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Math Help - Z_{1}=z_{2}

  1. #1
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    Z_{1}=z_{2}

    Suppose z_1=r_1(\cos(\theta_1)+i\sin(\theta_1)) and z_2=r_2(\cos(\theta_2)+i\sin(\theta_2)). If z_1=z_2, then how are r_1 \ \mbox{and} \ r_2 related? How are \theta_1 \ \mbox{and} \ \theta_2 related?

    If z_1=z_2, then a_1=a_2 \ \mbox{and} \ b_1=b_2. Therefore, r_1=r_2 and \theta_1=\theta_2 since a^2+b^2=r^2 \ \mbox{and} \ \theta=\tan\frac{b}{a}.

    Is this correct?
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  2. #2
    A Plied Mathematician
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    I don't think your answer takes into account fully the ambiguities inherent in polar coordinates. You have negative radii possibilities, as well as changing the angle by certain well-defined amounts. How does this change your picture?
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  3. #3
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    But by definition, z_1=z_2 iff. a_1=a_2 \ \mbox{and} \ b_1=b_2. Why can't I use that fact?
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  4. #4
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    Sure, but that's cartesian coordinates. As soon as you start talking about radii and angles (i.e., polar coordinates), the game is different. Example:

    -2\cos(\pi/4)=2\cos(5\pi/4).

    In fact, in polar coordinates, the point -2\,\text{cis}\,\pi/4 is the same point as 2\,\text{cis}\,5\pi/4.

    So from this, you can see that setting

    r_{1}\cos(\theta_{1})=r_{2}\cos(\theta_{2}) is simply not going to imply that r_{1}=r_{2} and \theta_{1}=\theta_{2}.

    Even if you throw in the other equation (the y equation), you're not going to resolve all the ambiguities.
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