# The Universe of Discourse

Thu, 26 Jan 2006

More irrational numbers
Gaal Yahas has written in with a delightfully simple proof that a particular number is irrational. Let x = log2 3; that is, such that 2x = 3. If x is rational, then we have 2a/b = 3 and 2a = 3b, where a and b are integers. But the left side is even and the right side is odd, so there are no such integers, and x must be irrational.

As long as I am on the subject, undergraduates are sometimes asked whether there are irrational numbers a and b such that ab is rational. It's easy to prove that there are. First, consider a = b = √2. If √2√2 is rational, then we are done. Otherwise, take a = √2√2 and b = √2. Both are irrational, but ab = 2.

This is also a standard example of a non-constructive proof: it demonstrates conclusively that the numbers in question exist, but it does not tell you which of the two constructed pairs is actually the one that is wanted. Pinning down the real answer is tricky. The Gelfond-Schneider theorem establishes that it is in fact the second pair, as one would expect.