RATIONAL NUMBER APPROXIMATION IN HIGHER RADIX FLOATING POINT SYSTEMS

Recent proposals have suggested that suitably encoded non-binary floating point representations might offer range and precision comparable to binary systems of equal word size. This is of obvious importance in that it allows computation to be performed on decimal operands without the overhead or error of base conversion while maintaining the error performance and representational characteristics of more traditional encodings. There remains, however, a more general question on the effect of the choice of radix on the ability of floating point systems to represent arbitrary rational numbers. Mathematical researchers have long recognized that some bases offer some representational advantages in that they generate fewer nonterminate values when representing arbitrary rational numbers. Base twelve, for example, has long been considered preferred over base ten because of its inclusion of three in its primary factorization allowing finite representation of a greater number of rational numbers. While such results are true for abstract number systems, little attention has been paid to machine based computation and its finite resources. In this study, such results are considered in an environment more typical of computer based models of number systems. Specifically, we consider the effect of the choice of floating point base on rational number approximation in systems which exhibit the typical characteristics of floating point representations-normalized encodings, limited exponent range and storage allocated in a fixed number of 'bits' per datum. The frequency with which terminate and representable results can be expected is considered for binary, decimal, and other potentially interesting bases.