H2 Maths 4.1 - Plotting Complex Numbers on the Argand Diagram

Step 1 - The Argand diagram

An Argand diagram is a coordinate plane with two axes.

Step 1 - The Argand diagram

The horizontal axis is the real axis, and the vertical axis is the imaginary axis.

Step 1 - The Argand diagram

Any complex number z equals a plus b i is placed as the point with horizontal coordinate a and vertical coordinate b.

Step 2 - Plot z = 3 − 4i

Take z equals three minus four i.

Step 2 - Plot z = 3 − 4i

The real part is 3 and the imaginary part is −4.

Step 2 - Plot z = 3 − 4i

Count three units along the real axis and four units down the imaginary axis.

Step 2 - Plot z = 3 − 4i

The point lies in the fourth quadrant.

Step 3 - A second example: z = −1 + 2i

Now take z equals negative one plus two i.

Step 3 - A second example: z = −1 + 2i

The real part is negative one, so the point is one unit to the left of the origin.

Step 3 - A second example: z = −1 + 2i

The imaginary part is two, so go two units up.

Step 3 - A second example: z = −1 + 2i

This point lies in the second quadrant, showing that any quadrant is possible depending on the signs of the real and imaginary parts.

Step 4 - The complex conjugate

The complex conjugate of z = 3 − 4i is z^* = 3 + 4i.

Step 4 - The complex conjugate

Only the sign of the imaginary part changes.

Step 4 - The complex conjugate

On the Argand diagram, z^* is the reflection of z in the real axis.

Step 5 - Summary and common pitfall

To plot any complex number on an Argand diagram, use the real part as the horizontal coordinate and the imaginary part as the vertical coordinate.

Step 5 - Summary and common pitfall

The conjugate z^* is always the mirror image in the real axis.

Step 5 - Summary and common pitfall

Common pitfall: do not swap the axes. The real axis is always horizontal and the imaginary axis is always vertical.