# Foundations of Finite Precision Rational Arithmetic

## Abstract

Finite precision fraction number systems are characterized and their number theoretic foundations are developed. Closed approximate rational arithmetic in these systems is obtained by the natural canonical rounding obtained using the continued fraction theory concept of best rational approximation. These systems are shown to be natural finite precision number systems in that they are essentially independent of the apparatus of the representation. The specific fixed-slash and floating-slash fraction number systems are described and their feasibility and convenience for Computer implementation are discussed. The foundations of adaptive variable precision are explored. The overall goal is to better understand the inherent mathematical properties of finite precision arithmetic and to provide a most natural and convenient computation system for approximating real arithmetic on a Computer.

## Keywords

Number System Double Precision Simple Fraction Finite Precision Simple Chain## Preview

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