How to obtain all the rational numbers without repetitions?

Solution 1:

This should ideally be a comment but I can't comment yet.

The Wilf-Calkin tree also enumerates positive rationals.

Wikipedia: Calkin-Wilf tree

Solution 2:

There's an easy method: just take the reduced fractions.

Another easy method is to note the diagonal argument has already given you an ordering on all of the positive fractions. So to take each positive rational exactly once, all you have to do is take only those fractions that don't have an equal value appearing earlier in the sequence.

There are more elegant schemes too: for example, Farey sequences can give you rationals from 0 to 1, and the the related Stern-Brocot tree for all positive rationals.