This paper provides some basic definitions and results concerning the representation of positive integers in arbitrary number systems. Here a number system N is of the form N = (n,m₁,. . .,mK number system N is of the form N = (n,m₁,. . .,m) where k :3W 1, n :3W 2 and 1 :3W m₁ :3W . . . :3W m_K. Words over the m_i represent integers obtained by considering the m_i as digits over the base n in the standard way. The generality is obtained by allowing the m_i to be larger than n or to be negative.

Sets of integers representable by some such number system N are called RNS sets. The paper defines and investigates the notions of equivalence, completeness, and ambiguity of number systems, and uses a translation from number systems to regular expressions to obtain various decidability and undecidability results. For example, equivalence and ambiguity are decidable, but the RNS-ness property of an recursively enumerable set is not.