# Euler's formula - Question

## Main Question or Discussion Point

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

mathwonk
Homework Helper
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.

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

Bystander
Homework Helper
Gold Member
Look at the Taylor's Series for eix, cos x, and sin(ix).

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

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