LCF: A Lexicographic Binary Representation of the Rationals

Peter Kornerup (Dept. of Mathematics and Computer Science, Odense University, Denmark)

David W. Matula (Dept. of Computer Science and Engineering, Southern Methodist University, USA)

Abstract: A binary representation of the rationals derived from their continued fraction expansions is described and analysed. The concepts "adjacency", "mediant" and "convergent" from the literature on Farey fractions and continued fractions are suitably extended to provide a foundation for this new binary representation system. Worst case representation-induced precision loss for any real number by a fixed length representable number of the system is shown to be at most 19% of bit word length, with no precision loss whatsoever induced in the representation of any reasonably sized rational number. The representation is supported by a computer arithmetic system implementing exact rational and approximate real computations in an on-line fashion.

Keywords: Computer arithmetic, continued fractions, lexicographic, number systems, number theory, rational numbers.

Categories: B.5.1, E.2, G.1.0