I’ve seen the following before but it is so cool I have to write it down:

The derivation of the formula below is really cool but this post is not about the derivation but an application of the formula:

[1]  $e^{ix} =$ $\cos x + i\sin x$

If we let $x = \pi$ we get

[2] $e^{i\pi} =$ $\cos \pi + i\sin \pi$

Simplify this by recalling that $\cos \pi = -1$ and $\sin \pi = 0$ we get:

[3] $e^{i\pi} = -1$

This is all really straight forward.  Here is the cool part.  What if we let $x = \pi/2$ then we get:

[4]  $e^{i\pi/2} =$ $\cos \pi/2 + i\sin \pi/2$

Since $\cos \pi/2 = 0$ and $\sin \pi/2 = i$ then this simplifies to be

[5]  $e^{i\pi/2} = i$

Finally if we raise both sides of the equation to the $i$ power we get:

[6] ${(e^{i\pi/2}})^i = i^i$

Combining the exponents on the left and noting that $(i)(i) =$ $i^2$ and $i^2 = -1$ we get

[7] $e^{-\pi/2} = i^i$

or

[8] $\dfrac{1}{e^{\pi/2}} = i^i$

or

[9] $\dfrac{1}{\sqrt{e^\pi}} = i^i$

Since the left hand side no longer has $i = \sqrt{-1}$ then if $e^{\pi}$ is a real number then $i^i$ is also a real number!   And, it is a real number see, for example, the articles “http://en.wikipedia.org/wiki/Gelfond%27s_constant” and “http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem

If you go to WolframAlpha.COM and enter the sting “evaluate $i^i$” it actually gives you a  number, marked as transcendental and it’s value is:

[10] $i^i =$ $0.207879576350761908546955 ...$

Finally I wrote this post after reading today “Saturday Morning Breakfast Cereal” cartoon.  WARNING: This day’s cartoon is most definitely N.S.F. (Not suitable for work!)