# Euler's formula - Question

1. May 22, 2006

### Markjdb

I was reading several articles on Euler's identity, which is:
e^(i*x) = cos x + i*sin x

I understand what this formula describes: the unit circle on the complex plane, but I never really understood why e is there from a geometric point of view. So my question would be, what relationship does e have with the unit circle and why is it, as a result, a part of the formula?

Any links would also be greatly appreciated.

- Mark

2. May 22, 2006

### dav2008

Last edited: May 22, 2006
3. May 22, 2006

### mathwonk

it has to do with the fact that exponentiation turns addition into multiplication, and the unit circle consists of the multiplicative group of complex numbers of length one.

so e^it transforms the adition on the real numbers t into multiplication of complex numbers of length one.

4. May 22, 2006

### Tide

It simply means that the infinitesimal change in z along a circular path is directly proportional to z: $dz = dx + i dy = i z d\theta$

5. May 23, 2006

### Bystander

Look at the Taylor's Series for eix, cos x, and sin(ix).

6. May 23, 2006

### Kanfoosh

there is no speceifc connection with it being he unit circle, because |a^ix|=1 for any a
the reason why it's e and not any other number is that (e^x)'=e^x
it just makes everything much more comfortable

7. May 23, 2006

### Markjdb

Thanks to all who responded to my post. I...I get it now!