  This paper extends earlier work on quantum theory representations of natural
numbers N, integers I, and rational numbers Ra to describe a space of these
representations and transformations on the space. The space is parameterized by
4-tuple points in a parameter set. Each point, (k,m,h,g), labels a specific
representation of X = N, I, Ra as a Fock space F^{X}_{k,m,h} of states of
finite length strings of qukits q and a string state basis B^{X}_{k,m,h,g}. The
pair (m,h) locates the q string in a square integer lattice I \times I, k is
the q base, and the function g fixes the gauge or basis states for each q. Maps
on the parameter set induce transformations on on the representation space.
There are two shifts, a base change operator W_{k',k}, and a basis or gauge
transformation function U_{k}. The invariance of the axioms and theorems for N,
I, and Ra under any transformation is discussed along with the dependence of
the properties of W_{k',k} on the prime factors of k' and k. This suggests that
one consider prime number q's, q_{2}, q_{3}, q_{5}, etc. as elementary and the
base k q's as composites of the prime number q's.
