[saltcured]

Well, if I remember correctly, a product of an irrational and a rational is still irrational, right?

If so, here's a simple argument: If you have N rational numbers and I have K irrationals to start with, I can produce roughly N*K additional irrationals. Since we know at least two irrationals (pi and e) we can make at least twice as many irrationals as rationals.

[jjgreen → saltcured]

Unfortunately this argument fails -- there are twice as many whole numbers as there are odd numbers right? but they are in 1:1 correspondence

0 <-> 1

1 <-> 3

2 <-> 5

   :

so the odd and whole numbers have the same cardinality: there are the same number of each.