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Thread: Z_{1}=z_{2}

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

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

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

    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, $\displaystyle z_1=z_2$ iff. $\displaystyle a_1=a_2 \ \mbox{and} \ b_1=b_2$. Why can't I use that fact?
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  4. #4
    A Plied Mathematician
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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:

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

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

    So from this, you can see that setting

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

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