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Indescribable cardinal
An indescribable cardinal is a large cardinal whose higher-order statements about Vκ already hold in some smaller Vα, introduced by Hanf and Scott in 1961.
Indian numbering system
The Indian numbering system, also called the South Asian numbering system, is used in India, Pakistan, Bangladesh, Nepal, and Sri Lanka, grouping digits in twos after the first three and naming lakh (100,000) and crore (10,000,000).
Induced representation
An induced representation is a representation of a group G built from a representation of a subgroup H, with Frobenius reciprocity making induction adjoint to restriction.
Inequality (mathematics)
An inequality, or mathematical inequality, is a mathematical relation stating that one quantity is greater than or less than another, written with the symbols , ≤, and ≥.
Infinite set
An infinite set is a set that is not finite, so its elements cannot be counted with a natural number; Cantor showed in 1874 that infinities come in different sizes.
Infinitesimal
An infinitesimal, or infinitesimal number, is a nonzero quantity smaller in absolute value than every positive real number, realized rigorously in the hyperreal numbers introduced by Abraham Robinson in 1960.
Infinity
Infinity is the condition of being without any finite bound in size, extent, or quantity, appearing in mathematics as both endless process and completed infinite objects.
Infinity symbol
The infinity symbol (∞), also known as the sideways eight or infinity sign, is a mathematical sign for infinite quantity, introduced by English mathematician John Wallis in 1655.
Injective dimension
Injective dimension, denoted id(M), is the least number of injective modules needed to resolve a module over a ring, measured via Ext; finiteness characterizes Gorenstein and regular rings.
Injective hull
An injective hull, also called an injective envelope, is the smallest injective module containing a given module M, unique up to isomorphism; every module has one.
Injective module
An injective module is a module J over a ring R such that every homomorphism from a submodule into J extends to the whole module, the dual of projectivity.
Inner automorphism
An inner automorphism of a group is conjugation by a fixed element, sending x to gxg⁻¹; the inner automorphisms form a normal subgroup of the automorphism group.
Inner model
An inner model is a transitive class of sets containing all the ordinals and satisfying ZF, introduced by Kurt Gödel, whose constructible universe L gave the first relative consistency proofs in set theory.
Integer
An integer, also called an integer number, is a whole number that may be positive, negative, or zero, and the set ℤ is closed under addition, subtraction, and multiplication.
Integer factorization
Integer factorization, also called prime factorization, is the problem of decomposing a composite integer into smaller factors, best known as the basis of RSA cryptography.
Integer overflow
Integer overflow, also called bit overflow, is the condition where an arithmetic operation on a fixed-width integer produces a result the type cannot represent, potentially causing memory-safety vulnerabilities.
Integral basis
An integral basis of a number field K of degree n is a set of n algebraic integers that spans the ring of integers as a free module over the integers.
Integral domain
An integral domain, also called an entire ring, is a commutative ring in which ab = 0 forces a = 0 or b = 0; the integers are the standard example.
Integral element
An integral element of a ring S over a base ring R satisfies a monic polynomial with coefficients in R, generalizing algebraic elements of field extensions.
Integrally closed domain
An integrally closed domain is an integral domain containing every element of its fraction field integral over it, that is, satisfying a monic polynomial over the domain.
Inter-universal Teichmüller theory
Inter-universal Teichmüller theory (IUT) is a mathematical framework developed by Shinichi Mochizuki at Kyoto University and released in 2012, claiming a proof of the abc conjecture that remains disputed.
Interior algebra
An interior algebra, also known as a closure algebra or topo-boolean algebra, is a Boolean algebra with an interior operator; McKinsey and Tarski introduced the structure in 1944.
International Mathematical Olympiad
The International Mathematical Olympiad, or IMO, is an annual mathematics competition for pre-university students, first held in Romania in 1959, with teams of up to six contestants per country.
Intersection theory
Intersection theory is the branch of algebraic geometry that defines intersection multiplicities and numbers, forming an intersection product on the Chow ring, with roots in Bézout's theorem.
Interval arithmetic
Interval arithmetic, also known as interval analysis, is a numerical method in which quantities are carried as closed intervals, guaranteeing that results enclose the true value despite rounding errors.
Invariant subspace
An invariant subspace is a subspace of a vector space that a linear operator maps into itself, the concept underlying eigenvectors, the Jordan canonical form, and a famous open problem.
Invariant theory
Invariant theory is the branch of mathematics that studies polynomial functions left unchanged by a group action, classically begun in Arthur Cayley's 1845 paper and central to Hilbert's finiteness theorem.
Inverse eigenvalue problem
The inverse eigenvalue problem asks whether a matrix with prescribed eigenvalues exists within a given structural class, such as symmetric or nonnegative matrices, and how to construct it.
Inverse Galois problem
The inverse Galois problem asks whether every finite group occurs as the Galois group of a finite Galois extension of the rational numbers; it remains open in general.
Inverse iteration
Inverse iteration is an eigenvalue algorithm that finds the eigenvector whose eigenvalue lies nearest a chosen shift, working by the power method applied to the inverse shifted matrix.