Understanding Euler's Formula: The Geometric Connection to e

[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

 

[dav2008]

Whenever you see e in an equation it generally comes from the fact that $$\int\frac{1}{x}dx=ln|x|+c$$

ln, of course, is log base e.

http://en.wikipedia.org/wiki/Euler's_formula--Scroll down to the Proofs section. The "Using calculus" is a bit easier to follow I think.

or http://mathworld.wolfram.com/EulerFormula.html

 

[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.

 

[Tide]

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]

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

 

[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

 

[Markjdb]

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