Complex-number arithmetic | Year 11

01 / LEARN

The idea, step by step

Addition and subtraction

Add real components to real components and imaginary components to imaginary components. The calculation $(a + b i) + (c + d i)$ becomes $(a + c) + (b + d) i$ .

When subtracting, distribute the minus sign through the whole second number. Writing an intermediate bracket prevents the common error of changing only its real part.

Multiplication and powers

Expand products in the same way as real algebra, then replace every $i^{2}$ by $- 1$ . Multiplication mixes the real and imaginary components, so multiplying corresponding components alone is not valid.

The powers repeat: $i, i^{2}, i^{3}, i^{4} = i, - 1, - i, 1$ . Reduce a large non-negative exponent modulo four, or group powers into blocks of $i^{4} = 1$ .

A common trap

Claiming $(a + b i) (c + d i) = a c + b d i$ .

Instead: Expand all four products. The product of the imaginary terms contributes $- b d$ to the real part.

02 / WORKED EXAMPLES

See the thinking

Example 1

A product with four terms

Calculate $(2 + i) (3 - 2 i)$ .

Expand: $6 - 4 i + 3 i - 2 i^{2}$ .
Use $i^{2} = - 1$ to replace $- 2 i^{2}$ by $2$ .
Collect terms: $6 + 2 - i = 8 - i$ .

8 - i

Example 2

Use the power cycle

Calculate $i^{37} + i^{18}$ .

$37 = 4 (9) + 1$ , so $i^{37} = i$ .
$18 = 4 (4) + 2$ , so $i^{18} = i^{2} = - 1$ .
Add and write in Cartesian form.

- 1 + i

Check your understanding

Calculate ${(1 - 3 i)}^{2}$ .

Show answer

$- 8 - 6 i$ .

Expansion gives $1 - 6 i + 9 i^{2} = 1 - 6 i - 9$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Calculate $(5 + i) - (2 - 4 i)$ .
2 marks
Calculate $(2 - 3 i) (- 1 + 2 i)$ .
3 marks
Expand ${(3 + 2 i)}^{2}$ .
2 marks
Find $i^{2026}$ .
2 marks
Solve $2 z + 3 i = 8 - i$ .
2 marks
Evaluate ${(1 + i)}^{4}$ without using a calculator.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $(5 + i) - (2 - 4 i)$ .
2 marks
$5 + i - 2 + 4 i = 3 + 5 i$ .
$3 + 5 i$ .
Calculate $(2 - 3 i) (- 1 + 2 i)$ .
3 marks
Expand to $- 2 + 4 i + 3 i - 6 i^{2}$ .
Replace $i^{2}$ by $- 1$ and collect.
$4 + 7 i$ .
Expand ${(3 + 2 i)}^{2}$ .
2 marks
$9 + 12 i + 4 i^{2} = 9 + 12 i - 4$ .
$5 + 12 i$ .
Find $i^{2026}$ .
2 marks
$2026 = 4 (506) + 2$ , so $i^{2026} = i^{2} = - 1$ .
$- 1$ .
Solve $2 z + 3 i = 8 - i$ .
2 marks
$2 z = 8 - 4 i$ .
Divide both components by $2$ .
$z = 4 - 2 i$ .
Evaluate ${(1 + i)}^{4}$ without using a calculator.
3 marks
${(1 + i)}^{2} = 1 + 2 i + i^{2} = 2 i$ .
Square again: ${(2 i)}^{2} = 4 i^{2} = - 4$ .
$- 4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, real and complex numbers: Cartesian arithmetic, conjugates, quadratic roots and Argand geometry. Polar form and roots of unity are developed in Year 12.

Official curriculum source ↗

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