The Modulus (Leaving Cert Mathematics): Revision Notes

The Modulus

What is the modulus?

The modulus of a complex number is a way of measuring its "size" or distance. Think of it as finding how far a complex number is from the starting point (origin) when you plot it on a coordinate system called an Argand diagram.

When we have a complex number written in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, the modulus tells us the straight-line distance from the origin $(0, 0)$ to the point $(a, b)$ on the Argand diagram.

Formula and notation

The modulus of a complex number $z = a + bi$ is calculated using the formula:

$|z| = |a + bi| = \sqrt{a^2 + b^2}$

Key points about the notation:

• We write the modulus using absolute value bars: $|z|$ or $|a + bi|$
• The result is always a positive real number (or zero)
• This formula comes from Pythagoras' theorem

Geometric interpretation

On an Argand diagram, the modulus represents the length of the line segment from the origin to the point representing the complex number. This forms the hypotenuse of a right triangle where:

• The horizontal side has length $|a|$
• The vertical side has length $|b|$
• The hypotenuse has length $\sqrt{a^2 + b^2}$ (the modulus)

Calculating the modulus step by step

To find the modulus of any complex number $a + bi$:

1. Identify the real part $a$ and imaginary part $b$
2. Square both parts: $a^2$ and $b^2$
3. Add the squares together: $a^2 + b^2$
4. Take the square root: $\sqrt{a^2 + b^2}$

Worked examples

Important properties

Triangle inequality

Key exam tips

Remember!