Chapter 1 What is the number line?

When we visualise the integers $\mathbb{Z}$ on a number line, we see them as well spaced points: there are large holes or gaps in-between. When we consider the rationals $\mathbb{Q}$, the points on our number line become much more dense, but there are still lots of holes: for instance, there are holes where we expect $\sqrt{2}$, $\pi$ and $e$ to be. By contrast, our idea of the real numbers is that they form a continuous, unbroken number line (a continuum). But how do we make this idea of an unbroken number line precise?

There is a single, almost magical, property of the real numbers is called the completeness axiom which encodes all we need to know about the continuum. This property is as important to numbers as the basic laws of addition and multiplication you learned in your early school years. It gives a precise meaning to the idea of a continuous, unbroken real number line. You should think of the adjective complete here as meaning ‘no holes’ or ‘no gaps’, so the name is very appropriate.

In this section we introduce the completeness axiom and use it to define the real numbers. Unlike the basic laws of arithmetic, the completeness axiom is a lot less intuitive or familiar, so it will take time for us to develop an understanding of what it means and how it can be used. However, everything we do in this course will ultimately rely on the completeness axiom.

Learning outcomes.

• •
  ​

  Explain key quantified statements such as the definition of an upper bound or the supremum of a set, in both intuitive and formal mathematical terms. Apply and illustrate these concepts through worked examples.

• •
  ​

  Apply theory (such as the approximation property) to determine the existence of and compute suprema/infima.

• •
  ​

  Use abstract arguments to establish properties of real numbers such as the existence of radicals and density of rationals/irrationals.

• •
  ​

  Create examples to illustrate concepts such as the distinction between the supremum and maximum of a set.