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Math Help - quick complex number questions

  1. #1
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    quick complex number questions

    Hi
    Need help on the following:
    1)if z is any cube root of unity then 1+z+z^2 must be.

    A - 0
    B - 1
    C - 3
    D - 0 or 3
    E - 1 or 3

    2) For any complex numbers z with conjugate \overline{z}, which of the following is not necessarily true?

    A - |z|=|\overline{z}|

    B - \frac{z}{\overline{z}} is real

    C - \frac{z}{\overline{z}} is real

    D - z+\overline{z} is real

    E - |z/\overline{z}| is real

    P.S
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  2. #2
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    You can have a go yourself

    say z = a+bi

    Quote Originally Posted by Paymemoney View Post

    A - |z|=|\overline{z}|
    Find

    |z| = |a+bi| = \sqrt{a^2+(b)^2} and |\overline{z}|=  |a-bi| = \sqrt{a^2+(-b)^2}

    are they the same?


    Quote Originally Posted by Paymemoney View Post



    B - \frac{z}{\overline{z}} is real

    Is \frac{a+bi}{a-bi} real?




    Quote Originally Posted by Paymemoney View Post


    C - \frac{z}{\overline{z}} is real

    not sure how this one is different to the previous option?!


    Quote Originally Posted by Paymemoney View Post


    D - z+\overline{z} is real
    evaluate a+bi + (a-bi) is it real?




    Quote Originally Posted by Paymemoney View Post

    E - |z/\overline{z}| is real
    Is \left|\frac{a+bi}{a-bi}\right| real?
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  3. #3
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    Quote Originally Posted by Paymemoney View Post
    Hi
    Need help on the following:
    1)if z is any cube root of unity then 1+z+z^2 must be.

    A - 0
    B - 1
    C - 3
    D - 0 or 3
    E - 1 or 3

    [snip]
    Note that z^3 = 1 \Rightarrow z^3 - 1 = 0 \Rightarrow (z - 1)(z^2 + z + 1) = 0 ....
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