# 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 $ℤ +$ denote the set of positive integers, i.e., $ℤ + = { n ∈ ℤ : n > 0 }$. Let $∼$ be the following relation defined on $ℤ × ℤ +$: 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 $ℤ × ℤ +$.

Proof.

Exercise.

Definition.

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

Proposition.

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

Proof.

Let $n , m ∈ ℕ$ 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 $n ∈ ℕ$ with its image $i ( n )$ under the map $i$. In particular, $0$ is the element $0 / 1 ∈ ℚ$, and $1$ is $1 / 1 ∈ ℚ$.

Definition.

We define addition, multiplication and order relations on $ℚ$ 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 $ℚ$ 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 $i : ℤ → ℚ$ 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 $n , m ∈ ℤ$.

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.