Mathematics

Differential equation: definitions, types, history, and applications Differential equation An overview of differential equations: definition, classification (ODE/PDE, linear/nonlinear), historical development, solution approaches, applications, and key distinctions.

Differential geometry: shapes, curvature, and applications Differential geometry Branch of mathematics applying calculus and linear algebra to study curves, surfaces, and higher-dimensional spaces; explores curvature, geodesics, and metrics with many scientific…

Differential operators: definitions, orders, linearity, and applications Differential operator Overview of differential operators: definitions, order and linearity, ordinary vs partial cases, domains and examples, and their roles in differential equations, physics, and appli…

Differentiation: mathematical, biological, educational, and commercial meanings Differentiation Survey of differentiation in calculus, cell biology, teaching practice, and business strategy: definitions, key ideas, history, methods, examples, and how the senses differ.

Dijkstra's algorithm A single-source shortest-path algorithm for graphs with nonnegative edge weights, used in routing, mapping, and network optimization. Explains operation, complexity, history, uses,…

Dimension: Concepts, Types, and Applications in Mathematics and Physics Dimension An accessible overview of what dimensions are, how they are defined in mathematics and physics, their historical development, and common uses from coordinates to fractals and space…

Dimensional analysis: units, consistency, and scaling in science Dimensional analysis A clear guide to dimensional analysis: treating units as algebraic factors, converting measurements, checking equations, and deriving dimensionless groups used in physics, chemistr…

Dimensionless quantity A pure number without physical units, formed by cancelling units in ratios or products. Central to dimensional analysis, similarity, nondimensionalization and physical constants.

Diophantine approximation Study of how closely real numbers can be approximated by rational numbers; key results, methods, history, and applications in number theory and beyond.

Diophantine equation: integer solutions to polynomial equations Diophantine equation An introduction to Diophantine equations: definitions, types, historical highlights, methods of solution, important examples and why some cases are decidable while others are not.

Diophantus of Alexandria Diophantus Ancient Greek mathematician known for Arithmetica and pioneering study of equations that led to the modern concept of Diophantine problems and influenced later number theory.

Dirac delta function Overview of the Dirac delta: informal definition, rigorous status as a distribution, main properties, common representations, and applications in mathematics, physics, and engineer…

Discrete cosine transform: concept, properties, and common applications Discrete cosine transform Overview of the discrete cosine transform (DCT), its mathematical role in frequency analysis, main properties and types, historical development, and typical uses such as image and…

Discrete mathematics Study of mathematical structures with distinct, separate values—integers, graphs, logical formulas—central to combinatorics, computer science, cryptography and discrete models; con…

Disjoint sets (mathematics) Disjoint sets Sets are disjoint when they share no elements. This article explains the definition, notation, common variations (pairwise, disjoint unions), examples, and applications across math…

Disjunction: logical 'or' (inclusive and exclusive) and biological separation Disjunction Disjunction denotes 'or' in logic—inclusive or exclusive—and the physical separation of chromosomes in biology. This article explains its types, notation, properties, history, uses…

Displacement vector A displacement vector represents the straight-line change in position between two points. It has magnitude and direction, differs from distance, and is central to kinematics and ve…

Disquisitiones Arithmeticae — Gauss's Foundations of Modern Number Theory Disquisitiones Arithmeticae Overview of Carl Friedrich Gauss's Disquisitiones Arithmeticae: scope, main results such as quadratic reciprocity, organization, historical context, and lasting influence on number…

Distribution (mathematics): generalized functions and their uses Distribution (mathematics) A distribution is a generalized function defined as a continuous linear functional on test functions. Introduced by Laurent Schwartz, distributions are key in PDEs, Fourier analysi…

Distributive property in algebra Distributive property The distributive property describes how one operation distributes over another (typically multiplication over addition). It enables expansion, factoring, and underpins ring and Boo…

Divergence (vector calculus) Divergence Divergence is a differential operator that assigns a scalar to each point of a vector field, quantifying net outflow or inflow. It is fundamental in fluid mechanics, electromagneti…

Division Division denotes separation, allocation, or classification across many fields — from mathematics and biology to organizations, geography, law, music and horticulture.

Division (mathematics): concepts, methods, and examples Division (mathematics) Division is a fundamental arithmetic operation that partitions quantities. This article explains its components, algorithms, historical notes, examples, and special cases like inte…

Division by Two (Halving): Concept, History, and Practical Uses Division by two Division by two, or halving, is the operation that splits a quantity into two equal parts. This article explains its mathematical properties, historical notes, computing applicatio…

Division by zero Explanation of why division by zero is undefined in standard arithmetic, distinctions between 0/0 and a/0, behavior in limits and extended number systems, and practical implication…

Divisor (number theory) Divisor A divisor (or factor) of an integer is an integer that divides it exactly. This article explains definitions, notation, types of divisors, basic properties, related concepts and co…

Dodecagon (12-sided polygon) Dodecagon A dodecagon is a twelve-sided polygon. This article explains regular and irregular forms, key measurements (angles, diagonals, area), symmetry, construction, uses, and related star…

Dodecahedron: geometry, types, history and uses Dodecahedron An introduction to the dodecahedron: its geometric properties, historical context, common variants, and applications in science, art and education.

Dot product (scalar product) Dot product The dot product is an operation on two vectors that yields a scalar equal to the product of their magnitudes and the cosine of the angle between them; used for angles, projections,…

Double: meanings, uses, and distinctions Double Overview of the word “double” covering arithmetic doubling, body doubles in performance, and doppelgängers; explains senses, examples, and how they differ from similar terms.

Double check (chess) Double check A double check occurs when the king is attacked by two pieces at once, typically via a discovered check. It is a powerful tactical motif that normally forces the king to move.

Double helix A double helix is two congruent helices sharing an axis and offset along it; the shape is best known as the structural form of DNA and other paired nucleic strands.

Dozen — the unit of twelve, its history, uses and related measures Dozen A dozen means twelve items. This article explains its origin, mathematical convenience, cultural uses (eggs, trade), related units (gross, great gross, baker's dozen) and links to…

Dynamic programming (algorithmic method for optimization and decomposition) Dynamic programming Dynamic programming is a method for solving complex problems by decomposing them into simpler overlapping subproblems and combining their solutions; widely used in CS, math and eco…

Dynamical systems theory: concepts, history, and applications Dynamical systems theory An overview of dynamical systems theory: definitions, core concepts (flows, maps, stability, attractors, bifurcations), historical roots, methods, and common applications across sc…

e (mathematical constant) e (mathematical constant) e is the base of natural logarithms and a fundamental mathematical constant (~2.71828) that arises in calculus, exponential growth, compound interest, complex analysis and many are…

Edward Kasner: mathematician, educator and the creator of 'googol' Edward Kasner Biography of Edward Kasner (1878–1955), his work in mathematics and education, invention of the words googol and googolplex, and his legacy in popularizing large numbers.

Statistical Efficiency: concepts, measures, and practical implications Efficiency (statistics) Statistical efficiency quantifies how well an estimator or test uses data to achieve precision or power. It compares variances, sample sizes, and asymptotic behavior to guide metho…

Eigenvalues and Eigenvectors Core linear-algebra concepts describing vectors that retain direction under a linear map and the scalars by which they are stretched; includes definitions, properties, history, com…

Elementary algebra: fundamentals, history, and real-world uses Elementary algebra An accessible overview of elementary algebra: variables, operations, equations, functions, historical development, and the subject's practical roles in science, finance, and educat…

Ellipse: definition, properties, history, and applications Ellipse An ellipse is a planar closed curve (a type of conic) defined by two foci; this article explains its geometry, equations, history, and common uses in astronomy, optics, and enginee…

Ellipsoid A three-dimensional analogue of an ellipse: a closed quadric surface defined by three principal semi-axes, used in geometry, geodesy, physics and data analysis.

Elliptic curve A smooth projective cubic curve with a group law; central in number theory, algebraic geometry and modern cryptography, usually given by a Weierstrass equation y² = x³ + ax + b.

Emmy Noether — Architect of Modern Abstract Algebra Emmy Noether Overview of Emmy Noether (1882–1935), her life, key mathematical contributions (Noether's theorem, ring theory), teaching, and lasting influence on mathematics and physics.

Empty set In set theory, the empty set is the unique set that contains no elements. This article explains its notation, formal properties, historical notes, common examples, and its role in…

Empty string (empty word, ε) Empty string The empty string is the unique sequence of zero characters. It has length zero, acts as the identity for concatenation in formal languages, and is distinct from the empty set and t…

English numerals: words, rules, and common forms English numerals How numbers are written and spoken in English: basic words, compound numbers, large-scale conventions, regional differences, and examples of usage and spelling rules.

Entscheidungsproblem: the decision problem in logic and computation Entscheidungsproblem The Entscheidungsproblem asks whether a general algorithm can decide the truth of any statement in a formal language. Hilbert posed it; Church and Turing showed no such algorithm e…