Key Concepts and Formulas

• Polar Form of a Complex Number: A complex number $z$ can be represented as $z = re^{i\theta}$, where $r = |z|$ is the modulus and $\theta = \text{Arg}(z)$ is the argument.
• Complex Conjugate: If $w = re^{i\theta}$, then its complex conjugate is $\overline{w} = re^{-i\theta}$.
• Euler's Formula: $e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1$.

Step-by-Step Solution

1. Expressing z and w in Polar Form

Why this step? Representing complex numbers in polar form is essential when dealing with magnitudes and arguments. It allows us to directly utilize the given information in our calculations.

Let $|z| = |w| = r$ and $\text{Arg}(z) = \theta$, $\text{Arg}(w) = \phi$. Then we can write:

$z = re^{i\theta} \quad \text{(Equation 1)}$ $w = re^{i\phi} \quad \text{(Equation 2)}$ Since $z$ and $w$ are non-zero, $r > 0$.

2. Using the Argument Condition

Why this step? This step utilizes the given relationship between the arguments of $z$ and $w$ to express one argument in terms of the other. This will allow us to substitute and eventually relate $z$ directly to $w$.

Given $\text{Arg}(z) + \text{Arg}(w) = \pi$, we have: $\theta + \phi = \pi$ Solving for $\theta$: $\theta = \pi - \phi \quad \text{(Equation 3)}$

3. Substituting and Simplifying z

Why this step? Substituting the expression for $\theta$ into the polar form of $z$ allows us to express $z$ in terms of $\phi$, which is related to $w$. This is a crucial step in linking $z$ and $w$.

Substitute Equation 3 into Equation 1: $z = re^{i(\pi - \phi)}$ Using exponent rules: $z = re^{i\pi}e^{-i\phi}$

4. Applying Euler's Formula

Why this step? Applying Euler's formula simplifies the expression by replacing $e^{i\pi}$ with a real number, making it easier to identify the relationship between $z$ and $w$.

Using Euler's formula, $e^{i\pi} = -1$: $z = r(-1)e^{-i\phi}$ $z = -re^{-i\phi} \quad \text{(Equation 4)}$

5. Relating z to the Conjugate of w

Why this step? This step connects our derived expression for $z$ to the complex conjugate of $w$, leading us to the final answer.

The complex conjugate of $w$ is: $\overline{w} = re^{-i\phi}$ Comparing this with Equation 4: $z = -re^{-i\phi} = -\overline{w}$

Thus, $z = -\overline{w}$.

Common Mistakes & Tips

• Remember the polar form $re^{i\theta}$ and how to manipulate it.
• Euler's formula is crucial for simplifying expressions involving $e^{i\pi}$.
• Be careful with signs when dealing with complex conjugates and Euler's formula.

Summary

By expressing $z$ and $w$ in polar form, using the given condition $\text{Arg}(z) + \text{Arg}(w) = \pi$, and applying Euler's formula, we found that $z = -\overline{w}$. This demonstrates how polar form and Euler's formula can simplify complex number problems.

The final answer is \boxed{-\overline \omega}, which corresponds to option (B).