- Oops, wrong mathemathics used -

e^(pi i) does not equal e^(pi)*e^(i)

@ruler501, check this link: http://en.wikipedia.org/wiki/Euler's_formula#Proofs It's a bit hard to understand without a backing in calculus though

Generally you can't really conceptualize raising a number to an imaginary power; the main method of going about this sort of thing is creating a power series (a polynomial) that mimicks the behavior of e^x. e^x's power series (which, when taken to an infinite amount of terms is perfectly accurate) then can be used with imaginary values for x and still make sense.
e^x's power series looks like this:

and so e^(ix)'s power series looks like this:

You'll have to take my word for it that that series is actually cosx+i*sinx. The rest of the proof goes from there.

Ah yes!  That has to be the most beautiful mathematical statement that exists!

Although I still sometimes refuse to believe in its beauty, this still eludes me more!

Although I still sometimes refuse to believe in its beauty, this still eludes me more!

Whoa, what?

And slightly off-topic: the Wikipedia URL is awesome too

How about 1+2+3+4+....=-1/12

Or 1+2^2+3^2+4^2+.....=0 O.o

Where'd you get that?

EDIT: Fixed the quote

Think about how you would do something like raising something to an imaginary power You can set up ratios and stuff and do more stuff and stuff. Also, e^(ix)=cos(x)+isin(x), so 3^i would be cos(ln(3))+isin(ln(3))

EDIT: so you can say with that the idea that e^(i2pi)-1=0 (that is i times 2 times pi)

So how would you calculate n^i with any real number for n
and then from that how would you calculate i^(ni)
None of these things work on my calculator so I'm wondering
@squidgetx I know that that factors into the two polynomial approximations for the trigonometric functions.

Does anyone one know anything that will help me learn more about/grasp imaginary numbers and things like raising things to imaginary powers?

Sorry I deleted my post, as it didn't make much sense. Yeah, i'm not quite sure how you would do it

Wiki
Also, 1+1+1+1...=-1/2

Math makes no sense. That's why I love it