Year 12: Complex numbers

Complex numbers extend familiar algebra into a new number system where $i^2 = - 1$ . Most questions are straightforward if you stay organised with real and imaginary parts.

Treat complex numbers like ordered pairs of information: one real part and one imaginary part. Keep those parts aligned through every line of working.

What to master

Adding, subtracting and multiplying complex numbers cleanly.
Using powers of $i$ and spotting the repeating cycle.
Simplifying powers of more complicated complex expressions.
Writing final answers in the form $a + b i$ .

Exam habit: if you are raising

i

to a large power, reduce the exponent modulo 4 immediately. It saves time and avoids sign errors.

Complex Numbers tutorials

How to stay organised

Complex-number questions are usually more orderly than they first appear. The main rule is to keep real and imaginary parts separated. If you write each step in the form $a + b i$ , the algebra stays readable and errors are easier to catch. This also helps when powers of $i$ are involved, because you can simplify the imaginary part cycle without disturbing the real coefficients. Once the expression is organised, the topic is often simpler than it looks.

Worked example

Problem: Simplify

(2 + 3 i) (1 - 4 i)

Expand carefully: $2 (1) + 2 (- 4 i) + 3 i (1) + 3 i (- 4 i)$ .
This gives $2 - 8 i + 3 i - 12 i^{2}$ .
Use $i^{2} = - 1$ : $- 12 i^{2} = 12$ .
Combine real and imaginary parts: $14 - 5 i$ .

Answer:

(2 + 3 i) (1 - 4 i) = 14 - 5 i

Common traps

Replacing $i^{2}$ with $1$ instead of $- 1$ .
Leaving the final answer in an expanded but unsimplified form.
Reducing large powers of $i$ incorrectly by forgetting the 4-step cycle.

Study routine

A good check is to box the real part and underline the imaginary part during your working. That might feel old-fashioned, but it forces you to see whether every term has been classified correctly. In a short topic with repetitive structure, clean notation is a real advantage.

Another strong habit is to reduce powers of $i$ immediately. Because the pattern repeats every four powers, expressions that look intimidating often collapse quickly once that cycle is used. In practice, complex numbers reward calm bookkeeping more than deep theory, so presentation and consistency matter a great deal.

Revision focus

These topics reward careful structure more than fast calculation. In complex numbers, discrete mathematics, graph theory, and number theory, the key objects carry meaning that should stay visible all the way through the working. Real and imaginary parts, matrix entries, graph properties, and modular statements all behave best when they are written clearly and interpreted consistently.

A reliable way to revise is to explain each line of working in words as you go. If you can say what a power of i is doing, what a matrix row represents, what a graph property implies, or what a congruence statement means, then the notation is working for you rather than against you. That level of clarity usually prevents the small symbolic errors that turn easy marks into lost marks.