Complex arithmetic in both forms | Year 12

01 / LEARN

Understand the idea.

Addition follows coordinates

Use Cartesian form for addition and subtraction: combine real parts with real parts and imaginary parts with imaginary parts. Their coordinates add, but the lengths and angles of the corresponding arrows usually do not.

For products, expand brackets and replace $i^{2}$ by $- 1$ . For division by a nonzero complex number, multiply both numerator and denominator by its conjugate. The denominator then becomes $w \bar{w} = ∣ w ∣^{2}$ , a positive real number.

Multiplication follows length and angle

In polar form, multiplication scales lengths and adds rotations: multiply the moduli and add the arguments. Division reverses these operations: divide the moduli and subtract the denominator's argument. The divisor must be nonzero.

Choose a representation to suit the operation. A product of several polar numbers avoids repeated bracket expansion; a sum is usually clearer after conversion to Cartesian form. Reduce a final angle when a specified argument interval requires it.

\begin{array}{c} \frac{r cis α}{s cis β} = \frac{r}{s} cis (α - β) \\ s > 0 \end{array}

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Divide with a conjugate

Simplify $\frac{3 + i}{1 - 2 i}$ .

Worked solution

1
Multiply by the conjugate of the denominator.
$\frac{3 + i}{1 - 2 i} = \frac{(3 + i) (1 + 2 i)}{(1 - 2 i) (1 + 2 i)}$
The multiplier is a fraction equal to one, with a nonzero denominator.
2
Expand and replace the square of i.
$\begin{aligned} (3 + i) (1 + 2 i) & = 3 + 7 i + 2 i^{2} \\ = 1 + 7 i \end{aligned}$
The identity $i^{2} = - 1$ converts the quadratic term to a real number.
3
Evaluate the real denominator.
$(1 - 2 i) (1 + 2 i) = 1 - {(2 i)}^{2} = 5$
A complex number multiplied by its conjugate has squared modulus.
4
Separate real and imaginary parts.
$\frac{1 + 7 i}{5} = \frac{1}{5} + \frac{7}{5} i$
Division by a real scalar divides each component.

Answer

$\frac{1}{5} + \frac{7}{5} i$ .

EXAMPLE 02

Use exact polar multiplication

Calculate $(2 cis (\frac{π}{3})) (3 cis (- \frac{π}{6}))$ .

Worked solution

1
Multiply the moduli.
$r = 2 \times 3 = 6$
Lengths multiply under complex multiplication.
2
Add the arguments.
$θ = \frac{π}{3} - \frac{π}{6} = \frac{π}{6}$
Successive rotations combine by adding their signed angles.
3
Convert the resulting polar number.
$\begin{aligned} 6 cis \frac{π}{6} & = 6 (\frac{\sqrt{3}}{2} + \frac{1}{2} i) \\ = 3 \sqrt{3} + 3 i \end{aligned}$
The cosine gives the real component and the sine gives the imaginary component.

Answer

$6 cis (\frac{π}{6}) = 3 \sqrt{3} + 3 i$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $\frac{4 cis (\frac{π}{2})}{2 cis (\frac{π}{6})}$ .

Show answer & reasoning

$2 cis (\frac{π}{3})$ .

The quotient has modulus $4 / 2 = 2$ and argument $\frac{π}{2} - \frac{π}{6} = \frac{π}{3}$ , giving $2 cis (\frac{π}{3})$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Calculate $(2 - 3 i) + (1 + 5 i)$ .
2 marks
Expand $(2 + i) (3 - 2 i)$ .
2 marks
Simplify $\frac{2 - i}{1 + i}$ .
2 marks
Calculate $\frac{4 cis (\frac{2 π}{3})}{2 cis (- \frac{π}{6})}$ .
2 marks
Find $2 cis (\frac{π}{3}) + 2 cis (- \frac{π}{3})$ .
2 marks
If $z w = 6 i$ and $z = 1 + i$ , find $w$ .
3 marks

Read the worked solutions

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

Calculate $(2 - 3 i) + (1 + 5 i)$ .
2 marks
Add real parts $2 + 1 = 3$ and imaginary parts $- 3 + 5 = 2$ .
$3 + 2 i$ .
Expand $(2 + i) (3 - 2 i)$ .
2 marks
$6 - 4 i + 3 i - 2 i^{2} = 6 - i + 2 = 8 - i$ .
$8 - i$ .
Simplify $\frac{2 - i}{1 + i}$ .
2 marks
Multiply by $\frac{1 - i}{1 - i}$ : numerator $1 - 3 i$ , denominator 2.
$\frac{1}{2} - \frac{3}{2} i$ .
Calculate $\frac{4 cis (\frac{2 π}{3})}{2 cis (- \frac{π}{6})}$ .
2 marks
Divide moduli to get 2; subtract angles to get $\frac{5 π}{6}$ .
$2 cis (\frac{5 π}{6}) = - \sqrt{3} + i$ .
Find $2 cis (\frac{π}{3}) + 2 cis (- \frac{π}{3})$ .
2 marks
The terms are $1 + \sqrt{3} i$ and $1 - \sqrt{3} i$ , so the imaginary parts cancel.
$2$ .
If $z w = 6 i$ and $z = 1 + i$ , find $w$ .
3 marks
$w = \frac{6 i}{1 + i} = \frac{6 i (1 - i)}{2} = 3 + 3 i$ .
$3 + 3 i$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 Specialist Mathematics complex numbers: Cartesian and polar representations, de Moivre, roots and roots of unity, Argand geometry and loci, complex factorisation and conjugate roots of real polynomials. Real-polynomial techniques are taught in the separate Real Polynomials chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Addition follows coordinates

Multiplication follows length and angle

See the method in action.

Divide with a conjugate

Multiply by the conjugate of the denominator.

Expand and replace the square of i.

Evaluate the real denominator.

Separate real and imaginary parts.

Use exact polar multiplication

Multiply the moduli.

Add the arguments.

Convert the resulting polar number.

Try it yourself.

Check your understanding

Keep practising