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Is Q countable?

Solution:

Each rational number r in Q is a numerator in Z divided by a denominator in N. It is characterized then by the pair a,b such that r = a/b. If we look at all pairs (a, b) for a in Z and b in N, we will actually get every r over and over again, as a/b, (2a)/(2b), (3a)/(3b)...

Therefore if we can list all the pairs (a, b) for a in Z and b in N we can get a list of the elements of Q by throwing away duplicates on this list.

Imagine then that we have a vertical column for each b in N and that column consists of a list of the elements of  Z (as in Exercise 1.1) We can list the resulting pairs by going up every diagonal as in the illustration below. This will give a list of every pair (a, b) for a in Z and b in N.

You run through the elements of N much faster than you do the elements of  Q but again nobody cares about this fact.