Integers And Absolute Values
Division of Integers
let us see the rule given below which helps us in performing the division operation over integers.
Rule I :- The quotient of two integers whth the same sign is positive integer obtained by dividing the numerical value of the divided with the numerical.
Illustration:- i)\(
( + 12) \div ( + 3) = + \frac{{12}}
{3} = + 4
\)
(ii)\(
( - 25) \div ( - 5) = + \frac{{25}}
{5} = + 5
\)
Rule 2:- The quotient of ’2’ integers with different signs is the negative integer obtained by dividing the numerical value of the dividend with the numerical value of the divisor
Illustration:- i)\(\
( + 36) \div ( - 6) = - \frac{{36}}
{6} = - 6
\) ii)\(
( - 32) \div ( - 4) = \frac{{ - 32}}
{4} = - 8
\)
Now we shall study the properties of operations over integers.
Properties of Division of Integers
1. Closure property
Observe the following
i)\(
75 \div 25 = 3
\) ii) is not an integer
We observe that when an integer is divided by another integer then the result is not always an integer. We can say that the integers are not closed under division.
The closure property doesn’t hold good for integers over division
2.Associative property:
Observe the following
1) i)\(
(8 \div 4) \div 2 = 2 \div 2 = 1
\) ii)\(
8 \div (4 \div 2) = 8 \div 2 = 4
\)
2) i)\(
(36 \div 6) \div 3 = 6 \div 3 = 2
\) ii) \(
36 \div (6 \div 3) = 36 \div 3 = 18
\)
Ist Integer :- (2nd integer \(
\div
\) 3rd integer)\(
\ne
\) (1st integer\(
\div
\) 2nd integer) \(
\div
\) 3rd integer
Conclusion :-
\(
\therefore
\)Associative law doesnot hold good under division
\(
\therefore
\)Associative law doesnot hold good for division of integers
\(
\therefore \,\,\,\,a,b,c \in ,(a \div b) \div c \ne a \div (b \div c)
\)