Well, it turns out that these two dimensional numbers offer so much more than the means by which to solve cubic equations for which they were originally conceived. Now we know that they are fundamental to quantum theory, electrical engineering, fluid dynamics, and chaos theory.
|Part of the Mandelbrot set; a chaotic figure in which every point represents a different complex number.|
Complex numbers are made up of two ingredients; a real part and an imaginary part. The real part is just a number on the real number line. The imaginary part is another number from the real number line but multiplied by i, the square root of -1.
"What?", I might hear you cry, "You can't have a square root of a negative number!" Well, in a way, you would be correct; there is certainly no number on the real number line such that its square is -1. One of the joys of maths, though, is to invent such things and see where they take us.
The first thing we might note is that if i is the root of -1, then 2i must be the root of -4, 3i the root of -9, 4i the root of -16, and so on. How might we represent these numbers? Well, they aren't on the real number line so we'll draw another. It will make sense if these number lines cross at 0 since, if we are to have a system of multiplication for complex numbers, we should have 0i=0 (actually the reason this should be is not that straightforward; the field of Field Theory is what to study if you wish to learn more about this). If we have two crossing number lines, we might as well make them perpendicular. We can then represent every point in a plane by a complex number consisting of a unique combination of real part (Re) and imaginary part (Im).
|The corners of a square represented by complex numbers; they're a bit like coordinates.|
We can do lots of things with complex numbers that we can do with real numbers and that includes plugging them into functions (One thing we can't do is to put a handful of them in order. Can you see why?). The Mandelbrot set, for example, relies on the ability to square a complex number. The very fact that each is made of two elements, makes this a particular challenge to visualise though.
We can easily draw a picture of the squaring of real numbers. We put a number in, and we get its square out, and we can plot these pairs of numbers on a two-dimensional graph.
|It's easy to draw a picture of the squaring of a real number.|
With complex numbers however, we need to input two-dimensional information and we need to represent the output in two dimensions. We therefore need four spatial dimensions in total and, unfortunately, our universe equips us only with three (until string theorists get their way).
Here are a few ways we might try and bypass this inconvenience.
First, we might split up the outcome of squaring z into its real and imaginary parts. We can then plot a surface to represent each as follows:
|Surfaces representing the real (left) and imaginary (right) parts of the square of z.|
I find that these aren't particularly helpful though; the outcome of the function is split up over two graphs. It's very difficult to imagine how they combine to form one complex number z.
Instead we might use a single three dimensional graph and harness the dimension of time to vary the fourth element. Suppose that we want to square a complex number z=a+bi. Now, let's fix the value of b at any particular time and plot the relationship between the real part of z and both real and imaginary parts of its square. We can then alter b to see how the graph changes for various imaginary parts of z.
|Plotting the square of z for Im(z)= -1, 0, 1.|
You can click here to take you to an Autograph Activities page to allow you to experiment for yourself.
While slightly more enlightening than the first attempt, I still don't think this is the best way to visualise the squaring of a complex number. Strangely enough, I think it's most useful to use only two dimensions and adopt a "before and after" approach. Let's think about what happens to a geometric shape when all the complex numbers lying on its boundary are squared. The following diagrams show how our earlier square is transformed.
|All the complex numbers lying on the blue square (left) are squared and result in a new shape (right).|
Crikey! Can you tell which side has become which? You can find out and see the transformation animated by clicking this link and following the instructions.
What do you think will happen if we change the initial position of the square? What will happen if we start with a square of different size? What will happen if we start with a different shape altogether?