Real and imaginary parts | Year 11

01 / LEARN

Understand the idea.

Cartesian form

In Cartesian form, $z = a + b i$ with real $a, b$ . The real part is $Re (z) = a$ and the imaginary part is $Im (z) = b$ . Both parts are real numbers; $b i$ is the imaginary term, rather than the imaginary part.

A missing component has coefficient zero: $- 5 = - 5 + 0 i$ and $7 i = 0 + 7 i$ . Nonzero pure imaginary numbers have real part zero. Zero has both parts zero, so check whether a question explicitly excludes it.

Equality uses both components

Two complex numbers are equal exactly when their real parts match and their imaginary parts match. Thus $a + b i = c + d i$ gives two real equations, $a = c$ and $b = d$ . Solve both because each controls a separate component.

Expand and collect terms before comparing components. Multiplication by $i$ can move a contribution into the real part: $i (b i) = - b$ . After finding unknown real coefficients, substitute them back to check both components together.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Read a reordered expression

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

Worked solution

1
Rewrite with the real component first.
$z = 3 i - 8 = - 8 + 3 i$
Reordering addition makes the Cartesian components easy to identify.
2
Read the two real coefficients.
$Re (z) = - 8, Im (z) = 3$
The imaginary part is the coefficient of i, without i attached.

Answer

$Re (z) = - 8$ , $Im (z) = 3$ .

EXAMPLE 02

Find real unknowns

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

Worked solution

1
Compare real components.
$x + 2 = 7 ⟹ x = 5$
Equality requires the real parts on both sides to agree.
2
Compare imaginary coefficients.
$3 - y = - 4 ⟹ y = 7$
The coefficient of i on the left must equal negative four.
3
Check the combined expression.
$(5 + 2) + (3 - 7) i = 7 - 4 i$
Both component equations must be satisfied by the same pair of values.

Answer

$x = 5$ , $y = 7$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$0$ and $- 1$ .

Write $- i = 0 + (- 1) i$ . Therefore $Re (- i) = 0$ and $Im (- i) = - 1$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Cartesian form

Equality uses both components

See the method in action.

Read a reordered expression

Rewrite with the real component first.

Read the two real coefficients.

Find real unknowns

Compare real components.

Compare imaginary coefficients.

Check the combined expression.

Try it yourself.

Check your understanding

Keep practising