letters by 13 places. Used for light obfuscation (spoilers, jokes) and as a pedagogical example; it is not s…

Rounding: methods, purpose, and impact on numerical results Rounding Rounding replaces a numeric value by a simpler, approximate value. This article explains common rules, implications for computation and measurement, practical examples, and ways to…

RSA algorithm: public-key encryption, key generation, and history RSA algorithm RSA is the best-known public-key cryptosystem. This article explains its basic principles, key generation, uses, historical origin, and security considerations in accessible terms.

Ruler (measuring instrument): types, history, and common uses Ruler A ruler is a straight measuring tool marked with graduated units. This article describes its characteristics, varieties, historical development, typical uses, and practical tips fo…

Russell's paradox: the self-membership contradiction in naive set theory Russell's paradox A clear account of Russell's paradox: its formal statement, simple analogies, historical discovery by Bertrand Russell, and how it shaped modern axiomatic set theory and type theor…

S (letter) — nineteenth letter of the English alphabet S Overview of the letter S: form, sounds, historical origin, common uses in English and other fields, typographic variants and notable facts.

Sampi: the ancient Greek letter and numeral Sampi Sampi (also Disigma) was an archaic Greek letter used in some dialects and later a numeral for 900. This article summarizes its forms, sounds, history, decline, and the questions s…

Sample size (statistics): definition, determination, and practical guidance Sample size Overview of sample size in statistics: notation, why it matters, common rules of thumb, methods for calculating size (confidence intervals, power), and practical considerations.

Scalar: single-valued quantities in measurement and mathematics Scalar A scalar is a single numerical value that expresses magnitude only. Used in measurement, physics, mathematics and computing, it contrasts with vectors and matrices and appears in u…

Scale (ratio) Explanation of scale as the numerical ratio between measurements on a representation and real-world dimensions; forms, examples, effects on area/volume, and guidance for maps, mode…

Scatter graph (scatter plot) Scatter graph A scatter graph displays paired numerical data as points on two axes, revealing relationships, trends, clusters, and outliers between two variables.

Scheduling (computing): concepts, types, and algorithms Scheduling (computing) Overview of scheduling in computing and combinatorial optimization: goals, models, common algorithms, complexity, and practical uses in operating systems, clusters, and manufacturi…

Scheme (programming language) Scheme is a minimalist Lisp dialect known for simple syntax, lexical scoping, first-class procedures, and a long role in computer science education.

Scientific notation: definition, format, uses and examples Scientific notation Scientific notation expresses numbers as a coefficient times a power of ten, simplifying work with very large or very small values. Includes format, conversion steps, examples, use…

Second (unit of time) Second The second is the SI base unit of time, defined by an atomic transition in caesium and used worldwide for clocks, science, and daily timekeeping; it has subdivisions, multiples, an…

Second Normal Form (2NF): Definition, Criteria, and Practical Guide Second normal form Second normal form (2NF) is a database normalization rule that removes partial dependencies from tables in first normal form to reduce redundancy and update anomalies.

Selection sort A simple in-place comparison sorting algorithm that repeatedly selects the smallest (or largest) element and moves it to its final position. Commonly taught for its clarity despite…

Self-similarity Property of a shape or system that resembles a part of itself at different scales; common in fractals, natural forms, and physical phenomena with exact, statistical, and self-affin…

Semi-major and Semi-minor Axes of an Ellipse Semi-major and semi-minor axes Definitions, properties, and uses of an ellipse's semi-major and semi-minor axes, with key formulas (area, eccentricity) and examples including the circle and orbital applications.

Semicircle A semicircle is half of a circle: a plane figure bounded by a diameter and the corresponding arc, with characteristic geometry, formulas, and classical connections such as Thales'…