We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite 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�1] 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 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.

Abstract: The result of a simple oating-point computation can be in great error, even though no error is signaled, no coding mistakes are in the program, and the computer hardware is functioning correctly. This paper proposes a set of instructions appropriate for a general purpose microprocessor that can be used to improve the credibility and accuracy of numerical computations. Such instructions provide direct hardware support for monitoring events which may threaten computational integrity, implementing oating-point data types of arbitrary precision, and repeating calculations with greater precision. These useful features are obtained by the e cient implementation of high radix on-line arithmetic. The prevalence of super-scalar and VLIW processors makes this approach especially attractive.