# 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 (

,)

as the fraction

/

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., += 0 . Let be the following relation defined on + : we define ( 1, 1)( 2, 2) to mean 1 2= 2 1 .

Proposition.

is an equivalence relation on + .

Proof.

Exercise.

Definition.

We write

/

for the equivalence class of (

,) +

. The set is the set +/ of equivalence classes.

Proposition.

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

Proof.

Let , with ()=() . Then /1=/1 hence (,1)(,1) hence 1= 1 hence = .

Definition.

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

Definition.

We define addition, multiplication and order relations on by

• 1/1

+

2/2

=(

12+

21)/12

• 1/1

2/2

=

1

2/12

• 1/1

2/2

12

21

Proposition.

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

1,1),(

2,2)

.

Proof.

Exercise.

Proposition.

The embedding is a homomorphism respecting +, and :

• (+)=()+() ;
• ( )=() () ; and
• () () ;

for all , .

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.