YT Math: Seeing i to i

There are a lot of videos on this for some reason, even though an introduction to complex numbers is all you need to address it.

Begin by rewriting in polar form: . You can arrive at this by plotting on the complex plane or by noting and using Euler’s formula.

Now raise this quantity to the power:

This is now clearly a real number like the titles say, as it’s with a real exponent.

If we take , we’ll have , which we can expect as a sanity check to have a value on the interval . A calculator confirms this is the number in that thumbnail.