If `z = 1 - i sqrt(3)` , then | arg z| + ` | arg bar(z)|` equals

Options

A. `(pi)/(3)`
B. `(2 pi)/( 3)`
C. 0
D. `(pi)/(2)`

Answer

To solve the problem, we need to find \( | \arg z | + | \arg \bar{z} | \) for the complex number \( z = 1 - i \sqrt{3} \).

### Step-by-Step Solution:

1. **Identify the complex number**:
\[ z = 1 - i \sqrt{3} \]
Here, \( a = 1 \) and \( b = -\sqrt{3} \).

2. **Find the argument of \( z \)**:
The argument \( \arg z \) can be determined using the formula:
\[ \arg z = \tan^{-1} \left( \frac{b}{a} \right) \]
In our case:
\[ \arg z = \tan^{-1} \left( \frac{-\sqrt{3}}{1} \right) = \tan^{-1}(-\sqrt{3}) \]
Since \( a > 0 \) and \( b < 0 \), \( z \) lies in the fourth quadrant. Therefore:
\[ \arg z = -\frac{\pi}{3} \]

3. **Find the conjugate of \( z \)**:
The conjugate \( \bar{z} \) is given by:
\[ \bar{z} = 1 + i \sqrt{3} \]

4. **Find the argument of \( \bar{z} \)**:
For \( \bar{z} \):
\[ \arg \bar{z} = \tan^{-1} \left( \frac{\sqrt{3}}{1} \right) = \tan^{-1}(\sqrt{3}) \]
Since both \( a \) and \( b \) are positive, \( \bar{z} \) lies in the first quadrant:
\[ \arg \bar{z} = \frac{\pi}{3} \]

5. **Calculate the mod of the arguments**:
Now we compute:
\[ | \arg z | = | -\frac{\pi}{3} | = \frac{\pi}{3} \]
\[ | \arg \bar{z} | = | \frac{\pi}{3} | = \frac{\pi}{3} \]

6. **Sum the mod of the arguments**:
Finally, we find:
\[ | \arg z | + | \arg \bar{z} | = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Answer:
\[ | \arg z | + | \arg \bar{z} | = \frac{2\pi}{3} \]