## 27. i^i is real

A funny thing happens when you take the imaginary unit $$i$$, and raise it to the power of $$i$$; it becomes real. Again, $$i^i \in \mathbb{R}$$.

To start off, we take Euler’s Formula:

$e^{ix} = \cos x + i \sin x$

We make the RHS of the above $$i$$ by setting $$x=\pi/2$$:

$e^{i \pi/2} = \cos (\pi/2) + i \sin (\pi/2) = i$

Raising the above to $$i$$, we get:

$e^{- \pi/2} = i^i$

Since the LHS is real, the RHS must be too. So, $$i^i$$ is real.

Funnily enough, that isn’t the only real value for $$i^i$$. We plugged $$x=\pi/2$$ into Euler’s Formula, but $$x=2k\pi + \pi/2$$ would have worked just as well. So:

$e^{-(2k\pi + \pi/2)} = i^i, k \in \mathbb{Z}$

To summarise, $$i$$ is an imaginary number, but $$i^i$$ is real, and if that you’re not satisfied with just one real value for it, there are infinitely more to choose from.