New Representation for Exact Real Numbers 1

We develop the theoretical foundation of a new representation of real numbers based on the innnite composition of linear fractional transformations (lft), equivalently the innnite product of matrices, with non-negative coeecients. Any rational interval in the one point compactiication of the real line, represented by the unit circle S 1 , 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 innnite product of matrices with integer coeecients 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 coeecients and successively reene that interval. Based on the classiication 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 S 1 by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.