  For scientific computations on a digital computer the set of real number is
usually approximated by a finite set F of "floating-point" numbers. We compare
the numerical accuracy possible with difference choices of F having
approximately the same range and requiring the same word length. In particular,
we compare different choices of base (or radix) in the usual floating-point
systems. The emphasis is on the choice of F, not on the details of the number
representation or the arithmetic, but both rounded and truncated arithmetic are
considered. Theoretical results are given, and some simulations of typical
floating-point computations (forming sums, solving systems of linear equations,
finding eigenvalues) are described. If the leading fraction bit of a normalized
base 2 number is not stored explicitly (saving a bit), and the criterion is to
minimize the mean square roundoff error, then base 2 is best. If unnormalized
numbers are allowed, so the first bit must be stored explicitly, then base 4
(or sometimes base 8) is the best of the usual systems.
