# Properties of Division of Integers

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Drag and drop the given division expressions into the correct classification.

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What do you observe? We observe *that integers are not closed under division.*

We know that division is not commutative for whole numbers. Let us check it for integers also.

Is (– 9) ÷ 3 the same as 3 ÷ (– 9) ?

Is (– 30) ÷ (– 6) the same as (– 6) ÷ (– 30) ?

Can we say that division is commutative for integers?

Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero. Thus, for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0.

When we divide a whole number by 1 it gives the same whole number. But is it true for negative integers also?

Observe the following :

− 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**

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What happens when we divide any integer by (–1)?

Try completing the following table

What do we observe? We see that **if any integer is divided by (–1), it does not give the same integer.**

But, does the division of integers follow the associative rule? Let's find out.

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Take for example: [(–16) ÷ 4] ÷ (–2) and (–16) ÷ [4 ÷ (–2)]

- [(–16) ÷ 4] ÷ (–2) =
÷ (–2) = - Now, (–16) ÷ [4 ÷ (–2)] = (–16) ÷
= - We see that the result is different for the individual expressions.
- Thus, associative property is not valid for division of integers.

Can you say that division is associative for integers?