Theorem 1.4.4: The Rationals

Let A be the set N x N - {0} and define a relation r on N x N - {0} by saying that (a, b) is related to (a’, b’) if a * b’ = a’ * b. Then this relation is an equivalence relation.

If [(a, b)] and [(a’, b’)] denotes the equivalence classes containing (a, b) and (a’, b’), respectively, and if we define the operations

1. [(a, b)] + [(a', b')] = [(a * b' + a’ * b, b * b')]
2. [(a, b)] * [(a’, b’)] = [(a * a’, b * b’)]

then these operations are well-defined and the resulting set of all equivalence classes has all of the familiar properties of the rational numbers (it therefore serves to define the rationals based only on the natural numbers).

Context