Representations of Each Number Type that Differ by Scale Factors

TL;DR

This paper explores how different scaled versions of number structures are mathematically equivalent, focusing on the isomorphisms that preserve number axioms under arbitrary scale transformations.

Contribution

It characterizes the structures of various number types that differ by scale factors and establishes conditions for their isomorphism and axiom preservation.

Findings

Number structures are isomorphic if scaled operations satisfy the same axioms.

Scaling of number values requires corresponding scaling of operations and relations.

The framework applies to different number types, ensuring their structural consistency under scaling.

Abstract

For each type of number, structures that differ by arbitrary scaling factors and are isomorphic to one another are described. The scaling of number values in one structure, relative to the values in another structure, must be compensated for by scaling of the basic operations and relations (if any) in the structure. The scaling must be such that one structure satisfies the relevant number type axioms if and only if the other structure does.