2 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS Problem 1.5. If f : R R is a polynomial function such that f(Q) Q and f(R \ Q) R \ Q, show that f(x) = ax + b for some a, b Q. To begin this chapter, we assume that the reader is familiar with the integers Z as an ordered integral domain and the rational numbers Q as an ordered field that is the field of fractions of the integers Z (see Appendix A). Exercise 1.0.1. Prove that any ordered integral domain contains the integers. Exercise 1.0.2. Prove that any field that contains the integers contains the rationals as a subfield. In this chapter, we do several things. First, we introduce the real num- bers by adding the least upper bound property to the axioms for an ordered field. Second, despite Osgood, we construct the real numbers from the rational numbers by the method of Cauchy sequences. Third, we construct the complex numbers from the real numbers and prove a few useful theorems about them. Intermingled in all of this is a discussion of the fields of alge- braic numbers and real algebraic numbers. As a project at the end of the chapter, we lead the reader through a discussion of the construction of the real numbers via Dedekind cuts. In other projects, we study the convergence properties of infinite series and decimal expansions of real numbers. 1.1. The Least Upper Bound Property and the Real Numbers Definition 1.1.1. Let F be an ordered field. Let A be a nonempty subset of F . We say that A is bounded above if there is an element M F with the property that if x A, then x M. We call M an upper bound for A. Similarly, we say that A is bounded below if there is an element m F such that if x A, then m x. We call m a lower bound for A. We say that A is bounded if A is bounded above and A is bounded below. Examples 1.1.2. (i) Consider the subset A of Q: A = 1 + (−1)n n n N . Then A is bounded above by 3/2 and bounded below by 0. (ii) Let A = {x Q | 0 x3 27}. Then A is bounded below by 0 and bounded above by 3. Exercise 1.1.3. Let a be a positive rational number and let A = {x Q | x2 a}. Show that A is bounded in Q. Definition 1.1.4. Let F be an ordered field, and let A be a nonempty subset of F which is bounded above. We say that L F is a least upper bound for A if the following two conditions hold:
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