Mathematics

Inflection point: definition, detection, examples and applications Inflection point A location on a curve where concavity changes. Explains mathematical criteria, examples (x^3 vs x^4), detection methods, parametric cases, undulation points, and practical uses in…

Information entropy: measure, meaning, and applications Information entropy Information entropy quantifies uncertainty in outcomes. This article explains its definition, properties, historical origin, examples (coin toss, distributions), and applications i…

Information theory: fundamentals, measures, history and applications Information theory Overview of information theory: essential concepts (entropy, mutual information, capacity), history, principal uses in compression and communication, and notable distinctions.

Injective function (one-to-one) Injective function An injective (one-to-one) function maps distinct inputs to distinct outputs. This article explains definitions, equivalent formulations, basic properties, examples, non-examples, a…

Insertion sort — a simple in-place comparison sorting algorithm Insertion sort Insertion sort is a stable, in-place comparison algorithm that builds a sorted sequence one item at a time. It is efficient for small or nearly sorted arrays and commonly used as a…

Integer: definition, properties, notation and basic arithmetic Integer An integer is a whole number with no fractional part. This article explains definitions, notation, arithmetic properties, algebraic structure, ordering, examples, history and commo…

Integer overflow (computer arithmetic) Integer overflow Integer overflow happens when a computed integer value exceeds the range that a fixed-size data type can represent, causing wraparound, truncation, or runtime error with potential…

Integral (calculus): concepts, notation, and applications Integral An integral is a foundational calculus concept that accumulates quantities — areas, volumes, and totals — and serves as the inverse operation to differentiation with many extension…

International Mathematical Olympiad Annual international competition for pre-university students featuring six proof-based problems over two days; teams of six, national selection, and awards for top performers.

International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics SCAN is a biennial international symposium on numerical analysis and reliable computation, covering numerical integration, linear algebra, PDE methods, interval arithmetic and vali…

Interpolation: Methods, Uses, and Practical Considerations Interpolation Interpolation estimates values between known data points. This article summarizes main methods, history, common applications, and limitations for scientific, engineering, and compu…

Interval An interval is a contiguous range between two points or values. It appears across mathematics, music, timekeeping, statistics and computing, with specific types and notations depen…

Interval (mathematics) An interval is a connected subset of the real numbers containing every point between two endpoints. Intervals can be open, closed, half-open, bounded or unbounded and are fundament…

Interval arithmetic: verified numerical computation and error bounding Interval arithmetic An introduction to interval arithmetic: definitions, basic operations, extensions, uses for error bounding and verified computing, and key limitations including overestimation and…

Inverse (Inversion): meanings, types, and applications Inverse Overview of 'inverse' or 'inversion' as a reversal operation or phenomenon across mathematics, science, language and other fields, with types, examples and important distinctions.

Inverse (mathematics) Overview of mathematical inverses: definitions, common types (additive, multiplicative, functional, matrix), existence and uniqueness, computation, examples and applications.

Inverse element (algebraic concept) Inverse element An inverse element reverses the effect of an operation: additive inverse negates, multiplicative inverse reciprocates, and more generally an inverse in a group undoes an element to…

Inverse function A function that reverses the mapping of another: definition, notation, existence conditions (injective/surjective), methods to compute inverses, common examples and graphical and c…

Invertible matrix An invertible matrix is a square matrix that has a two-sided inverse; this article explains definitions, equivalent conditions, methods to compute inverses, applications, and commo…

Irrational number A real number that cannot be expressed as a ratio of two integers; its decimal expansion is nonterminating and nonrepeating. Examples include √2, π and e. Distinctions: algebraic v…

Irula numerals: counting, structure and cultural context Irula numerals Overview of the numeral system and counting practices of the Irula people, covering structure, origins, uses, relationships to other Dravidian systems, and research status.

ISO 8601 — International standard for representing dates and times ISO 8601 ISO 8601 specifies an unambiguous, machine-friendly format for dates, times, combined date-times, time zones and week or ordinal dates used widely in computing and data exchange.

Isometry: distance-preserving transformations in geometry Isometry A map that preserves distances between points. In Euclidean geometry isometries include translations, rotations, reflections and glide reflections; more generally they are distance…

Isosceles triangle A triangle with two equal sides; this article explains definitions, key properties, basic formulas, historical background and common uses in geometry, architecture and polyhedral g…

Iteration in Computing, Mathematics, and Design: Repetition, Loops, and Iterative Methods Iteration Overview of iteration across computing, mathematics, and design: repeated steps, loop execution, iterative algorithms, numerical methods, fractals, and cycles of testing and refine…

Jacques Tits Jacques Tits (1930–2021) was a Belgium-born French mathematician whose work on group theory and incidence geometry — notably buildings, the Tits alternative, and the Tits group — r…

John Tate: influential American number theorist and algebraic geometer John Tate Overview of John T. Tate (1925–2019), his main contributions to number theory and arithmetic geometry, biography, major concepts (Tate module, Tate conjecture, Tate cohomology), an…

K — the eleventh letter of the English alphabet K Profile of the letter K: its place in the alphabet, name and sounds, early history from Phoenician to Latin, common uses and symbols (e.g., potassium, Kelvin, kilo), and notable fe…

Kalman filter: recursive state estimation for noisy systems Kalman filter A recursive algorithm that estimates the state of a dynamic system from noisy measurements. Widely used in navigation, tracking, control and signal processing; optimal for linear G…

Kantorovich theorem: convergence conditions for Newton's method Kantorovich theorem A clear overview of the Kantorovich (Newton–Kantorovich) theorem: its statement, assumptions, historical origin, and applications in numerical analysis and validated numerics.

Kepler conjecture (densest packing of equal spheres in three dimensions) Kepler conjecture A classical problem in discrete geometry about the densest arrangement of congruent spheres in 3D space; conjectured by Kepler and proven with extensive computer-aided case checkin…

Key (cryptography) A cryptographic key is information that governs encryption and decryption. This article explains types, properties, history, uses, management and notable distinctions.

Key schedule (cryptography) Key schedule Algorithm that derives per-round subkeys from a master key for iterative ciphers; covers purpose, common designs, security trade-offs, and notable examples such as DES and AES.

Key whitening (cryptography) Key whitening Key whitening strengthens block ciphers by mixing plaintext or ciphertext with subkey material (often via XOR) before and/or after the main rounds; it raises brute‑force cost but h…

Keystream — pseudorandom sequences used to encrypt data Keystream Explanation of keystreams: definition, formats, generation, historical context, how they’re used in stream and block ciphers (CTR), security considerations and common applications.

Kilosecond (1000 seconds) Kilosecond A kilosecond is 1,000 seconds — about 16 minutes 40 seconds. It is an SI-prefixed time unit used in scientific and technical contexts; a 24-hour day contains 86.4 kiloseconds.

King's Gambit (chess opening) King's Gambit The King's Gambit is an aggressive chess opening starting with 1.e4 e5 2.f4. White sacrifices a pawn to accelerate development and attack the black king; it has rich theory, named…

Klein bottle A Klein bottle is a closed, non‑orientable surface with no boundary that cannot be embedded in ordinary 3D space without self‑intersection; it is an important example in topology a…

Knight's tour (chess puzzle and Hamiltonian path) Knight's tour A sequence of knight moves that visits every square of a chessboard exactly once. Studied as a puzzle, a graph-theory problem, and an algorithmic exercise with open and closed vari…

Knot (unit of speed) Knot (speed) A knot is a unit of speed equal to one nautical mile per hour, commonly used in maritime, aviation, and meteorology. Abbreviations include kn and kt; it is a non‑SI unit accepted f…

Knot theory Knot theory is the branch of topology that studies closed loops embedded in three-dimensional space, their classification, invariants, and applications in science and mathematics.

Known-plaintext attack (KPA) Known-plaintext attack A known-plaintext attack is a cryptanalytic scenario in which an adversary has pairs of plaintext and matching ciphertext and uses them to deduce keys, keystreams, or weaknesses in…

Knuth's up-arrow notation: expressing extremely large integers Knuth's up-arrow notation Explanation of Knuth's up-arrow notation, its recursive definition, examples, relation to hyperoperations and Graham's number, and brief comparison with related notations.

Kruskal's algorithm: greedy method for minimum spanning trees Kruskal's algorithm Kruskal's algorithm is a greedy method to compute a minimum spanning tree (or forest) of a weighted, undirected graph by adding edges in increasing weight while avoiding cycles.

L (the letter) L L is the twelfth letter of the Latin alphabet, named “el.” It represents the lateral consonant /l/ in many languages and appears in typography, numerals, units, and varied symbolic…

Lagrange's theorem (group theory) A foundational result in finite group theory: the order of any subgroup divides the order of the group, and the quotient gives the number of cosets. Consequences include constraint…

Lakh (100,000): South Asian numeric unit Lakh Lakh is a counting unit equal to 100,000 widely used in South Asia. Covers definition, notation (1,00,000), relation to crore, usage in finance and media, etymology, and internatio…

Lambda (Greek letter Λ, λ) — forms, history and common uses Lambda Lambda (Λ, λ) is the 11th Greek letter. It denotes the 'l' sound, has numeric value 30, and appears across mathematics, physics, computing, and typography.