1.3. RATIONAL APPROXIMATION 5 Proof. Let S = {n ∈ Z | n a}. Then by the Archimedean property, S = ∅. The set S is bounded below by a, so by the well-ordering principle, S has a least element N. Then N − 1 / S, so N − 1 ≤ a N. We now show that there is a rational number between any two real numbers. Theorem 1.2.6. If a and b are real numbers with a b, then there exists a rational number r = p/q such that a r b. Proof. From the Archimedean property of R (Corollary 1.2.2) there exists q ∈ N such that 1/q b−a. Now consider the real number qa. By Theorem 1.2.5, there exists an integer p such that p − 1 ≤ qa p. It follows that p−1 q ≤ a p q . This implies that p q − 1 q ≤ a, that is, a p q ≤ a + 1 q b. Definition 1.2.7. A subset A of R is said to be dense in R if for any pair of real numbers a and b with a b, there is an r ∈ A such that a r b. Corollary 1.2.8. The rational numbers are dense in the real numbers. How do the irrational numbers behave? Exercise 1.2.9. (i) Show that any irrational number multiplied by any nonzero rational number is irrational. (ii) Show that the product of two irrational numbers may be rational or irrational. Next we show that there is an irrational number between any two real numbers. Corollary 1.2.10. The irrational numbers are dense in R. Proof. Take a, b ∈ R such that a b. We know that √ 2 is irrational and greater than 0. But then a 2 b 2 . By Corollary 1.2.8, there exists a rational number p/q with p = 0 such that a 2 p q b 2 . Thus a √ 2p/q b, and √ 2p/q is irrational. The real numbers are the union of two disjoint sets, the rational numbers and the irrational numbers, and each of these sets is dense in R. Density implies nothing about cardinality since the rationals are countable and the irrationals are not, as shown in Appendix A. 1.3. Rational Approximation We have just shown that both the rational numbers and the irrational num- bers are dense in the real numbers. But, really, how dense are they? It is reasonable to think that proximity for rational numbers can be measured in terms of the size of the denominator. To illustrate this, we ask the ques- tion, “How close do two rational numbers have to be in order to be the

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