Fixed-slash and floating-slash rational arithmetic

A finite precision rational number system provides for representation of a collection of rational numbers subject to limitations on numerator and denominator magnitude. In fixed-point and floatingpoint radix number systems only rationals of the form i/βj, where β is the base, can be realized. In contrast, a finite precision rational number system will allow representation of practically all simple fractions encountered in applications. In this preliminary report we first propose two types of finite precision rational number systems which we term fixed-slash and floating-slash systems [2]. We then consider the conversion (rounding) problem, that is, the determination of a number satisfying the numerator and denominator constraints approximating a given non representable real value. We show that the rounding problem is solvable by an efficient procedure, which we term mediant conversion, that derives from the theory of continued fractions.