The use of a negative base did not appear until the 1950s when several authors independently introduced the concept. Complement representation also became much discussed in this period as an alternative to sign magnitude for designing the arithmetic unit of a computer. The arithmetic of numbers represented in positional notation has a firm foundation derived from the theory of polynomial arithmetic that readily allows these extensions to negative bases and/or negative digit values, complement representation, and digit values in excess of the base. Our primary concern in this paper is the characterization and computation of those integral valued base and digit set pairs that provide complete and unique finite radix representation of the integers.