# Math Help - Complex numbers

1. ## Complex numbers

There's another question that I'm -completely- stuck on, I can't even think of how to start solving this one:

Find all complex numbers z for which $z^3 = -4 \bar z$

If someone is able to explain that to me, thanks.

2. Originally Posted by Wedric
Find all complex numbers z for which $z^3 = -4 \bar z$
Hello Wedric

first write z in "mod/arg" form. $z = Re^{i \theta }$ so $z^3 = R^3e^{i 3\theta }$ and $\bar z = Re^{-i \theta}$

$z^3 = -4 \bar z \ \ \Rightarrow \ \ R^3e^{i 3\theta } = -4Re^{-i \theta}$

$R^2e^{i 4\theta } = -4$

$R^2 \cos 4\theta + R^2 i \sin 4 \theta = -4$

Require $\sin 4 \theta = 0$
and $\cos 4\theta = -1$
$R=2$
$4 \theta = 2 \pi n +\pi$
$\theta = \frac{ \pi }{4} (1+ 2n) \ \ \forall n \in \mathbb{Z}$

$\therefore z = 2e^{i\frac{ \pi }{4} (1+ 2n)}$

Bobak

3. Thanks for the reply, I'll mull over that for a while.