Argand diagrams | Year 11 | Ascent Library

01 / LEARN

Understand the idea.

Two axes for one number

An Argand diagram plots $z = a + b i$ at the point $(a, b)$ . The horizontal axis is the real part and the vertical axis is the imaginary part. Both coordinates are real numbers; do not place $b i$ as the vertical coordinate.

The point also defines a position vector from the origin. Label the axes Re and Im, and use a consistent scale to interpret distances. A complex number on the horizontal axis has imaginary part zero.

Argand grid with real horizontal axis and imaginary vertical axis; z = 3 + 2i at (3,2); w = -2 - 3i at (-2,-3) The axes use equal physical scales.

Read geometric changes

Conjugation maps $(a, b)$ to $(a, - b)$ , a reflection in the real axis. Negation maps $(a, b)$ to $(- a, - b)$ , a half-turn about the origin. Compare which coordinates change to distinguish these operations.

The displacement from $z_{1}$ to $z_{2}$ is $z_{2} - z_{1}$ : final point minus initial point. Its components give horizontal and vertical movement. Use Pythagoras for its length, and use the diagram to check the direction and signs.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Plot and reflect

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

Argand grid with real horizontal axis and imaginary vertical axis; z at (-2,3); conjugate z at (-2,-3); -z at (2,-3) The axes use equal physical scales.

Worked solution

1
Read the original point.
$z = - 2 + 3 i (- 2, 3)$
The real coordinate is negative two and the imaginary coordinate is three.
2
Reflect by conjugating.
$\bar{z} = - 2 - 3 i (- 2, - 3)$
Reflection in the real axis reverses only the vertical coordinate.
3
Find the opposite point by negating.
$- z = 2 - 3 i (2, - 3)$
A half-turn about the origin reverses both coordinates.

Answer

$(- 2, 3)$ , $(- 2, - 3)$ and $(2, - 3)$ .

EXAMPLE 02

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.

Worked solution

1
Subtract the initial point from the final point.
$\begin{aligned} B - A & = (4 + 2 i) - (1 - 2 i) \\ = 3 + 4 i \end{aligned}$
Displacement from A to B is final position minus initial position.
2
Read the movement components.
$B - A (3, 4)$
The movement is three units right and four units upward.
3
Calculate the displacement length.
$∣ B - A ∣ = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5$
The horizontal and vertical displacements are perpendicular.

Answer

Displacement $3 + 4 i$ ; length $5$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$- 4 + 2 i$ .

The coordinates give real part negative four and imaginary coefficient two, so the represented complex number is $- 4 + 2 i$ .

Keep practising

6 worksheet questions · 17 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Two axes for one number

Read geometric changes

See the method in action.

Plot and reflect

Read the original point.

Reflect by conjugating.

Find the opposite point by negating.

Find a displacement

Subtract the initial point from the final point.

Read the movement components.

Calculate the displacement length.

Try it yourself.

Check your understanding

Keep practising