Thread: E to the i Pi

Plain and simple...

Why does e to the i pi equal negative one

That is e to the power of i times pi equals negative one.

My friend and I have been wondering.

Did you start from the definition of for ? Unless you know what the definition is, you are not likely to get very far at all. :?

One way is to define as the complex power series , which is analytic on the whole complex plane. Well? That probably won’t help in the discussion between you and your friend, so let’s try another one …

An alternative and intuitively simpler definition is this: writing , where x and y are real, define .

For , just substitute into the formula.

Originally Posted by JaneBennet

Did you start from the definition of for ? Unless you know what the definition is, you are not likely to get very far at all. :?

One way is to define as the complex power series , which is analytic on the whole complex plane. Well? That probably won’t help in the discussion between you and your friend, so let’s try another one …

An alternative and intuitively simpler definition is this: writing , where x and y are real, define .

For , just substitute into the formula.

The best way is to start with your power series and then substitute ix for z. Then notice that the power series naturally separates into a real and an imaginary part which are recognizable as the power series for cos(x) and i sin (x). You don't need the machinery of complex variables and analytic functions to do that.

There is another post about this, where Serpicojr (and Magimaster) helped me go through the derivations. You might find it helpful:

http://www.thescienceforum.com/viewt...umber+division

I didn't understand your explanations. I'll show what you guys said to my Calculus teacher, so your time spent answering wasn't pointless. Hopefully she'll be able to explain it to me.

Thanks

Originally Posted by SteveC

I didn't understand your explanations.

The whole point is that you need to know what it means to raise a number to a complex (or pure imaginary) power. Once you get that, e^(i*pi)=-1 follows naturally.

Cheers, Leszek.

Originally Posted by Leszek Luchowski

Originally Posted by SteveC

I didn't understand your explanations.

The whole point is that you need to know what it means to raise a number to a complex (or pure imaginary) power. Once you get that, e^(i*pi)=-1 follows naturally.

Cheers, Leszek.

Actually it helps to have defeined e^z via the power series before you try to define arbitrary complex numbers (or arbitrary real numbers for that matter) to a complex power. The reason is that to do the general case you need the notion of a logarithm, which takes some work in the complex case, but the exponential function is easily and globally defined by the power series.