Advanced Mathematics

4.1Construction of the Real Numbers

Purpose of Section

Starting with the natural numbers, we logically construct the integers as equivalence classes of pairs of natural numbers. We then logically construct the rational numbers as equivalence classes of pairs of integers. When we arrive at the real numbers, we use the Dedekind cut to define the real numbers in terms of rational numbers.

4.1.1 Introduction

No doubt, most readers of this book think of real numbers as values on a number line, which has long been accepted by scientists and engineers as a model for measurements of length, mass, and time.

Although there is nothing wrong with this intuitive interpretation, it is the goal of this section to show how the real numbers can be logically created from more primitive number systems like the natural numbers, as well as introducing aspects of the real numbers decimal expansions, the least upper bound property, types of real numbers like rational, irrational, algebraic, and transcendental, and completeness properties.

Without going into the history of how numbers went from 1, 2, 3, … to the real numbers, there are two fundamental approaches to how to define the real numbers. First, we can state axioms that we believe characterize our interpretation of the real numbers. This is the here they are, the real numbers. This approach is called the synthetic approach, whereby axioms hopefully embody what we believe a “continuum” should be.

On the other hand, we can “construct” ...