Claim:
There exist irrational numbers p and q such that p^q is a rational number.

Proof:
Consider the irrational number x = sqrt(2) [easy to prove this is irrational]. Now consider the number y = x^x. If y is rational, we're done. If not, then y is irrational. In that case, consider the number y^x.

y^x = (x^x)^x
= x^(x times x)
= sqrt(2) ^ (sqrt(2) times sqrt(2) )
= sqrt(2) ^ (2)
= 2 [a rational number]

Therefore, there must exist irrational numbers p and q such that p^q is rational. This proof is valid even though an explicit pair of numbers with this property was never found.

Lemma:
sqrt(2) is irrational.

Proof:
Suppose sqrt(2) = p/q for some integers p and q that do not share prime factors. Then

2 = p^2 / q^2
2 times q^2 = p^2

So p^2 is even. This implies p is even as well since an odd number squared is odd. We can write p = 2r for some integer r.

sqrt(2) = 2r / q
2 = 4 r^2 / q^2
q^2 = 2 r^2

Therefore q^2 is even, so q is also even by the same reasoning. But we stipulated that p and q shared no prime factors, so this is a contradiction. Therefore, sqrt(2) cannot be written as the quotient of two integers, i.e. it is irrational.

Enk would u please explain that math xD? Calcytron

The first post proves that certain numbers exist without giving any way of actually producing an example of them. The second post is just a supporting proof of one of the facts used in the first post.

The possibility of non-constructive proofs is a big deal philosophically. Most people aren't comfortable with the idea that you can prove something exists without being able to actually give an example of it.

This is beautiful, here it is explained in a video i found after i looked it up in the interwebs.

skipped to where he explains it:
http://www.youtube.com/watch?feature=pl ... nHI#t=354s

Nice link. The guy does the same exact proof (of course, it's pretty standard).

Screw you guys!

Screw me too!

don't forget to screw me

I feel left out now.

I feel sad for you people who dislike math

Enks, that was one of the questions I had to prove in my final maths exam for Year 12

In college you tend to hate actual numbers, because then you have to carry them with you through all the derivatives/integrals

This is what I try to tell my students but some of them just refuse to work in terms of variables. Algebra, people!