Density of the rational numbers

One of the tutors has asked me to put in a quick post about a homework question  you had on rational numbers which many people had trouble with (Sheet 2, Q4). It’s a good question to go through because it’s one of your first tastes of actual analysis, and involves proving something about all rational numbers at once, so it’s good proof practice.

The question was

Prove that for any , with , we can find some with .

We call this property of the rationals being “dense”, because it means that you can find rational numbers in even the smallest gaps on the number line. The question is asking, if I give you two different rational numbers, can you find another rational number that sits between them.

The first thing to say is that it isn’t enough to check this for some special cases. An answer like the following

Let and , then

,

so the result is true.

will score you no marks because it only deals with one case, and since there are infinitely many cases, you really haven’t made much of a dent! That said, checking a few examples is always a good way to get your head round things.

 

So how would you go about proving the statement for all the infinitely many cases? Well the best way is to take two rational numbers called and and see if you can find a rational number which is between them. Once you’ve found this number, you’ll need to prove that

1. is a rational number.
2. does sit between and .

So what number is always between and ? How about the average of and , which is . Now that we have a candidate, let’s check that it satisfies 1. and 2. above.

1. Is rational? To prove it is, we need to write it as for two integers and . Well and are both rational, so we can write them that way. How about we say and . So
   
   But are all integers, so and are also integers. So we’ve written in the form we wanted. The only other thing we need to check that the bottom of the fraction isn’t zero, as that would mess us up, but this is fine, because both and come from the bottom of other rational numbers, so neither of them could be zero.
2. Is between and ? Well , so . So we have
   .
   For the very same reason, we get
   .
   So indeed .

So there you go, a proof for infinitely many things at once, which touches on some the things you’ll be doing when you move onto analysis proper, later in the course. Remember: when you’ve finished a proof, try and ask yourself “have I covered all the possible cases?” and “have I proved everything I need to prove?”. In our example here, the first question means checking we’ve done all the infinitely many rationals, which we did. The second means checking the number we found was both rational and inbetween and , which it was.

If you have any questions, feel free to comment.

Good luck!