Plot the following numbers on a complex number plane and find their absolute values:
(a) 5
(b) 2i
(c) (i) `4-3i`
(ii) `(sqrt(3))/(2)+(1)/(2)i`

Answer

To solve the problem, we will plot the given complex numbers on a complex number plane and then calculate their absolute values step by step.

### Step 1: Plot the Complex Numbers

1. **Complex Number (a): 5**
- This number is purely real. On the complex plane, it is represented as the point (5, 0).
- **Plot**: Move 5 units along the real axis (x-axis).

2. **Complex Number (b): 2i**
- This number is purely imaginary. On the complex plane, it is represented as the point (0, 2).
- **Plot**: Move 2 units along the imaginary axis (y-axis).

3. **Complex Number (c)(i): 4 - 3i**
- This number has both real and imaginary parts. It is represented as the point (4, -3).
- **Plot**: Move 4 units along the real axis and 3 units down along the imaginary axis.

4. **Complex Number (c)(ii): (√3)/2 + (1/2)i**
- This number also has both real and imaginary parts. It is represented as the point (√3/2, 1/2).
- **Plot**: Move approximately 0.866 units along the real axis and 0.5 units up along the imaginary axis.

### Step 2: Calculate the Absolute Values

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

1. **For (a): 5**
- Here, \( x = 5 \) and \( y = 0 \).
\[ |5| = \sqrt{5^2 + 0^2} = \sqrt{25} = 5 \]

2. **For (b): 2i**
- Here, \( x = 0 \) and \( y = 2 \).
\[ |2i| = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]

3. **For (c)(i): 4 - 3i**
- Here, \( x = 4 \) and \( y = -3 \).
\[ |4 - 3i| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

4. **For (c)(ii): (√3)/2 + (1/2)i**
- Here, \( x = \frac{\sqrt{3}}{2} \) and \( y = \frac{1}{2} \).
\[ \left| \frac{\sqrt{3}}{2} + \frac{1}{2}i \right| = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]

### Summary of Results
- **(a)**: \( |5| = 5 \)
- **(b)**: \( |2i| = 2 \)
- **(c)(i)**: \( |4 - 3i| = 5 \)
- **(c)(ii)**: \( \left| \frac{\sqrt{3}}{2} + \frac{1}{2}i \right| = 1 \)