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Properties of Division of Integers

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Properties of Division of Integers:

1. Integers are not closed under division.

Statement

Inference

(– 8) ÷ (– 4) = 2 Result is an integer
(– 4) ÷ (– 8) = `(– 4)/(– 8)` Result is not an integer
(– 8) ÷ 3 = `(– 8)/3` Result is not an integer
3 ÷ (– 8) = `3/(– 8)` Result is not an integer

2. The division is neither commutative for whole numbers nor commutative for integers.

For example,

(– 8) ÷ (– 4) ≠ (– 4) ÷ (– 8).

(– 9) ÷ 3 ≠ 3 ÷ (– 9)

(– 30) ÷ (– 6) ≠ (– 6) ÷ (– 30).

3. Division is not associative for integers.

We know that [(– 16) ÷ 4] ÷ (– 2) = (– 4) ÷ (– 2) = 2.

and (– 16) ÷ [4 ÷ (–2)] = (– 16) ÷ (– 2) = 8

So [(– 16) ÷ 4] ÷ (– 2) ≠ (– 16) ÷ [4 ÷ (–2)]

We can say that division is not associative for integers.

4. For any integer a, we have a ÷ 0 is not defined and zero divided by an integer other than zero is equal to zero.

For any integer a, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero i.e., for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.

For example, (-12) ÷ 0 = not defined and 0 ÷ (–12) = 0 

5. For any integer a, we have a ÷ 1 = a.

When we divide a whole number by 1 it gives the same whole number.

(– 8) ÷ 1 = (– 8)

(–11) ÷ 1 = – 11

(–13) ÷ 1 = – 13

(–25) ÷ 1 = - 25

(–37) ÷ 1 = - 37

(– 48) ÷ 1 = - 48

This shows that negative integer divided by 1 gives the same negative integer.

So, any integer divided by 1 gives the same integer.

In general, for any integer a, a ÷ 1 = a.

(– 8) ÷ (–1) = 81

1 ÷ (– 1) = – 1

113 ÷ (–1) = - 113

(–25) ÷ (–1) = 25

(–37) ÷ (–1) = 37

48 ÷ (–1) = - 48

We can say that if any integer is divided by (–1) it does not give the same integer.

In general, for any integer a, a ÷ (-1) = - a.

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Closure Property Of Division [00:05:49]
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