Irrationals and the Fundamental Theorem of Arithmetic

Abstract

Recent issues of the MONTHLY contain articles [1, 2] that discuss approaches to the of the square roots of nonsquare natural numbers using arguments designed to avoid any mention of unique factorization. The argument in [2] is incomplete but, as the article points out, the important objective when introducing a topic such as the existence of irrationals is to present arguments that the students find convincing and many students have difficulty with reasoning involving the unique factorization aspect of the Fundamental Theorem of Arithmetic. The following discussion is designed to give students some appreciation of the significance of the property of unique factorization by considering the irrationality of square roots in systems where this property does not hold. Ideally it should come immediately after the presentation of a plausibility argument such as that in [2] showing the students that square roots of primes are irrational. Given a subset A of the natural numbers that is closed under multiplication, we say an element c of A is A-composite provided c = ab where a and b are elements in A with a # 1 and b 0 1. An element c in A is called A-prime provided c 0 1 and c is not A-composite. For example, if A is the set of all multiples of 4, then 16, 48 and 80 are A-composites and 4, 40 and 100 are A-primes. We say a real number x is A-rational provided x can be expressed in the form a/b where a and b are elements of A. Note that each element c in A is A-rational since c can be expressed as C2/C. This is analogous to the fact that every natural number is rational. However, there are properties of the natural numbers that do not extend to any set A. For example, let A be the set of all multiples of 4 and let c 100. Since c is an A-prime, by analogy with the natural numbers its square root should not be A-rational, but Vc = a/b where a = 40 and b = 4. Such a situation is possible since in the set A the number a2 can be factored into A-primes as either a * a or c * b * b. There is nothing special about the multiples of 4 in the above example. A useful exercise is to have each student pick a subset of the natural numbers that is closed under multiplication and investigate the irrationality of roots in this system. This serves to illustrate the rarity of the unique factorization property and the crucial role played by this property in the study of algebraic irrationals.