1.4. Natural Numbers, Integers, and Rational Numbers

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).


The proof is much similar to proving the similar statement regarding the integers. The details are left as an exercise.

Notice the requirement that (a,b) N x N - {0} i.e. the integer b can not be zero. If we did allow both a and b to become zero, the relation would not be an equivalence relation any longer. As a hint: what pairs (a, b) would be related to (0,0) ?

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