Hi Everyone,

I'm preparing a discussion to be given at school on complex numbers. The discussion is going to be led by four of us and I am allocated to talk about the uses of complex numbers. I know a fair bit about their manipulation etc and have been following the imaginary numbers discussion at this site, but I am still stuck for much to say about uses. Any ideas would be much appreciated.
Many thanks.

Ella

Hi Ella,
the most important use is solving polynomials like

z² +1=0

This seems obvious, but is the main motivation for the complex numbers. There is a theorem (The Fundamental Theorem of Algebra) stating that EVERY polynomial has solutions if complex numbers are used.

Jonathan

Thanks,

It would be very helpful if you would be willing to expand on the differential equations. I have looked at very basic ones, but do not know too much about them, especially concerning complex numbers.

Ella

Another thing, of course I appreciate using complex numbers to solve polynomials, and I do understand that this is a wonderful thing to be able to do, as it expands the mathematics we can deal with, and that in itself is of great intellectual value. But does solving polynomials using complex numbers actually have any practical value?

Ella

Of course, one of the main uses of complex numbers is to produce fractals, which make wonderful screensavers.

What is the connection between complex numbers and fractals? I know a little about fractals, but in what I have read and been told, complex numbers were never mentioned.

Ella

Well, the most well known fractal is the Mandelbrot set, which you've probably heard of and seen pictures of (it's beetle-like). Well, to find whether a point in the plane is inside the mandelbrot set or not, you use the following rule. Let c=x+iy, the complex number corresponding to the point in the plane. Now, let z=c. Now work out z² +c, and put this value into z, we write this as z -> z² +c (z maps to z² +c). Do this forever (well, 100 times is probably enough), if the value of z is zooming off to infinity, the point isn't inside the set, otherwise it is inside the set. If it is inside, we draw a black dot, otherwise we draw a white dot.

Hi Ella!

Complex numbers are usually used in the definition of fractals. For example, the "most famous" fractal, the Mandelbrot set is defined as follows.

Pick a number on the complex plane, say c
Set z=0
Then set z=z² +c
Then keep doing this and see if either z diverges to infinity, or it stays inside quite a small boundary, say modulus z < 3. You can't prove it if will diverge, but you can try working out z=z² +c about 500 times or something, and see if it diverged then. If it hasn't diverged, plot a black dot. It is has diverged, then you can either a) plot a white dot, or to get the exciting coloured fractals you can plot a different colour based on how quickly the point diverged.

If you'd like to investigate this, could I advise the excellent fractint from this site . You'll have to put bit of work into getting it to work perfectly, but it is the best fractal program I've come across. The windows version isn't quite as advanced but is probably more than advanced enough, and also is nicer to use.

While on the subject of uses for complex numbers, there are quite a few. Unless you have a lot of time they are difficult to explain, but there is something known as residue therem which gives you a simple(ish) way of working out some integrals you would never have a chance of doing, like say (sin(x))/x between -infinity and infinity by considering it as an integral in the larger complex plane. I can give you a slightly better outline of this, but it's quite difficult to do.

Also in quite a lot of physics problems, there can be problems which appear to have no relation to complex numbers but by far the easiest way to solve them is to use complex numbers. For example the way that airoplane wings are created to give the best uplift uses complex numbers and is very difficult to do without them, and there are many others.

Sorry if that got a bit technical, one of the problems with complex number theory is that there is quite a lot of difficult theory to be learnt before it becomes really useful.

Also complex numbers are a natural extension of real numbers.U see , U can represent all real no as a point on a line.So it is natural to ask what about a plane?? Now it happens that complex no can be represented as a point on a plane.
There are also other generalization for higher dimensions... but none for 3 .