Question

If $z\ne 0$ is a complex number,then prove that $Re\left(z\right)=0⟹Im\left(z^2\right)=0$.

Answer 1

If $z\ne 0$, let $z=x+iy$.
Then, $z^2=(x+iy)(x+iy)$

$\Rightarrow z^2=x^2-y^2+i\left(2xy\right)$
$\because Re\left(z\right)=0$

$⟹x=0$

$⟹Im\left(z^2\right)=2xy=0$