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.