Unumber

Unumber is a term used in speculative mathematics and puzzle literature to denote a member of a nonstandard arithmetic system. It is not a standard mathematical object and has no universal definition; instead, a discussion of unumbers typically begins with a chosen binary operation ⊗ on the nonnegative integers and then defines an unumber as any n ≥ 0 such that n ⊗ n = n.

Etymology and scope: The name combines the idea of a nonstandard or “unusual” number with the notion of a number itself. Because there is no single canonical formulation, the exact meaning of an unumber varies between authors and contexts.

Framework and properties: In common expositions, an unumber is defined relative to a specific operation ⊗. Authors may require ⊗ to be associative or commutative, but these conditions are not universal. The set of unumbers thus depends on the chosen operation and can range from a large collection to just a handful of fixed points.

Examples: Under ordinary multiplication, the equation n ⊗ n = n becomes n^2 = n, yielding unumbers n = 0 or 1. Under the operation n ⊗ m = max(n, m), every nonnegative integer satisfies n ⊗ n = n, so all are unumbers. Other definitions produce different fixed-point sets and illustrate how arithmetic changes affect structure.

Relation to broader concepts: The idea connects to fixed points and idempotent elements in universal algebra and is often discussed in the context of recreational mathematics and thought experiments.

Usage: Unumber is used mainly as a conceptual tool to explore how altering arithmetic rules influences properties like growth, convergence, and fixed points. It is not part of formal mathematics but appears in fiction, puzzles, and expository writing.

See also: fixed point, idempotent element, nonstandard arithmetic, recreational mathematics.