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]

If we let we get

[2]

Simplify this by recalling that and we get:

[3]

This is all really straight forward. Here is the cool part. What if we let then we get:

[4]

Since and then this simplifies to be

[5]

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

[6]

Combining the exponents on the left and noting that and we get

[7]

or

[8]

or

[9]

Since the left hand side no longer has then if is a real number then 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 ” it actually gives you a number, marked as transcendental and it’s value is:

[10]

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!)