April 13, 2012
Yesterday we saw that we could raise a rational number to a rational power and get an irrational number.
This fact shouldn’t have shocked you.
Today we’ll see that we can raise an irrational number to an irrational power and get a rational number.
The result is surprising but the proof is, in my opinion, downright delightful.
I first read this proof in an article by Paul Halmos.
Yesterday we showed that the square root of two is irrational. What about the square root of two raised to the square root of two power?
If that was a rational number then we’d be done.
If not, take this number (the square root of two raised to the square root of two power) and raise it to the square root of two power. This is a rational number – it’s the number two. (a^b)^c = a ^(bc). Therefore we’re done.
I love this proof.
What I love the most is that we don’t know whether the square root of two raised to the square root of two is rational or not but in either case the theorem is proved.