A Combinatorial Approach to Positional Number Systems

TL;DR

This paper introduces a combinatorial method to construct infinitely many binary radix systems from pairs of binary strings, expanding the understanding of positional number systems with potential applications in number representation.

Contribution

It presents a novel combinatorial framework for generating binary radix systems from binary string pairs, broadening the scope of positional number system constructions.

Findings

Constructed infinitely many binary radix systems from string pairs

All minimal-criteria binary radix systems can be generated this way

Provides a new perspective on binary number representations

Abstract

Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in simple situations. This paper concerns binary radix systems, i.e., positional number systems with digits 0 and 1. Our combinatorial approach is to construct infinitely many binary radix systems, each one from a single pair of binary strings. Every binary radix system that satisfies even a minimal set of conditions that would be expected of a positional number system can be constructed in this way.