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Phase of a complex number

  1. Dec 27, 2008 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    Is the phase of a complex number always taken with respect to the real, positive axis? I mean, is it always the direction as shown here: http://theories.toequest.com/content_images/4/argand.gif

    Thanks in advance.
     
  2. jcsd
  3. Dec 27, 2008 #2

    HallsofIvy

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    Yes. Any complex number, a+ bi, can be written, in "polar coordinates", as [itex]r (cos(\theta)+ i sin(\theta))= re^{i\theta}[/itex] where r is the distance from (0, 0) (= 0+ i0) to (a,b) (= a+ bi) and [itex]\theta[/itex] is the angle the line from (0,0) to (a, b) makes with the positive x- axis.

    Note that because cosine, sine and [itex]e^{i\theta}[/itex] are all periodic with period [itex]2\pi[/itex] we can add any multiple of [itex]2\pi[/itex] to theta: [itex]a+ bi= r (cos(\theta+ 2n\pi)+ i sin(\theta+ 2n\pi)= re^{i(\theta+ 2n\pi)}[/itex] for n any integer. However, that angle is still measured from the positive x-axis.
     
    Last edited: Dec 27, 2008
  4. Dec 27, 2008 #3
    Thanks. You have helped me alot lately.

    Merry Christmas.
     
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