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Euler's formula - Question

  1. May 22, 2006 #1
    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. jcsd
  3. May 22, 2006 #2


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


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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.
  5. May 22, 2006 #4


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    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]
  6. May 23, 2006 #5


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    Look at the Taylor's Series for eix, cos x, and sin(ix).
  7. May 23, 2006 #6
    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
  8. May 23, 2006 #7
    Thanks to all who responded to my post. I...I get it now! :smile:
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