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.

i^i Derivation

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?

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