Phine Numbers

A.J.C. Hurkens
Radboud University Nijmegen


The binary number system is based on the classical fact that the unit interval [0, 1] is the union of two intervals of size
1/2 . But constructing a binary representation of 0.0101 . . . + 0.00101 . . . is a problem: one needs overlapping subintervals. If one takes as size the reciprocal phi of the golden ratio Phi (the positive numbers given by phi = 1 / Phi and
Phi = 1 + phi), then the size of the overlap is phi^3. This leads to the phinary number system in which sums can be calculated from left to right using only a finite number of states. Each natural number has a finite representation in base Phi.

In this article we introduce the phine number system. It is based on a variant with three subintervals of size phi^2. In base Phi^2, one needs an extra digit: either 2 or -1. In both variants, each natural number has a symmetrical representation. Addition can be defined on such sequences of signed bits in a symmetrical way. The signed bit of this sum at some position only depends on the original signed bits at positions up to distance 4.

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