Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be expressed as $z = x + iy$, where $x = \operatorname{Re}(z)$ is the real part and $y = \operatorname{Im}(z)$ is the imaginary part, and $x, y \in \mathbb{R}$.
• Complex Conjugate: The complex conjugate of $z = x + iy$ is $\bar{z} = x - iy$.
• Zero Complex Number: A complex number $a + ib = 0$ if and only if $a = 0$ and $b = 0$.

Step 1: Expressing the Given Equation in Terms of Real and Imaginary Parts

We are given the equation $z^2 + \bar{z} = 0$. We want to express this equation in terms of the real and imaginary parts of $z$ by substituting $z = x + iy$ and $\bar{z} = x - iy$. First, we compute $z^2$: $z^2 = (x + iy)^2 = x^2 + 2ixy + (iy)^2 = x^2 + 2ixy - y^2 = (x^2 - y^2) + i(2xy)$ Now, substitute $z^2$ and $\bar{z}$ into the original equation: $(x^2 - y^2) + i(2xy) + (x - iy) = 0$ Group the real and imaginary terms: $(x^2 - y^2 + x) + i(2xy - y) = 0$

Explanation: We rewrite the complex equation in terms of its real and imaginary components, allowing us to separate it into two real equations.

Step 2: Forming a System of Real Equations

For the complex number $(x^2 - y^2 + x) + i(2xy - y)$ to be equal to zero, both its real and imaginary parts must be zero. This gives us the following system of equations:

1. Real part: $x^2 - y^2 + x = 0$
2. Imaginary part: $2xy - y = 0$

Explanation: This step transforms the problem into solving a system of two real equations, which is a more familiar task.

Step 3: Solving the System of Equations to Find $x$ and $y$

We solve the second equation first, as it's simpler: $2xy - y = 0$ Factor out $y$: $y(2x - 1) = 0$ This implies either $y = 0$ or $2x - 1 = 0$.

Case 1: $y = 0$ Substitute $y = 0$ into the first equation ($x^2 - y^2 + x = 0$): $x^2 - (0)^2 + x = 0$ $x^2 + x = 0$ Factor out $x$: $x(x + 1) = 0$ This gives two possible values for $x$: $x = 0$ or $x = -1$.

• If $x = 0$ and $y = 0$, then $z_1 = 0 + i(0) = 0$.
• If $x = -1$ and $y = 0$, then $z_2 = -1 + i(0) = -1$.

Case 2: $x = \frac{1}{2}$ Substitute $x = \frac{1}{2}$ into the first equation ($x^2 - y^2 + x = 0$): $\left(\frac{1}{2}\right)^2 - y^2 + \frac{1}{2} = 0$ $\frac{1}{4} - y^2 + \frac{1}{2} = 0$ $\frac{3}{4} - y^2 = 0$ $y^2 = \frac{3}{4}$ Taking the square root of both sides: $y = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2}$

• If $x = \frac{1}{2}$ and $y = \frac{\sqrt{3}}{2}$, then $z_3 = \frac{1}{2} + i\frac{\sqrt{3}}{2}$.
• If $x = \frac{1}{2}$ and $y = -\frac{\sqrt{3}}{2}$, then $z_4 = \frac{1}{2} - i\frac{\sqrt{3}}{2}$.

Explanation: By considering each possible case for $y(2x-1)=0$, we find all possible solutions for $x$ and $y$ that satisfy the system of equations.

Step 4: Identifying the Complex Numbers in Set $S$

Combining the solutions from both cases, the set $S$ of complex numbers satisfying $z^2 + \bar{z} = 0$ is: $S = \left\{0, -1, \frac{1}{2} + i\frac{\sqrt{3}}{2}, \frac{1}{2} - i\frac{\sqrt{3}}{2}\right\}$

Step 5: Calculating the Required Sum

We need to find the sum $\sum_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$. Let's calculate $(\operatorname{Re}(z)+\operatorname{Im}(z))$ for each element in $S$:

1. For $z_1 = 0$: $\operatorname{Re}(z_1) + \operatorname{Im}(z_1) = 0 + 0 = 0$
2. For $z_2 = -1$: $\operatorname{Re}(z_2) + \operatorname{Im}(z_2) = -1 + 0 = -1$
3. For $z_3 = \frac{1}{2} + i\frac{\sqrt{3}}{2}$: $\operatorname{Re}(z_3) + \operatorname{Im}(z_3) = \frac{1}{2} + \frac{\sqrt{3}}{2}$
4. For $z_4 = \frac{1}{2} - i\frac{\sqrt{3}}{2}$: $\operatorname{Re}(z_4) + \operatorname{Im}(z_4) = \frac{1}{2} - \frac{\sqrt{3}}{2}$

Now, sum these values: $\sum_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z)) = 0 + (-1) + \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)$ $= -1 + \frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{1}{2} - \frac{\sqrt{3}}{2}$ $= -1 + \left(\frac{1}{2} + \frac{1}{2}\right) + \left(\frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2}\right)$ $= -1 + 1 + 0$ $= 0$

Explanation: We compute the real and imaginary parts for each solution and sum them as requested.

Step 6: Identifying the Error

The problem statement said the correct answer is 2. Let us review the solution and identify the error.

Going back to step 5, the sum is incorrectly calculated. The problem asks for $\sum_{z \in S} (\operatorname{Re}(z) + \operatorname{Im}(z))$. We have $S = \{0, -1, \frac{1}{2} + i\frac{\sqrt{3}}{2}, \frac{1}{2} - i\frac{\sqrt{3}}{2}\}$.

So we need to sum the real and imaginary parts of each complex number: \begin{itemize} \item $z = 0$: $\operatorname{Re}(0) + \operatorname{Im}(0) = 0 + 0 = 0$ \item $z = -1$: $\operatorname{Re}(-1) + \operatorname{Im}(-1) = -1 + 0 = -1$ \item $z = \frac{1}{2} + i\frac{\sqrt{3}}{2}$: $\operatorname{Re}(\frac{1}{2} + i\frac{\sqrt{3}}{2}) + \operatorname{Im}(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = \frac{1}{2} + \frac{\sqrt{3}}{2}$ \item $z = \frac{1}{2} - i\frac{\sqrt{3}}{2}$: $\operatorname{Re}(\frac{1}{2} - i\frac{\sqrt{3}}{2}) + \operatorname{Im}(\frac{1}{2} - i\frac{\sqrt{3}}{2}) = \frac{1}{2} - \frac{\sqrt{3}}{2}$ \end{itemize}

Then the sum is $0 + (-1) + (\frac{1}{2} + \frac{\sqrt{3}}{2}) + (\frac{1}{2} - \frac{\sqrt{3}}{2}) = 0 - 1 + \frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{1}{2} - \frac{\sqrt{3}}{2} = -1 + \frac{1}{2} + \frac{1}{2} = -1 + 1 = 0$

Let's check the original equation. $z^2 + \bar{z} = 0$ $z^2 = -\bar{z}$ Let $z = x+iy$, so $(x+iy)^2 = -(x - iy) = -x + iy$. $x^2 - y^2 + 2ixy = -x + iy$ So $x^2 - y^2 = -x$ and $2xy = y$. From $2xy = y$, we get $y(2x - 1) = 0$. Thus $y = 0$ or $x = 1/2$.

If $y = 0$, then $x^2 = -x$ so $x(x+1) = 0$, thus $x = 0$ or $x = -1$. This gives $z = 0$ and $z = -1$. If $x = 1/2$, then $1/4 - y^2 = -1/2$, so $y^2 = 1/4 + 1/2 = 3/4$. Thus $y = \pm \frac{\sqrt{3}}{2}$. So $z = \frac{1}{2} \pm i \frac{\sqrt{3}}{2}$.

We have $z \in \{0, -1, \frac{1}{2} + i \frac{\sqrt{3}}{2}, \frac{1}{2} - i \frac{\sqrt{3}}{2}\}$. For $z = 0$, $\operatorname{Re}(z) + \operatorname{Im}(z) = 0$ For $z = -1$, $\operatorname{Re}(z) + \operatorname{Im}(z) = -1$ For $z = \frac{1}{2} + i \frac{\sqrt{3}}{2}$, $\operatorname{Re}(z) + \operatorname{Im}(z) = \frac{1}{2} + \frac{\sqrt{3}}{2}$ For $z = \frac{1}{2} - i \frac{\sqrt{3}}{2}$, $\operatorname{Re}(z) + \operatorname{Im}(z) = \frac{1}{2} - \frac{\sqrt{3}}{2}$ So $\sum (\operatorname{Re}(z) + \operatorname{Im}(z)) = 0 - 1 + \frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{1}{2} - \frac{\sqrt{3}}{2} = -1 + 1 = 0$

There is an error in the problem. $z^2 + \bar{z} = 0$ gives the sum as 0.

However, let's assume the question meant $z^2 + z = 0$. Then $z(z+1) = 0$ so $z = 0$ or $z = -1$. The sum is $0 - 1 = -1$.

Let's assume the question meant $\bar{z}^2 + z = 0$. Let $z = x + iy$ so $\bar{z} = x - iy$. Then $(x-iy)^2 + x + iy = 0$. $x^2 - y^2 - 2ixy + x + iy = 0$ So $x^2 - y^2 + x = 0$ and $-2xy + y = 0$. So $y(1 - 2x) = 0$, so $y = 0$ or $x = 1/2$. If $y = 0$, then $x^2 + x = 0$ so $x = 0, -1$. If $x = 1/2$, then $1/4 - y^2 + 1/2 = 0$ so $y^2 = 3/4$ and $y = \pm \frac{\sqrt{3}}{2}$. $z = 0, -1, \frac{1}{2} \pm i \frac{\sqrt{3}}{2}$. $0 - 1 + \frac{1}{2} + \frac{\sqrt{3}}{2} + \frac{1}{2} - \frac{\sqrt{3}}{2} = 0$

Let's assume the question meant $z + \bar{z}^2 = 0$. This is equivalent to $\bar{z} + z^2 = 0$, which is the original question.

There must be an error in the problem. The answer is 0.

Common Mistakes & Tips

• Careful Expansion: Ensure $(x+iy)^2$ is expanded correctly as $x^2 - y^2 + 2ixy$. A common error is forgetting $(iy)^2 = -y^2$.
• Separating Real and Imaginary Parts: Precisely identify and group all real terms and all imaginary terms. Any mixing will lead to incorrect equations.
• Solving Systems of Equations: When faced with a product equaling zero, like $y(2x-1)=0$, remember to consider all possible cases (here, $y=0$ or $2x-1=0$). Missing a case will lead to an incomplete set of solutions.
• Verification: After finding the complex solutions, it's good practice to quickly substitute them back into the original equation $z^2 + \bar{z} = 0$ to ensure they satisfy it.

Summary

We transformed the complex equation into a system of real equations by substituting $z = x + iy$, separating the real and imaginary parts, and then equating each part to zero. Solving this system yields the real and imaginary components of all complex solutions. The final step involves carefully summing the required expressions for each solution. The sum $\sum_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to 0. There seems to be an error in the problem statement because the correct answer should be 0, and not 2.

Final Answer The final answer is \boxed{0}.