Let
be the set of integers. We will consider the set
. This set will be our "proto-rationals". Define the equivalence relation on
given by
. That this is an equivalence relation is obvious. To each equivalence class there is a set of equivalence classes. It is these equivalence classes which will constitute the rational numbers.
Let
and call
the set of rational numbers. We will show that this
has all the familiar properties of rationals. However, since we defined it using equivalence classes, we will have to show invariance with respect to the representative element at each step. Denote the equivalence class of
by
. Before we begin, define the following operations on
:
- i) Addition: If
, define
.
- ii) Multiplication: If
, define
.
is obviously closed under these operations.
- 1. Let
and
. Then
.
Proof: Since
and
, then
, so
, showing the theorem.
- 2. Let
and
, then
.
Proof: Since
and
, then
, showing the theorem.
Also, since the above arguments used arbitrary representatives for the equivalence classes, addition and multiplication on equivalence classes is well-defined. Thus we can transfer, or project, the operations onto
. In the following, write
. Then, if
, we define
- i) their sum
, and
- ii) their product as
.
Under these operations it can readily be checked that
is a commutative ring. It has additive identity
and multiplicative identity
, written
and
, respectively. In general, we write any element
simply as
.
We will now show that it is a field.
Let
such that
. Then there exists an element
such that
, the multiplicative identity, showing that multiplicative inverses exist for nonzero elements, and therefore that it is a field, as promised. With this we have shown that
has all the properties of the rationals and therefore is the set of all rational numbers.