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


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


[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!)