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Math Help - How to Computing Square root of a Complex number...

  1. #1
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    How to Computing Square root of a Complex number...

    Hi,

    How do I compute square root of a complexed number without a calculator?
    In other words how should I think and break down the following with just pen and paper?

    \displaystyle\sqrt{8i}

    I would appriciate any guidance.
    Thank you
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  2. #2
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    Convert it to exponential form.

    \displaystyle \sqrt{8i} = \sqrt{8e^{\frac{\pi i}{2}}}

    \displaystyle = 2\sqrt{2}(e^{\frac{\pi i}{2}})^{\frac{1}{2}}

    \displaystyle = 2\sqrt{2}e^{\frac{\pi i}{4}}

    \displaystyle = 2\sqrt{2}\left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right)

    \displaystyle = 2\sqrt{2}\left(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}i}{2}\right)

    \displaystyle = 2 + 2i.


    Note that this only gives one of the square roots, but you can find the other using the fact that there are always 2 square roots and they are evenly spaced around a circle.
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  3. #3
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    Hello, 4Math!

    Here is a primitive method . . .


    Find: . \sqrt{8i}

    Let: a + bi \:=\:\sqrt{8i} . where \,a and \,b are real.

    Square both sides: . (a + bi)^2\;=\;(\sqrt{8i})^2

    And we have: . (a^2-b^2) + (2ab)i \;=\;8i


    Equate real and imaginary components: . \begin{array}{cccc}a^2-b^2 &=& 0 & [1] \\ 2ab &=& 8 & [2]\end{array}

    From [2] we have: . b \:=\:\frac{4}{a}\;\;[3]

    Substitute into [1]: . a^2 - \left(\tfrac{4}{a}\right)^2 \:=\:0 \quad\Rightarrow\quad a^4 \:=\:16

    . . Hence: . a \:=\:\pm2

    Substitute into [3]: . b \:=\:\dfrac{4}{\pm2} \:=\:\pm2


    Hence: . a + bi \;=\;\pm2 \pm 2i


    Therefore: . \sqrt{8i} \;=\;\pm2(1 + i)
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