A Peculiar, Non-Constructive Proof

A friend of mine loves to trot out this oddity, and I always manage to forget it afterwards. Now that I have it in mind, I will take a minute to write it here for posterity.

Theorem: There exist a,bR, irrational, such that abQ.

In other words, the theorem asserts that an irrational raised to an irrational may be rational. This isn’t especially interesting and, while surprising, I don’t claim to have any sort of intuition violated here. However, the proof is slippery and clever.

Proof: I will take for granted that 2 is irrational, which is a standard first theorem in an analysis course.

Consider the quantity x=22.

There are two cases to be considered here:

  1. x is rational, so the theorem is true (with a=b=2), or

  2. x is irrational, and we can compute x2=22=2

which also would show that the theorem holds (with a=x, b=2).

Therefore the theorem is proven either way, and we remain ignorant of whether x=22 is rational or not.