Properties of proper rational numbers

TL;DR

This paper explores elementary properties of proper rational numbers, including conditions for their sums and products to be integers, and establishes that two proper rationals cannot both have integer sum and product.

Contribution

It provides precise theorems characterizing when sums and products of proper rationals are integers, and proves that two proper rationals cannot both have integer sum and product.

Findings

Conditions for sum of two proper rationals to be an integer

Conditions for product of two proper rationals to be an integer

No two proper rationals can both have integer sum and product

Abstract

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.