Rationally biased arithmetic

One can naively view a computer number system as a pair (F, P) consisting of a finite set F of real numbers and a rounding rule P. One such number system is a hyperbolic rational number system which has as F a finite set of rational numbers and as P the so-called mediant rounding rule. In this paper we demonstrate how one can simulate a hyperbolic rational number system in any high level language that supports floating point computation. From this simulation we infer that hyperbolic rational number systems form viable alternatives to traditional binary floating point number systems. Many properties of hyperbolic rational number systems are derived from the relationship of their rounding rule to the well-developed theory of best rational approximation.