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Math Help - Complex number

  1. #1
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    Unhappy Complex number

    Find the square roots of the following complex numbers:
    a)- i
    b) x+ i(x^4+x^2+1)^1/2

    Thanks
    Last edited by matsci0000; August 13th 2009 at 09:30 PM. Reason: wrong information
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  2. #2
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    Quote Originally Posted by matsci0000 View Post
    Find the square roots of the following complex numbers:
    a) i
    well i = \sqrt{-1} so therefore

    \sqrt{i} = \sqrt{\sqrt{-1}} = \sqrt[4]{-1}
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  3. #3
    MHF Contributor red_dog's Avatar
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    a) Let z=-i. We have to find the complex numbers w=a+bi such as z=w^2

    -i=a^2-b^2+2abi and |z|=|w|^2. Then

    \left\{\begin{array}{ll}a^2-b^2=0\\2ab=-1\\a^2+b^2=1\end{array}\right.

    From the first and the tird equation we get (using the sign for a and b from the second equation)

    a=\pm\frac{\sqrt{2}}{2}, \ b=\mp\frac{\sqrt{2}}{2}

    Then, w_1=\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}i, \ w_2=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i

    b) Using the same method we have

    \left\{\begin{array}{ll}a^2-b^2=x\\2ab=\sqrt{x^4+x^2+1}\\a^2+b^2=x^2+1\end{arr  ay}\right.

    We get a=\pm\sqrt{\frac{x^2+x+1}{2}}, \ b=\pm\sqrt{\frac{x^2-x+1}{2}}

    Then, w_1=\sqrt{\frac{x^2+x+1}{2}}+i\sqrt{\frac{x^2-x+1}{2}}, \ w_2=-\sqrt{\frac{x^2+x+1}{2}}-i\sqrt{\frac{x^2-x+1}{2}}
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