Complex-number arithmetic | Year 11

01 / LEARN

Understand the idea.

Addition and subtraction

Add complex numbers by collecting like components: $(a + b i) + (c + d i) = (a + c) + (b + d) i$ . Real and imaginary terms stay separate during addition, just as coefficients of the same algebraic term combine.

For subtraction, put brackets around the second number before distributing the minus sign. For example, subtracting $c + d i$ subtracts both $c$ and $d i$ . Then combine the two real terms and the two imaginary terms.

Multiplication and powers

For multiplication, expand all four products, then replace $i^{2}$ by $- 1$ and collect components. In particular, $(b i) (d i) = - b d$ contributes to the real part. Multiplying only corresponding components leaves out the cross terms.

The powers of $i$ cycle every four: $i, - 1, - i, 1$ . For a non-negative integer exponent, separate complete groups of four and retain the remainder. A remainder of zero gives one, not zero.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A product with four terms

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

Worked solution

1
Expand all four products.
$(2 + i) (3 - 2 i) = 6 - 4 i + 3 i - 2 i^{2}$
Every term in the first bracket multiplies every term in the second.
2
Replace the square of i.
$6 - 4 i + 3 i - 2 i^{2} = 6 - 4 i + 3 i + 2$
Since $i^{2} = - 1$ , the term $- 2 i^{2}$ becomes positive two.
3
Collect real and imaginary terms.
$(6 + 2) + (- 4 + 3) i = 8 - i$
The real components add to eight and the imaginary coefficient is negative one.

Answer

$8 - i$ .

EXAMPLE 02

Use the power cycle

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

Worked solution

1
Reduce the first exponent by groups of four.
$37 = 4 (9) + 1, i^{37} = {(i^{4})}^{9} i = i$
Each complete group contributes a factor of one.
2
Reduce the second exponent.
$18 = 4 (4) + 2, i^{18} = {(i^{4})}^{4} i^{2} = - 1$
The remainder two selects the value $i^{2} = - 1$ .
3
Add in Cartesian form.
$i^{37} + i^{18} = i - 1 = - 1 + i$
Write the real component before the imaginary component.

Answer

$- 1 + i$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$- 8 - 6 i$ .

Expand the full square: ${(1 - 3 i)}^{2} = 1 - 6 i + 9 i^{2}$ . Replacing $i^{2}$ by $- 1$ gives $1 - 6 i - 9 = - 8 - 6 i$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Addition and subtraction

Multiplication and powers

See the method in action.

A product with four terms

Expand all four products.

Replace the square of i.

Collect real and imaginary terms.

Use the power cycle

Reduce the first exponent by groups of four.

Reduce the second exponent.

Add in Cartesian form.

Try it yourself.

Check your understanding

Keep practising