The set of real numbers are infinite. They are so many that real numbers can't even be enumerated. However, unlike real numbers the set of integers is countably infinite as a mapping from 1,2,3,4,5 .... to 0,1,-1,2,-2,3,-3 can be made easily.

Surprisingly, rational Numbers are countably infinite too !!, meaning they can be enumerated. A diagonalization argument introduced by George Cantor easily shows this. Recollect that rational numbers are those numbers that can be represented in the form p/q. http://en.wikipedia.org/wiki/Rational_numbers

ex: The first ten rational numbers under this enumeration are 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 3/2, 2/3, 1/4.

Find the index of a positive rational number enumerated this way. ie the index of 1/3 is 6

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