Complex Numbers for Leaving Cert Maths: A Clear Guide
Quick answer: A complex number has the form z = a + bi, where i² = −1. Add and subtract by combining real and imaginary parts. Multiply like brackets, then replace i² with −1. To divide, multiply top and bottom by the conjugate of the bottom. The modulus |a + bi| = √(a² + b²) is the distance from the origin on the Argand diagram.
The basics
i² = −1, so √−9 = 3i.
In z = a + bi, a is the real part and b is the imaginary part.
On the Argand diagram, plot a across and b up.
Adding and subtracting
(3 + 2i) + (1 − 5i)
Real: 3 + 1 = 4. Imaginary: 2 − 5 = −3. Answer: 4 − 3i.
Multiplying
(2 + 3i)(1 − i)
= 2 − 2i + 3i − 3i²
= 2 + i + 3 (since −3i² = +3)
= 5 + i
The conjugate
The conjugate of a + bi is a − bi. Multiplying a number by its conjugate always gives a real number:
(3 + 4i)(3 − 4i) = 9 + 16 = 25.
Dividing
(5 + i) ÷ (1 − i)
Multiply top and bottom by the conjugate, 1 + i:
Top: (5 + i)(1 + i) = 5 + 5i + i + i² = 4 + 6i
Bottom: (1 − i)(1 + i) = 1 + 1 = 2
Answer: 2 + 3i. (Check: (2 + 3i)(1 − i) = 5 + i, from the example above.)
Modulus
Find |3 + 4i|.
√(3² + 4²) = √25 = 5.
Quadratics with complex roots
Solve x² + 4x + 13 = 0.
Discriminant: 16 − 52 = −36. √−36 = 6i.
x = (−4 ± 6i) ÷ 2 = −2 ± 3i.
Complex roots of a quadratic with real coefficients always come as a conjugate pair.
Exam tips
Replace i² with −1 straight away.
Check divisions by multiplying back.
Label the axes on an Argand diagram: Re and Im.
Tips are MMS editorial guidance.
Common mistakes
Leaving i² in the answer.
Multiplying only the top by the conjugate. Do both.
Forgetting the square root in the modulus.
When extra support might help
Support is worth considering when algebra with i feels confusing, or before the mocks.
A grind can't guarantee a grade, but can build method and confidence. Math Make Smart offers online maths grinds and Leaving Cert support, and you can book a free grind. See also our Leaving Cert probability guide.
Frequently asked questions
What is i in maths?
The number whose square is −1.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom.
What is the modulus of a complex number?
Its distance from 0 on the Argand diagram: √(a² + b²).