Complex Numbers (Leaving Cert Higher Level): Argand Diagram & De Moivre

Complex numbers extend the real numbers using i = √(-1). The Higher course covers the Argand diagram, modulus, polar form and De Moivre's theorem, all on Paper 1.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

A complex number has the form a + bi, where i² = -1.
The Argand diagram plots complex numbers like coordinates, with real on the x-axis and imaginary on the y-axis.
De Moivre's theorem is the fastest way to find powers and roots of complex numbers.
Complex numbers are examined on Paper 1.

Complex Numbers explained

What are Imaginary Numbers?

The imaginary unit $i$ is defined as:

$i = \sqrt{-1}$

This means: $i^2 = -1$

Why do we need this? Equations like $x^2 + 1 = 0$ have no real solutions, but with $i$ , we can write $x = \pm i$ .

Powers of i: - $i^1 = i$ - $i^2 = -1$ - $i^3 = -i$ - $i^4 = 1$ (and the pattern repeats)

Complex Numbers Definition

A complex number has the form:

$z = a + bi$

where: - $a$ is the real part (written as $\text{Re}(z)$ ) - $b$ is the imaginary part (written as $\text{Im}(z)$ ) - $a$ and $b$ are real numbers

Examples: - $3 + 4i$ (real part: 3, imaginary part: 4) - $-2 + 5i$ (real part: -2, imaginary part: 5) - $7$ (real part: 7, imaginary part: 0)

Example: Powers of i

Simplify $i^{17}$ :

Since $i^4 = 1$ , we divide the exponent by 4: $17 = 4 \times 4 + 1$

So: $i^{17} = i^{4 \times 4 + 1} = (i^4)^4 \cdot i^1 = 1^4 \cdot i = i$

Pattern: Divide exponent by 4, use remainder: - Remainder 0 → $i^0 = 1$ - Remainder 1 → $i^1 = i$ - Remainder 2 → $i^2 = -1$ - Remainder 3 → $i^3 = -i$

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Key formulas

Name	Formula	Description
Imaginary Unit	$i^2 = -1$	Definition of the imaginary unit i
Complex Number Form	$z = a + bi$	Standard form with real part a and imaginary part b
Modulus	$\|z\| = \sqrt{a^2 + b^2}$	Modulus (length) of complex number a+bi
Conjugate	$\bar{z} = a - bi$	Conjugate of complex number a+bi
Product with Conjugate	$z \cdot \bar{z} = \|z\|^2$	Product of complex number and its conjugate
Polar Form	$z = r(\cos\theta + i\sin\theta)$	Polar form with modulus r and argument θ

Worked examples

Worked example 1

What is $i^{50}$ ?

Divide 50 by 4: 50 = 4×12 + 2. The remainder is 2, so i^50 = i^2 = -1

Answer:

-1

Worked example 2

What is the modulus of $z = 5 + 12i$ ?

Use the formula |z| = √(a² + b²). Here: |5 + 12i| = √(5² + 12²) = √(25 + 144) = √169 = 13

Answer:

Where students lose marks

Forgetting that i² = -1 when multiplying out brackets.
Errors converting between rectangular (a + bi) and polar form.
Getting the argument in the wrong quadrant on the Argand diagram.

Frequently asked questions

When do I use De Moivre's theorem on the Leaving Cert?

De Moivre's theorem lets you raise a complex number in polar form to a power, and find its roots, quickly. It is essential for the harder complex number questions on Paper 1.

What is the modulus of a complex number?

The modulus is the distance of the complex number from the origin on the Argand diagram, found using Pythagoras' theorem on its real and imaginary parts.

Are complex numbers on Paper 1 or Paper 2?

Complex numbers are examined on Paper 1 of Leaving Cert Higher Level Maths.

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