14.2 The Argand diagram

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 $x$ and a $y$ co-ordinate, we can say that we have a real ($\Re$) and an imaginary ($Im$) co-ordinate.

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

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

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 $z\in\mathbb{C}$ we know (or at least our intuitive notion of length lines up with) that the \saylength of $z$ is equal to $\left\lvert z\right\rvert$. We also know that for two vectors $a$ and $b$, the vector between $a$ and $b$ is given by $-a+b$, and thus the distance between $a$ and $b$ can be written as $\left\lvert a-b\right\rvert$.

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

more examples

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