Construction of the rationals

Our next task is to define the set of rational numbers from the integers using equivalence classes of pairs of integers. The idea is clear: we think of a pair of integers ( p , q ) as the fraction p / q and use an equivalence relation to identify fractions that should have the same values.

1. Getting the rationals from the integers

Definition.

Let + denote the set of positive integers, i.e., + = { n : n > 0 } . Let be the following relation defined on × + : we define ( x 1 , y 1 ) ( x 2 , y 2 ) to mean x 1 · y 2 = x 2 · y 1 .

Proposition.

is an equivalence relation on × + .

Proof.

Exercise.

Definition.

We write p / q for the equivalence class of ( p , q ) × + . The set is the set × + / of equivalence classes.

Proposition.

The function i defined by i ( n ) = n / 1 is a one-to-one function mapping into .

Proof.

Let n , m with i ( n ) = i ( m ) . Then n / 1 = m / 1 hence ( n , 1 ) ( m , 1 ) hence n · 1 = m · 1 hence n = m .

Definition.

We identify each n with its image i ( n ) under the map i . In particular, 0 is the element 0 / 1 , and 1 is 1 / 1 .

Definition.

We define addition, multiplication and order relations on by

  • p 1 / q 1 + p 2 / q 2 = ( p 1 q 2 + p 2 q 1 ) / q 1 q 2
  • p 1 / q 1 · p 2 / q 2 = p 1 p 2 / q 1 q 2
  • p 1 / q 1 < p 2 / q 2 p 1 q 2 < p 2 q 1

Proposition.

The operations +,· and relation on are well-defined, i.e., the definitions above do not depend on the particular choice of representatives ( p 1 , q 1 ) , ( p 2 , q 2 ) .

Proof.

Exercise.

Proposition.

The embedding i : is a homomorphism respecting +,· and :

  • i ( n + m ) = i ( n ) + i ( m ) ;
  • i ( n · m ) = i ( n ) · i ( m ) ; and
  • n < m i ( n ) i ( m ) ;

for all n , m .

Proof.

Exercise.

2. The ordered field structure

The key axioms for the rationals have already been given. They form an Archimedean ordered field, in fact a minimal Archimedean ordered field in the sense that no proper subset is also an Archimedean ordered field.

Theorem.

The rationals with + , · , < as defined here satisfy the axioms of an Archemedean ordered field.

Proof.

A rather long exercise.