Xnumbers

Xnumbers are a conceptual extension of numerical values that pair a numeric magnitude with an extra context tag. They are designed to encode not only the value itself but also ancillary information such as units, provenance, or qualitative context, allowing arithmetic that respects this additional structure.

An Xnumber is formally a pair (n, t), where n is a numeric value and t is a tag drawn from a predefined tag set T. Arithmetic rules extend conventional operations with tag-aware constraints. Addition and subtraction require compatible tags (t must match or be defined by a permissible tag fusion); multiplication and division combine tags via a defined tag algebra, producing new tags (for example, units multiply to yield derived units). Projection maps can recover the underlying numeric value by discarding the tag, while tag-aware arithmetic propagates or transforms tags as dictated by the tag algebra.

The collection of Xnumbers can form algebraic structures such as rings or semirings under suitable tag operations. Variants exist for different base numeric types, including Xintegers and Xreals, each with its own tag semantics. The framework often emphasizes type-safety and error prevention by design, reducing unit inconsistencies or provenance ambiguities in computations.

Applications include units-aware numerical calculations, dimensional analysis, and provenance tracking in simulations. They provide a formal mechanism to enforce contextual constraints during computation and to propagate metadata alongside values, enabling safer and more transparent numerical workflows.

See also: units, dimensional analysis, unit-safe arithmetic, provenance tracking.