The set of real numbers is one of the one of the most fundamental objects in mathematics. In order to describe the real numbers we will introduce notation for some related objects. Let Z denote the set of integers. That is,

Z = {...,-3,-2,-1,0,1,2,3,...}.

Then let Q denote the set of rational numbers. Thus Q consists of all fractions where p and q are integers and q 0. There is also an equivalence relation in the field Q which we use automatically when doing arithmetic. Two fractions and are equal in Q, that is they determine the same rational number, whenever the identity ps = rq holds. As usual we regard Z as a subset of Q by recognizing that each integer n can be written as n/ 1, and so the integer n may be viewed as a rational number. We assume that the reader is familiar...