Real Numbers to Irrational Powers

Question

In a related question we discussed raising numbers to powers.

I am interested if anybody knows any results for raising numbers to irrational powers.

For instance, we can easily show that there exists an irrational number raised to an irrational power such that the result is a rational number. Observe ${\sqrt 2 ^ {\sqrt 2}}$. Since we do not know if ${\sqrt 2 ^ {\sqrt 2}}$ is rational or not, there are two cases.

1. ${\sqrt 2 ^ {\sqrt 2}}$ is rational, and we are finished.

2. ${\sqrt 2 ^ {\sqrt 2}}$ is irrational, but if we raise it by ${\sqrt 2}$ again, we can see that $$\left ( \sqrt 2 ^ \sqrt 2 \right ) ^ \sqrt 2 = \sqrt 2 ^ {\sqrt 2 \cdot \sqrt 2} = \sqrt 2 ^ 2 = 2.$$

Either way, we have shown that there exists an irrational number raised to an irrational power such that the result is rational.

Can more be said about raising irrational numbers to irrational powers?

Answer 1

Some examples:

• Eulers Identity $e^{i \pi} + 1 = 0$

• Gelfond-Schneider Constant: $2^\sqrt{2} = 2.66514414269022518865\cdots$ is trancendental by Gelfond-Schneider.

• $i^i = 0.2078795763507619085469556198\cdots$ is also trancendental

• Ramanujan constant: $e^{\pi \sqrt{163}} = 62537412640768743.999999999999250072597\cdots$.

• $e^\gamma$ where $\gamma$ is the Euler–Mascheroni constant. (okay nobody has proved it's irrational yet, but surely is)

Answer 2

Forgive my lack of knowledge of how to correctly implement equations in TeX! I always particularly liked that...

$e^{\pi} - \pi = 19.9990999791894757672664\cdots$

I think it would also be possible to prove that result is transcendental as well. The two cases being $e^{\pi}$ is transcendental or $e^{\pi}$ isn't. The difference of a non-transcendental number and a transcendental number is transcendental, and the only time the difference of two transcendental numbers wouldn't be transcendental is when you could extract the second from the first - That is to say, you could find some non-transcendental $x$ that would satisfy $e^{\pi} = x + \pi$

Unfortunately, my formal proof skills are not what they once were. I'm sure this is trivial, and really the result isn't even that close to 20. But I always liked it :)