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AS Level · Further Pure Mathematics 1 WFM01

Operations with Complex Numbers

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Edexcel IAL · Further Pure 1

Operations with Complex Numbers

Edexcel International AS · Further Maths: Further Pure 1

Big idea: When real numbers aren't enough to solve equations, we invent i = √(−1), and every number becomes a pair (real part, imaginary part) that we can add, multiply, divide, and plot.

Chapter Overview

Imaginary Unit

i² = −1 solves x² = −1; pure imaginary numbers live on the Im axis.

Arithmetic

Add, subtract, multiply and divide complex numbers using algebra rules plus i² = −1.

Conjugates

z* flips the sign of the imaginary part. Use it to realise (clear) complex denominators.

Argand Diagrams

Plot z = x + yi as the point (x, y) on a 2-D grid — real axis horizontal, imaginary vertical.

Modulus & Argument

|z| = distance from origin; arg z = angle to positive real axis (−π < arg z ≤ π).

Polar Form

z = r(cos θ + i sin θ) re-expresses a complex number using its size and direction.

Equating Parts

If two complex numbers are equal, their real parts must match AND their imaginary parts must match.

1 · Introduction to Complex Numbers

Why do we need complex numbers?

Consider x² = −9. Squaring any real number gives a positive result, so this equation has no real solutions — but it does have solutions if we allow a new kind of number.

i² = −1 ⟹ i = √(−1)

The imaginary unit

Using this: x² = −9 = 9 × i², so x = ±3i. Problem solved.

Don't confuse: i itself is not a real number. Writing 25i can look like 25 — space your working clearly or write (25)i instead.

Cartesian Form: z = x + yi

A complex number has two parts that sit side-by-side:

Real part — written Re(z) = x
Imaginary part — written Im(z) = y (note: y itself is a real number; it's multiplied by i to make it imaginary)

z = x + yi

Standard Cartesian Form (x, y ∈ ℝ)

If y = 0, then z = x is a real number. So ℝ ⊂ ℂ (reals are a subset of complex numbers).

If x = 0, then z = yi is a pure imaginary number, like 4i or −3i.

Arithmetic with Complex Numbers

Adding / Subtracting: combine real parts together, imaginary parts together.

Worked Example

(3 + 4i) + (2 + 8i) and (3 + 4i) − (2 + 8i)

Add: (3+2) + (4+8)i = 5 + 12i

Subtract: (3−2) + (4−8)i = 1 − 4i

Multiplying / Dividing by a real number: apply it to both parts separately.

Worked Example

10(3 + 4i) and (3 + 4i) ÷ 10

Multiply: 30 + 40i

Divide: 0.3 + 0.4i (or 3/10 + 4/10 i)

Practice Question 1

Given that z₁ = p + 2i and z₂ = −7 + qi, where p and q are real constants, and that z₁ + 2z₂ = 4 − 8i, find the values of p and q.

2 · Multiplying Complex Numbers

Expanding brackets — always use i² = −1

Multiply like any two brackets (FOIL), then replace every i² with −1.

(a + bi)(c + di) = (ac − bd) + (ad + bc)i

General multiplication rule (derived from i² = −1)

Special case: (a + bi)(a − bi) = a² + b² (always a real, positive result — no i terms!)

Worked Example — (4 + i)(2 + 9i)

Expand: 8 + 36i + 2i + 9i²

Replace i² = −1: 8 + 38i + 9(−1) = 8 − 9 + 38i

Worked Example — (3 − 4i)²

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Also in the full note

Powers of i — a cycling pattern
The Complex Conjugate z*
Dividing Complex Numbers — "Realising the Denominator"
Plotting complex numbers as points
Modulus — how far from the origin?
Useful modulus rules
Argument — which direction from the origin?
Converting to Polar Form

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Operations with Complex Numbers

Operations with Complex Numbers

Why do we need complex numbers?

Cartesian Form: z = x + yi

Arithmetic with Complex Numbers

Expanding brackets — always use i² = −1

More Further Pure Mathematics 1 topics