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Math Help - Sqaure roots of complex number

  1. #1
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    Sqaure roots of complex number

    Hi,

    The question i have is "Find the sqaure roots of \sqrt{3} + i "

    The book gives the answer:
    \sqrt{2}e^{(\frac{1}{2} + k)\pi i}
    = \pm \sqrt{2}(cos(\frac{\pi}{12}) + isin(\frac{\pi}{12}))

    I can see how they get that but I thought you have to plug in k = 0, 1(n-1) and then use the the new arguements given from that to find them.

    I get arg = \frac{\pi}{6}, \frac{7\pi}{6}

    Which leads my answer to be:

    \pm(\frac{\sqrt{6}}{2} + i\frac{\sqrt{2}}{2})


    Could anyone clear up my grave error?

    Many thanks
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  2. #2
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    Maybe

    Im not sure of this will work but use Moivre´s.
    Wright the complex number in polar form and set that to be z^2
    Then solve for Z....
    Hope it was helpful..
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  3. #3
    MHF Contributor Calculus26's Avatar
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    Simply factor out a sqrt(2) from

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  4. #4
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    Turns out like this for me

    : sqrt(3) +i
    abs:z = 2
    1/sqrt3) = 30deg" alt="argz = atan1/sqrt3) = 30deg" />

    => 2(cos(pi/6)+isin(pi/6)
    Set that to be z^2

    z = sqrt(2)(cos(pi/12) +isin(pi/12)) + the other root that gave the +/- sign to the equation...
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  5. #5
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    Thanks for the tips,

    When I take out the factor of root 2 I have :

    <br />
\pm \sqrt{2}(cos(\frac{\pi}{6}) +isin(\frac{\pi}{6}))<br />

    Where apparently im supposed to have \theta = \frac{\pi}{12}
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  6. #6
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    Divide the argument

    Need to devide the argument (pi/6) by 2. Z^n has n roots, the arguments of the roots are evently spread around a circle.

    To find roots of z^2 devide the argument by 2 => (pi/6)2 and that gives you pi/12.... the other roots are found by:

     <br />
(2*pi*n)/2 = pi*n<br />
where your n can be 1;2
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  7. #7
    MHF Contributor Calculus26's Avatar
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    Perhaps the following attachment may help
    Attached Thumbnails Attached Thumbnails Sqaure roots of complex number-complex.jpg  
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  8. #8
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    Great, thanks for all the help.

    As a side note could anyone recommend a book/website with exercises to do with complex number? I have 2 already but they lack exercises.

    Thanks
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