Chapter 1
The real numbers

From the considerations in the Introduction, it is clear that in order to have a firm foundation of Calculus, one needs to study the real numbers carefully. We will do this in this chapter. The plan is as follows:

1. 1. An intuitive, visual picture of : the number line. We will begin our understanding of intuitively as points on the 'number line'. This way, we will have a mental picture of , in order to begin stating the precise properties of the real numbers that we will need in the sequel. It is a legitimate issue to worry about the actual construction of the set of real numbers, and we will say something about this in Section 1.8.
2. 2. Properties of . Having a rough feeling for the real numbers as being points of the real line, we will proceed to state the precise properties of the real numbers we will need. So we will think of as an undefined set for now, and just state rigorously what properties we need this set to have. These desirable properties fall under three categories:
   1. the field axioms, which tell us about what laws the arithmetic of the real numbers should follow,
   2. the order axiom, telling us that comparison of real numbers is possible with an order > and what properties this order relation has, and
   3. the Least Upper Bound Property of , which tells us roughly that unlike the set of rational numbers, the real number line has 'no holes'. This last property is the most important one from the viewpoint of Calculus: it is the one which makes Calculus possible with real numbers. If rational numbers had this nice property, then we would not have bothered studying real numbers, and instead we would have just used rational numbers for doing Calculus.
3. 3. The construction of . Although we will think of real numbers intuitively as 'numbers that can be depicted on the number line', this is not acceptable as a rigorous mathematical definition. So one can ask:

   Is there really a set that can be constructed which has the stipulated properties (2)(a), (b), and (c) (and which will be detailed further in Sections 1.2, 1.3, 1.4)?

The answer is yes, and we will make some remarks about this in Section 1.8.

1.1 Intuitive picture of as points on the number line

In elementary school, we learn about

Incidentally, the rationale behind denoting the rational numbers by is that it reminds us of 'quotient', and for integers comes from the German word 'zählen' (meaning 'count'). In the above,

represents a whole family of 'equivalent fractions'; for example,

We are accustomed to visualising these numbers on the 'number line'. What is the number line? It is any line in the plane, on which we have chosen a point O as the 'origin', representing the number 0, and chosen a unit length by marking off a point on the right of O, where the number 1 is placed. In this way, we get all the positive integers, by repeatedly marking off successively the unit length towards the right, and all the negative integers by repeatedly marking off successively the unit length towards the left.

Just like the integers can be depicted on the number line, we can also depict all rational numbers on it as follows. First of all, here is a procedure for dividing a unit length on the number line into d () equal parts, allowing us to construct the rational number on the number line. See Figure 1.1.

Figure 1.1 Construction of rational numbers: in the above picture, given the length 1 (that is, knowing the position of B), we can construct the length 1/5, and so the point A corresponds to the rational number 1/5.

The steps are as follows: Let the points O and B correspond to the numbers 0 and 1.

1. Take any arbitrary length along a ray starting at O in any direction other than that of the number line itself.
2. Let be a point on the ray such that .
3. Draw parallel to to meet the number line at A.

Conclusion: From the similar triangles and , we see that the length .

Having obtained , we can now construct on the number line for any , by repeating the length n times towards the right of 0 if , and towards the left times from 0 if n is negative.

Hence, we can depict all the rational numbers on the number line. Does this exhaust the number line? That is, suppose that we start with all the points on the number line being coloured black, and suppose that at a later time, we colour all the rational ones by red: are there any black points left over? The answer is yes, and we demonstrate this below. We will show that there does 'exist', based on geometric reasoning, a point on the number line, whose square is 2, but we will also argue that this number, denoted by , is not a rational number.

First of all, the picture below shows that exists as a point on the number line. Indeed, by looking at the right angled triangle , Pythagoras's Theorem tells us that the length of the hypotenuse OA satisfies

and so is a number, denoted say by , whose square is 2. By taking O as the centre and radius , we can draw a circle using a compass that intersects the number line at a point C, corresponding to the number . Is a rational number? We show below that it isn't!

Exercise 1.1.

Depict and on the number line.

Theorem 1.1 (An 'origami' proof of the irrationality of ).

There is no rational number such that .

Proof.

Suppose that is a rational number. Then some scaling of the triangle

by an integer will produce a similar triangle, all of whose sides are integers. Choose the smallest such triangle, say , with integer lengths , and , . Now do the following origami: fold along a line passing through A so that B lies on AC, giving rise to the point on AC. The 'crease' in the paper is actually the angle bisector AD of the angle .

In , , . So is an isosceles right triangle. We have , while

So is similar to the triangle

has integer side lengths, and is smaller than , contradicting the choice of . So there is no rational number q such that .

A different proof is given in the exercise below.

Exercise 1.2 (*).

We offer a different proof of the irrationality of , and en route learn a technique to prove the irrationality of 'surds'.1

1. Prove the Rational Zeros Theorem: Let be integers such that and are not zero. Let where are integers having no common factor and such that . Suppose that r is a zero of the polynomial . Then q divides and p divides .
2. Show that is irrational.
3. Show that is irrational.

Thus, we have seen that the elements of can be depicted on the number line, and that not all the points on the number line belong to . We think of as all the points on the number line. As mentioned earlier, if we take out everything on the number line (the black points) except for the rational numbers (the red points), then there will be holes among the rational numbers (for example, there will be a missing black point where lies on the number line). We can think of the real numbers as 'filling in' these holes between the rational numbers. We will say more about this when we make remarks about the construction of . Right now, we just have an intuitive picture of the set of real numbers as a bigger set than the rational numbers, and we think of the real numbers as points on the number line. Admittedly, this is certainly not a mathematical definition, and is extremely vague. In order to be precise, and to do Calculus rigorously, we just can't rely on this vague intuitive picture of the real numbers. So we now turn to the precise properties of the real numbers that we are allowed to use in developing Calculus. While stating these properties, we will think of the set as an (as yet) undefined set containing which will satisfy the properties of

1. the field axioms (laws of arithmetic in ),
2. the order axioms (allowing us to compare real numbers with ), and
3. the Least Upper Bound Property (making Calculus possible in ),

stipulated below.

It is a pertinent question if one can construct (if there really exists) such a set satisfying the above properties (1-3). The answer to this question is yes, but it is tedious. So in this first introductory course, we will not worry ourselves too much with it. It is a bit like the process of learning physics: typically one does not start with quantum mechanics and the structure of an atom, but with the familiar realm of classical mechanics. To consider another example, imagine how difficult it would be to learn a foreign language if one starts to painfully memorise systematically all the rules of grammar first; instead a much more fruitful method is to start practicing simple phrases, moving on to perhaps children's comic books, listening to pop music in that language, news, literature, and so on. Of course, along the way one picks up grammar and a formal study can be done at leisure later resulting in better comprehension....