Euler's Formula

Okay, so we all know Euler’s Identity that says for any Real number , and where (if you don't know about Euler's Identity, here is a quck explanation from MathWorld. The article also contains two seperate proofs). Let me just say that Euler's Identity is absolutely remarkable, probably because it seems so unlikely. But I was playing around with this identity one night, and I came across something strange mathematics…

First of all, we can see that by taking (this fact is absolutely amazing by the way).

Now let’s see what happens if . Then:

___ (since )

___ (since )

___ (multiplied by 2)

Now this is obviously not true since . But what is most concerning for me is that this implies that is a REAL number, which is just crazy.

It actually took me a while to figure out what the problem was in my steps, but I found it eventually.

Can you see where the problem is? Let me know what you think.