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hscscience Ext 2 · Y12
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Module 12 · L05 of 16 ~40 min ⚡ +90 XP available

The Argand Diagram

A complex number $z = a + bi$ carries two pieces of information, a real part and an imaginary part. Plotting it on the Argand diagram (the complex plane) turns arithmetic into geometry: addition becomes a parallelogram, subtraction becomes a directed segment, and the modulus becomes a length. Every later technique in Module 12, polar form, multiplication by rotation, loci, sits on top of this picture.

Today's hook, Let $z_1 = 3 + 2i$ and $z_2 = -1 + 4i$. Before reading on, sketch both on an Argand diagram. Then sketch $z_1 + z_2$ and $z_1 - z_2$ as vectors. Predict: which of these has the longer length? Compare your answer after card 05.
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Recall, your gut answer first

+5 XP warm-up You already know how to plot the point in the Cartesian plane.

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Recall, your gut answer first
+5 XP warm-up

You already know how to plot the point $(3, 2)$ in the Cartesian plane. Before checking describe how plotting the complex number $z = 3 + 2i$ on the Argand diagram is similar, and how (if at all) it is different. What do the two axes represent?

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The two moves for the Argand diagram

+5 XP to read Every Argand-diagram question rewards two habits: read $z$ as an ordered pair $(\text{Re}\,z,\;\text{Im}\,z)$, then decide whether you want the point picture or the vector picture.

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The two moves for the Argand diagram
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Every Argand-diagram question rewards two habits: read $z$ as an ordered pair $(\text{Re}\,z,\;\text{Im}\,z)$, then decide whether you want the point picture or the vector picture. The point picture is best for plotting and loci; the vector picture is best for addition, subtraction and translation.

The point-or-vector reading: (1) extract $a = \text{Re}\,z$ and $b = \text{Im}\,z$, (2) plot the point $(a, b)$ on axes labelled Re and Im, (3) optionally draw the arrow from $0$ to that point, the position vector of $z$.

$z = a + bi \;\leftrightarrow\; (a,b) \;\leftrightarrow\; \vec{OZ}$

The complex number z = a + bi is both the point (a, b) on the Argand plane and the position vector from the origin to it.
$z = a + bi \;\leftrightarrow\; (a, b)$
Re horizontal, Im vertical
By NESA convention the horizontal axis is the real axis and the vertical axis is the imaginary axis. Label both, examiners look for this.
Addition is the parallelogram
$z_1 + z_2$ is the diagonal of the parallelogram with sides $\vec{OZ_1}$ and $\vec{OZ_2}$. Components add: $(a_1 + a_2, b_1 + b_2)$.
Subtraction is the segment
$z_1 - z_2$ is the vector from $Z_2$ to $Z_1$ (translated to start at $O$). Its length is the distance between the two points.
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What you'll master
Know

Key facts

  • The Argand diagram has a real axis (horizontal) and an imaginary axis (vertical)
  • $z = a + bi$ corresponds to the point $(a, b)$ and the position vector $\vec{OZ}$
  • $z_1 + z_2$ adds componentwise; geometrically it is the parallelogram law
  • $z_1 - z_2$ is the vector from $Z_2$ to $Z_1$
Understand

Concepts

  • Why complex numbers can be represented as points OR as vectors
  • Why $|z_1 - z_2|$ is the distance between two complex points
  • Why a complex number and its conjugate are mirror images in the real axis
Can do

Skills

  • Plot any complex number on an Argand diagram with labelled axes
  • Draw $z_1 + z_2$ and $z_1 - z_2$ as vectors and verify with components
  • Identify the geometric relationship between $z$, $-z$, $\bar z$ and $-\bar z$
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Key terms
Argand diagramThe plane used to represent complex numbers: horizontal axis is the real axis, vertical axis is the imaginary axis. Also called the complex plane.
Real part ($\text{Re}\,z$)For $z = a + bi$, $\text{Re}\,z = a$. It is the horizontal coordinate of $z$ on the Argand diagram.
Imaginary part ($\text{Im}\,z$)For $z = a + bi$, $\text{Im}\,z = b$ (note: $b$, not $bi$). It is the vertical coordinate of $z$.
Position vector $\vec{OZ}$The arrow from the origin $O$ to the point $Z = (a, b)$ representing $z = a + bi$. Often used to depict addition.
Modulus $|z|$The length of the position vector: $|z| = \sqrt{a^2 + b^2}$. Geometrically it is the distance from $O$ to $Z$.
Conjugate $\bar z$For $z = a + bi$, $\bar z = a - bi$. On the Argand diagram, $\bar z$ is the reflection of $z$ in the real axis.
MEX-N1NESA outcome (Complex Numbers I): uses the Argand diagram to represent complex numbers as points and vectors and to perform addition and subtraction geometrically.
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The complex plane and plotting points

The Argand diagram is the Cartesian plane with one relabelling: the horizontal axis is the real axis () and the vertical axis...

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The complex plane and plotting points
core concept

The Argand diagram is the Cartesian plane with one relabelling: the horizontal axis is the real axis ($\text{Re}$) and the vertical axis is the imaginary axis ($\text{Im}$). A complex number $z = a + bi$ is plotted at the point with horizontal coordinate $a$ and vertical coordinate $b$, i.e., at $(a, b)$.

  • Pure real numbers ($b = 0$) lie on the real axis.
  • Pure imaginary numbers ($a = 0$) lie on the imaginary axis.
  • The origin is $0 = 0 + 0i$.
  • The conjugate $\bar z = a - bi$ is the reflection of $z$ in the real axis.
  • The negative $-z = -a - bi$ is the reflection of $z$ through the origin.

Worked through the hook: $z_1 = 3 + 2i$ plots at $(3, 2)$, three units right, two units up. $z_2 = -1 + 4i$ plots at $(-1, 4)$, one unit left, four units up.

Connecting to later work. Every result in Module 12, modulus, argument, polar form, de Moivre, is read off the Argand diagram. Investing time in clean, labelled diagrams pays off in every later question.

Axes: horizontal = Re, vertical = Im (label both) · $z = a + bi$ is the point $(a, b)$ · $\bar z$ reflects $z$ in the real axis; $-z$ reflects through the origin · Pure real on the Re axis; pure imaginary on the Im axis

Pause, copy the Argand-plane axis labels (Re horizontal, Im vertical), the reflection rules for $\bar{z}$ and $-z$, and the positions of pure real and pure imaginary numbers into your book.

Quick check: On the Argand diagram, the complex number $z = -2 + 5i$ is plotted at which point?

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Vectors: addition and subtraction

We just saw that maps to the point in the Argand plane, with reflected in the real axis and reflected through the origin.

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Vectors: addition and subtraction
core concept

We just saw that $z = a+bi$ maps to the point $(a,b)$ in the Argand plane, with $\bar{z}$ reflected in the real axis and $-z$ reflected through the origin. That raises a question: how does complex addition look geometrically on the Argand plane? This card answers it → $z_1 + z_2$ obeys the parallelogram law (components add), and $z_1 - z_2$ is the directed segment from $Z_2$ to $Z_1$.

Every complex number $z$ can also be drawn as the position vector $\vec{OZ}$, an arrow from the origin to the point representing $z$. The vector picture is what makes complex arithmetic geometric.

  • Addition. $z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i$, components add. Geometrically, draw $\vec{OZ_1}$ and $\vec{OZ_2}$; complete the parallelogram. The diagonal from $O$ is $z_1 + z_2$.
  • Subtraction. $z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i$, components subtract. Geometrically, $z_1 - z_2$ is the vector pointing from $Z_2$ to $Z_1$ (translated to start at $O$).

Distance. Because $z_1 - z_2$ is the displacement from $Z_2$ to $Z_1$, the modulus $|z_1 - z_2|$ is the distance between the two points. This is the most-used identity in Module 12 loci.

$$z_1 \pm z_2 = (a_1 \pm a_2) + (b_1 \pm b_2)i \qquad |z_1 - z_2| = \text{distance between } Z_1 \text{ and } Z_2$$
Common mistake. The arrow for $z_1 - z_2$ as drawn on the diagram goes from $Z_2$ to $Z_1$, not from $Z_1$ to $Z_2$. The subtraction arrow points "to the first, from the second".

$z_1 + z_2$: parallelogram law, components add · $z_1 - z_2$: directed segment from $Z_2$ to $Z_1$, components subtract · $|z_1 - z_2|$ = distance between $Z_1$ and $Z_2$ · Direction of the subtraction arrow: "to the first, from the second"

Pause, copy the parallelogram law for addition, the subtraction-as-directed-segment rule (from $Z_2$ to $Z_1$), and the distance formula $|z_1-z_2|$ into your book.

Did you get this? True or false: if $z_1 = 4 + i$ and $z_2 = 1 + 3i$, then on the Argand diagram $z_1 - z_2$ is the vector pointing from $Z_2$ to $Z_1$, and $|z_1 - z_2| = \sqrt{13}$.

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Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · PLOT A COMPLEX NUMBER AND ITS CONJUGATE

Plot $z = 4 - 3i$ and $\bar z$ on the same Argand diagram. State the geometric relationship between the two points.

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Identify real and imaginary parts: $\text{Re}\,z = 4$, $\text{Im}\,z = -3$. So $z$ plots at $(4, -3)$, four units right, three units down.
$\text{Im}\,z$ is the coefficient of $i$ without the $i$; the sign matters. Negative imaginary parts plot below the real axis.
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PROBLEM 2 · ADDITION AS A PARALLELOGRAM

Let $z_1 = 3 + i$ and $z_2 = 1 + 2i$. Find $z_1 + z_2$ algebraically. Then describe how the same answer appears on the Argand diagram as a parallelogram.

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Add components: $z_1 + z_2 = (3 + 1) + (1 + 2)i = 4 + 3i$.
Real parts add to real parts; imaginary parts add to imaginary parts. Algebra mirrors the geometry.
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PROBLEM 3 · SUBTRACTION AS DISTANCE

Let $z_1 = 5 + 4i$ and $z_2 = 2 + 8i$. Find $z_1 - z_2$ and show that $|z_1 - z_2|$ is the distance between $Z_1$ and $Z_2$ on the Argand diagram.

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Subtract components: $z_1 - z_2 = (5 - 2) + (4 - 8)i = 3 - 4i$.
As with addition, subtraction acts componentwise. The result is itself a complex number with its own point/vector picture.
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Fill the gap: On the Argand diagram, the horizontal axis is the axis and the vertical axis is the axis. The complex number $z = a + bi$ plots at the point $(a, b)$.

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Misconceptions to fix · the 3 traps that cost marks

Misconceptions to fix · the 3 traps that cost marks

Misconceptions to fix · the 3 traps that cost marks

Trap 01
Plotting $b$ instead of $bi$ as the vertical coordinate (or vice versa)
For $z = a + bi$, the imaginary part is the real number $b$, not $bi$. The vertical coordinate is $b$, not $bi$. Common error: confusing the symbol $\text{Im}\,z$ with $bi$ and plotting incorrectly.
Trap 02
Wrong direction for the subtraction vector
$z_1 - z_2$ is the displacement from $Z_2$ to $Z_1$, not from $Z_1$ to $Z_2$. Reversing this flips the sign of every component and is a common locus mistake.
Trap 03
Forgetting to label the axes "Re" and "Im"
An Argand diagram with unlabelled axes, or labelled $x$ and $y$, is treated as a Cartesian sketch. NESA markers expect explicit Re/Im labels and a clearly marked origin.

Did you get this? True or false: the imaginary part of $z = 5 - 7i$ is $-7i$.

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Revisit your thinking

Activities · practice with the ideas

Activities · practice with the ideas

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Plot $z_1 = 2 + 3i$, $z_2 = -3 + i$, $z_3 = -2 - 2i$, $z_4 = 4 - i$ on a single Argand diagram. State which axis each pure real or pure imaginary number would lie on, and verify by inspection that none of these four are pure.

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For $z = -3 + 4i$, plot $z$, $\bar z$, $-z$ and $-\bar z$ on the same Argand diagram. Describe the symmetries: which axis or point relates which pair?

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Given $z_1 = 1 + 2i$ and $z_2 = 3 - i$, compute $z_1 + z_2$ and $z_1 - z_2$ algebraically. Then sketch both as vectors on an Argand diagram, marking the parallelogram for the sum.

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Compute $|z_1 - z_2|$ for $z_1 = 7 + i$ and $z_2 = 3 + 4i$. Then verify your answer is the distance between the two points $(7, 1)$ and $(3, 4)$ using the distance formula.

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Three complex numbers $z_1, z_2, z_3$ form the vertices of a triangle on the Argand diagram. Express the lengths of the three sides using moduli of differences. Which identity could you use to test whether the triangle is isosceles?

Odd one out: Three of these statements about the Argand diagram are correct. Which one is NOT?

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Revisit your thinking

Earlier you sketched $z_1 = 3 + 2i$ and $z_2 = -1 + 4i$ and predicted which of $z_1 + z_2$ or $z_1 - z_2$ has the longer length.

Algebraically: $z_1 + z_2 = 2 + 6i$ with modulus $\sqrt{4 + 36} = \sqrt{40} \approx 6.32$; $z_1 - z_2 = 4 - 2i$ with modulus $\sqrt{16 + 4} = \sqrt{20} \approx 4.47$. The sum has the longer length here. Geometrically, the sum is the parallelogram diagonal, when the two vectors point in similar directions, the diagonal is large; when they oppose, it is small. This is the same intuition you used in Year 11 vector addition, now applied to complex numbers.

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Show what you have learned

Multiple choice, then short answer under exam conditions.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

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Short answer
ApplyBand 32 marks

Q1. On the Argand diagram, plot $z_1 = 4 + 2i$ and $z_2 = 1 + 5i$. Hence find $z_1 - z_2$ algebraically and state which point $Z_2 \to Z_1$ vector is equivalent to. (2 marks)

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ApplyBand 43 marks

Q2. Given $z = -2 + 3i$, plot $z$, $\bar z$ and $-z$ on the same Argand diagram. Describe each transformation $z \to \bar z$ and $z \to -z$ geometrically. (3 marks)

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AnalyseBand 53 marks

Q3. Three complex numbers are given: $z_1 = 1 + i$, $z_2 = 5 + 4i$, $z_3 = 4 + 7i$. (a) Plot $Z_1, Z_2, Z_3$ on an Argand diagram. (b) Find the side lengths $|z_2 - z_1|$, $|z_3 - z_2|$, $|z_3 - z_1|$. (c) Hence classify the triangle $Z_1 Z_2 Z_3$. (3 marks)

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Comprehensive answers (click to reveal)

Activity answers:

1. $z_1 = (2,3)$ Q1; $z_2 = (-3, 1)$ Q2; $z_3 = (-2,-2)$ Q3; $z_4 = (4,-1)$ Q4. Pure real lies on Re axis; pure imaginary on Im axis. None of these are pure because each has both a non-zero real and a non-zero imaginary part.

2. $z = (-3, 4)$; $\bar z = (-3, -4)$; $-z = (3, -4)$; $-\bar z = (3, 4)$. Reflection in Re axis: $z \leftrightarrow \bar z$ and $-z \leftrightarrow -\bar z$. Reflection in Im axis: $z \leftrightarrow -\bar z$ and $\bar z \leftrightarrow -z$. Reflection through origin: $z \leftrightarrow -z$ and $\bar z \leftrightarrow -\bar z$.

3. $z_1 + z_2 = 4 + i$, diagonal of parallelogram on $\vec{OZ_1}, \vec{OZ_2}$. $z_1 - z_2 = -2 + 3i$, arrow from $Z_2 = (3, -1)$ to $Z_1 = (1, 2)$, components $(-2, 3)$, matches.

4. $z_1 - z_2 = 4 - 3i$; $|z_1 - z_2| = 5$. Distance from $(7,1)$ to $(3,4)$: $\sqrt{16 + 9} = 5$. Identical.

5. Side lengths: $|z_1 - z_2|, |z_2 - z_3|, |z_1 - z_3|$. Isosceles iff at least two are equal, set any pair of moduli equal and solve/check.

Q1 (2 marks): $Z_1 = (4, 2)$, $Z_2 = (1, 5)$ plotted with Re and Im axes labelled [1]. $z_1 - z_2 = 3 - 3i$; the displacement from $Z_2$ to $Z_1$ is $(3, -3)$, equivalent to $z_1 - z_2$ [1].

Q2 (3 marks): $z = (-2, 3)$, $\bar z = (-2, -3)$, $-z = (2, -3)$ plotted [1]. $z \to \bar z$ is reflection in the real axis [1]. $z \to -z$ is rotation by $180°$ about the origin (equivalently, point-reflection through $O$) [1].

Q3 (3 marks): (a) Points plotted [1]. (b) $|z_2 - z_1| = |4 + 3i| = 5$; $|z_3 - z_2| = |-1 + 3i| = \sqrt{10}$; $|z_3 - z_1| = |3 + 6i| = \sqrt{45} = 3\sqrt{5}$ [1]. (c) All three sides differ, so the triangle is scalene. Check: $5^2 + (\sqrt{10})^2 = 25 + 10 = 35 \neq 45$, so not right-angled at $Z_2$ [1].