numerikim

Numerikim is a theoretical construct used in speculative mathematics and educational contexts to explore how numeral systems and base representations interact. It is not an established mathematical object in mainstream literature, but rather a framework for thinking about numbers that carry multiple base representations at once and how those representations relate to a single underlying value.

Definition and concept: A numerikim pairs a numerical value with a set of representations across different bases. For a given family of bases B, a numerikim stores, for each base b in B, a digit string that represents the same value in base b. There is a canonical value that the various representations evaluate to, and there are conversion operations that map between the base-specific representations and the common value. In this sense, numerikim provides a concrete way to study base-change behavior, cross-base arithmetic, and the notion of numerical equality across numeral systems.

Formal framework (conceptual): A numerikim can be modeled as a triple consisting of a value set, a collection of base representations, and evaluation/conversion procedures. The key idea is that arithmetic performed by operating on the representations (after translating to a common form) should yield results that are consistent with the shared underlying value, thereby illuminating base-dependence and base-independence in computations.

Examples and scope: The ordinary integers can be viewed as a simple embedded class of numerikim where a single canonical base suffices. More intricate examples come from constructing nonstandard base families that reveal quirks of cross-base conversions or pedagogical demonstrations of numeral systems.

Applications and relation: Numerikim serves as a didactic tool for teaching base systems, a conceptual aid in multibase algorithm design, and a way to frame questions about base-invariant properties and equivalence of numeration methods in theoretical discussions.