Consider x=√2 (the square root of 2, a well known irrational). Then
(xx)x = xx*x = x2 = 2
is definitely a rational number
, while x is an irrational number
. Now, if y = xx
, we're finished -- take a=b=x, and ab
=y is rational. Otherwise, y is irrational
, and we can take a=y, b=x.
The worrying thing about this proof is that it proves a specific fact without giving an example of it (it is possible, with lots more work, to come up with known irrational a,b for which ab is rational). This sort of proof lead the intuitionists and constructivists to reject the principle of the excluded middle in mathematical logic. But most mathematicians are much too pragmatic to reject these proofs, prefering merely to point them out as oddity.