Given the complex number `z= (-1 + sqrt3i)/(2) and w= (-1- sqrt3i)/(2)` (where `i= sqrt-1`)
Represent z and w accurately on the complex plane.

Answer

To represent the complex numbers \( z \) and \( w \) accurately on the complex plane, we will follow these steps:

### Step 1: Identify the complex numbers
We are given:
\[ z = \frac{-1 + \sqrt{3}i}{2} \]
\[ w = \frac{-1 - \sqrt{3}i}{2} \]

### Step 2: Simplify the complex numbers
We can rewrite \( z \) and \( w \) in the standard form \( a + bi \):
- For \( z \):
\[ z = \frac{-1}{2} + \frac{\sqrt{3}}{2}i \]
- For \( w \):
\[ w = \frac{-1}{2} - \frac{\sqrt{3}}{2}i \]

### Step 3: Identify the real and imaginary parts
From the standard form, we can identify:
- For \( z \):
- Real part \( a = -\frac{1}{2} \)
- Imaginary part \( b = \frac{\sqrt{3}}{2} \)

- For \( w \):
- Real part \( a = -\frac{1}{2} \)
- Imaginary part \( b = -\frac{\sqrt{3}}{2} \)

### Step 4: Plot the points on the complex plane
The complex plane has a horizontal axis (real part) and a vertical axis (imaginary part).

- For \( z \):
- The point is \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \)
- The real part \( -\frac{1}{2} \) is located on the negative x-axis.
- The imaginary part \( \frac{\sqrt{3}}{2} \) is approximately \( 0.866 \), which is between \( 0.5 \) and \( 1 \) on the positive y-axis.

- For \( w \):
- The point is \( \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \)
- The real part \( -\frac{1}{2} \) is the same as for \( z \).
- The imaginary part \( -\frac{\sqrt{3}}{2} \) is approximately \( -0.866 \), which is between \( -0.5 \) and \( -1 \) on the negative y-axis.

### Step 5: Draw the points on the complex plane
1. Draw the complex plane with the x-axis (real part) and y-axis (imaginary part).
2. Mark the point for \( z \) at \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \).
3. Mark the point for \( w \) at \( \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \).

### Final Representation
- The point \( z \) is located in the second quadrant.
- The point \( w \) is located in the third quadrant.