Signed-Bit Representations of Real Numbers

TL;DR

This paper explores signed-bit representations of real numbers, extending traditional binary representations by including -1, and applies this to constructively define real numbers and analyze homomorphisms in Riesz spaces.

Contribution

It develops signed-bit equivalents of Dedekind cuts, Cauchy sequences, and regular sequences, and applies this to homomorphisms of Riesz spaces into the reals.

Findings

Signed-bit representations effectively model real numbers constructively.

Extended notions of Dedekind cuts, Cauchy sequences, and regular sequences are formulated.

Application to homomorphisms in Riesz spaces demonstrates the utility of signed-bit representations.

Abstract

The signed-bit representation of real numbers is like the binary representation, but in addition to 0 and 1 you can also use -1. It lends itself especially well to the constructive (intuitionistic) theory of the real numbers. The first part of the paper develops and studies the signed-bit equivalents of three common notions of a real number: Dedekind cuts, Cauchy sequences, and regular sequences. This theory is then applied to homomorphisms of Riesz spaces into the reals.