We wish to define a way of ordering the fractions. We can do this by consider the fractions as the ordered pairs

, such that

, and with the equivalence

for any

.

In

Exercise 16.8 we found a 1-1 mapping

. Thus there exists a well-defined inverse function

.

We first construct the sequence

via:

Note that we needed to include the negative values because the fractions take both positive and negative values. Now we edit the sequence to remove duplicates and values which do not represent fractions. Namely we go through the sequence

term by term starting from

. If

or

for some

then we delete the term. Otherwise we leave it in, identifying

with the fraction

.

After relabelling indices, this provides a new sequence

which defines an ordering for the fractions.