IB Maths AI HLComplex NumbersPaper 1 & 2Complex plane~6 min read
Introduction to Argand Diagrams
A complex number has two parts — so it can be drawn on a 2D grid. The Argand diagram turns every complex number into a point, or an arrow, on the complex plane: algebra becomes geometry.
📘 What you need to know
The complex plane (Argand plane) is a 2D grid for picturing complex numbers.
The horizontal axis is the real axis (Re); the vertical axis is the imaginary axis (Im).
z = x + yi is plotted at the point (x, y) — just like Cartesian coordinates.
It can be shown as a point (×) or as a vector arrow from the origin to (x, y).
The conjugate z* is the reflection of z in the real axis.
The distance from the origin to the point is the modulus |z|.
The complex plane
The complex plane, also called the Argand plane, is a 2D grid that works just like Cartesian coordinates — but the two axes carry the two parts of a complex number. The horizontal axis is the real axis, labelled Re, and the vertical axis is the imaginary axis, labelled Im.
Any complex number z = x + yi has a real part x and an imaginary part y, so it locates a single point on this plane — the same way (x, y) locates a point on an ordinary grid.
Plotting complex numbers: points and vectors
An Argand diagram is a geometrical picture of complex numbers on the complex plane. The number z = x + yi can be represented in two equivalent ways: as the point (x, y), usually marked with a cross, or as a vector — an arrow from the origin (0, 0) pointing out to (x, y).
Each complex number is placed by its real part (horizontal) and imaginary part (vertical). z1 is drawn as a vector from the origin; z2, z3, z4 as points — both are valid.
Sketching tip: when a question says sketch an Argand diagram, you do not need graph paper or a full grid — a rough set of axes with the points in roughly the right places, clearly labelled, is enough.
Reading a diagram and the conjugate’s reflection
The process also runs in reverse: given a point on an Argand diagram, read off its horizontal coordinate as the real part and its vertical coordinate as the imaginary part to recover the complex number.
The diagram also makes the conjugate visual. Since z* = x − yi keeps the real part but flips the imaginary part, z and z* are mirror images in the real axis. Likewise −z is z rotated 180° about the origin.
Argand representationz = x + yi → point (x, y) or vector from O
distance from O = |z| = √(x² + y²)z* is the reflection of z in the real axis
🧠Recipe — plotting a complex number on an Argand diagram
Draw the complex plane: a horizontal real axis (Re) and a vertical imaginary axis (Im).
Read off the parts of z = x + yi: real part x, imaginary part y.
Move x along Re and y along Im to locate the point (x, y).
Mark it — a cross for a point, or an arrow from the origin for a vector.
Label the point with the complex number, and repeat for any others.
Worked examples
WE 1
Plotting points
State the coordinates and the quadrant for each number when plotted on an Argand diagram: z1 = 4 + 3i, z2 = −2 + 5i, z3 = −3 − 4i.
real part → x-coordinate, imaginary part → y-coordinatez₁ = 4 + 3i → (4, 3), x > 0, y > 0z₂ = −2 + 5i → (−2, 5), x < 0, y > 0z₃ = −3 − 4i → (−3, −4), x < 0, y < 0z₁: quadrant 1 · z₂: quadrant 2 · z₃: quadrant 3the signs of the two parts fix the quadrant straight away.
WE 2
Reading numbers off a diagram
On an Argand diagram, point A is at (−4, 6) and point B is at (5, 0). Write down the complex numbers represented by A and B.
x-coordinate → real part, y-coordinate → imaginary partA = (−4, 6) → −4 + 6iB = (5, 0) → 5 + 0iA = −4 + 6i · B = 5B sits on the real axis — zero imaginary part means a purely real number.
WE 3
Identifying the quadrant
Without drawing axes, state which quadrant of the Argand diagram each number lies in: z1 = 7 − 2i, z2 = −5 − 8i, z3 = −1 + 6i.
check the sign of each partz₁: x > 0, y < 0 → quadrant 4z₂: x < 0, y < 0 → quadrant 3z₃: x < 0, y > 0 → quadrant 2z₁: Q4 · z₂: Q3 · z₃: Q2quadrants run anticlockwise from the top-right, exactly as in Cartesian work.
WE 4
The conjugate as a reflection
For z = 3 + 5i, write down z* and −z, and describe where each lies relative to z on an Argand diagram.
conjugate: keep real part, flip imaginary partz* = 3 − 5inegative: flip both parts−z = −3 − 5iz* = 3 − 5i · −z = −3 − 5iz* is z reflected in the real axis; −z is z rotated 180° about the origin.
WE 5
The length of the vector
The complex number z = −6 + 8i is drawn as a vector on an Argand diagram. Find the length of that vector.
vector length = distance from origin = |z||z| = √(x² + y²) = √((−6)² + 8²)= √(36 + 64) = √100length = 10the vector’s length is just the modulus — geometry meets algebra.
WE 6
Full question: points on a diagram
Points P, Q, R represent zP = 2 + i, zQ = −3 + 4i, zR = −1 − 5i. (a) State the quadrant of each point. (b) Write down zQ* and state where it lies. (c) Find the distance of R from the origin.
(a) check signs of the partsP (2, 1) → Q1; Q (−3, 4) → Q2; R (−1, −5) → Q3(b) conjugate of z_Q: flip imaginary partz_Q* = −3 − 4i — reflection of Q in the real axis(c) distance = |z_R| = √((−1)² + (−5)²)= √(1 + 25) = √26(a) Q1, Q2, Q3 · (b) −3 − 4i (in Q3) · (c) √26 ≈ 5.10reflecting Q in the real axis moves it from quadrant 2 down into quadrant 3.
💡 Top tips
Treat the Argand diagram like ordinary (x, y) coordinates — real across, imaginary up.
A point and a vector are equally correct — use whichever the question asks for.
For a sketch, label the points clearly — no grid or graph paper needed.
The conjugate z* is always the mirror image in the real axis.
The length of the vector from the origin is the modulus |z|.
âš Common mistakes
Swapping the axes — the real part is horizontal, the imaginary part vertical.
Plotting the i — for 3 + 4i the height is 4, not 4i.
Reflecting z* in the wrong axis — it is the real axis, not the imaginary one.
Drawing a vector with no arrow — the arrow shows direction away from the origin.
Forgetting purely real or imaginary numbers sit on an axis, not in a quadrant.
That completes the Complex Numbers chapter — Cartesian form, the four operations, complex roots of quadratics, modulus & argument, and the Argand diagram. You can now move freely between a complex number’s algebra and its geometry.
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