1.7 Recap and reflection

In this chapter, we introduced the concept of a least upper bound (supremum) and the completeness axiom. We described how these ideas give a rigorous meaning to the idea that the real numbers $\mathbb{R}$ form a continuous, unbroken line.

We used the completeness axiom to derive some basic properties of real numbers. In particular, we showed that, in contrast with the rationals, there exists some $s\in\mathbb{R}$ such that $s^{2}=2$. In other words, we can interpret $\sqrt{2}$ as a number in our real number line. This answers one of the questions set out in the introduction to the course.

However, it is too early to celebrate. So far, our only description of $\sqrt{2}$ is that it equals the supremum of the set $\{a\in\mathbb{Q}:a^{2}\leq 2\}$. This is rather abstract and does not appear related to our usual understanding of numbers through decimal expansions.

We need to build on our definition of $\mathbb{R}$ – the system of numbers that satisfies the completeness axiom – to understand how to work with real numbers in practice. This will lead us to study sequences and series of numbers, and the notion of a limit. These concepts will provide us with a fuller understanding of decimal expansions such as $\sqrt{2}=1.414213\dots$, how to compute them, and how carry out arithmetic computations over the real line.