A Constructive Realization of the Rational, Real, and Complex Numbers via Sequences over Finite Fields of Increasing Order

Based on the representation of finite field elements by fractions, an algorithmic procedure is introduced to define a field Q(p)=Qp,+,⋅ of discrete rational numbers. The set Qp consists of fractions m/n with m and n∈Z such that m<p and 0<n<p, where p is a prime number. An isomorphism with the Galois field Fp is established. Sequences qn such that n∈N, qn∈Qp(n), and p(n) denotes the increasing sequence of primes, which are constant or convergent under suitable definitions, are used to construct fields isomorphic to Q and R, respectively. Suitable addition and multiplication operations are defined on Qp′2 with p′, a non-Pythagorean prime, such that Cp′=Qp′2+,⋅ is a field. Sequences zn such that n∈N and zn∈Qp′n2, together with a suitably defined convergence criterion, are then used to construct a field isomorphic to C.