A combinatorial approach to binary positional number systems

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 the binary case – digits 0 and 1. A binary positional number system (binary radix system) with base equal to the golden ratio \((1+\sqrt{5}\,)/2\) is fairly well known. The main result of this paper is a construction of infinitely many binary radix systems, each one constructed combinatorially 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.