This is the second of two posts about the number line and I will assume that you’ve at least skimmed through the first. In this post I will be getting to the root of the awesomeness of the number line. If you follow it until the end you will genuinely have understood more than just the gist of the idea.
Collections of Numbers
Picture the number line. Let’s think about numbers which are on the left hand side of the number ‘two’. The number ‘one’ is on the left-hand side of two. As is ‘three-quarters’. As is ‘negative fourteen point three’ etc. Numbers that are to the left of ‘two’ are said to be ‘less’ than two. So, we are currently thinking about numbers which are less than two. Just so we’re all agreed and we know what we’re dealing with, imagine colouring a number blue if it is less than two and black if it isn’t. A portion of your number line should now looks like this:
What is there to say about this collection of numbers, the blue portion of our number line? Well, it goes on forever to the left but it doesn’t go on forever to the right because it stops at two. There is another way of thinking about this fact, which I’ll now illustrate: Is there a point on the number line, to the right of which there are no blue bits? The answer is ‘yes’. Think about the position of the number ‘three’. There aren’t any blue bits to the right of ‘three’. Similarly, there aren’t any blue bits to the right of ‘two’. Is there a point on the number line, to the left of which there aren’t any blue bits? The answer is ‘no’. Bear these ideas in mind, as we will return to them.
Usually, such a collection of numbers (i.e. one that doesn’t go on forever to the right) is said to be “bounded above”. This is one of the more sensible terms from the nomenclature. In a similar fashion, we might consider the collection of numbers which are to the right of ‘zero’. I won’t draw a coloured picture but I’d still like you to imagine what the picture would look like. These numbers are said to be ‘greater’ than zero. Being greater than zero is synonymous with being ‘positive’ and this collection of numbers, being one that does not go on forever to the left, is “bounded below”.
Here is another collection of numbers:
This collection doesn’t go on forever in either direction. It is entirely within a portion of the number line. Is there a point on the number line, to the left of which there are no blue bits? Yes. There are no blue bits to the left of ‘zero’. Is there a point on the number line, to the right of which there aren’t any blue bits? Yes, there are no blue bits to the right of, say, ‘four’. Now, the answer is ‘yes’ in both cases. Such a collection of numbers is simply said to be “bounded”.
So, what we’ve done so far is discuss the conceptual distinctions between sets of numbers which are bounded, just bounded above, just bounded below or simply not bounded at all, like say ‘the collection of all numbers on the number line’.
Doing The Best We Can
We won’t be concerning ourselves with sets which are not bounded at all (i.e. unbounded). The fundamental property of the number line which I am making it my goal to explain is an idea involving sets which are bounded above.
Think back to our example of a set which was bounded above: The set of numbers which are less than ‘two’. One way in which we thought about the fact that this set was bounded was to observe on our picture that there was a number (well, it is obvious there are loads of such numbers) to the right of which there were no blue bits.
Suppose now, just hypothetically, that you were presented with a set of numbers which you suspect is bounded above and challenged to demonstrate that this were so. You can provide concrete evidence, in the form of a number, by simply creating the picture of the number line and pointing to a position on the number line, to the right of which there are no blue bits. For example,
This concrete evidence is more satisfying mathematically than saying “Oh look it doesn’t go on forever”.
Here comes the important part: The important part is that there are lots of numbers which serve your purpose when you are looking for a number to point to to say “There are no blue bits to the right of here”. You can choose one number from many at your disposal. Of all the numbers you can choose from, I want you to think about the one which is leftmost. This number will be just next to the blue bits. It will be the smallest of the numbers from which you can choose. This simple idea is what this post is about. Let’s go through that again.
The first collection of numbers I got you to think about was the set of numbers that are less than ‘two’. Have another look at the first picture if you need to. If you were challenged to prove to me that this set were bounded above, there are again lots of numbers you could use as evidence in the same way as before (i.e. by saying “There are no blue bits to the right of this number”). Which is the smallest such number? The answer is definitely ‘two’. This number is called (for obvious reasons) the “least upper bound” of the set. So, the least upper bound of the set of numbers which are less than ‘two’ is ‘two.
Every set of numbers which is bounded above has a least upper bound.
The fact that every set of numbers which is bounded above has a least upper bound is of fundamental importance to mathematical analysis. It is something which one would typically learn about during the first year of an undergraduate degree in maths in the UK. When I learnt about it and people told me that it was important, I could not see why (however, by that stage I had already committed to a degree in maths). The point is that if you, on some level, have understood what I’ve been saying, then you have begun to get to grips with a real piece of maths from a real maths degree. Naturally, there have been omissions of details and it would not have been presented to you in the manner in which it has been, but the simplicity of the underlying idea is undeniable.
Do you remember how we used the number line at the end of the previous post? We used it to demonstrate why there must be a number whose square is six. One can perform a similar demonstration using the ideas of this post: Just think about the set of numbers which square to give you a number less than six. This is bounded above because there are no such numbers to the right of the ‘three’ position (after all, a number greater than three squares to give you a number greater than nine). We therefore know that there must be a least upper bound. What is the square of this least upper bound? Indeed it is six. How do I know? Well, in order to finish the argument off properly, one would normally reason that the square of the least upper bound cannot be less than six and then that it cannot be greater than six either. I have chosen to omit the details of this, but it can be reasoned through without using anything more advanced than things like being to the left or right of other things etc.
I’d like quickly to draw your attention to the fact that we are already paying the price of not having proper notation and terminology and rigour. While it does in theory keeps things more accessible, without such things, the mathematics is at times unconvincing and rather wordy (leading me to feel that I ought to omit things like finishing off the argument at the end of the previous paragraph). I have left out little details along the way and the exposition as a whole doesn’t really stand up to close scrutiny on its own. If you were very very sceptical and questioned everything I said, then you could pick holes in my logic. Thus, although it can look formidable and confusing, the formality which proper mathematical notation affords is genuinely desirable, to shore up our reasoning, not least for elegance and succinctness.