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.
Categories: quant-ph hep-th math-ph math.MP