Is it possible with a given amount of paint, to paint intervals around all rational numbers on the real number axis?

Dear Ward

You can indeed prove this, but it can only be called ‘easy’ if you have already developed some mathematical material that is needed for this. Building/proving it all mathematically very neatly would lead us too far here, but I can go over the idea in a nutshell. We do that in two steps; first we show that you can ‘number’ all rational numbers.

Part of the answer lies in the fact that there are “much fewer” rational numbers than irrational numbers, which together fill the entire real number axis. There are of course infinitely many in both cases, so there is a need for a mathematically more precise way to compare those ‘sizes’.

There are infinitely many natural numbers, but you can name them successively and thus ‘count’: 0, 1, 2, 3, … There are infinitely many, but we call that ‘countable (infinite)’ just because you she can list that way. Any (infinite) set for which this succeeds is called countable (infinite). Or to put it another way: a set is countably infinite if we can link every element of that set to a natural number and vice versa (more precisely: a bijection can be made between the natural numbers and the elements of that set).

Perhaps not very intuitive, but with this convention the integers are also countable and so in that sense the set of the integers is ‘the same size’ as the set of the natural numbers. For example, a possible listing would be: 0, 1, -1, 2, -2, 3, -3, 4, … Obviously, every integer will appear in exactly one place in this list. This means that we can ‘number’ the integers with the natural numbers, we can speak of a first, second, third, etc. integer in a given listing and by going through them in this way, we go over all integers.

Even less intuitive is the fact that you can also list all rational numbers in such a way! That set is also countable and we can speak of the first, second, third, etc. rational number and run through them all in that way. The formal details (laying such a bijection) can be read via the attached link, but the attached image makes it visually quite clear how you could go through all (positive) rational numbers and thus also ‘number’ them.

Now that we have numbered the rational numbers, we can return to the real number axis and the paint analogy for the second step. The real number axis is full of rational numbers, but of course there are also irrational numbers between rational numbers. The idea is to choose an open interval around each rational number and then we will add up the lengths of all those intervals (of course there are infinitely many!). If we can keep that total sum of lengths finite, then, as it were, a finite amount of paint suffices to color all those intervals and then all rational numbers are also ‘covered’ with (a finite amount of) paint.

We are going to make use of the fact that we have been able to number the rational numbers: I can now conveniently r₁r₂r₃, … to call. Suppose we presuppose a certain positive number epsilon (I shorten to e) in order to keep the total sum of the lengths of the intervals less than this e. around r₁ we take an interval of length e/4, e.g. nicely symmetrical the interval (r₁-e/8,r₁+e/8). Then we take around r₂ again such an interval but of length e/8, so halved. If we continue to halve in this way, infinitely many intervals arise whose lengths are: e/4, e/8, e/16, e/32, … Around every rational number, in this way a interval around that rational number, with decreasing lengths of the intervals.

Now we’re there, if you’ve already studied geometric series? After all, for the sum of the lengths of these intervals we look for e/4 + e/8 + e/16 + e/32 + … and that is the sum of a geometric sequence. If you haven’t seen this yet, you can also visualize this: start from a circle and shade a quarter, then shade another 1/8th, then 1/16th, etc. It should be clear then when adding ‘ all these terms’, the sum will never exceed 1/2. We say that the sum of all the lengths is equal to e/2 and that is clearly smaller (because exactly half!) of the finite e (epsilon) from which we started.

Hopefully you were able to follow this a bit. A few more notes, in case you’d like to look up more about this:

– The trick with integers and rational numbers doesn’t work; you can prove that the irrational numbers, for example, cannot be counted, so there are ‘too many’ of them. The consequence of this is that the real numbers are also uncountable; we call such sets ‘uncountable’.

– The trick with decreasing intervals to get a finite sum is related to the mathematical concept of ‘measure’. We say that the rational numbers have “measure 0” precisely because we can create intervals around them whose sum total of lengths is less than any positive number (previously called epsilon) that we choose beforehand.

regards

Tom