Topology/Real analysis/Constructing the real numbers

In the previous section we showed that not all numbers are rational(I.E. can be expressed as a/b, where a,b are both integers). For instance $\sqrt{2}$ is not a rational number. In this section we shall show how we can formally define the real numbers.

Numbers and Algorithms

However, before we get into the details of constructing the real numbers, it will be useful to review what we mean by a number. In algebra, you generally think of a number by its operations. For instance, $6=2*3$ or $10=5+5$. We then abstract from this notion, and we can say things like $x^2=25$, and then solve for x. In this case, we are talking about the number x, which satisfies a certain property. However, in the case of $x^2=5$, how do we find x? The ancient Greeks, realized x couldn't be expressed as fraction, which means it also can't be expressed as a decimal.

In normal practice, we might say x is about 2.23607, but 2.23607 does not equal the square root of 5. We could just stick with $\sqrt{5}$, but then we couldn't do much with it. For instance, how would we answer the question is $\sqrt{7}-\sqrt{5}>1$? The only way to deal with these questions is to deal with approximates, but mathematics is supposed to be precise. Analysis resolves these issues by formally constructing the real numbers.

Cauchy Sequences

We assume given the rational numbers, denoted $\mathbb{Q}$. For now we may only work with them. As (to be) mentioned above, we can think of the real numbers as limits of converging sequences of approximating rational numbers. There is a big roadblock to this approach: how does one define convergence without explicit reference to the very limit that we are trying to define? Cauchy's acheivement, however, was to find the way to formalize this concept without any mention of the limit.

We begin with a few premilinaries about sequences, which will be treated more fully in the next chapter. We define a sequence of rational numbers to be a map $a:{\mathbb{Z}}^{+}\to \mathbb{Q}$. It is an infinite list of rational numbers, denoted $\{a_1,a_2,a_3,\dots \}$ or, more compactly, $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$. Sometimes we may choose the subscript to start at zero, and when the context is clear we may just write $\{a_n\}$ for brevity.

A sequence of rational numbers, $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$, is said to be a Cauchy sequence if for any $ϵ>0$ rational, there exists $N\in {\mathbb{Z}}^{+}$ such that for all $m,n\ge N$ we have $|a_m-a_n|<ϵ$.