Associative Property Of Rational Numbers
Associative Property Of Rational Numbers
We know that the rational numbers are the numbers which can be written in the
form of p/q, where p and q are integers and the denominator q <> 0.
All natural numbers, whole numbers, integers and all fraction numbers with their
additive inverse as the elements of the set of rational numbers.
There are different properties of rational numbers. In this session we will learn about
the Associative Property of Rational Numbers.
We will be checking that if the Associative property holds true for all the
mathematical operators namely Addition, subtraction, multiplication and division or
not .
Let us first define what is the Associative property of Rational numbers. According to
the Associative Property of Rational Numbers, we mean that:
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if p1/q1 , p2/q2 and p3/q3 are any three rational numbers, then we have Associative
Property of Addition of Rational Numbers, by which we have (p1/q1 + p2/q2) + p3/q3
= p1/q1 + (p2/q2 + p3/q3). Eg if we have p1/q1 = 1/7 , p2/q2 = 3/7, and p3/q3 = 2/7
then we have
( p1/q1 + p2/q2 ) + p3/q3 = (1/7 + 3 / 7) + 2/7 = 4/7 + 2/7 = 6/7
Also we have p1/q1 + (p2/q2 + p3/q3) = 1/7 + (3/7 + 2/7) = 1/7 + 5/7 = 6/7
In both the cases we get the same result. Thus we can say that the Associative
property of Addition holds true for the rational numbers.
if p1/q1 , p2/q2 and p3/q3 are any two rational numbers, then we have Associative
Property of Multiplication of Rational Numbers, by which we have (p1/q1 * p2/q2) *
p3/q3 = p1/q1 * (p2/q2 * p3/q3). Eg if we have p1/q1 = 1/7 , p2/q2 = 3/7, and p3/q3
= 2/7 then we have
( p1/q1 * p2/q2 ) * p3/q3 = (1/7 * 3 / 7) * 2/7 = (3/49) * (2/7) = ( 6 / 343)
Also we have p1/q1 * (p2/q2 * p3/q3) = 1/7 * (3/7 * 2/7) = (6 / 343)
In both the cases we get the same result. Thus we can say that the Associative
property of Multiplication holds true for the rational numbers.
On the another hand if we try the Associative property for the subtraction and division
of the rational numbers
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if p1/q1 , p2/q2 and p3/q3 are any two rational numbers, then we have Associative
Property of subtraction of Rational Numbers, by which we have (p1/q1 - p2/q2) -
p3/q3 = p1/q1 - (p2/q2 - p3/q3). Eg if we have p1/q1 = 1/7 , p2/q2 = 3/7, and p3/q3
= 2/7 then we have
( p1/q1 - p2/q2 ) - p3/q3 = (1/7 - 3 / 7) - 2/7 = (-2/7) - (2/7) = ( -4/ 7)
Also we have p1/q1 - (p2/q2 - p3/q3) = 1/7 - (3/7 - 2/7) = 1/7 -1/7 = 0
The two results are not equal. Similarly it is not true for division.
Thus we come to a conclusion that the Associative property does not hold true for
Division.


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