IB Maths AA HLTopic 1 — Number & AlgebraPaper 1 & 2~8 min read
Introduction to Argand Diagrams
So far complex numbers have been objects of algebra — symbols on the page. The Argand diagram changes that. It’s a 2D picture where every complex number gets its own location, just like every coordinate has its own dot on a graph. The real part runs along the horizontal axis; the imaginary part runs up the vertical one. Suddenly you can see what 3 + 4i looks like, what its conjugate looks like, and why complex numbers behave the way they do. Once you’ve drawn a few, the geometry of the topic starts to click — and the rest of complex numbers becomes much more intuitive.
📘 What you need to know
The Argand diagram (or complex plane) is a 2D grid for plotting complex numbers — like a Cartesian xy-plane, but with different axis labels.
The horizontal axis is the real axis (labelled Re). The vertical axis is the imaginary axis (labelled Im).
The complex number z = x + yi is plotted at the point (x, y).
You can also represent z as a vector from the origin (0, 0) to (x, y), with an arrow pointing away from the origin.
Real numbers sit on the horizontal axis (imaginary part = 0). Purely imaginary numbers sit on the vertical axis (real part = 0).
The complex conjugatez* is the reflection of z in the real axis. Same horizontal position, opposite vertical position.
For sketches in exams, you don’t need a full grid — just labelled axes and the correct points or vectors with their values clearly written.
The complex plane
You already know how to plot points on a Cartesian xy-plane: the horizontal coordinate goes on the x-axis, the vertical on the y-axis. The Argand diagram works exactly the same way, but the axes mean something slightly different.
Horizontal axis
Re (real)
Position on this axis = the real part of z
Vertical axis
Im (imaginary)
Position on this axis = the imaginary part of z
A complex number z = x + yi is plotted at the point (x, y). The real part is the horizontal coordinate; the imaginary part is the vertical one (just the number, without the i). It’s that simple.
The Argand diagram is named after Jean-Robert Argand, who came up with this picture in 1806. Once you see complex numbers as points (or arrows), nearly everything in this topic — addition, multiplication, conjugates, modulus — turns into a piece of geometry instead of just symbols on a page.
Plotting a complex number
Two ways to draw a complex number on an Argand diagram, and questions usually accept either:
As a point
× at (x, y)
A small cross or dot at the location, with the number labelled next to it.
As a vector
→ arrow from O
An arrow from the origin (0, 0) to (x, y) with the head at the endpoint.
The vector picture is often more useful because it gives a visual sense of direction and distance — both of which become important in the next note (modulus and argument).
Four complex numbers, one in each quadrant
When sketching an Argand diagram for an exam, you don’t need a graph paper grid. Just draw clearly labelled axes (Re and Im), put crosses or vectors at roughly the right positions, and write the value of each complex number next to its mark. A neat freehand sketch with the right structure scores full marks.
Where do special complex numbers sit?
Some kinds of complex numbers always end up in the same place on the diagram:
Purely real numbers (like 5 or −2) sit on the real axis — the horizontal line. Purely imaginary numbers (like 3i or −4i) sit on the imaginary axis — the vertical line. 0 sits at the origin.
🤔 Why does the conjugate look like a reflection?
The conjugate of z = a + bi is z* = a − bi. The real part stays at a (same horizontal position). The imaginary part flips from b to −b (mirror image vertically). That’s exactly what reflection in the real axis does to a point: keep x, flip y. So z and z* are always mirror images across the horizontal axis.
How to sketch an Argand diagram in an exam
🧭 Recipe — sketching an Argand diagram
Draw two perpendicular axes. Horizontal first, then vertical, both with arrows.
Label the axes. Real axis is “Re”; imaginary axis is “Im”.
Mark each complex number as either a point (×) or a vector (arrow from origin), and label it with its value (e.g., “z1 = 2 + 3i”).
Mark a few scale points on each axis (just enough to make positions clear) — you don’t need a full grid.
Make the picture roughly to scale. Don’t put 1 + 100i in the first quadrant next to a tiny 1 + i — relative positions matter.
Worked examples
WE 1
Plot complex numbers on an Argand diagram
Sketch the complex numbers z1 = 3 + i, z2 = −2 + 4i and z3 = −1 − 3i on the same Argand diagram.
Step 1: Identify the quadrant for each complex numberz1 = 3 + i → real > 0, imag > 0 → Q1 (top right)z2 = −2 + 4i → real < 0, imag > 0 → Q2 (top left)z3 = −1 − 3i → real < 0, imag < 0 → Q3 (bottom left)Step 2: Sketch each as a vector from the originall three plotted ✓labelled axes + arrows from origin + value labels — that’s all the marks need
WE 2
Read complex numbers from a diagram
Two points P and Q are plotted on the Argand diagram below. P is at coordinates (4, 2) and Q is at coordinates (−3, −5). Write down the complex numbers represented by each point.
Step 1: Recall the conventionpoint (x, y) on Argand diagram → complex number x + yiStep 2: Read off each pointP at (4, 2) → real part 4, imaginary part 2Q at (−3, −5) → real part −3, imaginary part −5P represents 4 + 2i, Q represents −3 − 5ion an Argand diagram, the y-coordinate IS the coefficient of i — don’t add an extra i in the answer
WE 3
Sketch a number and its conjugate
Sketch z = 5 − 4i and its complex conjugate z* on the same Argand diagram. Comment on the geometric relationship between them.
Step 1: Find the conjugate by flipping the imaginary signz = 5 − 4i → z* = 5 + 4iStep 2: Plot bothStep 3: Comment on the geometryz* is the reflection of z in the real axis — same horizontal position, flipped vertical positionz and z* are reflections of each other in the real axis ✓conjugates always sit symmetrically above and below the Re-axis
WE 4
Adding complex numbers as vectors
Let z1 = 2 + i and z2 = 1 + 3i. Compute z1 + z2, and describe how this addition would look as a vector operation on an Argand diagram.
Step 1: Compute the sumz1 + z2 = (2 + 1) + (1 + 3)i = 3 + 4iStep 2: Geometric interpretationimagine placing the two vectors tip-to-tailvector for z1 goes 2 right + 1 upvector for z2 goes 1 right + 3 uptotal displacement: 3 right + 4 up → endpoint at (3, 4)z1 + z2 = 3 + 4i — same as adding the vectors tip-to-tailcomplex addition behaves exactly like 2D vector addition — the real and imaginary parts are just the components
WE 5
Where do these special complex numbers sit?
For each of the following complex numbers, state where it sits on the Argand diagram (which axis or quadrant):
(a) z = 7 (b) w = −5i (c) u = 3 − 2i (d) v = 0
Step 1: Identify the real and imaginary parts of eachz = 7 = 7 + 0i → real part 7, imaginary part 0w = −5i = 0 − 5i → real part 0, imaginary part −5u = 3 − 2i → real part 3, imaginary part −2v = 0 = 0 + 0i → both parts zeroStep 2: Locate eachz: imag = 0 → on the real axis (positive side)w: real = 0 → on the imaginary axis (negative side)u: real > 0, imag < 0 → Quadrant 4v: both zero → at the originz on Re-axis, w on Im-axis, u in Q4, v at originwhenever a part is zero, the number lives on an axis — not in a quadrant
💡 Top tips
Always label axes Re and Im — not “x” and “y”. Examiners look for these labels specifically.
Use vectors (arrows from origin), not just dots, when sketching for a question about modulus or argument. The arrow shows direction and length, both of which carry meaning.
The complex conjugate is a reflection in the real axis. Internalise this — it makes conjugate-related questions almost free.
Roughly to scale is fine. You don’t need precision — just plausible relative positions.
Mark a couple of scale points on each axis (e.g., 1, −1) so the examiner can see your positions are correct.
Real numbers sit on the horizontal axis, purely imaginary numbers on the vertical. Test yourself with a few — 5, −7, 3i, −2i — make sure you can place them instantly.
The origin is 0 + 0i — it represents the complex number 0 itself.
⚠ Common mistakes
Labelling the axes “x” and “y”. They’re “Re” and “Im”. The labels matter for marks.
Including the i when reading off a position. Point (4, 3) represents 4 + 3i — the y-coordinate is just 3, not 3i.
Plotting purely imaginary numbers in a quadrant. Numbers like 5i sit on the imaginary axis, not in a quadrant.
Mixing up conjugate with negation. The conjugate of a + bi is a − bi (reflect in real axis). The negative is −a − bi (rotate 180° around origin). Different operations, different pictures.
Drawing arrows that don’t start at the origin. Vectors representing complex numbers always begin at O. If your arrow starts somewhere else, it’s wrong.
Forgetting to label each plotted number with its value. A cross at (3, 4) is meaningless without “z = 3 + 4i” written beside it.
Putting the imaginary axis horizontal and the real axis vertical. Real is horizontal; imaginary is vertical. Always.
The Argand diagram is the bridge between algebra and geometry in this topic. Once you can picture a complex number as a point or an arrow, the next ideas — modulus and argument — become very natural. Modulus is just the length of the arrow; argument is the angle it makes with the positive real axis. That’s the focus of the next note. Get comfortable with the diagram first, and the rest of the topic falls into place.
Need help with Argand Diagrams?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
