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₊

_{2_{1)/_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.