Given the complex number `z= (-1 + sqrt3i)/(2) and w= (-1- sqrt3i)/(2)` (where `i= sqrt-1`)
Calculate the modulus and argument of `(w)/(z)`

Answer

To solve the problem, we need to calculate the modulus and argument of the complex number \( \frac{w}{z} \), where \( z = \frac{-1 + \sqrt{3}i}{2} \) and \( w = \frac{-1 - \sqrt{3}i}{2} \).

### Step-by-Step Solution:

1. **Write the Complex Numbers:**
\[ z = \frac{-1 + \sqrt{3}i}{2}, \quad w = \frac{-1 - \sqrt{3}i}{2} \]

2. **Calculate \( \frac{w}{z} \):**
\[ \frac{w}{z} = \frac{\frac{-1 - \sqrt{3}i}{2}}{\frac{-1 + \sqrt{3}i}{2}} = \frac{-1 - \sqrt{3}i}{-1 + \sqrt{3}i} \]

3. **Rationalize the Denominator:**
To rationalize, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{w}{z} = \frac{(-1 - \sqrt{3}i)(-1 - \sqrt{3}i)}{(-1 + \sqrt{3}i)(-1 - \sqrt{3}i)} \]

4. **Calculate the Denominator:**
\[ (-1 + \sqrt{3}i)(-1 - \sqrt{3}i) = (-1)^2 - (\sqrt{3}i)^2 = 1 + 3 = 4 \]

5. **Calculate the Numerator:**
\[ (-1 - \sqrt{3}i)(-1 - \sqrt{3}i) = (-1)^2 + 2(-1)(-\sqrt{3}i) + (-\sqrt{3}i)^2 = 1 + 2\sqrt{3}i - 3 = -2 + 2\sqrt{3}i \]

6. **Combine Results:**
\[ \frac{w}{z} = \frac{-2 + 2\sqrt{3}i}{4} = \frac{-1 + \sqrt{3}i}{2} \]

7. **Identify Real and Imaginary Parts:**
\[ x = -\frac{1}{2}, \quad y = \frac{\sqrt{3}}{2} \]

8. **Calculate the Modulus:**
\[ | \frac{w}{z} | = \sqrt{x^2 + y^2} = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

9. **Calculate the Argument:**
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(-\sqrt{3}) \]
Since \( x < 0 \) and \( y > 0 \), the angle is in the second quadrant:
\[ \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Results:
- **Modulus:** \( | \frac{w}{z} | = 1 \)
- **Argument:** \( \arg\left(\frac{w}{z}\right) = \frac{2\pi}{3} \)