  Earlier work on modular arithmetic of k-ary representations of length L of
the natural numbers in quantum mechanics is extended here to k-ary
representations of all natural numbers, and to integers and rational numbers.
Since the length L is indeterminate, representations of states and operators
using creation and annihilation operators for bosons and fermions are defined.
Emphasis is on definitions and properties of operators corresponding to the
basic operations whose properties are given by the axioms for each type of
number. The importance of the requirement of efficient implementability for
physical models of the axioms is emphasized. Based on this, successor
operations for each value of j corresponding to addition of k^{j-1} if j>0 and
k^{j} if j<0 are defined. It follows from the efficient implementability of
these successors, which is the case for all computers, that implementation of
the addition and multiplication operators, which are defined in terms of
polynomially many iterations of the successors, should be efficient. This is
not the case for definitions based on the successor for j=1 only. This is the
only successor defined in the usual axioms of arithmetic.
