Foundations of finite precision arithmetic

Abstract

Completeness and uniqueness properties for the representation of the base 㬲 digital numbers by finite length radix polynomials with various digit sets are studied. The digit sets guaranteeing completeness and uniqueness are characterized. A digital conversion algorithm is introduced for determining a base β radix polynomial with digits from a specified set D having a particular value whenever such a radix polynomial exists. The notion of precision of a radix polynomial is formalized, and the determination of the precision from the given base β, digit set D, and real value a of the radix polynomial is investigated.