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 $\overline{z}$, 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

2. You can have a go yourself

say $z = a+bi$

Originally Posted by Paymemoney

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?

Originally Posted by Paymemoney

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

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

Originally Posted by Paymemoney

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

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

Originally Posted by Paymemoney

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

Originally Posted by Paymemoney

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