Computing with Exact Real Numbers in a Radix-r System

Abstract

This paper investigates an arithmetic based upon the representation of computable exact real numbers by lazy infinite sequences of signed digits in a positional radix-r system. We discuss advantages and problems associated with this representation, and develop well-behaved algorithms for a comprehensive range of numeric operations, including the four basic operations of arithmetic.The author gratefully acknowledges the financial support of the Computing Laboratory at the University of Kent at Canterbury, and the Committee ofVice-Chancellors and Principals of the Universities of the United Kingdom. Partial support for this research was also provided by grants from the Ian Karten Charitable Trust and the Anglo-Jewish Association.Credit for some of the ideas put forward in this paper must go to Carl Pixley, whose early unpublished work at Burroughs Corporation's Austin Research Centre was a great source of inspiration for this research. My thanks are also due to Professor David Turner for his numerous suggestions and guidance in this project all the way from inception to completion. Finally, the author wishes to thank the referee for his careful reading of the manuscript and pointing out a number of errors and obscurities.