  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.
