Properties of proper rational numbers

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1 arxiv: v1 [math.gm] 29 Sep 2011 Properties of proper rational numers Konstantine Zelator Mathematics, Statistics, and Computer Science 212 Ben Franklin Hall Bloomsurg University of Pennsylvania 400 East 2nd Street Bloomsurg, PA USA and P.O. Box 4280 Pittsurgh, PA s: konstantine 1 Introduction Novemer 6, 2018 The set of rational numers can e thought of as the disjoint union of two of its main susets: the set of integers and the set of proper rationals. Definition 1: A proper rational numer is a rational numer which is not an integer. The aim of this work is simple and direct. Namely, to explore some of the asic or elementary properties of the proper rationals. We will make use of the standard notation (u,w) denoting the greatest common divisor of two integers u and w. Also, the notation u w to denote that u is a divisor of w. 1

2 Proposition 1. Let r e a proper rational numer. Then r can e written in the form, r = c where c and are relatively prime integers; (c,) = 1, and with 2. Proof. Since r is a proper rational, it cannot, y definition, e zero. Hence r = A, for some positive integers A and B; if r > 0. If, on the other hand, B r < 0, then r = A, A,B eing positive integers. Let d = (A,B), then B A = da, B = d, for relative prime positive integers a and. We have, r = A = da = a, for r > 0. Clearly, cannot equal 1, for then r would equal B d a, an integer, contrary to the fact that r is a proper rational. Hence, 2. If, on the other hand, r < 0, r = A = a = c with c = a, and B 2. Definition 2: A proper rational numer r is said to e in standard form if it is written as r = c, where c and are relatively prime integers and 2 2 The reciprocal of a proper rational We state the following result without proof. We invite the interested reader tofill in the details. Theorem 1. Let r = c (i) If c = 1, the reciprocal 1 r = c e a proper rational in standard form. is an integer. (i) If c 2, the reciprocal 1 r is a proper positive rational. (iii) If c 2 and c < 0, the reciprocal 1 is negative proper rational with r the standard form eing 1 = d, where d =. r c 3 An ovious property Is the sum of a proper rational and an integer always a proper rational? The answer is a rather ovious yes. Theorem 2. Suppose that r = c is a proper rational in standard form; and d and integer. Then the sume r+d is a proper rational. Proof. If, to the contrary, r+d = i, for some i Z, then r = i d, an integer contradicting the fact that r is a proper rational. 2

3 4 A lemma from numer theory We will make repeated use of the very well known, and important, lemma elow. For a proof of this lemma, see reference [1]. It can e found in just aout every elementary numer theory ook. Lemma 1. (Euclid s lemma) (i) (Standard version) Let m,n,k e positive integers such that m is a divisor of the product n k; and suppose that (m,n) = 1. Then m is a divisor of k. (ii) (Extended version) Let m, n, k e non-zero integers such that m nk and (m,n) = 1. Then m k. 5 A slightly less ovious property When is the product of a proper rational with an integer, an integer? A proper rational? Theorem 3. Let r = c integer. e a proper rational in standard form and i an (a) The product r i is an integer if, and only if, i. () The product r i is a proper rational if, and only if, is not a divisor of i. Proof. () This part is logically equivalent to part (a). (a) If divides i, then i = q, an integer. so we have r i = cq = c q, an integer. Now the converse. Suppose that r i is an integer t: r i = t, which yields, c i = t (1) Since r is a proper rational, (,c) = 1 y defintion. Equation (1) shows that c i; and since (,c) = 1. Lemma 1 implies that must divide i. We are done. 3

4 6 The sum of two proper rationals An interesting equation arises. When is the sum of two proper rationals also a proper rational? When is it an integer? There is no ovious answer here. Theorem 4. Let r 1 = c 1 and r 2 = c 2 2 e proper rationals in standard form. Then, (i) The sum r 1 +r 2 is an integer if, and only if, = 2 and is a divisor of the sum c 1 +c 2. (ii) The sum r 1 + r 2 is a proper rational if, and only if, either 2 or = 2 ut with not eing a divisor of c 1 +c 2. Proof. (ii) This part is logically equivalent to part (i). (i) If = 2 and (c 1 +c 1 ), then r 1 +r 2 = c 1+c 2, is oviously an integer. Next, let us prove the converse statement. Suppose that r 1 +r 2 = i, an integer. Some routine algera produces or equivalently c 1 2 +c 2 = i 2 (2) c 1 2 = (i 2 c 2 ). (3) According to (3), c 1 2 ; and since (,c 1 ) = 1, Lemma 1 implies that 2. A similar argument, using equation (2), once more estalishes that 2. Clearly, since the two positive integers and 2 are divisors of each other, they must e equal; = 2 (an easy exercise in elementary numer theory). From = 2 and (2), we otain c 1 +c 2 = i i ; and thus it is clear that (c 1 +c 2 ). 7 The product of two proper rationals Theorem 5. Let r 1 = c 1 and r 2 = c 2 2 e proper rationals in standard form. (a) The product r 1 r 2 is an integer if, and only if, c 2 and 2 c 1. () The product r 1 r 2 is a proper rational if, and only if, is not a divisor of c 2 ; or 2 is not a divisor of c 1. 4

5 Proof. () This part is logically equivalent to part (a). (a) Suppose that c 2 and 2 c 1 ; then c 2 = a and c 1 = 2 d where a and d are (non-zero) integers. We have r 1 r 2 = c 1c 2 2 = ad 2 2 = ad, an integer. Conversely, suppose that r 1 r 2 = i, an integer. Then c 1 c 2 = i 2 (4) Since (,c 1 ) = 1 = (c 2, 2 ), (4), in conjunction with Lemma 1, imply that c 2 and 2 c 1. We are done. 8 One more result and its corollary In Theorem 4 part (i), gives us the precise conditions for the sum of two proper rationals to e an integer. Likewise, Theorem 5 part (a) gives us the exact conditions for the product of two proper rationals to e an integer. Naturally, the following question arises. Can we find two proper rational numers whose sum is an integer; and also whose product is aninteger? Theorem7provides ananswer inthenegative. Theorem 7isadirect consequence of Theorem 6 elow. Theorem 6. If oth the sum and the product of two rational numers are integers, then so are the two rationals, integers. Proof. Let r 1,r 2 e the two rationals, and suppose that r 1 +r 2 = i 1 r 1 r 2 = i 2 i 1,i 2 Z If either of r 1,r 2 is an integer, then the first equation in (5) implies that the other one is also an integer. So we are done in this case. So, assume that neither of r 1,r 2 is an integer; which means that they are oth proper rationals. Let then r 1 = c 1, r 2 = c 2 2 e the standard forms of r 1 and r 2. That is, (c 1, ) = 1 = (c 2, 2 ), 2, 2 2 and, of course, c 1 c 2 0. Comining this information with (5), we get { } c1 2 +c 2 = i 1 2 (6) c 1 c 2 = i 2 2 (5) 5

6 From the first equation in (6) we otain c 1 2 = (i 1 2 c 2 ), which shows that c 1 2. This, comined with (c 1, ) = 1 and Lemma 1 allow us to deduce that 2. Similarly, using the first equation in (6), we infer that 2 which implies = 2. Hence, the second equation of (6) gives, c 1 c 2 = i (7) By virtue of (,c 1 ) = (,c 2 ) = 1, equation (7) implies = 1; = 2 = 1. Therefore r 1 and r 2 are integers. We have the immediate corollary. Theorem 7. There exist no two proper rationals oth of whose sum and product are integers. 9 A closing remark Theorem 7 can also e proved y using the well known Rational Root Theorem for polynomials with integer coefficients. The Rational Root Theorem implies that if a monic (i.e., leading coefficient is 1) polynomial with integer coefficients has a rational root that root must e an integer. Every rational of such a monic polynomia must e an integer (equivalently, each of its real roots, if any, must e either an irrational numer or an integer). Thus, in our case, the rational numers r 1 and r 2 are the roots of the monic trinomial, t(x) = (x r 1 )(x r 2 ) = x 2 i 1 x+i 2 ; a monic quadratic polynomial with integer coefficients i 1 and i 2. Hence, r 1 and r 2 must e integers. For more details, see reference [1]. References [1] 1 Kenneth H. Rose, Elementary Numers Theory and Its Applications, fifth edition, 2005, Pearson-Addison-Wesley. For Lemma 1 (Lemma 3.4 in the aove ook), see page 109 for the Rational Root Theorem, see page