Argand diagrams | Year 11 | Ascent Library

01 / LEARN

The idea, step by step

Two axes for one number

On an Argand diagram, $a + b i$ is plotted at the point $(a, b)$ . The horizontal axis records the real component and the vertical axis records the imaginary component. A point is not drawn at $(a, b i)$ ; both coordinates are real numbers.

The same diagram can represent a position vector from the origin. Label the axes Re and Im so a reader knows what the coordinates mean.

Read geometric changes

Conjugation changes $(a, b)$ to $(a, - b)$ , reflecting in the real axis. Negation changes it to $(- a, - b)$ , a half-turn about the origin. These are different operations.

To move from $z_{1} = a + b i$ to $z_{2} = c + d i$ , the horizontal change is $c - a$ and the vertical change is $d - b$ . A diagram helps check signs; an exact calculation establishes the result.

A common trap

Using an ordinary real number line to order all complex numbers from smallest to largest.

Instead: Complex numbers have two components. An Argand diagram locates them, but there is no usual less-than ordering for arbitrary complex numbers.

02 / WORKED EXAMPLES

See the thinking

Example 1

Plot and reflect

For $z = - 2 + 3 i$ , give coordinates for $z$ , $\bar{z}$ and $- z$ .

Read $z$ directly as real/imaginary coordinates.
Conjugate by reversing only the vertical coordinate.
Negate by reversing both coordinates.

(- 2, 3)

(- 2, - 3)

and

(2, - 3)

Example 2

Find a displacement

Point $A$ represents $1 - 2 i$ and $B$ represents $4 + 2 i$ . Find the displacement from $A$ to $B$ and its length.

Subtract initial from final: $(4 + 2 i) - (1 - 2 i) = 3 + 4 i$ .
The coordinate changes are $(3, 4)$ .
Pythagoras gives length $\sqrt{3^{2} + 4^{2}} = 5$ .

Displacement

3 + 4 i

; length

5

Check your understanding

What number is represented by $(- 4, 2)$ on an Argand diagram?

Show answer

$- 4 + 2 i$ .

The first coordinate gives the real part; the second is the coefficient of $i$ .

03 / WORKSHEET

Your turn

6 questions · 17 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

On an Argand diagram, plot $2 + 3 i$ , $- 4 + i$ and $- 2 i$ . State their coordinates.
3 marks
A point is three units left of the origin and four units below it. Write the complex number.
2 marks
For $z = 1 - 4 i$ , state the coordinates of $\bar{z}$ and $- z$ .
3 marks
Find the distance from $A = - 1 + i$ to $B = 3 + 4 i$ .
3 marks
A rectangle has vertices $0$ , $4$ , $4 + 3 i$ and $3 i$ . Find its perimeter and area.
3 marks
Describe the transformation that maps every point $z$ to $- \bar{z}$ .
3 marks

Read the worked solutions

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

On an Argand diagram, plot $2 + 3 i$ , $- 4 + i$ and $- 2 i$ . State their coordinates.
3 marks
Use the real part for the horizontal coordinate and the imaginary coefficient for the vertical coordinate.
$(2, 3)$ , $(- 4, 1)$ and $(0, - 2)$ .
A point is three units left of the origin and four units below it. Write the complex number.
2 marks
Left is negative real; below is negative imaginary.
$- 3 - 4 i$ .
For $z = 1 - 4 i$ , state the coordinates of $\bar{z}$ and $- z$ .
3 marks
Conjugation reverses the imaginary coordinate.
Negation reverses both coordinates.
$(1, 4)$ and $(- 1, 4)$ .
Find the distance from $A = - 1 + i$ to $B = 3 + 4 i$ .
3 marks
The horizontal and vertical differences are $4$ and $3$ .
Distance is $\sqrt{4^{2} + 3^{2}} = 5$ .
$5$ .
A rectangle has vertices $0$ , $4$ , $4 + 3 i$ and $3 i$ . Find its perimeter and area.
3 marks
The side lengths are $4$ and $3$ .
Perimeter $= 2 (4 + 3) = 14$ and area $= 4 \times 3 = 12$ .
Perimeter $14$ units; area $12$ square units.
Describe the transformation that maps every point $z$ to $- \bar{z}$ .
3 marks
If $z = a + b i$ , then $- \bar{z} = - a + b i$ .
The horizontal coordinate reverses while the vertical coordinate stays fixed.
Reflection in the imaginary axis.

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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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