# PROPERTIES OF DIVISION OF INTEGERS

## About "Properties of division of integers"

Properties of division of integers :

In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z  =  {... - 2, - 1,0,1,2, ...}, is the set of all integers

Here, we are going to see the following four properties of division of of integers.

(i)  Closure property

(ii)  Commutative property

(iii)  Associative property

Let us look at these properties of subtraction of integers in detail.

## Closure property of division of integers

Observe the following examples :

(i)  15 ÷ 5  =  15/5  =  3

(ii)  (-3)  ÷  9  =  -3/9  =  -1/3

(i)  7 ÷ 4  =  7/4  =  1.75

(ii)  1  ÷  2  =  1/2  =  0.5

From the above examples we observe that integers are not closed under division.

## Commutative property of division of integers

Observe the following examples :

15 ÷ 5  =  15/5  =  3

÷ 15  =  5/15  =  1/3

Therefore, 15 ÷ 5  ≠  ÷ 15

From the above example, we observe that integers are not commutative under division.

## Associative property of division of integers

Observe the following examples :

12 ÷ (6 ÷ 2)  =  12 ÷ 3  =  4

(12 ÷ 6) ÷ 2  =  2 ÷ 2  =  1

Therefore, 12 ÷ (6 ÷ 2)  ≠  (12 ÷ 6) ÷ 2

From the above example, we observe that integers are not associative under division. ## Some more properties of division of integers

We know that division is the inverse operation of multiplication.

Positive integer / Positive integer  =  Positive number

Negative integer / Negative integer  =  Positive number

Negative integer / Positive integer  =  Negative number

Positive integer / Negative integer  =  Negative number

For example,

250 / 50  =  5

(-144) / (-12)  =  12

(-120) / 20  =  -6

100 / (-25)  =  -4

## Division by zero

Division of any number (except 0) by zero  is meaningless because division by zero is not defined.

After having gone through the stuff given above, we hope that the students would have understood "Associative property of division of integers".

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