  A quantum theoretic representation of real and complex numbers is described
here as equivalence classes of Cauchy sequences of quantum states of finite
strings of qubits. There are 4 types of qubits each with associated single
qubit annihilation creation (a-c) operators that give the state and location of
each qubit type on a 2 dimensional integer lattice. The string states, defined
as finite products of creation operators acting on the qubit vacuum state
correspond to complex rational numbers with real and imaginary components.
These states span a Fock space. Arithmetic relations and operations are defined
for the string states. Cauchy sequences of these states are defined, and the
arithmetic relations and operations lifted to apply to these sequences. Based
on these, equivalence classes of these sequences are seen to have the requisite
properties of real and complex numbers. The representations have some
interesting aspects. Quantum equivalence classes are larger than their
corresponding classical classes, but no new classes are created. There exist
Cauchy sequences such that each state in the sequence is an entangled
superposition of the real and imaginary components, yet the sequence is a real
number. Also, except for coefficients of superposition states, the construction
is done with no reference to the real and complex number base, R and C, of the
Fock space.
