numerre

Numerre is a term used in some educational materials to denote a simple abstraction for modular arithmetic, serving as a didactic stand-in for residue classes. It is not a formal object in standard mathematics, but it is often employed to simplify explanations of congruence and arithmetic under a modulus.

A numerre is defined as a pair (a, m) where m is an integer greater than 1 and a is an integer with 0 ≤ a < m. The numerre represents the residue class a modulo m. Operations on numerres are performed by applying ordinary integer operations and then reducing the result modulo m. Specifically, addition and multiplication are defined by (a, m) + (b, m) = ((a + b) mod m, m) and (a, m) × (b, m) = ((a · b) mod m, m). An inverse of a numerre exists when gcd(a, m) = 1, in which case (a, m) has a multiplicative inverse (modulo m).

Canonical form is achieved by reducing a to the least nonnegative remainder 0 ≤ a < m. For example, with m = 5, the numerre (2, 5) plus (3, 5) yields (0, 5), while their product is (1, 5), since 2·3 ≡ 1 (mod 5).

Usage and limitations: numerres are a teaching device to illustrate residue classes, closure properties, and inverses in a concrete format. In formal mathematics, one would refer directly to residue classes modulo m rather than the numerre construct.

See also: modular arithmetic, residue class, congruence, group of integers modulo m. References: standard texts on modular arithmetic and contemporary teaching resources discuss residue classes and their arithmetic; numerre as a term appears primarily in classroom and online educational contexts.