Question

Find all complex numbers $z$ which satisfy the following equation $z^2=\overline{z}$

• A. $0,-1,-\omega ,-{\omega }^2.$
• B. $0,-1,\omega ,{\omega }^2.$
• C. $0,1,\omega ,{\omega }^2.$
• D. $0,1,-\omega ,-{\omega }^2.$

Answer 1

The correct option is C $0,1,\omega ,{\omega }^2.$

$z^2=\overline{z}$

Or
$z^2=\frac{1}{z}$

Or
$z^3=1$

Or
$(z^3-1)=0$

$(z-1)(z^2+z+1)=0$

$z=1$ and $z^2+z+1=0$

Hence

$z=w,w^2$ where $w,w^2$ are cube roots of unity.