A New Representation for Exact Real Numbers 1

We develop the theoretical foundation of a new representation of real numbers based on the in nite composition of linear fractional transformations (lft), equivalently the in nite product of matrices, with non-negative coe cients. Any rational interval in the one point compacti cation of the real line, represented by the unit circle S1, is expressed as the image of the base interval [0;1] under an lft. A sequence of shrinking nested intervals is then represented by an in nite product of matrices with integer coe cients such that the rst so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative integer coe cients and successively re ne that interval. Based on the classi cation 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 =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.