Finite segment p-adic number systems with applications to exact computation

Abstract

A fractional weighted number system, based on Hensel's p-adic number system, is proposed for constructing a unique code (called Hensel's code) for rational numbers in a certain range. In this system, every rational number has an exact representation. The four basic arithmetic algorithms that use the code for the rational operands, proceed in one direction, giving rise to an exact result having the same code-wordlength as the two operands. In particular, the division algorithm is deterministic (free from trial and error). As a result, arithmetic can be carried out exactly and much faster, using the same hardware meant for p-ary systems. This new number system combines the best features and advantages of both the p-ary and residue number systems. In view of its exactness in representation and arithmetic, this number system will be a very valuable tool for solving numerical problems involving rational numbers, exactly.