Animated Solution for Mathematics - Complex Numbers: Let $S=\{z\in \mathbb{C}:z^2+\bar{z}=0\}$. Then ${\sum }_{z\in S}(\text{Re}(z)+\text{Im}(z))$ is equal to

Imagine you are standing on the Argand plane, looking at the equation $z^2+\bar{z}=0$. We are looking for all complex numbers $z$ that satisfy this condition.

To solve this, we peel back the layers of the complex number $z=x+iy$, where $x$ and $y$ are real numbers. Substituting this into our equation, we obtain:

Expanding the square and recalling that $i^2=-1$, the expression becomes:

Grouping the real and imaginary parts, we arrive at:

For this equation to hold true, both the real part and the imaginary part must independently equal zero.

Focusing on the imaginary part, we have $2xy-y=0$, which factors into:

This yields two distinct cases.

If $y=0$, our complex number is purely real. Substituting $y=0$ into the real part equation $x^2-y^2+x=0$, we get:

Thus, $x=0$ or $x=-1$. This provides two roots: $z_1=0$ and $z_2=-1$.

Substituting $x=\frac{1}{2}$ into the real part equation $x^2-y^2+x=0$, we get:

This simplifies to:

This gives $y=\pm \frac{\sqrt{3}}{2}$. Consequently, our remaining two roots are:

We have identified the set of solutions:

The final step is to sum the real and imaginary parts of these roots. The sum of the real parts is:

The sum of the imaginary parts is:

The total sum of the roots is 0. The symmetry of these roots on the Argand plane is remarkable, as they are perfectly balanced, leading to a complete cancellation.