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 (
,)
/
Definition.
Let
Proposition.
Proof.
Exercise.
Definition.
We write
/
,)
Proposition.
The function defined by
(
Proof.
Let
Definition.
We identify each
Definition.
We define addition, multiplication and order relations
on
_{1/1}
_{2/2}
_{12+}
_{21)/12}
_{1/1}
_{2/2}
_{1}
_{2/12}
_{1/1}
_{2/2}
_{12}
_{21}
Proposition.
The operations +,
_{1,1),(}
_{2,2)}
Proof.
Exercise.
Proposition.
The embedding
for all
Proof.
Exercise.
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 +,
Proof.
A rather long exercise.