Let S = z ∈ ℂ : z 2 + z ¯ = 0 . Then ∑ z ∈ S Re z + Im z is equal to _______.

Given,

$S=\left\{z\in C:z^2+\overline{z}=0\right\}$

Let $z=x+iy$

$z^2=x^2-y^2+2ixy$

$\overline{z}=x-iy$

Now putting the value in $z^2+\overline{z}$ we get,

$z^2+\overline{z}=x^2-y^2+x+i\left(2xy-y\right)=0$

Now on comparing both side we get,

$\Rightarrow x^2+x-y^2=0$ and $2xy-y=0$

$\Rightarrow y=0$ or $x=\frac{1}{2}$

Now if $y=0; x=0,-1$

If $x=\frac{1}{2} \text{then }y=\frac{\sqrt{3}}{2},\frac{-\sqrt{3}}{2}$

So, $\sum _{z\in S}\left(Re\left(z\right)+Im\left(z\right)=\left(0-1+\frac{1}{2}+\frac{1}{2}\right)+0+0+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}\right)=0$