14.2 The Argand diagram

14.2.1 Plotting complex numbers

Complex numbers can be interpreted in a Cartesian fashion33 3 Where every point in the plain is uniquely specified by numerical co-ordinates. Instead of an xx and a yy co-ordinate, we can say that we have a real (\Re) and an imaginary (ImIm) co-ordinate.

For example, if we have the expression 3+21=3+2i3+2\sqrt{-1}=3+2i, we could say that the co-ordinates are (3,2)(3,2), and plot these on a graph.

\Im axis\Re axis

We label the vertical axis as \Im (or Im) - for \sayimaginary - and the horizontal axis as \Re (or Re) - for \sayreal.

14.2.2 Loci on the Argand diagram

These are the kind of questions which certain examiners love to pose. There are two ways to go about this; geometrically or algebraically. Ordinarily I would always pick the option with lots of algebra over the option with lots of geometry. Unfortunately, in this case the option with lots of algebra also includes pretty much the same amount of geometry.

In essence, problems about loci on the Argand diagram are about finding some complex numbers which satisfy a set of constraints. There are a few very important facts to know, with which most of these questions can be tackled.

The key fact to remember is that complex numbers can be treated as vectors. For example, for any zz\in\mathbb{C} we know (or at least our intuitive notion of length lines up with) that the \saylength of zz is equal to |z|\left\lvert z\right\rvert. We also know that for two vectors aa and bb, the vector between aa and bb is given by a+b-a+b, and thus the distance between aa and bb can be written as |ab|\left\lvert a-b\right\rvert.

Example 14.2.1

Sketch the complete set of points which satisfy

|z9i|=|z12|\left\lvert z-9i\right\rvert=\left\lvert z-12\right\rvert (14.35)

We can apply the trick of reading this aloud. We know that |z9i|\left\lvert z-9i\right\rvert gives us the distance between zz and the complex number 9i9i. Furthermore, 2|z12|2\left\lvert z-12\right\rvert means the distance between zz and 1212, so overall what this means is \saywe want zz such that the distance between zz and 9i-9i is the same as the distance between zz and 1212. If we sketch this, this will be the perpendicular bisector of the line between 1212 and 9i9i (through the middle of the line).

more examples

14.2.3 Polar co-ordinates and complex numbers

We can also define complex numbers differently (i.e. not using Cartesian co-ordinate system). Instead of identifying each complex number using an xx (real) and yy (imaginary) co-ordinate, we can instead define it using an angle (θ\theta) and a distance from the origin (rr).

14.2.4 Graphical interpretation of operations on complex numbers