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Math Help - Complex numbers polar form - 2 other answers?

  1. #1
    Junior Member Cthul's Avatar
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    Complex numbers polar form - 2 other answers?

    The question asks me to find the solution to z^{3}=3^{\frac{1}{2}}+i in polar form.

    I got
    1^{\frac{1}{3}}cis(\frac{\pi}{18})
    Which is correct, but there 2 other answers but I can't seem to get it which are
    1^{\frac{1}{3}}cis(\frac{13\pi}{18})
    And
    1^{\frac{1}{3}}cis(\frac{-11\pi}{18})

    There's also another question I don't understand:
    "How many degrees apart are two consecutive roots of z^{8}=1"
    Last edited by mr fantastic; January 15th 2011 at 01:54 AM. Reason: Another question
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  2. #2
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    Use

    \displaystyle w_k=r^{1/n}\left(\cos{\left(\frac{\theta+2\pi k}{n}\right)}+i\sin{\left(\frac{\theta+2\pi k}{n}\right)}\right)
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  3. #3
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    There are three cube roots, they are all of the same magnitude and are evenly spaced around a circle.

    So they differ by an angle of \displaystyle \frac{2\pi}{3}.
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  4. #4
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    The point is that adding 2\pi to the argument of a complex number just goes around the "circle" exactly once so that you are back to exactly the same place and same complex number:
    re^{i\theta}= re^{i(\theta+ 2\pi)}= re^{i(\theta+ 4\pi)}= re^{i(\theta+ 6\pi)}= re^{i(\theta+ 8\pi)}

    But when you divide the argument by 3 to find cube root, that changes:
    r^{1/3}e^{i\theta/3}\ne r^{1/3}e^{i(\theta/3+ (2/3)\pi)}\ne r^{1/3}e^{i(\theta/3+ (4/3)\pi)

    Notice, however, that continuing to 6\pi, 8\pi, etc. does NOT give us anything new:
    r^{1/3}e^{i(\theta/3+ (6/3)\pi)}= r^{1/3}e^{i\theta/3+ 2\pi i}= r^{1/3}e^{i\theta}
    r^{1/3}e^{i(\theta/3+ (8/3)\pi)}= r^{1/3}e^{i(\theta+ (2/3)\pi)+ 2\pi i}= r^{1/3}e^{i(\theta+ (2/3)\pi)}
    r^{1/3}e^{i(\theta/3+ (10/3)\pi}= r^{1/3}e^{i(\theta+ (4/3)\pi)+ 2\pi i}= r^{1/3}e^{i(\theta+ (2/3)\pi)}
    so we get exactly 3 distinct third roots.

    Show should be able to see that adding any number of multiples of 2\pi to the argument will always give exactly n distinct nth roots of a complex number.
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    Quote Originally Posted by Cthul View Post

    There's also another question I don't understand:
    "How many degrees apart are two consecutive roots of z^{8}=1"
    \displaystyle\ z^8=1\Rightarrow\ z=\sqrt[8]{1+0i}

    There are 8 "8th roots" of 1, all evenly spaced around the circle.

    Hence, the roots are all seperated by \displaystyle\frac{2{\pi}}{8}\;radians=\frac{360^o  }{8}
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