Best Answer:
It turns out that there are actually more irrational numbers than rational numbers. The rationals are COUNTABLY infinite; the irrationals are UNCOUNTABLY infinite. This means that the set of irrational numbers has a cardinality called the "cardinality of the continuum," which is strictly greater than the cardinality of the set of natural numbers (i.e., the set {1,2,3,4,...}). The set of rational numbers has the same cardinality (number of elements) as the set of natural numbers, so there are more irrationals (numbers like pi and e) than rationals (numbers like 1/2, 3/4, etc).

There is a famous proof, called Cantor's diagonalization argument, which shows that the set of irrational numbers (non-terminating, non-repeating decimals) is uncountably infinite. You should be able to Google "Cantor's diagonalization argument" and get several hits that will run through the very brief, very intuitive proof.

To see a proof that the rationals are countably infinite, go here: http://everything2.com/index.pl?node_id=1259484.

An interesting lesson to take away from this discussion is that there are different "levels" of infinity! Just because two sets have infinitely many elements does NOT mean that they are the same size!

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