## Did You Know: i^i Is Real?

In my last post, A REALLY Brief Intro to Complex Numbers, I talked about a few necessary definitions to know what Complex numbers $\mathbb{C}$ are.  Complex numbers are really intriguing and necessary for our electronic devices to work.  You might be reading this on a computer, smartphone, tablet, etc., but none of the advances in technology would be possible without use of complex numbers.  Creating wonders from wonders.

I’m not sure what sparked the thought but I was curious what kind of number $i^i$ is.  Since $i= \sqrt{-1}$, then it is the same to ask “What is the value of $\sqrt{-1}^{\sqrt{-1}}$?  It definitely raises some interesting questions.  Is it real number?  Is it another complex number?  What does it mean to raise a number to the power of $\sqrt{-1}$?

To find out the value of $i^i$, I needed a starting point.  Without getting into the derivation, we can use a really handy formula — Euler’s Formula!  Euler’s Formula says $e^{ix} = \cos (x) +\imath \cdot \sin (x)$.  To try out this formula, we will substitute $x$ for $\pi$.  This means,

We can now use this equation to find out what the value of $i^i$.

Really?! $i^i$ is a real number?! Yes!  It is a transcendental number but a real number nonetheless!  An imaginary raised the imaginary power!

We can even look on a graph to convince us even more.  The function $e^x$ has a domain of all real numbers and a range of real numbers from $(0,\infty)$.  Take a look below: (click the picture if it isn’t displayed large enough)

Pretty neat, huh?