# E to the pi i

1. Feb 4, 2005

### sebasalekhine7

ok, here it goes, why is e^(pi.i)=-1 ?

Last edited: Feb 4, 2005
2. Feb 4, 2005

### Integral

Staff Emeritus
It comes form Eulers relationship

$$e^{ix} = \cos(x) + i \sin(x)$$

Edit: LOL, I fixed it already!
Now I am moving this to Math.

Last edited: Feb 4, 2005
3. Feb 4, 2005

### freemind

Any complex number $$a + bi$$ can be written in the form $$r(\cos{\theta} + i\sin{\theta})$$. De Moivre's theorem states that this in turn can be written in the form $$re^{i\theta}$$. That is, $$re^{i\theta} = r(\cos{\theta} + i\sin{\theta})$$. Plug in $$\theta = \pi$$ and behold the magic!

4. Feb 4, 2005

### sebasalekhine7

Was it Euler? well, in that case u would have to use Taylor's expansion series and yes, it would work. Why is this important at all then?

5. Feb 4, 2005

### freemind

It is commonly regarded to be one of the most (ahem) beautiful and elegant mathematical relationships in our universe (yes, our ). C'mon, wouldn't you agree that it is beautiful, succintly relating the 5 most important numbers in mathematics?!?! ($$e^{\pi i} + 1 = 0$$)

6. Feb 5, 2005

### Integral

Staff Emeritus
In addition it uses each of the fundamental mathematical operations, addition, multiplication, exponentiation, and equality. It is consider mathematical poetry.