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"If $z$ and $w$ are the two complex numbers such that $z = \overline{w}$ and $\arg\left(\frac{z}{w}\right) = \pi$, then $\arg(z)$ is:"

Solution

Given the conditions:

1. $z = \overline{w}$
2. $\arg\left(\frac{z}{w}\right) = \pi$

We need to find $\arg(z)$.

Steps to Solve

1. Understanding $z = \overline{w}$:

   • $\overline{w}$ is the complex conjugate of $w$.
   • If $w = a + bi$, then $z = a - bi$.

2. Using the argument condition $\arg\left(\frac{z}{w}\right) = \pi$:

   • $\arg\left(\frac{z}{w}\right)$ implies that $\frac{z}{w}$ is a negative real number because its argument is $\pi$.

Let's express $z$ and $w$ in polar form:

• $w = r e^{i\theta}$
• $z = r e^{-i\theta}$ (since $z = \overline{w}$)

Now, compute $\frac{z}{w}$: $\frac{z}{w} = \frac{r e^{-i\theta}}{r e^{i\theta}} = e^{-i\theta} e^{-i\theta} = e^{-2i\theta}$

For the argument of $\frac{z}{w}$ to be $\pi$: $\arg\left(e^{-2i\theta}\right) = \pi$

This implies: $-2\theta = \pi$ $\theta = -\frac{\pi}{2}$

3. Finding $\arg(z)$:
   • $z = r e^{-i\theta}$
   • From the above, $\theta = -\frac{\pi}{2}$
   • Thus, $\arg(z) = -\theta = -(-\frac{\pi}{2}) = \frac{\pi}{2}$

Conclusion

The argument of $z$ is $\frac{\pi}{2}$.

Would you like further details or have any questions?

Related Questions

1. How would the solution change if $z$ and $w$ were not conjugates?
2. Can you explain the concept of complex conjugates in more detail?
3. How do arguments of complex numbers help in solving equations?
4. What happens if the argument condition was $\frac{\pi}{2}$ instead of $\pi$?
5. How can we visually interpret the arguments on the complex plane?
6. What are the properties of complex conjugates in general?
7. Can you demonstrate solving a similar problem with different conditions?
8. How do we handle complex numbers with real and imaginary parts separately?

Tip

When dealing with arguments of complex numbers, remember that the argument of the product or quotient of two complex numbers is the sum or difference of their individual arguments.