Complex Numbers & the Argand Diagram

What Are Complex Numbers?

Complex numbers extend the real number system by introducing the imaginary unit $i$, defined by $i^2 = -1$. A complex number has the form:

where $a$ is the real part and $b$ is the imaginary part.

They were invented to solve equations like $x^2 + 1 = 0$, but they turn out to be essential in electrical engineering, quantum mechanics, signal processing and fluid dynamics.

Learning Goals: By the end of this guide, you should be able to:

1. Perform arithmetic (add, subtract, multiply, divide) with complex numbers.
2. Represent complex numbers on the Argand diagram.
3. Convert between Cartesian and modulus-argument (polar) form.
4. Apply de Moivre's theorem to find powers and roots.

Arithmetic Operations

Key identity: $z \cdot \overline{z} = a^2 + b^2 = |z|^2$

The Argand Diagram

The Argand diagram represents complex numbers as points (or position vectors) in a 2D plane:

• Horizontal axis = real part
• Vertical axis = imaginary part

The complex number $z = 3 + 4i$ corresponds to the point $(3, 4)$.

Modulus-Argument (Polar) Form

Every complex number can be written as:

where:

• $r = |z| = \sqrt{a^2 + b^2}$ is the modulus (distance from origin)
• $\theta = \arg(z) = \arctan\frac{b}{a}$ is the argument (angle with positive real axis)

Euler's Formula

So $z = re^{i\theta}$. This connects complex numbers to trigonometry and exponentials.

De Moivre's Theorem

For any integer $n$:

Or equivalently: $(re^{i\theta})^n = r^n e^{in\theta}$

Application: Finding $n$th Roots

The $n$th roots of a complex number $z = re^{i\theta}$ are:

These $n$ roots are equally spaced around a circle of radius $r^{1/n}$ on the Argand diagram.

Worked Examples

Example 1: Multiplication in Polar Form

Multiply $z_1 = 2(\cos 30° + i\sin 30°)$ and $z_2 = 3(\cos 45° + i\sin 45°)$.

Multiplication multiplies moduli and adds arguments.

Example 2: Cube Roots of $-8$

Write $-8 = 8e^{i\pi}$. The three cube roots are:

• $z_0 = 2e^{i\pi/3} = 1 + \sqrt{3}i$
• $z_1 = 2e^{i\pi} = -2$
• $z_2 = 2e^{i5\pi/3} = 1 - \sqrt{3}i$

Common Mistakes

1. Forgetting that $i^2 = -1$ — When expanding $(a+bi)(c+di)$, the $bd \cdot i^2$ term becomes $-bd$, not $+bd$.
2. Wrong quadrant for the argument — $\arctan\frac{b}{a}$ gives the reference angle, but you must adjust for the correct quadrant using the signs of $a$ and $b$.
3. Missing roots — The $n$th root of a complex number has $n$ distinct values, not just one.

Exam Tips (A-Level / AP / IB)

• To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.
• When plotting on an Argand diagram, clearly mark the modulus and argument.
• For de Moivre problems, convert to polar form first — it makes powers and roots straightforward.

Frequently Asked Questions

Why are complex numbers called "imaginary"?

The term is historical and somewhat misleading. Complex numbers are just as "real" as real numbers — they are a valid mathematical extension. "Imaginary" simply refers to the $i$ component.

What is the complex conjugate used for?

The conjugate $\overline{z} = a - bi$ is used to: (1) rationalise denominators in division, (2) find the modulus ($|z|^2 = z\overline{z}$), and (3) reflect points across the real axis on the Argand diagram.

Related Topics

• Trigonometric Graphs — Euler's formula unifies trig functions and complex exponentials.
• Vector Operations — Complex multiplication is equivalent to vector rotation and scaling.
• Probability Distributions — The characteristic function uses complex exponentials.