Mathematics

Ordered pair An ordered pair is a 2‑tuple (a, b) in which order matters: (a, b) = (c, d) iff a = c and b = d. It underlies Cartesian products, relations and functions and can be encoded in set…

Orders of Magnitude of Time Orders of magnitude (time) A clear guide to the concept of orders of magnitude for time intervals: definition, representative scales from the Planck time to cosmic epochs, examples, history and practical use…

Ordinal number An ordinal number indicates position or rank in an ordered sequence. Discusses grammar, formation rules, common uses, and the mathematical concept of ordinals including infinite or…

Ordinary differential equation (ODE) Ordinary differential equation An ordinary differential equation (ODE) relates a function of a single independent variable to its derivatives. ODEs classify by order and linearity and appear across science and e…

Ordinary Least Squares (OLS): method, properties, and applications Ordinary least squares Ordinary least squares (OLS) is the standard method for estimating linear model parameters by minimizing squared residuals. Overview, assumptions, computation, uses, limitations, a…

Orientability in topology and geometry Orientability Orientability describes whether a surface or manifold admits a consistent choice of 'direction' (orientation). It distinguishes orientable shapes from non‑orientable ones like the…

Orthographic projection Orthographic projection: a parallel projection method that maps three-dimensional objects onto a two-dimensional plane using perpendicular view rays. Common in technical drawing, C…

Orthonormality in Inner Product Spaces Orthonormality Orthonormality: definition, properties, and applications of orthonormal vectors and bases in real and complex inner product spaces.

Oscillation Repetitive back-and-forth or cyclical motion seen in physical, biological, and engineered systems; characterized by amplitude, frequency, phase and affected by damping and forcing.

Ounce (unit of mass and weight) Ounce A traditional unit used for mass, weight, and volume in imperial and US customary systems. Includes the avoirdupois and troy ounces, plus the distinct fluid ounce for volume.

P versus NP: the central question in computational complexity P versus NP problem An overview of the P versus NP problem: definitions, intuitive examples, historical milestones, consequences of either outcome, and current research directions in complexity theory…

Padding (cryptography) Techniques that add extra data to messages to meet block sizes or conceal known plaintext; common padding schemes, purposes, risks (padding oracle), and mitigation strategies.

Palindrome: definition, types, history and examples Palindrome A palindrome is a sequence—word, sentence, or number—that reads the same backward and forward. This article explains forms, rules, history, examples, and cultural and technical use…

Palm (unit of length) Palm (unit) The palm is a traditional anthropic unit based on the breadth of the human hand, commonly equal to about three inches (≈7.62 cm). It underlies several related measures such as digi…

Parabola: definition, properties, history and applications Parabola A parabola is a plane curve and conic section defined by a focus and directrix or by a quadratic equation. This article summarizes its geometry, algebra, history, properties, and c…

Parallel: meanings, examples, and distinctions across disciplines Parallel Comprehensive overview of the term “parallel”, explaining its meanings in geometry, geography, grammar, science, music and everyday use, with examples and key distinctions.

Parallel (geometry) Parallel describes lines or planes that do not meet: in Euclidean space they remain a constant distance apart. The concept appears in axioms, vector calculus, design, and non-Eucli…

Parallel postulate (Euclid's fifth postulate) Parallel postulate The parallel postulate: Euclid's fifth axiom that distinguishes Euclidean geometry. Covers its statement and equivalent forms, history of attempts to prove it, development of non‑E…

Parallel projection (geometry and linear algebra) Parallel projection Parallel projection maps points to a subspace along a fixed direction. Covers definitions, algebraic form, types (orthographic, oblique), properties, history, and applications in d…

Parallelepiped A parallelepiped is a six-faced polyhedron whose faces are parallelograms; this article explains its definitions, geometry, volume, special cases, and roles in mathematics and appl…

Parallelogram: Definition, Properties, History and Uses Parallelogram A parallelogram is a four-sided polygon with two pairs of parallel sides. This article explains its defining properties, area formulas, diagonal behavior, history and common applic…

Parameter: definition, uses, and distinctions in math, statistics, and general contexts Parameter An overview of parameters: their meaning in mathematics and statistics, how they differ from variables and constants, common examples, parameter estimation, and related terms like…

Parametric statistics Branch of statistics that models data using fixed-form probability distributions defined by a finite set of parameters; includes estimation, hypothesis testing, and common methods…

Parser (computer science) Parser A parser analyzes text to determine its structure according to a grammar. This article explains what parsers do, their components, common algorithms and types, and where they are u…

Partial derivative (multivariable calculus) Partial derivative Partial derivative: definition, notation, computation rules, higher-order and mixed derivatives, gradient and Hessian, examples, applications in optimization, PDEs, physics and eco…

Partial Differential Equation An overview of partial differential equations: definition, classification, main examples, solution methods, applications, and historical context in mathematics and science.

Parts per million (ppm): definition, conversions and applications Parts per million Concise encyclopedia entry explaining parts per million (ppm): meaning, common conversions, contexts of use (chemistry, environment, engineering), measurement considerations and re…

Pascal's simplex A multidimensional generalization of Pascal's triangle producing multinomial coefficients; useful in combinatorics, probability, and discrete geometry.

Pascal's triangle A triangular arrangement of numbers in which each entry is the sum of the two above. Encodes binomial coefficients, combinatorial counts, and many classical numeric patterns and id…

Pattern: concept, recognition, types and applications Pattern A pattern is an abstract model or recurring arrangement used to organize, interpret, or create structure across arts, science, technology and everyday life.

Pattern matching: concepts, techniques, and applications Pattern matching Overview of pattern matching across text and images: core methods (template matching, feature matching, regular expressions), image pyramids, uses, history, and practical considera…

Pentachoron (4‑simplex) — the regular 5‑cell of four‑dimensional space Pentachoron A pentachoron (5‑cell, pentatope or 4‑simplex) is the simplest regular convex polytope in 4D, made of five tetrahedral cells arranged with full symmetry and self‑duality.

Pentacontagon (50-sided polygon) Pentacontagon A pentacontagon is a polygon with fifty sides. This article explains its geometry, formulas for angles and area, symmetry, constructibility, etymology, and notable combinatorial pr…

Pentadecagon (15-sided polygon) Pentadecagon A pentadecagon is a polygon with 15 sides and vertices. The regular pentadecagon has interior angles of 156°, central angle 24°, is constructible by classical methods, and has dihe…

Pentagon: five-sided polygon — geometry, forms, and uses Pentagon Concise overview of pentagons: definition, regular pentagon properties (angles, diagonals, golden ratio), classifications (convex, concave, star), occurrences in nature, chemistry…

Pentagram A pentagram is a five-pointed star polygon with deep roots in geometry, religion, magic and art. This article explains its form, historical development, meanings and common distinc…

Pentahedron: Polyhedron with Five Faces Pentahedron A pentahedron is any polyhedron that has five faces. This article explains the two convex types, basic properties from Euler's formula, common variations, examples and notable dist…

Pentomino: the five-square polyomino and its puzzles Pentomino A pentomino is a plane figure formed by five equal squares joined edge-to-edge. This article covers its shapes, naming, mathematical properties, tiling puzzles, history and cultura…

Per mille (‰): meaning, notation, history and common uses Per mille Per mille (‰) denotes parts per thousand. This entry explains the symbol, numeric conversions, history, notation variants, and common applications such as salinity, blood alcohol,…

Percentage (percent) — definition, calculation, and common uses Percentage Percentage expresses a part per hundred using the symbol %, used to compare ratios, measure change, and report rates in finance, statistics, and everyday life.

Percentile (statistics) — definition, calculation, uses, and distinctions Percentile Definition and explanation of percentiles: how they are computed, common methods, historical notes, uses in testing and growth charts, and differences from related measures.

Perfect: meanings and uses across disciplines Perfect Overview of 'perfect' as a general term and its technical senses in philosophy, law, grammar, mathematics, biology and music, with examples and notable distinctions.

Perfect number A perfect number equals the sum of its proper positive divisors. This article explains the definition, key properties, history, examples, and open problems around even and (unknown…

Perimeter (geometry) — length around a plane figure Perimeter Perimeter is the total length of the boundary of a flat shape. This article gives definitions, common formulas, measurement methods, history, practical uses, and notable mathematic…

Periodic function A mathematical function that repeats values at regular intervals. Covers definition, fundamental period, examples (trigonometric, waves), key properties, history, and applications.

Permutations — ordered arrangements, counting formulas, and common representations Permutation Overview of permutations: ordered arrangements of distinct items, counting formulas (n!, multinomial), transpositions, cycle and list representations, and common examples in math,…

Permutation box (P-box) in symmetric cryptography Permutation box A permutation box (P-box) is a reversible bit-permutation component used in block ciphers to spread input bits across outputs, increasing diffusion and complicating the plaintext–c…

Peter L. Montgomery (mathematician) Peter Montgomery (mathematician) American mathematician and cryptographer known for Montgomery reduction, Montgomery curves and the Montgomery ladder; influential in elliptic-curve factorization and efficient modu…