In this section, focus into the Argand diagram, a crucial tool for visualizing and understanding complex numbers. This diagram not only represents complex numbers graphically but also provides a clear insight into their operations, making it an essential concept for students.

**Argand Diagram Basics**

**1. Argand Diagram:** A 2D plane to represent complex numbers, where the horizontal axis (Real Axis) shows the real part and the vertical axis (Imaginary Axis) shows the imaginary part of a complex number.

**2. Real Axis: **The 'Re' axis. In a complex number $z = a + bi$, 'a' is plotted here.

**3. Imaginary Axis: **The 'Im' axis. In $z = a + bi$, 'bi' goes here.

**4. Point Representation:** Each complex number $z = a + bi$ is a unique point $a, b$ on the diagram, useful for visualizing complex numbers as vectors.

**5. Modulus and Argument:**

- Modulus $|z|$: Distance from the point to the origin.
- Argument $\arg(z)$: Angle from the positive real axis to the line connecting the point and the origin.

Image courtesy of Online Math Learning

## Operations on the Argand Diagram

**1. Addition:** Add complex numbers by adding their real and imaginary parts separately. Geometrically, it's like vector addition: the sum $z_1 + z_2$ is the diagonal of the parallelogram formed by $z_1$ and $z_2$.

**2. Subtraction:** Subtract by taking the difference of real and imaginary parts. Geometrically, it's the vector from the head of the subtracted number to the other number's head.

**3. Multiplication:** Multiply by rotating and scaling. Multiply the angle of $z_1$ by the argument of $z_2$ and the modulus of $z_1$ by the modulus of $z_2$. Results in rotating and stretching $z_1's$ vector.

**4. Division:** Inverse of multiplication. Divide the angle of $z_1$ by the argument of $z_2$ and the modulus of $z_1$ by the modulus of $z_2$. Leads to rotating and shrinking $z_1's$ vector.

**5. Conjugation:** Conjugate of $z = a + bi$ is $z* = a - bi$. Reflect the point $a, b$ across the real axis to get $a, -b$.

## Example Questions

### Question 1: Representation of $3 + 4i$ on an Argand Diagram

#### Solution:

**Understand:**$3 + 4i$ has real part 3, imaginary part 4i.**Plot:**Point $3, 4$ on diagram, 3 on real axis, 4 on imaginary axis.**Modulus:**Distance from origin. Calculate as $√3² + 4² = 5$.**Argument:**Angle with positive real axis. Calculate as $\tan^{-1}\left(\frac{4}{3}\right) ≈ 53.13^\circ$.

### Question 2: Adding $1 + 2i$ and $3 + 4i$ on the Argand Diagram

**Solution:**

**Understand:**Add real parts (1 and 3) and imaginary parts (2i and 4i) separately.**Plot:**Points (1, 2) and (3, 4) on diagram.**Parallelogram:**Draw using (1, 2) and (3, 4) as vertices.**Sum:**Diagonal from origin to opposite vertex of parallelogram. Algebraically, $4 + 6i$.**Interpretation:**Sum is like vector addition of (1, 2) and (3, 4).

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.