How Do You Use the Argand Diagram to Represent Complex Numbers?

A complex number $z=a+bi$ is represented on an Argand diagram by the point ((a,b)). The horizontal axis shows the real part $a$, while the vertical axis shows the imaginary part $b$.

This is HL-only content from AHL 1.12, Introduction to complex numbers. Assuming an HL Paper 1 short-response context, you should be able to plot, label, and interpret complex numbers using an Argand diagram.

How the Representation Works

For $z=a+bi$:

1. Move $a$ units along the real axis.
2. Move $b$ units parallel to the imaginary axis.
3. Plot and label the point $z$.

The complex number may also be represented by a position vector from the origin to ((a,b)).

Points on either axis are also valid complex numbers. If $b=0$, the number is purely real and lies on the real axis; if $a=0$, it is purely imaginary and lies on the imaginary axis. The origin represents $0+0i$. The conjugate $\bar{z}=a-bi$ appears as the reflection of in the real axis.

For example, $z=3-4i$ is plotted at $(3,-4)$: move three units right and four units down. Its modulus, meaning its distance from the origin, is

$$|z|=\sqrt{3^2+(-4)^2}=5.$$

Its argument is the directed angle from the positive real axis to the position vector. The signs of the coordinates must be used to identify the correct quadrant before stating this angle.

A common misconception is to plot $3-4i$ at $(-4,3)$. This incorrectly swaps the axes: the coefficient of $1$ is always the horizontal coordinate, and the coefficient of $i$ is always the vertical coordinate.

Exam Technique

For the command term Plot, mark the point accurately and label it. If asked to find the modulus or argument, show the relevant calculation and check the quadrant; a calculator angle without quadrant reasoning may produce an incorrect argument.