Mathematics

Semiprime number A semiprime is a natural number that equals the product of two primes (which may be equal). Semiprimes are a basic object in number theory and form the security basis for some cryp…

Sequence (ordered list of elements and mathematical concept) Sequence An ordered list of items or events. In mathematics a sequence is a function with natural numbers as domain; types include finite/infinite, arithmetic/geometric, recurrent; importan…

Series (mathematical sums, convergence, and applications) Series A series is the sum of terms taken in sequence; in mathematics it often means an infinite sum. Topics include notation, convergence and divergence, common examples, tests, history,…

Set (mathematics) Set A set is a basic mathematical object that collects distinct elements. This article explains definition, notation, operations, foundational issues, and common applications.

Set theory Set theory is the branch of mathematics that studies collections of distinct objects, their properties and relationships; it provides foundational language and tools used across ma…

Seven Bridges of Königsberg: Euler's problem and the birth of graph theory Seven Bridges of Königsberg A clear account of the Seven Bridges of Königsberg problem, Euler's 1735 solution, its abstraction into graph theory, criteria for Eulerian walks, and modern significance.

Sexagesimal (base 60) numeral system Sexagesimal Sexagesimal is a base‑60 positional numeral system developed in ancient Mesopotamia that survives today in measuring time, angles and geographic coordinates.

Secure Hashing Algorithms (SHA): overview, families, uses and security SHA hash functions Summary of the SHA family of cryptographic hash functions: design goals, main versions (SHA‑0, SHA‑1, SHA‑2, SHA‑3), typical uses, and current security guidance.

SHA-1 (Secure Hash Algorithm 1) SHA-1 SHA-1 is a 160-bit cryptographic hash function used for data integrity. Once widely adopted, it is now considered insecure against collision attacks and has been largely superseded…

Shape: definitions, classification, properties and applications Shape A shape is a mathematical or perceptual form defined by boundaries and dimensions. This article explains classification by dimension, properties, history, and common uses in scienc…

Shor's algorithm: quantum integer factoring and cryptography impact Shor's algorithm Shor's algorithm is a quantum algorithm that factors integers and finds discrete orders in polynomial time, with important implications for public-key cryptography and the developm…

Short ton: definition, conversions, history and uses Short ton The short ton is a U.S. customary unit of mass equal to 2,000 pounds (≈907.185 kg). This article explains its definition, conversions, history, common uses, and differences from th…

SI derived unit An SI derived unit is a measurement unit defined from the seven SI base units (or dimensionless); 22 of them have special names such as hertz, newton and pascal.

Sicilian Defence A comprehensive overview of the Sicilian Defence, a leading chess reply to 1.e4: strategic themes, principal variations, historical notes and practical considerations for club and…

Side (geometry) Side A side is a straight line segment that bounds a polygon or, in three dimensions, an edge or face of a solid; this article defines the term, contrasts related concepts, and gives ex…

Siegfried Gottwald (1943–2015), German mathematician and logician Siegfried Gottwald Concise encyclopedia entry on Siegfried Johannes Gottwald: his life, academic career at Leipzig University, contributions to set theory and logic, publications, and legacy.

Sieve of Eratosthenes — classical algorithm for finding primes Sieve of Eratosthenes A clear overview of the Sieve of Eratosthenes: how it works, why the p² optimization holds, its complexity, historical origin, common variants and practical uses for enumerating pr…

Sigma: the Greek letter Σ (sigma) — form, history and uses Sigma Sigma (Σ, σ, ς) is the 18th Greek letter, representing the /s/ sound. It has distinct medial and final lowercase forms, a numeric value of 200, and wide use in mathematics, science…

Sign (mathematics) In mathematics, sign indicates whether a number or quantity is positive, negative, or (in some conventions) zero; it also names the signum function and related uses across algebra…

Signed number representations in binary: sign-and-magnitude, ones' and two's complement Signed number representations How negative integers are encoded in fixed-width binary: common schemes (sign-and-magnitude, ones' complement, two's complement, biased), their ranges, arithmetic behavior, example…

Significant figures (significant digits) Significant figures Digits in a number that indicate the precision of a measurement or calculated value; rules determine which digits count and how to round results in arithmetic.

Similarity (mathematics, computing, and general usage) Similarity A clear, multi-disciplinary overview of similarity: geometric and algebraic definitions, computational measures, historical notes, applications, and distinctions from related conce…

Similarity (geometry) Relation between figures that have the same shape but possibly different sizes; defined by equal corresponding angles and proportional side lengths and realized by dilations, isome…

Simplex (n-dimensional simplex) Simplex An n-simplex is the simplest convex polytope in n dimensions: the convex hull of n+1 affinely independent points. It is fundamental in geometry, topology, numerical methods and com…

Simply connected space (topology) Simply connected space A simply connected space is a path-connected topological space in which every closed loop can be continuously contracted to a point, equivalently having trivial fundamental group.

Simpson's paradox (Yule–Simpson effect) Simpson's paradox A statistical phenomenon where an association that holds within separate groups reverses when the groups are combined, often caused by confounding or unequal group sizes.

Sine wave (sinusoid): definition, properties, and applications Sine wave A sine wave (sinusoid) is the fundamental smooth periodic oscillation described by the sine function; it characterizes simple harmonic motion and serves as a basis for Fourier anal…

Size (measure of extent and scale) Size Size denotes how much space an object occupies or how large it appears. This article explains definitions, units, scales from tiny to cosmic, measurement methods, uncertainty, and…

Skew lines (nonparallel, nonintersecting lines in 3D geometry) Skew lines Lines in three-dimensional space that neither intersect nor are parallel; defined by noncoplanarity. Article covers definition, algebraic tests, distance formula, examples and appl…

Soap bubble: structure, colours, behaviour and cultural meaning Soap bubble A soap bubble is a thin liquid film enclosing air that forms a near-spherical membrane. This article explains how bubbles form, why they show colours, how they behave, and their us…

Sobolev space — function spaces for weak derivatives Sobolev space Sobolev spaces are spaces of functions whose weak derivatives lie in L^p. They provide the natural framework for modern partial differential equations, variational methods, and app…

Solid geometry: study of three-dimensional shapes and measures Solid geometry Overview of solid geometry: definitions, common solids, measurement principles, historical development, applications and distinctions from plane and spherical geometry.

Sophie Germain prime A Sophie Germain prime is a prime p for which 2p+1 is also prime; the number 2p+1 is called a safe prime. They are named for Sophie Germain and are important in number theory and c…

Sorting algorithm A sorting algorithm arranges elements of a collection in a prescribed order. Covers types, properties (stability, in-place), common algorithms, complexity bounds, history, and prac…

Spache Readability Formula A readability measure designed for young readers that estimates grade level using sentence length and proportion of unfamiliar words; commonly used for children’s texts and simple…

Spearman's rank correlation coefficient A nonparametric statistic measuring the strength and direction of a monotonic relationship between two ranked variables. Useful for ordinal data and for assessing correlations with…

Spectrum of a Ring The spectrum of a ring is the set of its prime ideals endowed with the Zariski topology and a natural sheaf; it forms the basic building block of schemes linking algebra and geomet…

Sphere A sphere is the set of points at a fixed distance from a center in three-dimensional space; its surface, properties, history, and uses are central in geometry, physics and everyday…

Sphere packing (geometry and applications) Sphere packing Overview of sphere packing: arranging non-overlapping spheres to maximize occupied volume. Covers definitions, densities, classical and high-dimensional results, methods, related p…

Spherical coordinate system A three-number system for locating points in three-dimensional space using a radius and two angles; common in math, physics, astronomy and geodesy with several sign and notation co…

Spiral Curves: Definition, Types, Mathematical Properties and Applications Spiral Comprehensive guide to spiral curves covering definitions, major types (Archimedean, logarithmic, hyperbolic), core mathematical properties and practical applications in science, e…

Square (plane geometry) Square A square is a regular quadrilateral: four equal sides and four right angles. This article explains its defining properties, formulas, symmetries, relations to other polygons, histo…

Square (algebra): definition, properties, and uses Square (algebra) In algebra, squaring means multiplying a number by itself (raising to the power of two). This article explains notation, basic properties, identities, geometric connections and com…

Square (block cipher) Square (cipher) Square is a 1997 substitution–permutation block cipher that influenced the Rijndael design (AES). It introduced the integral or "Square" attack and uses 128-bit block and key sizes…

Square foot (unit of area) Square foot A square foot is a unit of area equal to a square one foot on a side. Common in the United States and used in construction, real estate and manufacturing; convertible to metric and…

Square kilometre (km²) Square kilometre A square kilometre (km²) is the area of a square with sides of one kilometre. It is used for measuring land and regions; equals 1,000,000 m², 100 hectares and about 0.386 square mi…

Square mile (unit of area) Square mile A square mile is a unit of area equal to a square with sides one statute mile. Widely used in land measurement; equals 640 acres and about 2.59 square kilometres (mi², sq mi).

Square number (perfect square) Square number A square number is an integer produced by multiplying a whole number by itself; this article explains definitions, common properties, history, examples and applications.