Real and imaginary parts | Year 11

01 / LEARN

The idea, step by step

Cartesian form

A complex number has the form $z = a + b i$ , where $a$ and $b$ are real. The real part is $Re (z) = a$ and the imaginary part is $Im (z) = b$ . The imaginary part is the real coefficient of $i$ , not the whole term $b i$ .

Real numbers have imaginary part zero. A nonzero pure imaginary number has real part zero. For example, $- 5 = - 5 + 0 i$ and $7 i = 0 + 7 i$ . Zero has both components zero; handle it separately when a question asks for a nonzero pure imaginary number.

Equality uses both components

If $a + b i = c + d i$ with real $a, b, c, d$ , then $a = c$ and $b = d$ . Comparing components turns one complex equation into two real equations.

First expand and simplify before comparing. In particular, a factor of $i$ can change both components because $i (b i) = - b$ .

A common trap

Writing $Im (2 - 5 i) = - 5 i$ .

Instead: The imaginary part is the coefficient $- 5$ ; the imaginary term is $- 5 i$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

Read a reordered expression

Find the real and imaginary parts of $z = 3 i - 8$ .

Rewrite as $z = - 8 + 3 i$ .
Read the constant component and the coefficient of $i$ separately.

Re (z) = - 8

Im (z) = 3

Example 2

Find real unknowns

Real numbers $x, y$ satisfy $(x + 2) + (3 - y) i = 7 - 4 i$ . Find $x, y$ .

Compare real parts: $x + 2 = 7$ , giving $x = 5$ .
Compare imaginary parts: $3 - y = - 4$ , giving $y = 7$ .
Substitution produces $7 - 4 i$ , as required.

x = 5

y = 7

Check your understanding

Find $Re (- i)$ and $Im (- i)$ .

Show answer

$0$ and $- 1$ .

Write $- i = 0 + (- 1) i$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State the real and imaginary parts of $- 6 + 9 i$ .
2 marks
State the real and imaginary parts of $4 - 3 i + 2 i$ .
2 marks
Classify $- 8$ , $6 i$ and $2 + i$ as real, pure imaginary, or neither.
3 marks
For real $x, y$ , solve $x + (y - 2) i = 3 + 5 i$ .
2 marks
For real $x, y$ , solve $(4 - 2 i) + (x + y i) = 7 + 3 i$ .
3 marks
If $z = 3 - 4 i$ , find the real and imaginary parts of $i z$ .
3 marks

Read the worked solutions

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

State the real and imaginary parts of $- 6 + 9 i$ .
2 marks
Compare with $a + b i$ .
Real part $- 6$ ; imaginary part $9$ .
State the real and imaginary parts of $4 - 3 i + 2 i$ .
2 marks
$- 3 i + 2 i = - i$ , so the number is $4 - i$ .
Real part $4$ ; imaginary part $- 1$ .
Classify $- 8$ , $6 i$ and $2 + i$ as real, pure imaginary, or neither.
3 marks
A real number has coefficient of $i$ equal to zero.
A pure imaginary number has real part zero.
$- 8$ is real; $6 i$ is pure imaginary; $2 + i$ is neither.
For real $x, y$ , solve $x + (y - 2) i = 3 + 5 i$ .
2 marks
Real parts give $x = 3$ .
Imaginary parts give $y - 2 = 5$ .
$x = 3$ , $y = 7$ .
For real $x, y$ , solve $(4 - 2 i) + (x + y i) = 7 + 3 i$ .
3 marks
Real parts: $4 + x = 7$ .
Imaginary parts: $- 2 + y = 3$ .
$x = 3$ , $y = 5$ .
If $z = 3 - 4 i$ , find the real and imaginary parts of $i z$ .
3 marks
$i z = 3 i - 4 i^{2} = 4 + 3 i$ .
Read the simplified Cartesian form.
Real part $4$ ; imaginary part $3$ .

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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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