e^(pi*i)=-1

e^(i Theta) = cos Theta + i sin Theta.

That is true is apparent from the Taylor expansions of e^x, cos x and sin x, but I don't know who originally proved it or even saw the link. From there, you can see that if Theta=pi then the sin term is zero and you are left with -1.

What is really cool is using this to prove some of the basic trig formulas. For example, finding the cos (a+b).

e^i(a+b) =
(e^ia)(e^ib) =
(cos a + i sin a)(cos b + i sin b) =
(cos a)(cos b) + i(sin a)(cos b) + i(cos a)(sin b) - (sin a)(sin b) =
[(cos a)(cos b) - (sin a)(sin b)] + i[(sin a)(cos b) + (cos a)(sin b)]

e^i(a+b) = cos(a+b) + i sin(a+b)

Setting the real and imaginary parts equal to each other,

cos (a+b) = [(cos a)(cos b) - (sin a)(sin b)]
sin(a+b) = [(sin a)(cos b) + (cos a)(sin b)]

http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2046%20e%20pi%20and%20i.pdf

That was a nice read. Thanks for the link.

You're welcome. There's a regular column called "How Euler Did It by Ed Sandifer" at the maa.org site. It's updated roughly every month, and I think you can get to archived articles. The other regular columns are often interesting too.

It's amazing that euler did so much that there can be a regular column devoted to his work. I'd be happy to achieve half that.

It was voted one of science's top 3 formulea - over over 100, like e = mc^2 (simplified form) because it relates all mathematics constants e, i, 0, 1 and pi into one equation.

only 5. It missed the golden ratio