6 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS same rational number?” This is not a trick question—it is designed to il- lustrate the principle mentioned above. In particular, if a/b, c/d Q and |a/b c/d| 1/bd, then a/b = c/d. This idea can be encapsulated in the following theorem. Throughout this section, we shall assume that the denominator of a rational number is a positive integer and that the numerator and denominator are relatively prime. Theorem 1.3.1. If a/b is a fixed rational number and p/q is a rational number such that 0 |p/q a/b| 1/mb for some positive integer m, then q m. Proof. Easy. We now present several facts on rational approximation. For α in various subsets of the real numbers, we prove results which give an idea of the degree of accuracy with which α may be approximated. The results take the following form: Given a real number α, (1) positive real numbers c(α) and t exist so that there are infinitely many rational numbers p/q with p/q| c(α)/qt (2) positive real numbers c(α) and t exist so that there are only finitely many rational numbers p/q with p/q| c(α)/qt (3) for δ 0, there exist real numbers c(α, δ) and t so that p/q| c(α, δ)/qt+δ for all rational numbers p/q. To begin, we present an exercise which follows easily from elementary number theory. Exercise 1.3.2. Let a and b be relatively prime integers. Show that the equation ax + by = 1 has infinitely many solutions (x, y) with x and y relatively prime. Theorem 1.3.3. Let α = a/b with a and b relatively prime and b = 1. Then there exist infinitely many p/q Q such that |a/b p/q| 1/q. Proof. Let (x, y) = (q, −p) be a solution to the equation ax+by = 1. Then q = 0 since b = 1. We may assume q 0. We then have |a/b p/q| = 1/bq 1/q. Remark 1.3.4. If b = 1, then the same result holds with replaced by ≤. The next theorem characterizes rational numbers in terms of rational approximation. We first need the following exercise. Exercise 1.3.5. Let α be a real number, and let η and t be positive real numbers. Show that there exists only a finite number of rational numbers p/q with q η which satisfy p/q| 1/qt. Theorem 1.3.6. Let α = a/b Q. Then there are only finitely many p/q so that |a/b p/q| 1/q2.
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