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

Answer

To solve the problem of finding the modulus and argument of the complex numbers \( z \) and \( w \), we proceed as follows:

Given:
\[ z = \frac{-1 + \sqrt{3}i}{2} \]
\[ w = \frac{-1 - \sqrt{3}i}{2} \]

### Step 1: Calculate the modulus of \( z \)

The modulus of a complex number \( z = x + yi \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]

For \( z \), we have:
- \( x = -\frac{1}{2} \)
- \( y = \frac{\sqrt{3}}{2} \)

Now, we calculate:
\[ |z| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 2: Calculate the argument of \( z \)

The argument of a complex number \( z = x + yi \) is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

For \( z \):
\[ \theta_z = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(-\sqrt{3}) \]

The value of \( \tan^{-1}(-\sqrt{3}) \) corresponds to an angle in the second quadrant (since \( x < 0 \) and \( y > 0 \)):
\[ \theta_z = \frac{2\pi}{3} \quad \text{(in radians)} \]

### Step 3: Calculate the modulus of \( w \)

Using the same formula for modulus:
\[ |w| = \sqrt{x^2 + y^2} \]

For \( w \):
- \( x = -\frac{1}{2} \)
- \( y = -\frac{\sqrt{3}}{2} \)

Now, we calculate:
\[ |w| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(-\frac{\sqrt{3}}{2}\right)^2} \]
\[ = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \]

### Step 4: Calculate the argument of \( w \)

For \( w \):
\[ \theta_w = \tan^{-1}\left(\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right) = \tan^{-1}(\sqrt{3}) \]

The value of \( \tan^{-1}(\sqrt{3}) \) corresponds to an angle in the first quadrant:
\[ \theta_w = \frac{\pi}{3} \quad \text{(in radians)} \]

### Final Results

- Modulus of \( z \): \( |z| = 1 \)
- Argument of \( z \): \( \theta_z = \frac{2\pi}{3} \)
- Modulus of \( w \): \( |w| = 1 \)
- Argument of \( w \): \( \theta_w = \frac{\pi}{3} \)