This short article is aimed at educators and teachers of mathematics.Its goal is simple and direct:to explore some of the basic/elementary properties of proper rational numbers.A proper rational number is a rational which is not an integer. A proper rational r can be written in standard form: r=c/b,where c and b are relatively prime integers; and with b greater than or equal to 2. There are seven theorems, one proposition, and one lemma; Lemma1, in this paper. Lemma1 is a very well known result, commonly known as Euclid's lemma.It is used repeatedly throughout this paper, and its proof can be found in reference[1]. Theorem4(i) gives precise conditions for the sum of two proper rationals to be an integer.Theorem5(a) gives exact conditions for the product to be an integer. Theorem7 states that there exist no two proper rationals both of whose sum and product are integers.This follows from Theorem6 which states that if two rational numbers have a sum being an integer; and a product being an integer;then these two rationals must both be in fact integers.
The set of rational numbers can be thought of as the disjoint union of two of its main subsets: the set of integers and the set of proper rationals.
Definition 1: A proper rational number is a rational number which is not an integer.
The aim of this work is simple and direct. Namely, to explore some of the basic 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.
Is the sum of a proper rational and an integer always a proper rational? The answer is a rather obvious yes.
Theorem 2. Suppose that r = c b 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 = id, an integer contradicting the fact that r is a proper rational.
We will make repeated use of the very well known, and important, lemma below. For a proof of this lemma, see reference [1]. It can be found in just about every elementary number theory book.
(ii) (Extended version) Let m, n, k be non-zero integers such that m|nk and (m, n) = 1. Then m|k.
When is the product of a proper rational with an integer, an integer? A proper rational?
Theorem 3. Let r = c b be a proper rational in standard form and i an integer.
(a) The product r • i is an integer if, and only if, b|i.
(b) The product r • i is a proper rational if, and only if, b is not a divisor of i.
Since r is a proper rational, (b, c) = 1 by defintion. Equation (1) shows that b|c • i; and since (b, c) = 1. Lemma 1 implies that b must divide i. We are done.
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 obvious answer here. Suppose that r 1 + r 2 = i, an integer. Some routine algebra produces
or equivalently
According to (3), b 1 |c 1 b 2 ; and since (b 1 , c 1 ) = 1, Lemma 1 implies that b 1 |b 2 . A similar argument, using equation ( 2 We have
ad, an integer. Conversely, suppose that r 1 r 2 = i, an integer. Then 4), in conjunction with Lemma 1, imply that b 1 |c 2 and b 2 |c 1 . We are done.
One more result and its corollary
In Theorem 4 part (i), gives us the precise conditions for the sum of two proper rationals to be an integer. Likewise, Theorem 5 part (a) gives us the exact conditions for the product of two proper rationals to be an integer. Naturally, the following question arises. Can we find two proper rational numbers whose sum is an integer; and also whose product is an integer? Theorem 7 provides an answer in the negative. Theorem 7 is a direct consequence of Theorem 6 below.
Theorem 6. If both the sum and the product of two rational numbers are integers, then so are the two rationals, integers.
Proof. Let r 1 , r 2 be the two rationals, and suppose that
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 both proper rationals. Let then
Combining this information with (5), we get
From the first equation in ( 6) we obtain
This, combined with (c 1 , b 1 ) = 1 and Lemma 1 allow us to deduce that b 1 |b 2 . Similarly, using the first equation in ( 6), we infer that b 2 |b 1 which implies b 1 = b 2 . Hence, the second equation of ( 6) gives,
Therefore r 1 and r 2 are integers.
We have the immediate corollary.
Theorem 7. There exist no two proper rationals both of whose sum and product are integers.
Theorem 7 can also be proved by 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 be an integer. Every rational of such a monic polynomia must be an integer (equivalently, each of its real roots, if any, must be either an irrational number or an integer). Thus, in our case, the rational numbers r 1 and r 2 are the roots of the monic trinomial, t(x) = (xr 1 )(xr 2 ) = x 2i 1 x + i 2 ; a monic quadratic polynomial with integer coefficients -i 1 and i 2 . Hence, r 1 and r 2 must be integers.
For more details, see reference [1].
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