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Absolute value of complex numbers

  1. Mar 2, 2009 #1
    When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
     
  2. jcsd
  3. Mar 2, 2009 #2

    CompuChip

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    -1 is not the complex conjugate of 1.
     
  4. Mar 2, 2009 #3
    What I'm asking is, are there any conditions on a and b such that the above statement is true?
     
  5. Mar 2, 2009 #4

    CompuChip

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    Yes, but they are all trivial rewritings of |a| = |b|.
    If it were as simple as "a = b or a = b*", for example, we wouldn't need the bar notation, would we :smile:

    Some of those rewritings / interpretations are:
    * a / b lies on the unit circle
    * a and b lie on a circle in the complex plane
    * the distance of a and b to the origin is the same
    * Re(a)^2 - Re(b)^2 = Im(b)^2 - Im(a)^2
    * There exists [itex]r > 0, \theta_1, \theta_2 \in [0, 2\pi)[/itex] such that [itex]a = r e^{i\theta 1}[/itex], [itex]b = r e^{i\theta_2}[/itex] (and r = |a| = |b|)
    * There exists [itex]\theta \in [0, 2\pi)[/itex] such that [itex]a = e^{i \theta} b[/itex].
     
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