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Year 11 Specialist (Unit 1 & 2) Complex numbers

Argand diagrams

20 practice questions 0 video lessons Theory + worked examples

Explore Argand diagrams for Year 11 Specialist Mathematics in Queensland (QCAA). An Argand diagram is the plane where every complex number becomes a point, with the real part across and the imaginary part up, turning complex arithmetic into pictures.

You will learn to plot complex numbers, measure the modulus as a distance, locate conjugates and negatives, add by the parallelogram rule, and see multiplication by \(i\) as a rotation — the geometric foundation for polar form and regions of the complex plane.

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Theory

An Argand diagram pictures every complex number \(z=a+bi\) as the point \((a,b)\) on a plane in Year 11 Specialist Mathematics (QCAA, Queensland). The real part runs along the horizontal axis and the imaginary part up the vertical axis, so that adding, conjugating and multiplying complex numbers all become geometry you can see.

An Argand diagram (or complex plane) is a set of axes for plotting complex numbers. The horizontal axis is the real axis and the vertical axis is the imaginary axis.

The complex number \(z=a+bi\) is plotted as the single point \((a,b)\), where \(a=\operatorname{Re}z\) is the real part and \(b=\operatorname{Im}z\) is the imaginary part. The arrow from the origin to that point is the position vector of \(z\).

The modulus \(|z|\) is the length of that position vector — the distance of the point from the origin. The conjugate \(\bar{z}=a-bi\) is the reflection of \(z\) in the real axis, and \(-z\) is a half-turn of \(z\) about the origin.

Because \(z=a+bi\) behaves like the vector \(\begin{pmatrix}a\\b\end{pmatrix}\), addition is vector addition (the parallelogram rule) and multiplication by \(i\) rotates a point \(90^\circ\) anticlockwise about the origin.

Plotting z = 3+4i on an Argand diagramReal axis horizontal, imaginary axis vertical, unit grid. A navy position vector runs from the origin to the point (3,4); dashed lines drop to 3 on the real axis and across to 4 on the imaginary axis. The vector length is the modulus, 5. Re Im z = 3+4i
Plotting \(z=3+4i\) as the point \((3,4)\); the position-vector length is the modulus \(|z|=5\).
Conjugate and negative of z = 3+2iThe point z at (3,2) in navy; its conjugate at (3,-2) in red is the mirror image in the real axis; its negative at (-3,-2) in green is a half-turn about the origin. Dashed lines join z to its conjugate and to its negative. Re Im z = 3+2i 3−2i −3−2i
The conjugate \(\bar{z}=3-2i\) reflects \(z=3+2i\) in the real axis; \(-z=-3-2i\) is a half-turn about the origin.

A complex number is plotted using its parts as Cartesian coordinates:

\[ z=a+bi \ \longrightarrow\ (a,b)=(\operatorname{Re}z,\ \operatorname{Im}z) \]
z=a+bi→(a,b)

The modulus is the distance of the point from the origin:

\[ |z|=\sqrt{a^2+b^2} \]
|z|=a2+b2

Conjugate (reflection in the real axis) and negative (half-turn):

\[ \bar{z}=a-bi, \qquad -z=-a-bi \]
z¯=a−bi

Addition combines the parts; the distance between two points is a modulus:

\[ z_1+z_2=(a_1+a_2)+(b_1+b_2)i \]
\[ \text{distance}=|z_1-z_2|=\sqrt{(a_1-a_2)^2+(b_1-b_2)^2} \]
|z1−z2|
Modulus = distance. \(|z|\) is how far the point is from the origin, and \(|z_1-z_2|\) is the distance between the two points. Multiplying by \(i\) rotates a point \(90^\circ\) anticlockwise about the origin.

How to work on the Argand plane

  1. Read off the parts. Write \(z=a+bi\) and identify \(a=\operatorname{Re}z\) (horizontal) and \(b=\operatorname{Im}z\) (vertical).
  2. Plot the point \((a,b)\) and, if asked, draw the position vector from the origin to it.
  3. Apply the geometry: conjugate = reflect in the real axis; negative = half-turn about the origin; multiply by \(i\) = rotate \(90^\circ\) anticlockwise; add = vector (parallelogram) addition.
  4. Measure with the modulus: use \(|z|=\sqrt{a^2+b^2}\) for distance from the origin, or \(|z_1-z_2|\) for the distance between two points, leaving surds exact.
Example 1 — Plot and measure
Plot \(z=-8+6i\) on an Argand diagram, write its coordinates, and find \(|z|\).
Solution

Read off the real and imaginary parts:

\(z\)\(=\)\(-8+6i\)
\(\operatorname{Re}z\)\(=\)\(-8\)
\(\operatorname{Im}z\)\(=\)\(6\)
\((a,b)\)\(=\)\((-8,6)\)

Now the modulus is the distance of that point from the origin:

\(|z|\)\(=\)\(\sqrt{(-8)^2+6^2}\)
\(=\)\(\sqrt{64+36}\)
\(=\)\(\sqrt{100}\)
\(=\)\(10\)

The point is \((-8,6)\) and \(|z|=10\).

Example 2 — Conjugate and negative
For \(z=5-3i\), write the conjugate \(\bar{z}\) and the negative \(-z\), and describe each geometrically.
Solution

Conjugate: reflect in the real axis, so keep the real part and change the sign of the imaginary part:

\(z\)\(=\)\(5-3i\)
\(\bar{z}\)\(=\)\(5-(-3)i\)
\(=\)\(5+3i\)

Negative: a half-turn about the origin, so change the sign of both parts:

\(-z\)\(=\)\(-(5-3i)\)
\(=\)\(-5+3i\)

\(\bar{z}=5+3i\) (mirror in the real axis); \(-z=-5+3i\) (half-turn about the origin).

Example 3 — Add by the parallelogram rule
Let \(z_1=2+5i\) and \(z_2=4-2i\). Find \(z_1+z_2\) and describe it on the Argand diagram.
Solution

Add the real parts and the imaginary parts separately:

\(z_1+z_2\)\(=\)\((2+5i)+(4-2i)\)
\(=\)\((2+4)+(5-2)i\)
\(=\)\(6+3i\)

Geometrically this is vector addition — the sum is the diagonal of the parallelogram with sides \(z_1\) and \(z_2\).

\(z_1+z_2=6+3i\), the fourth vertex of the parallelogram \(0,\,z_1,\,z_2\).

Adding z1 = 2+5i and z2 = 4-2i by vector additionz1 in navy and z2 in red drawn from the origin; the gold resultant z1 + z2 = 6+3i is the diagonal of the parallelogram whose sides are z1 and z2. Dashed lines complete the parallelogram from each vector tip to the sum. Re Im z₁ = 2+5i z₂ = 4−2i z₁+z₂ = 6+3i
Example 4 — Multiply by \(i\) (rotation)
Given \(z=4-i\), find \(iz\) and state the transformation it represents on the Argand diagram.
Solution

Multiply out and replace \(i^2\) with \(-1\):

\(iz\)\(=\)\(i(4-i)\)
\(=\)\(4i-i^2\)
\(=\)\(4i-(-1)\)
\(=\)\(1+4i\)

Compare the points: \((4,-1)\) maps to \((1,4)\), a \(90^\circ\) anticlockwise turn about the origin.

\(iz=1+4i\); multiplying by \(i\) rotates the point \(90^\circ\) anticlockwise about the origin.

Common pitfalls

Swapping the axes. The real part is the horizontal coordinate and the imaginary part is the vertical one, so \(z=a+bi\) is \((a,b)\) — not \((b,a)\). Plot \(-3+2i\) at \((-3,2)\), left and up.
Mishandling the double negative. The conjugate of \(7-2i\) is \(7+2i\): only the sign of the imaginary part flips. Subtracting \(z_2=2-4i\) becomes \(\dots-(2-4i)=\dots-2+4i\).
Decimalising an exact modulus. \(|2+3i|=\sqrt{13}\); leave surds exact and do not round unless a decimal is asked for.

Frequently asked questions

What is an Argand diagram?

It is a plane for plotting complex numbers: the real part \(a\) of \(z=a+bi\) is the horizontal coordinate and the imaginary part \(b\) is the vertical coordinate, so \(z\) is the point \((a,b)\).

How do you plot a complex number on an Argand diagram?

Write \(z=a+bi\), then plot the point \((a,b)\): go \(a\) across on the real axis and \(b\) up on the imaginary axis. You may draw the position vector from the origin to that point.

What does the modulus mean on an Argand diagram?

The modulus \(|z|=\sqrt{a^2+b^2}\) is the distance of the point from the origin — the length of the position vector.

Where is the conjugate on an Argand diagram?

The conjugate \(\bar{z}=a-bi\) is the reflection of \(z\) in the real axis, directly below (or above) \(z\) at the same horizontal position.

How do you add complex numbers geometrically?

Addition is vector addition: draw \(z_1\) and \(z_2\) from the origin and complete the parallelogram; the sum \(z_1+z_2\) is the diagonal, found by adding the real and imaginary parts separately.

What does multiplying by \(i\) do on the Argand plane?

Multiplying a complex number by \(i\) rotates its point \(90^\circ\) anticlockwise about the origin, for example \(i(4-i)=1+4i\).

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