Kongruence

Kongruence is a term used in mathematics to denote a relation of equivalence that preserves a chosen structure. In many contexts, Kongruence is equivalent to the standard concept of congruence. The word appears as a direct translation of Kongruenz in German and is sometimes used in multilingual mathematical literature. The most common usage is in modular arithmetic and geometry.

In modular arithmetic, two integers a and b are said to be congruent modulo n when their difference is divisible by n; written a ≡ b (mod n). This relation is an equivalence relation, and it partitions the integers into residue classes modulo n. Arithmetic operations are well-defined on these classes: if a ≡ b (mod n) and c ≡ d (mod n), then a+c ≡ b+d (mod n) and ac ≡ bd (mod n). Congruences are used to perform calculations with large numbers and to study periodic patterns.

In geometry, congruence refers to the notion that two figures have the same size and shape, obtainable from one by an isometry (rotation, reflection, or translation). Triangles are congruent if they satisfy any of the standard criteria (SSS, SAS, ASA, AAS, RHS). Beyond Euclidean geometry, congruence concepts extend to algebra via congruence relations: two elements are congruent modulo an ideal, yielding quotient structures that preserve algebraic operations.

Kongruence thus encompasses several related ideas about when two objects can be considered equal in a structure-preserving sense. See also congruence, modular arithmetic, isometry, and quotient structures.