Definition 2

Let $\mathbf{R}$ denote the topological group which is the completion of the additive group $\mathbf{Q}$ of the rational line. The elements of $\mathbf{R}$ are called real numbers; as a topological space, $\mathbf{R}$ is called the real line; as a topological group, $\mathbf{R}$ is called the additive group of the real line.

We shall always identify $\mathbf{Q}$ with the dense subgroup of $\mathbf{R}$ to which it is canonically isomorphic. With this convention, every rational number is a real number. Every real number which is not rational is said to be irrational; we have seen in Chapter II, § 3, no. 3 that such numbers exist (we shall show this in another way in § 3, no. 3 of this chapter; see also Exercise 2 to § 2); hence (Chapter III, § 2, no. 1) the set $\mathbf{CQ}$ of irrational numbers is dense in $\mathbf{R}$.

We shall show that the order structure of $\mathbf{Q}$ can be extended to $\mathbf{R}$ in such a way that the extended ordering is still compatible with the additive group structure of $\mathbf{R}$: