A New Representation for Exact Real Numbers

Abstract

We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infinite product of matrices, with non-negative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S1, is expressed as the image of the base interval [0, ∞] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative integer coefficients and successively refine that interval. Based on the classification of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S1 by π/4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.