Real Numbers

In this chapter, we will construct the set of real numbers from the set of rational numbers through equivalence classes. We will begin by discussing the motivation for defining real numbers. We will discuss convergent rational number sequences and some of their properties that we will apply in the study of real numbers. We will define real numbers and discuss general properties of real numbers. We will then define and derive general properties of real number operations, including addition, subtraction, negative operator, identity operator, multiplication, and division. As we proceed in this chapter, we will show that fractional real numbers are isomorphic to rational numbers.