A short sneak preview of the full-length ‘Mandelbulbs’ video currently in production. A Koch Snowflake is an example of a 2D fractal with infinite perimeter but finite area. Full details of the calculation in the final video… COMING SOON!
Complex Numbers – they don’t have to be complex!
The idea of complex numbers stems from a question that bugged mathematicians for thousands of years: what is the square root of -1? That is, which number do you multiply by itself to get -1?
Such a simple question has blossomed into a vast mathematical theory, for the simple reason that the answer isn’t real! It can’t be 1, as 1 * 1 = 1; it can’t be -1, as -1 * -1 = 1; whichever number you multiply by itself, you can’t get a negative number. Up until the 16th century, almost everyone ignored this issue; perhaps they were afraid of the implications it could bring. But then, gradually, people began to realise that there was a whole new world of mathematics waiting to be discovered if they faced up to the question.
In order to explain this apparent gap in maths, the idea of an ‘imaginary’ number was introduced. The prolific Swiss mathematician Leonhard Euler first used the letter i to represent the square root of -1, and as with most of his ideas, it stuck. Now i isn’t something that you’ll see in everyday life in relation to physical quantities, such as money. If you’re lucky enough to have money in your bank account, then you’ll see a positive number on your bank statement. If, as is the case for most students, you currently owe money to the bank (for example, if you have an overdraft), then your statement will display a negative number. However, because i is an ‘imaginary’ unit, it is neither ‘positive’ nor ‘negative’ in this sense, and so it won’t crop up in these situations.
Helpfully, you can add, subtract, multiply and divide using i in the same way as with any other numbers. By doing so, we expand the idea of imaginary numbers to the idea of complex numbers.
Take two real numbers a and b – these are the type that we’re used to dealing with.
They could be positive, negative, whole numbers, fractions, whatever.
A complex number is then formed by taking the number a + b * i. Let’s call this number z.
We say that a is the real part of z, and b is the imaginary part of z.
Any number that you can make in this way is a complex number.
For example, let a = -3 and b = 2; then -3 + 2*i, which we write as -3 + 2i, is a complex number.
As we saw before, complex numbers don’t actually pop up in ‘real-life’ situations. So why do we care about them? The reason is that complex numbers have some very neat properties that allow them to be used in all sorts of mathematical contexts. So even though you may not see the number i in everyday life, it’s very likely that there are complex numbers involved behind the scenes wherever you look. Let’s have a quick glance at some of these properties.
The key observation is that the square of i is -1, that is, i * i = -1.
We can use this fact to multiply complex numbers together.
Let’s look at a concrete example: multiply 2 + 2i by 4 – 3i.
We use the grid method for multiplying out brackets:
4 | -3i | |
2 | 2 * 4 = 8 | 2 * -3i = -6i |
+2i | 4 * 2i = 8i | 2i * -3i = -6 * i * i = -6 * -1 = 6 |
Adding the results together, we get (2 + 2i)(4 – 3i) = 8 + 6 – 6i + 8i = 14 + 2i.
Therefore, multiplying two complex numbers has given us another complex number!
This is true in general, and it turns out to be very handy. In fact, Carl Friedrich Gauss proved a very famous result – known as the Fundamental Theorem of Algebra because it’s so important – that effectively tells us that the solutions to all equations can be written as complex numbers. This is extremely useful because we know that we don’t have to go any ‘deeper’ into numbers; once you’ve got your head around complex numbers, you can proudly declare that you’ve mastered them all!
Because of this fundamental theorem, our little friend i pops up all over the place in physics, engineering, computer science, and of course, in all sorts of areas of maths. While it may only be imaginary, its applications can be very real, from air traffic control, to animating characters in films. It plays a really important role in much of theoretical mathematics, which in turn is used in almost every scientific discipline. And to think, all of this stemmed from an innocent-looking question about -1; what were they so scared of?!
Kai Laddiman