Chapter 10 - The Real Numbers

Abstract

This chapter provides an overview on the real numbers in the number system. A “number” means an element of a field; the two fields most commonly used in analysis are the real number system R and the complex number system C. Both of these fields are introduced in this chapter. The chapter also describes that the real number system R is a Dedekind complete, chain ordered field or, in the terminology of some mathematicians, a complete ordered field. If R is also a field, it is called a “chain ordered field.” Some mathematicians call them an ordered ring and an ordered field, respectively. The chapter considers the real number system as a model for the set of all points on a Euclidean straight line. In addition to this the chapter also describes the hyperreal numbers that are also called “infinitesimal.” The set of all infinitesimals is an ordered ring (without unit). Some mathematicians exclude 0 when they define infinitesimal, but that definition has the disadvantage that the resulting set of infinitesimals does not have such a nice algebraic structure.