GCSEs are a type of public exam taken by school leavers in the UK when they are around 16 years of age. Included in the standard syllabus is some familiarity with a thing called the number line. It is an object which harbours some ideas which are very important to mathematics at a higher level, but these ideas would not generally have not been taught to GCSE candidates.
I’m going to try to explain what the number line is and why I think it is so great. In this first post (hopefully one of two), we will construct the number line and talk about it a bit. In the second we’ll get to the root of its awesomeness. We’ll start very slowly, try to keep things simple and we’ll try to keep things abstract!
So, imagine a line. What have you got in your head? My line is horizontal. And it keeps going forever to the right and to the left. Once your line looks like this, we’ll be able to continue. The idea will be that every position on the line corresponds to a number, much like a ruler. We just need to decide which numbers go where.
The important thing is, we are absolutely free to decide which numbers go where. We created this idea in our minds and we can do whatever the hell we want with it. So, pick a point on the line, anywhere at all, it doesn’t matter, but in your mind just pick one point. And now pick a number, any number, and label that point with that number. We are going to do this twice more, so make sure you remember your numbers. You might like to write them down, but in a moment I will draw a picture with example numbers so you could just wait until then.
Now pick a point on the line to the right of the first point you picked. Choose a number for that point. Finally, pick another point to the right of the second one. Choose a number. If, for example, I chose the numbers five, thirteen and two, a portion of my line might look something like this:
These numbers are not ‘wrong’ in any way (it’s just a random piece of maths which we created, after all), but there are some valid criticisms of our approach so far. Firstly, we could go on picking points one-by-one forever and we would never accomplish our goal of making sure every point on the line corresponds to a number, because, as we said at the start, the line goes on forever. Also, there is something jarring about the way in which these numbers are not in order. We all know that two comes before five which in turn comes before thirteen, making our placement look haphazard and silly. There is a much more favourable approach, with which I am sure you are familiar.
Scrap the old choices and start again by picking a point, anywhere at all. Label this point ‘zero’. Now pick a point to the right of zero and label it ‘one’. Now, the important fact is that there is a sense in which these two choices can determine which numbers go where. To illustrate my point: Now that a portion of our line looks like this:
Where does the number ‘two’ go now? Well, where ought it to go? Taking into account where ‘one’ and ‘zero’ are, where is the most natural place for it to be? Where does the number ‘one half’ go? Where does negative one go? Hopefully you are beginning to get the point. It is my intention that you have visualized something like this:
Now, every number has its place. The criticisms of the previous method vanish. We have prescribed the place of every number all at once and they are all arranged in order. This is a much more pleasing state of affairs. We have created what is known as ‘the’ number line, the use of the definite article being testament to the fact that, despite initially having the freedom to choose where numbers went, there is something optimal about this particular arrangement.
Using the number line
Now that we have constructed the number line, it immediately repays us by being helpful for understanding addition and negative numbers and the order of numbers and all the other things underpinning how it is used at GCSE level.
What can we actually use the number line for though? Once you understand things at the GCSE level, is it actually of any further use? It certainly doesn’t appear in the A-level syllabi which I learnt…
I’ll now describe a rather famous application of the number line in order to try to illustrate its merit as a mathematical object. Do you know what it means to square a number? It means to multiply it by itself. For example when you square two you get four. Often one hears things like ‘two squared is four’ or ‘the square of two is four’. Now, I have a question. Is there a number whose square is six? (I apologise if you feel that I have unduly anthropomorphised my number in the previous sentence. I assure you that it’s quite common.)
Make sure you understand what I’m asking. I’m wondering whether or not such number exists. For example, is there a number bigger than all other numbers? The answer is no. No such number exists. I’m after a number such that when you square it, you get six.
You may feel that such a number obviously exists and simply reach for a calculator and press the appropriate buttons. Your calculator then throws out a long decimal, the details of which are irrelevant. Or, if your calculator is very sophisticated it might simply display the notation for ‘the square root of six’. Neither is satisfying. The first is not because if you multiply the decimal you have been presented with by itself, it does not give you six (your calculator will lie. Do you trust it? Do the calculation by hand if you like…). The second is obviously just a restatement of the input.
There are much more elegant ways to achieve our goal. I’ll describe one such way now. We know that two squared is four and that three squared is nine. This is evidence that some numbers (e.g. two) square to give numbers which are smaller than six and some numbers (e.g. three) square to give numbers which are bigger than six. This will be crucial.
By thinking about these facts and using the number line, we ought to be able to convince ourselves that there is indeed a number which squares to give six. Imagine putting your finger on the number line at the ‘two’ position. Now imagine slowly but surely sliding your finger to the right. Do it slowly, but with a constant gliding motion. Since bigger numbers square to give you bigger numbers, you’ll be passing over numbers whose squares are bigger than four. And then numbers whose squares are bigger still and bigger still, until, after a while, you arrive at the ‘three’ position. And as we agreed earlier, three squared is nine. At some point you must have passed over a number which squares to give you six, because six in in between four and nine (and there aren’t any gaps in the line or anything silly like that).
This is a perfectly valid way of convincing oneself that there is a number which squares to give you six. It is, well, mathematics. Notice how we know very little about this number. We know that it’s between two and three. But that’s about it. We needn’t be able to write it down or anything like that. That would actually be a ridiculous requirement.
However, you may still be left dissatisfied. Partly for the aforementioned reason that you don’t actually know the number and partly because it doesn’t feel like we’ve done any maths, really. This might leave you with nagging doubts about whether what we have done is ‘OK’. Does it really ‘work’? Although I certainly do consider what we’ve done to be a form of mathematical proof, it would not be accepted by the mathematical community at large, and for a very good reason. It lacks something called ‘rigour’, the application of which would help us to be much surer that our ‘proof’ did indeed ‘work’. However, that discussion is for another day.