numberits

Numberits are a class of abstract mathematical objects designed to generalize natural numbers by attaching contextual information to each value. A numberit consists of two parts: a magnitude, typically a nonnegative integer, and a tag-set drawn from a fixed universe of tags. The tag-set stores metadata such as provenance, type constraints, or contextual properties, and can be empty. When the tag-set is empty, the object corresponds to an ordinary natural number.

Operations on numberits are defined to preserve the intuitive interpretation of magnitude while combining contextual information. For two numberits (a, A) and (b, B), addition is defined as (a + b, A ∪ B) and multiplication as (a · b, A ∪ B). In this system, the tag-set is propagated through arithmetic via set union; the additive and multiplicative identities are (0, ∅) and (1, ∅), respectively. Some variants replace union with other tag-combining operations and extend the model to integers or real numbers by allowing signed magnitudes or alternative encodings of magnitude.

Numberits form a commutative semiring isomorphic to the product of the natural numbers with the power set of the tag universe, under the described operations. The concept is used in thought experiments and educational contexts to illustrate how numerical value and contextual information can be integrated in a single formal object, and it has parallels in tagged arithmetic and data-tag frameworks.

See also: tagged numbers, semirings, data provenance, context-aware arithmetic.