# Thread: quick complex number questions

1. ## 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

2. You can have a go yourself

say $\displaystyle z = a+bi$ Originally Posted by Paymemoney 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? Originally Posted by Paymemoney B - $\displaystyle \frac{z}{\overline{z}}$ is real

Is $\displaystyle \frac{a+bi}{a-bi}$ real? Originally Posted by Paymemoney C - $\displaystyle \frac{z}{\overline{z}}$ is real

not sure how this one is different to the previous option?! Originally Posted by Paymemoney D - $\displaystyle z+\overline{z}$ is real
evaluate $\displaystyle a+bi + (a-bi)$ is it real? Originally Posted by Paymemoney E - $\displaystyle |z/\overline{z}|$ is real
Is $\displaystyle \left|\frac{a+bi}{a-bi}\right|$ real?

3. Originally Posted by Paymemoney 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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