RealNumberIDs

RealNumberIDs is a concept used in mathematical databases and symbolic computation to assign stable identifiers to real numbers, enabling precise reference and cross-linking across systems. It is not a widely standardized term, but rather a framework for thinking about how real numbers can be identified and retrieved without losing equality or introducing ambiguity.

A RealNumberID typically represents a real number by an underlying constructive representation or a registry entry. Common approaches include encoding real numbers as Cauchy sequences of rationals, Dedekind cuts, continued fraction expansions, or symbolic descriptions such as algebraic numbers and named constants. In some designs, an ID corresponds to a canonical form, while in others it points to a registry that records the exact representation and a proof that the representation denotes the intended real number. Canonicalization is a central concern to avoid aliasing, where different representations denote the same real.

Key challenges arise from the mathematical nature of real numbers. RealNumberIDs often must distinguish between computable and non-computable reals, handle infinite data requirements, and address the fact that there is no simple finite encoding for arbitrary real numbers. Practical implementations typically restrict to computable reals or use layered representations that provide progressively accurate approximations. Canonical, provably equivalent mappings between representations are essential to preserve equality across systems.

Applications of RealNumberIDs include mathematical knowledge bases, theorem provers, and symbolic computation platforms, where exact numeric references improve provenance, reproducibility, and interoperability. The concept interacts with established notions such as computable analysis, algebraic numbers, and exact arithmetic, while remaining distinct from standard floating-point identifiers.