Complex numbers, integral to advanced mathematics, are numbers consisting of a real part and an imaginary part. They are pivotal in various scientific and engineering fields, offering deep insights into mathematical concepts.

## Addition and Subtraction of Complex Numbers

### Complex Number Form:

- $z = a + bi$, where '$a$' is the real part, '$bi$' is the imaginary part.

### Addition:

- For $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, add real and imaginary parts separately:
- $z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i$.

### Subtraction:

- $z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i$.
- Used in various mathematical and engineering contexts.

## Multiplication of Complex Numbers

### Distributive Property:

- Multiply $(a_1 + b_1i)(a_2 + b_2i)$ to get $a_1a_2 + a_1b_2i + b_1a_2i + b_1b_2i^2$.
- Simplify using $i^2 = -1 : a_1a_2 - b_1b_2 + (a_1b_2 + b_1a_2)i$.
- Key for understanding rotation and scaling in the complex plane.

## Division of Complex Numbers

### Using the Conjugate:

- The conjugate of $z = a + bi$ is $z^* = a - bi$.
- Divide $\frac{z_1}{z_2}$ by multiplying numerator and denominator with $z_2^*$ : $\frac{z_1}{z_2}$ = $\frac{z_1 \cdot z_2^*}{z_2 \cdot z_2^*}$.
- Eliminates the imaginary part in the denominator.
- Essential in complex equations and electrical engineering.

## Example Questions

### Addition Example

**Problem Statement:**

Compute $(3 + 4i) + (5 - 2i)$.

**Solution:**

**1. Separate Real and Imaginary Parts:**

- Real: $3 + 5$.
- Imaginary: $4i - 2i$.

**2. Combine and Simplify:**

- Real: $3 + 5 = 8$.
- Imaginary: $4i - 2i = 2i$.

**3. Result:**

- Sum is $8 + 2i$.

### Multiplication Example

#### Problem Statement:

Determine the product of $(2 + 3i)$ and $(4 - 5i)$.

#### Solution:

**1. Apply Distributive Property:**

- Perform $(2 + 3i)(4 - 5i)$.

**2. Multiply Each Pair of Terms:**

- Calculate $2 \cdot 4$, $2 \cdot -5i$, $3i \cdot 4$, and $3i \cdot -5i$.

**3. Combine Results:**

- Sum up $8 - 10i + 12i - 15i^2$.

**4. Use **$i^2 = -1$**:**

- Convert $-15i^2$ to $+15$.

**5. Combine Like Terms:**

- Add $8 - 10i + 12i + 15$ together.

**6. Final Product:**

- Result is $23 + 2i$.

### Division Example

#### Problem Statement:

Divide $(1 + 2i)$ by $(3 - 4i)$.

#### Solution:

**1. Multiply by Conjugate:**

- Use $\frac{3 + 4i}{3 + 4i}$ to clear the imaginary part in the denominator.
- Thus, $\frac{1 + 2i}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i}$.

**2. Simplify Numerator:**

- Calculate $(1 + 2i)(3 + 4i)$.
- Expand to get $3 + 4i + 6i + 8$ (using $i^2 = -1$).

**3. Simplify Numerator Further:**

- Combine like terms: $11 + 10i$.

**4. Simplify Denominator:**

- Calculate $(3 - 4i)(3 + 4i)$.
- Expand to $9 + 16$ (imaginary terms cancel out).

**5. Complete Division:**

- Divide $\frac{11 + 10i}{25}$.

**6. Final Answer:**

- Separate into real and imaginary parts: $\frac{11}{25} + \frac{10}{25}i$

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.