Many of us find anything to do with numbers complex, not to say downright confusing. We fret our way through school learning more arithmetic, algebra and geometry than most intelligent people would need in a lifetime. That's not to mention the seemingly useless trigonometry and calculus.
This Entry introduces a branch of mathematics which is even more detached from the real world. In fact, the first thing to say about complex numbers is that they are not all real.
Other numbers are real. Mathematicians use the word 'real' to describe numbers which you can measure. Three, -56.3, the square root of 42, and pi (3.14159...) are real. Complex numbers have a part which is real and a part which is not - we call this part 'imaginary'.
To try to see an imaginary number, take your calculator and try to find the square root of -1. You were probably told at school that you 'can't have' a square root of a negative number. Your poor calculator will try its hardest to give you an answer, but it will fail, maybe spectacularly, and it will display an error message which the calculator designer programmed into it for this purpose. Maybe it will say 'E' or 'Error', or fill the display with asterisks or something. It makes you wonder whether, if the programmer hadn't told it what to say, the calculator would now be ushering you into a window on a new world of imaginary numbers.
This square root of minus 1 is the basic imaginary number. Mathematicians1 call it i. (It's funny how the moment mathematicians discover an interesting number they immediately give it a letter instead; at least this one's not Greek.)
So there it is: i.
i squared is -1.
Figment of your imagination?
As we said earlier, a complex number has a real part and an imaginary part. There are a number of ways to write them, and here's one. This is called Cartesian notation, after René Descartes, who was somewhat suspicious about this strange mathematics.
(4 - i) is a complex number. The real part is four, and the imaginary part is -i.
(-7 + 3i) is another - the real part is minus 7 and the imaginary part is 3 times i.
To visualise complex numbers, we need to plot them on a special graph. The x-axis (the one which goes horizontally across the graph) is used to plot the real part of the number, and the vertical y-axis is used to plot the imaginary part - the number by which we multiply i. These axes meet at the origin, a point where the values of both are zero. So, a number like (2 + 4i) will be a point on this graph which is two units to the right of the origin, and four units above it. (-7 -3i) will be a point 7 units to the left of the origin and 3 units below it. This graph has a special name, the Argand Diagram, after the 19th Century French amateur mathematician Jean-Robert Argand, who published the idea in 18062. If you've ever studied vectors, then you will find this graph familiar. Numbers on it can also be described using 'polar representation', which makes use of their distance from the origin (the 'size' or 'modulus' of the number) and the angle of the line joining the number to the origin, as measured anticlockwise from the real axis (this is known as the 'argument').
So, whose idea was this?
We can blame the Italians for this one. Renaissance mathematicians were studying equations which didn't seem to have any solutions, when they came up with a couple of new concepts. One was negative numbers, the other was imaginary numbers, but the latter didn't really become accepted until the 19th Century, following work by German mathematicians Georg Cantor and Julius Dedekind.
What can you do with them?
Well, you can do most things that you can do with real numbers (with some exceptions), and some odd things besides.
You can add and subtract them - just add or subtract the real and imaginary parts separately. The sum of the two numbers above is (-3 + 2i). The difference is (11 - 4i).
You can multiply them together, by multiplying each element in one number by each element in the other, as if you were multiplying algebraic expressions in brackets. Just remember that when you get an i squared, you can replace it with minus 1.
(4 - i) x (-7 + 3i) = -28 + 12i + 7i + 3 = (-25 + 19i)
Dividing them is a bit trickier. You need to multiply the numerator and the denominator by what is called the 'complex conjugate' of the denominator. The conjugate of a complex number is the same number, but with the sign of the imaginary part reversed. The complex conjugate of (4 - i) is (4 + i) and that of (-7 - 3i) is (-7 + 3i). Doing this multiplication removes the i's in the denominator. We'll leave this for you to try at home. Have a go at dividing (1 + i) by (1 - i), using this method3.
Why are they so interesting?
As the Italians found, they let you solve 'impossible' problems!
Here's one - you have 20 metres of fencing, and you want to enclose a rectangular area of 40 square metres. How long are the sides? (Go on - try it.)
Given up? Well, you can do it with complex numbers, and the answer is that the long sides are (5 + the square root of minus 15) metres and the short sides are (5 - the square root of minus 15) metres.
OK, it's an impossible answer - if you don't believe it, just ring up B&Q4 and try to order the square root of minus 15 metres of larch lap - but the point is, it's an answer that fits the question. If you multiply the sides you get 40 square metres, and if you add them all you get 20 metres.
That doesn't sound useful to me
Newton, practical as ever, saw these numbers purely as an indication that a problem couldn't be solved, but in fact they have particular use in pure (theoretical) mathematics. The German mathematician Carl Friedrich Gauss studied them in the early 19th Century, and complex numbers which have integers for both the real and imaginary parts are known as Gaussian integers. The Swiss mathematician Leonard Euler derived the well-known Euler Equation which links complex numbers to trigonometrical functions.
For a more practical example, we'll look at a problem studied by the Italian mathematician Rafael Bombelli in the 16th Century.
This is quite involved, and we won't go into detail, but Bombelli was looking at problems with simultaneous equations like, for example:
x2 + y = 21 and xy = 20
These lead to the cubic equation5:
x3 - 21x + 20 = 0
Now, Bombelli knew a formula for solving equations of this form, and it had to be derived in two stages. The first stage was to find a number g such that...
3x = 3g1/3 - 7g-1/3 (Note: g1/3 is another way of saying the cube root of g.)
...but the number he found using his formula for g was complex - it involved the square root of a negative number. He then discovered that another complex number when cubed gave the required value for g - for this example g1/3 turns out to be (2 + the square root of minus 3) - but the clever thing was this: when he completed his formula using this number, the answer it gave at the end was x = 4. So x turned out to be a real number6, but in order to calculate it, he needed to use complex numbers.
So there you have it. Complex numbers are useful to mathematicians as they extend our mathematics and enable us to solve real problems which were previously unattainable through our restrictions on real solutions throughout. Not only this, but complex numbers provide a way of describing two-dimensional spaces, leading to their use in electrical engineering, control theory and quantum mechanics.