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Math Help - find a square root of the complex number

  1. #1
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    find a square root of the complex number

    Question. Find a square root of the complex number  i .

    Answer.
    Let i^2=w. By inspection (of i), the modulus of w is 1; we need to find \theta.

     (\cos(\frac{\theta}{n}+\frac{2kn}{\pi})+i\sin(\fra  c{\theta}{n}+\frac{2kn}{\pi}) where n=2 and k is 0 and 1 (from 0 to n-1).

    Therefore there are two numbers w that, when squared, give i:
    \theta_1=\frac{\pi}{4} and \theta_2=\frac{5\pi}{4}
    w_1=\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}
    w_2=-\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}}

    The textbook answer, however, only gives one value of w ( w_1). Am I wrong in thinking that there are two values of w? Or, the question has only asked for one of them ('a square root'), rather than 'find all square roots'?

    PS I understand that there are always two numbers (one positive and one negative) that, when squared, give the same number. I only wanted to check whether the same principle applies to the questions about complex numbers.
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  2. #2
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    You are correct to say there is two solutions here \displaystyle\pm\left(\frac{1}{\sqrt{2}}+\frac{i}{  \sqrt{2}}\right)

    Your book has given 'a' root, not all.
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  3. #3
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    Thanks for clarification, much appreciated!
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