  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.
