A rational number is a number which can be represented as a fraction involving 2 integers.
There are an infinite number of such rationals: take the arrangement:
1/2 1/3 2/3 1/4 2/4 3/4 1/5 2/5 3/5 4/5
.....
etc and you have clearly written down all the fractions that are possible. You can also place these fractions in correspondance with the integers (fraction No. 2 is 1/3; No 6 is 3/4 etc), so the number of fractions is the same as the number of integers (for every integer there is a fraction). And there are infinite fractions between each pair of integers and there are an infinite number of such intervals between integers. However in this case, infinity * infinity is still only infinity.