In my last post, A REALLY Brief Intro to Complex Numbers, I talked about a few necessary definitions to know what Complex numbers 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 is. Since
, then it is the same to ask “What is the value of
? 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
?
To find out the value of , I needed a starting point. Without getting into the derivation, we can use a really handy formula — Euler’s Formula! Euler’s Formula says
. To try out this formula, we will substitute
for
. This means,
We can now use this equation to find out what the value of .
Really?! 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 has a domain of all real numbers and a range of real numbers from
. Take a look below: (click the picture if it isn’t displayed large enough)
Pretty neat, huh?
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