Mathematics

Base (mathematics): radix, positional systems, and common numeral systems Base (mathematics) A mathematical base (radix) is the number of distinct digits in a positional numeral system. This article explains definitions, notation, history, examples (binary, decimal, hexade…

Basis (linear algebra): definition, properties, examples, and uses Basis (linear algebra) A basis of a vector space is a set of vectors that spans the space and is linearly independent. This article explains definitions, key properties, examples, history, and common app…

Bayes' theorem: updating probabilities and Bayesian inference Bayes' theorem Bayes' theorem relates conditional probabilities to reverse conditions, enabling updated belief (posterior) from prior information and new evidence; central to statistics, science,…

Bayesian network A Bayesian network is a probabilistic graphical model that represents variables and their conditional dependencies using a directed acyclic graph for reasoning, prediction, and dec…

Bayesian probability Bayesian probability: a method for updating degrees of belief about uncertain events using prior information and new evidence. Explains Bayes' theorem, components, history, applica…

Bean machine (Galton board or quincunx) Bean machine A mechanical demonstration device that produces a binomial/normal-shaped distribution of balls, illustrating probability, binomial coefficients and the central limit theorem.

Beaufort scale An observational scale that classifies wind force by effects on sea and land; developed for maritime use and still used in forecasts, shipping and weather reporting worldwide.

Bertrand's postulate (Bertrand–Chebyshev theorem) Bertrand's postulate A classical result in number theory asserting that for every integer n>1 there is at least one prime p with n

Besov space (function spaces of fractional smoothness) Besov space A family of Banach spaces measuring fractional smoothness of functions or distributions via differences, Fourier decompositions, or wavelet coefficients; important in PDEs, approxi…

Greek Letter Beta: History, Pronunciation, and Uses in Science and Finance Beta Overview of the Greek letter beta (Β, β): its origin, phonetic evolution and numeric value, descendants like Latin B/Cyrillic letters, and modern roles in science, mathematics, sta…

Big O notation: a practical guide to asymptotic growth Big O notation Big O notation describes the growth rate of functions and is used to compare algorithm efficiency, expressing upper bounds on time or space as input size increases.

Bijective function (bijection) Bijective function A bijection is a function that pairs each element of one set with exactly one element of another set and vice versa. It is both injective and surjective and therefore has an invers…

Billion (number) Billion A billion is a large integer with two common meanings: 10^9 (short scale) and 10^12 (long scale). The term’s use varies by country and context; scientific notation or SI prefixes a…

Binary (base-2 numeral system) Binary Binary is the positional numeral system using two symbols (0 and 1). It forms the foundation of modern digital electronics and represents data in bits and bytes for computation, st…

Binary number (base‑2) — definition, structure, history, and uses Binary number The binary numeral system (base‑2) uses two digits, 0 and 1. This article explains its positional structure, historical development, conversion examples, and its central role in co…

Binary operation A binary operation combines two elements of a set to produce a third. Fundamental in algebra, logic and computing, binary operations underlie structures such as groups, rings and m…

Binomial distribution A discrete probability model for the count of successes in a fixed number of independent yes/no trials with constant success probability; widely used in inference, quality control…

Binomial expansion and the binomial theorem Binomial expansion Explanation of the binomial expansion: formula, properties, historical background, common forms (finite and generalized), examples, and applications in algebra, combinatorics and p…

Block (general term) Block Overview of the word “block”: its meanings as a solid piece, an obstruction, a modular unit in technology and construction, and related uses in law, medicine, sport and social medi…

Bloom filter — probabilistic set-membership data structure Bloom filter A Bloom filter is a compact, probabilistic data structure that tests whether an element is a member of a set. It trades a small risk of false positives for substantial space and ti…

Body mass index (BMI) Body mass index Body mass index (BMI) is a simple ratio of weight to height used to screen for underweight, overweight and obesity in populations and in clinical practice, with known limitations.

Boolean algebra: structure, laws, history, and applications Boolean algebra Boolean algebra studies binary values and logical operations (AND, OR, NOT). It provides rules for simplifying logical expressions and underlies digital circuits, set operations, a…

Boolean expression (logic and programming) Boolean expression A Boolean expression is a formula that evaluates to true or false. Used in logic, mathematics, programming, and digital circuits, it combines variables and operators to make decisi…

Boolean satisfiability problem (SAT) Boolean satisfiability problem The decision problem of determining whether a propositional logic formula can be made true by some assignment of truth values; central to computational complexity and many practica…

Bracket: types, uses, and distinctions in writing, mathematics and computing Bracket A concise guide to brackets — their forms (round, square, curly, angle), naming differences, editorial and mathematical uses, programming roles, style considerations, and historica…

Brownian bridge A Brownian bridge is a Gaussian stochastic process obtained by conditioning Brownian motion to hit a specified endpoint; it is central in probability theory, statistics and simulat…

Brute force attack (cryptanalysis and password-guessing techniques) Brute force attack A brute force attack tries every possible key or password until success. Common in cryptanalysis and account compromise, its feasibility depends on keyspace size, computing power,…

Bubble sort — a simple comparison-based sorting algorithm Bubble sort Bubble sort is an elementary, in-place comparison sorting method that repeatedly swaps adjacent out-of-order items. It is stable and easy to teach but inefficient for large lists.

Burmese calendar — the traditional lunisolar calendar of Myanmar Burmese calendar A lunisolar calendar used in Myanmar for religious, agricultural and civil events. Months follow lunar cycles with occasional intercalation and a distinct era used for year numberi…

Caesar cipher A simple substitution cipher that shifts letters by a fixed amount. Originating in Roman times, it illustrates basic encryption, has variants like ROT13, and is useful for teaching…

Calabi–Yau manifold Calabi-Yau manifold A Calabi–Yau manifold is a compact Kähler manifold with vanishing first Chern class whose Ricci‑flat metrics and special holonomy make it central in algebraic geometry and string t…

Calculator — functions, history, types, and everyday uses Calculator Overview of calculators: what they do, how they developed, main types (mechanical, electronic, scientific, graphing), typical features and common applications in education, enginee…

Calculus: Differential and Integral Methods for Continuous Change Calculus Calculus is the mathematical study of continuous change, using limits, derivatives and integrals to model rates, accumulation, and change across science, engineering and beyond.

Calculus of Variations — theory, key concepts, history and applications Calculus of variations Branch of analysis concerned with finding extrema of functionals. Introduces Euler–Lagrange equations, boundary conditions and methods used across physics, geometry and optimizatio…

Cantor's diagonal argument: demonstrating larger infinities Cantor's diagonal argument An accessible account of Cantor's diagonal argument: the construction showing some infinite sets (notably the real numbers) are uncountable, the related power-set theorem, history,…

Carat — gemstone weight and karat gold-purity explained Carat Carat commonly refers to gemstone mass (1 carat = 200 mg) and—by different spelling, karat—to gold purity (24‑karat = pure gold). This article explains meanings, conversions, histo…

Cardinal number Cardinal numbers describe the size of a collection or set, from simple counting numbers to infinite cardinalities such as aleph‑null and the continuum; central to counting and set…

Cardinal numeral A cardinal numeral is a word or expression that denotes quantity for counting and measuring. This article explains forms, grammatical behavior, history, uses, and how cardinals dif…

Cardinality of sets: measuring the size of finite and infinite collections Cardinality Explore cardinality: how mathematicians define and compare the size of finite and infinite sets using bijections and injections, and classify sets as finite, countable, or uncounta…

Carl Friedrich Gauss: Life, Work, and Legacy (1777–1855) Carl Friedrich Gauss Overview of Carl Friedrich Gauss, the German mathematician and scientist, covering his major contributions in number theory, statistics, astronomy, geodesy, and his lasting scienti…

Carl Pomerance: contributions to number theory and integer factorization Carl Pomerance Profile of Carl Pomerance, an American mathematician known for the quadratic sieve and work in computational number theory, teaching at Dartmouth, authorship, and collaborations in…

Carmichael number A Carmichael number is a composite integer that satisfies Fermat-style congruences for all bases coprime to it, making it a universal Fermat pseudoprime with special algebraic stru…

Cartesian coordinate system A system that assigns ordered numerical coordinates to points using perpendicular axes, enabling algebraic description of geometric figures and extension to higher dimensions and a…

Cartesian product The Cartesian product of sets is the collection of ordered tuples formed by taking one element from each set. It underlies coordinates, relations, and many constructions across mat…

Catalan solids Catalan solid Catalan solids are the 13 convex duals of the Archimedean solids. They are face-transitive polyhedra with congruent (often nonregular) faces, named for mathematician Eugène Catalan…

Category theory: concepts, history, and applications Category theory Category theory organizes mathematical structures using objects and morphisms, emphasizing composition, universality, and mappings between theories; influential across algebra, top…

Catenary: the curve of a hanging chain and its properties Catenary A catenary is the curve formed by a flexible, uniform chain hanging under gravity. This article explains its equation, physical meaning, history, and common applications in enginee…

Cauchy distribution (Cauchy–Lorentz) Cauchy distribution A heavy-tailed continuous probability distribution with location and scale parameters; known for undefined mean and variance and frequent use in physics (spectral lines, resonance)…