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Thread: 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 $\displaystyle \overline{z}$, which of the following is not necessarily true?

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

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

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

    D - $\displaystyle z+\overline{z}$ is real

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

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

    say $\displaystyle z = a+bi$

    Quote Originally Posted by Paymemoney View Post

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

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

    are they the same?


    Quote Originally Posted by Paymemoney View Post



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

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




    Quote Originally Posted by Paymemoney View Post


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

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


    Quote Originally Posted by Paymemoney View Post


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




    Quote Originally Posted by Paymemoney View Post

    E - $\displaystyle |z/\overline{z}|$ is real
    Is $\displaystyle \left|\frac{a+bi}{a-bi}\right|$ 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 $\displaystyle z^3 = 1 \Rightarrow z^3 - 1 = 0 \Rightarrow (z - 1)(z^2 + z + 1) = 0 ....$
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