Rational numbers

Having talked about integers, it’s now time to turn to fractions. In order to discuss fractions, why don’t we order a pizza? Here it is. We want to share this pizza with another person. How to do this? One way would be to cut the pizza across at its widest point, thereby creating two equal pieces, give one to the other person, and keep that one for ourselves. This illustrates very clearly the concept one piece out of two, the fraction 1/2. However, a pizza can also be cut into four equal pieces by two cuts, as I’ve shown. In that case, you would want two of these pieces. If you took two of them, and kept them for yourself.

You would have two pieces out of four. Now two out of four, if you do the prime characterizations, factorizations– excuse me– of the number 2 and the number 4, you see that 2 cancels out and you get back to the fraction 1/2. In other words, two quarters is the same as 1/2 except the two quarters is a reducible form of the fraction, whereas 1/2 is an irreducible form. To see how to reduce more complicated fractions, consider this one. What we do is the following. We write both the numerator– that’s 30, the top part of the fraction– and the denominator, 132, the bottom part in their prime factorizations. And then we cancel out top and bottom, whatever we can.

In this case, the 3’s cancel out, and one of the 2’s. And you’re left with 5 over 22. That’s an irreducible fraction because 5 is a prime number. Another type of number we need is the negative integers. In our modern era, it’s probably easier to imagine negative numbers than it was in the distant past. Anyone who has a credit card or a bank balance knows all too well that numbers can be negative. Another type of number we wish to introduce then, a class of numbers, is the integers z. You put all the integers together the natural numbers, and the negative integers, and you have the set of all integers called z.

The set of rational numbers, denoted q, is the set of all fractions a over b where a and b are chosen among any integers. The only problem is you have to avoid 0 for b because division by 0 is not defined. Notice that in reading the name of the set q, the description, rather, of the set q, you say a over b such that and so forth. The colon is the two dots there, is a way of describing a set, and it stands for where or such that.

Now do the rational numbers suffice? That’s a question that worried the ancient Greek mathematicians for a long time. I mean by that, are the rational numbers enough to describe all kinds of reality? In order to answer that question, let’s, again, order a pizza, but this time, a square one. As you can see, the side of the square is one unit. It could be one foot. It could be 30.54 centimetres. It’s just one unit of length. And, again, we wish to divide it between two people. How do we do so? Well, we could just cut along the diagonal of the square. That would clearly be fair. And then we’d be left with a one triangular piece.

What about the length of the cut in this case, the number I’m calling here h? What can we say about h? Well, a lot of light is thrown on this question by recalling the classical theorem of Pythagoras, which says that in a right angle triangle, such as I’ve drawn here, the sum of the squares of the two shorter sides is equal to the square of the longer side, called hypotenuse. In other words, this formula holds. If we apply this fact to our piece of pizza, you see that h squared is equal 2. That is, h is a number, which when multiplied by itself gives you 2. We usually summarise that by saying that h is equal to root 2.

That’s the radical sign that I’ve put around the 2. Now, h is a very real number. It’s easy to see that it’s bigger than 1 and it’s less than 2. And the question is, is it a rational number? It turns out that the square root of 2 is not rational. And I’m going to prove it now. This is a challenge proof for you. You don’t need to follow this proof to follow this course. Why am I doing it? For the fun of it, and because it’s beautiful. Here’s the proof. We’re going to use a method of reasoning called reasoning by the absurd.

We’re going to suppose that root 2 is in fact a rational number of the form a over b. And we’re going to derive from that a contradiction. The existence of a contradiction will prove that root 2 could not have been rational. Now, in writing root 2 in the form a over b, we can certainly assume that the fraction a over b is irreducible. So there it is. Root 2 is given by that. If you square both sides, then you obtain this equation. That certainly implies that a squared is an even number because it’s equal to 2 times an integer. Therefore, a squared is divisible by 2.

That in turn implies that a must be divisible by 2, because if a had no 2 in its prime factorisation, then neither would a squared. If a is divisible by 2, then a squared is divisible by 4. That means 2b squared is divisible by 4, because 2b squared, we just saw above is equal to a squared. It follows then that b squared is divisible by 2. And by the same reasoning as for a above, therefore b is divisible by 2. Now, step back for a moment. We’ve proven here that a is divisible by 2 and b is divisible by 2. Well, there’s our contradiction, because we said that a over b was an irreducible fraction. That contradiction ends the proof.

That square box that has just appeared, by the way, is the way modern mathematicians mark the end of a proof. In the old days, you used to write QED for a certain Latin phrase. Now, we can locate numbers, such as root 2, on a real line in a useful way. Here’s how it works. We give ourselves an infinite horizontal axis. We’re going to be calling it the real line. We choose a point on this line where 0 will be fixed. Now, once we have selected a unit distance, one unit distance, we can place all the integers subsequently. We put the positive ones to the right and the negative ones to the left.

Once we have these numbers placed, we find it fairly easy to place any number on the axis in the following sense. If I want to know where root 2 should be, well I know it lies between 1 and 2– I’ve already made that observation, and it should be right about there– and I can place it by simply taking my piece of pizza, and having one end at 0, and the other end will fall at root 2. Of course, that could be a little messy. I don’t suggest you do that at home. Now, the set of all numbers on this real line is denoted R. It’s the set of real numbers. It is the end of our quest for numbers.

As you see, it’s the biggest class of numbers. It is a beautiful class of numbers in which to do much mathematics, for instance, calculus. And we’ll be talking about it in the next segments.