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Showing 121–150 of 1709 reference entries

Mathematics & logicZFC set-theory framework

Set existence is governed by explicit axioms and schemas.

DefinitionMathematics & logicCompleteness of the real numbers

Upper-bounded nonempty real sets have least upper bounds.

Mathematical statementMathematics & logicFinite and countable sets

Countability includes finite sets in this convention.

DefinitionMathematics & logicEquivalence relations and quotient classes

An equivalence relation partitions a set into classes.

DefinitionMathematics & logicModels, completeness and incompleteness

Semantic consequence and arithmetic completeness are distinct.

Mathematical statementMathematics & logicInteger divisibility and the Euclidean algorithm

Repeated division computes a greatest common divisor.

MethodMathematics & logicUnique prime factorization of integers

Positive integers greater than one have unique prime factors.

Mathematical statementMathematics & logicInteger arithmetic modulo n

Congruence identifies integers differing by a multiple of n.

DefinitionMathematics & logicChinese Remainder Theorem

Coprime moduli specify one combined residue.

Mathematical statementMathematics & logicGroups and abelian groups

A group has associative composition, identity and inverses.

DefinitionMathematics & logicGroup homomorphisms and kernels

Homomorphisms preserve the group operation.

DefinitionMathematics & logicRings and identity conventions

A ring combines addition with distributive multiplication.

DefinitionMathematics & logicFields and division

A field allows division by every nonzero element.

DefinitionMathematics & logicFundamental Theorem of Algebra

Nonconstant complex polynomials factor into linear terms.

Mathematical statementMathematics & logicMetric spaces

A metric supplies a distance satisfying three axioms.

DefinitionMathematics & logicTopological spaces

A topology specifies which subsets are open.

DefinitionMathematics & logicCompactness and finite subcovers

Compactness reduces every open cover to a finite one.

DefinitionMathematics & logicContinuity between topological spaces

Continuity preserves openness through inverse images.

DefinitionMathematics & logicEuclidean and curved-surface geometry

Triangle-angle relations depend on the geometry.

Mathematical statementMathematics & logicAbstract vector spaces

Vector-space laws extend beyond coordinate arrows.

DefinitionMathematics & logicBases and finite dimension

A basis spans without linear redundancy.

DefinitionMathematics & logicRank–nullity theorem

Kernel and image dimensions sum to domain dimension.

Mathematical statementMathematics & logicEigenvalues and eigenvectors

An eigenvector changes only by a scalar factor.

DefinitionMathematics & logicFinite-dimensional self-adjoint spectral theorem

Self-adjoint operators have orthonormal eigenbases.

Mathematical statementMathematics & logicReal singular value decomposition

Any real rectangular matrix has orthogonal factors.

Mathematical statementMathematics & logicConvergence and Cauchy sequences

Real convergence can be checked by distances between late terms.

Mathematical statementMathematics & logicInfinite series and absolute convergence

A series converges through its partial sums.

Mathematical statementMathematics & logicFinite Taylor approximation and remainder

The remainder quantifies a finite polynomial approximation.

Mathematical statementMathematics & logicUniform convergence of functions

One cutoff controls the error at every domain point.

Mathematical statementMathematics & logicLebesgue integral and integrability

Integration starts from measurable simple functions.

Definition
DefinitionSource checked

Integer arithmetic modulo n

Mathematics & logic

Congruence identifies integers differing by a multiple of n.

Key knowledge

For n ≥ 2, a ≡ b (mod n) means n divides a−b. Addition and multiplication descend to the n congruence classes; cancellation or division is not valid without invertibility.

Values and equations

a ≡ b (mod n) ⇔ n | (a−b)

a, b
Integer representatives
n
Integer modulus at least two

Conditions and scope

  • Operands are integers and the same modulus is used.
  • A multiplicative inverse exists only for a residue coprime to n.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Abstract Algebra, 2.1 Integers mod nThomas W. JudsonCourse notesopenmathbooks.orgRead source
  2. Theory of Numbers, Lecture 5Abhinav Kumar and Joseph Lee / MITCourse notes · PDFocw.mit.eduRead source

Evidence and review

Section 2.1.1: quotient classes, modular addition/multiplication and noninvertible examples. Simultaneous congruences and Chinese Remainder Theorem, physical pages 1–3.

2026-10-10 · Congruence classes and permitted modular operations.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–5 of 5 relationships

Scientific relationships · 5 / 5

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • For n ≥ 2, a ≡ b (mod n) means n divides a−b. Addition and multiplication descend to the n congruence classes; cancellation or division is not valid without invertibility.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • If positive moduli m,n are coprime, then any prescribed residues a,b determine exactly one residue x modulo mn satisfying x ≡ a (mod m) and x ≡ b (mod n).
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • A field has characteristic zero if no positive n satisfies n·1=0; otherwise its characteristic is a prime p. It contains a copy of Q in characteristic zero or Fₚ in characteristic p.
  4. Related to
    Relationship sources · 1

    Linked statements

    • Shor's factoring algorithm reduces factoring to order finding and uses modular exponentiation, quantum phase/Fourier processing and classical continued fractions. Its asymptotic cost is polynomial in the integer's bit length under a uniform circuit and arithmetic model. This establishes an algorithmic threat to factoring-based cryptography with sufficiently reliable quantum resources; it does not establish that a present noisy device can factor cryptographically sized integers.

    Order finding uses modular multiplication.

  5. Related to
    Relationship sources · 1

    Linked statements

    • For a primitive n-th root zeta-n, Gal(Q(zeta-n)/Q) is isomorphic to the units modulo n by a↦(zeta-n↦zeta-n^a), and the degree is Euler phi(n). For n=5 the group is cyclic of order four; for n=8 it is C2×C2. Cyclotomic polynomials are irreducible over Q; a different base field can have a smaller group.

    Automorphisms are indexed by invertible residue classes.

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