Euler's formula - Question

  • Thread starter Markjdb
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  • #1
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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
 

Answers and Replies

  • #3
mathwonk
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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
Tide
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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: [itex]dz = dx + i dy = i z d\theta[/itex]
 
  • #5
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Look at the Taylor's Series for eix, cos x, and sin(ix).
 
  • #6
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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
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Thanks to all who responded to my post. I...I get it now! :smile:
 

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