This is the introduction which I give to my first year mathematics class when they see imaginary numbers for the first time. I thought I’d type it up here as it’s received good reactions the two times that I’ve introduced it in this format. Note that this probably isn’t the canonical way to introduce complex numbers, but then most of my lectures don’t necessarily take the normal route…
Complex Numbers, a philosophical detour
Before we get on to talking about imaginary numbers and complex numbers, let’s try and break down our preconceptions about numbers in general.
We look at the world around us and see many things which we categorise. We see a computer, a piece of paper, we see other people, we see our hands. These are labels that we use to categorise the world, but these objects seem very physical and very real. We rarely question their existence, though if one wants to take the Cartesian view, we should also question the reality we are in. We are not going to go that far, but let’s try and ask about the existence of numbers.
I have definitely seen five pieces of paper, but I have never seen a five. I’ve seen the number written down, but I can write down anything I want and it doesn’t necessarily mean that it exists. I can write down a erga[oeiave21 but that doesn’t suddenly bring a erga[oeiave21 into existence. How about a -5? I’ve definitely never seen a -5 though I understand perfectly well what it means. The integers seem to be very good ways of describing, or more specifically counting objects and the negative numbers are a good way of keeping track the transfer of objects from one place to another. I can also ask you to give me 3 coffees, and here I am really asking you to apply 3 as an operation to the object coffee. 3 is acting almost more like a verb than it is a noun. When I describe that there are 30 people in a class, I am really thinking about this as a description, or an adjective. So in the real world, somehow numbers feel like verbs and adjectives. I certainly wouldn’t say that ‘heavy’ exists, but certainly a book which has been described as heavy does.
However, there is a world in which numbers really do seem to be more like nouns than they do in the world around us, and that is in the abstract world of mathematics. In the universe of equations, numbers somehow feel much more concrete and I can manipulate them and transmogrify them from one form to another using a set of mathematical operations which become more and more finely tuned and specialised as we learn more and more mathematics. I can take a 5 and I can apply a the sin function to it to give me another number. I think of this rather as taking an object and putting it in a machine which turns it into another object. Here 5 is very real, but so is -5 and so is pi/4 and so are all the numbers that we’ve ever used. They are simply the objects which are manipulated by our mathematical machinery. Whether or not they exist as objects around us isn’t very important for our use of them in the mathematical universe.
Incidentally, I have here separated the real universe from the more abstract, platonic, mathematical one, but it is fair to say that we have found mathematics as the best language with which to accurately describe the real universe. All of our models and precise descriptions of the universe are built using mathematics, and it acts as an incredible way of describing the laws of nature. Which came first, the mathematics or the universe? That is not a question I am going to get onto here, but it’s certainly a profound one!
A foray into a new number system
OK, so we have a mathematical world of numbers and we can manipulate them. Thus, we should be perfectly happy to have some more ingredients in that world, that don’t have such an obvious mapping to the things in the world around us. We will discover that actually they help us enormously in the things that we can do with the mathematical machinery. It’s like having a powerful car but not the right fuel to really take it up to top speed. We are about to find out what that fuel is and push the limits of what our car can do!
Previously, if we set up a certain type of quadratic equation and plugged it into our machinery to find a solution, the machinery would jam and we wouldn’t get an answer out. This was a real shame because it didn’t seem to be that much more of a complicated equation than any other that we had studied. We are perfectly happy with solving an equation like:
You can plot the graph of y=x^2-1 and see that it equals 0 at two points x=1 and x=-1. That’s fine, our mathematical machinery can deal with that fine, but when we ask to solve something so similar:
our traditional machinery comes juddering to a halt and we get an error message on the screen – you’re not allowed to take the square root of a negative number, says our program. In fact when we plot the graph of y=x^2+1 it’s clear that it doesn’t cross the x axis, so it can’t have a solution…can it? Maybe we’re not looking hard enough. Maybe our machinery is fine, but we’ve just fed it the wrong fuel. In fact, we can find the solution just fine. The solutions are:
x=√-1 and -√-1
You might look at this and go “Absolutely not!” You can’t take the square root of a negative number, but if you plug that into the equation, it works just fine and is a perfectly good solution. What is not true is that √-1 is like the normal numbers that we are used to using. In fact, let’s give this solution a name. We’ll call it i:
(Note that we are actually being mathematically sloppy here, but for a first pass, this will do – we can explain the subtleties later – in particular the domain of the square root is only the positive real numbers and thus we have to say what we mean by this function separately to deal with non integer powers of negative numbers. We should really think of i as that number which when squared gives -1, rather than it being one of the two roots of -1.)
What is i? It’s one of the solutions to the equation above. Plug it in, you’ll see that it works. There’s no funny business going on here. So what if it doesn’t correspond to an object in the world you see around you, nor does -76 but we don’t have a problem with using that number, do we? i stands for imaginary. So we call i the imaginary number, but in fact it is no more imaginary than most other numbers, it’s just a little harder to understand it because we are used to things which represent a size. Numbers which are not imaginary are called real, but again, this name is probably not a very good name as -32 is in some sense no more “real” than is i.
Once you have defined this new type of number – a number that squares to a negative number, we open up so many new possibilities. Things that previously would have driven our machinery to a halt are now very easily accessible. With this new number we’ve just upgraded our mathematical machinery so that it can handle so many more problems than it could before.
It might seem that i wouldn’t have anything to do with the real world and it’s true that any measurement of the real world will give us one of the real numbers that we are used to, but using i makes many things much more natural than they would be without it. This is true in many many fields of science and so having i at hand is absolutely indispensable when you want to describe the real world. Very often given a question about the real world, it is much easier to take a detour into the world of these imaginary numbers which gives us a shortcut to an answer, than taking the long route using only the real numbers.
What we’ve shown here is that we can deal with √-1 but we can quite happily extend this to the square root of any negative number (Again, we must note that actually this is not formally correct, but to build basic intuition it is enough – we need to go a little further to see where the subtelties lie). From now on √-b where b is a positive number can simply be written as i√b. So i can be multiplied by real numbers. We can take 2 lots of i and 3 lots of i and add them together to get 5 lots of i: 2i+3i=5i.
How about adding a real number onto this? Well it turns out that adding together 3+4i can’t be simplified any more. You can’t add apples and oranges and get pomegranates. What you end up with are a few apples and a few oranges, and that’s as simple as it gets. In fact any number of this form, which has a real part, and an imaginary part is known as complex, and these complex numbers form a whole new domain which will supercharge our mathematical toolkit. We will see the power it gives us in the coming lectures…