Question

The number of complex numbers which are conjugate of their own cube, is ___

Answer 1

Answer: 5

Let $z$ be such a complex number.
Then according to the question, we have
$z^3=\bar{z}....(1)$
$\Rightarrow |z|=0$
or $|z|=1$
$\Rightarrow z=0$
or $z=\cos \theta +i\sin \theta$
Putting $z=\cos \theta +i\sin \theta$ in (1),
we get $\cos 3\theta =\cos \theta$
and $\sin 3\theta =-\sin \theta$
$\Rightarrow \sin \theta =0$
or $\sin \theta =1$
or $\sin \theta =-1$
$\Rightarrow \theta =0$
or $\pi$, or $\frac{\pi }{2}$
or $-\frac{\pi }{2}$
$\Rightarrow z=1,-1,i,-i$
or $z=0$