What is Euler's formula e^(πi) = -1 and how does it work?

[DennisG]

ok, so I saw this thing one time that looked like this:

e^(pi*i) = - 1

can anyone tell me what this is, what it's used for, and especially how it works? A friend showed me the equation one time, but neither of us knew a thing about it.

Thanks for any help you can offer,
Dennis

 

[newPatrick]

Its Euler formula coming from the expansions of sinx, cosx, e^x
e(PI^i)=cis(PI)=-1

 

[Integral]

So as not to be to repetitive, see this thread

 

[Zurtex]

I'm going to assume from the way you asked the question that you don't know what e is in the 1st place? (Please don't take this as an offence if you do know it)

There are lots of definitions of e, it is probably the most important number in calculus. Here are a few definitions:

$$e = \sum_{n=0}^{\infty} \frac{1}{n!} = \lim_{x \rightarrow \infty} \left( 1 + \frac{1}{x} \right)^x$$

Also:

$$\frac{d}{dx} \left( e^x \right) = e^x$$

And:

$$\int_1^e \frac{dx}{x} = 1$$

Edit: Amended thanks to below post.

As well as:

$$e = 2.718281828459045235326 \ldots$$

As for the result:

$$e^{\pi i} = -1$$

This comes from some maths orientated around complex numbers which yields the formulae:

$$e^{\theta i} = \cos \theta + i \sin \theta$$

(More about the above result in a link given I believe)

Hope that helps.

 

[Integral]

I posted the link to the other thread in hopes that further posts on this topic would be in the existing thread.

EDIT: Oh yeah, I forgot to mention

$$\int^e _1 \frac {dx} x = 1$$ not e.