Did You Know: i^i Is Real?

August 19, 2013 MathyNick Blog, Journal, My Random Mathy Thoughts No comments

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?

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