Reminds me of a cool proof I saw recently that there are two numbers a and b such that a and b are both irrational, but a^b is rational: Take sqrt(2)^sqrt(2), which is either rational or not. If it's rational, we're done. If not, consider sqrt(2) ^ (sqrt(2) ^ sqrt(2)). Since (a^b)^c = a^bc, we get sqrt(2) ^ (sqrt(2))^2 = sqrt(2)^2 = 2, which is rational! It feels like a bit of a sleight of hand, since we don't actual…
I wonder what the easiest to prove example of a, b irrational with a^b rational is? The easiest I can think of offhand would be e^log(2). To prove that we need to prove that e is irrational and the log(2) is irrational. To prove log(2) is irrational one approach is to prove that e^r is irrational for rational r != 0, which would imply that if log(2) is rational then e^log(2) would be irrational. To prove that e^r is…
Re: A joke in approximating numbers raised to irrational powers
#41Also see https://math.andrej.com/2009/12/28/constructive-gem-irration... for a similar proof using 2^(log_2 3)