## Topic outline

• ### Division of Integers

##### Division is the inverse operation of multiplication.

Let us see an example for whole numbers.

Dividing 24 by 4 means finding an integer which when multiplied with 4 gives us 24, such an integer is 6.

Since 6 × 4 = 24

So 24 ÷ 6 = 4 and 24 ÷ 4 = 6

Therefore, for each multiplication statement of whole numbers there are two division statements.

##### Division of negative integer by positive integer

Steps:

1.   First divide them as whole numbers.

2.   Then put a minus sign (–) before the quotient. We, thus, get a negative integer.

Example: (–10) ÷ 2 = (– 5)

(–32) ÷ (8) = (– 4)

##### Division of a positive integer by a negative integer

Steps:

1.   First divide them as whole numbers.

2.  Then put a minus sign (–) before the quotient. That is, we get a negative integer.

Example: 81 ÷ (–9) = –9

60 ÷ (–10) = –6

In general, for any two positive integers a and b, a ÷ (– b) = (– a) ÷ b where b ≠ 0

##### Divide a negative integer by a negative integer

Steps:

1.   Divide them as whole numbers.

2.   Then put a positive sign (+). That is, we get a positive integer.

Example: (–15) ÷ (– 3) = 5

(–21) ÷ (– 7) = 3

In general, for any two positive integers a and b, (– a) ÷ (– b) = a ÷ b where b ≠ 0

### Properties of Division of Integers

##### Closure under Division

Division of integers doesn’t follow the closure property.

Let’s consider the following pairs of integers.

(-12) x (-6) = 2 (Result is an integer)

(-5) x (-10) = -1/2 (Result is not an integer)

We observe that integers are not closed under division.

##### Commutative property of Division

Division of integers is not commutative for integer.

Let’s consider the following pairs of integers.

(– 14) ÷ (– 7) = 2;

(– 7) ÷ (– 14) = 1/2

(– 14) ÷ (– 7) ≠ (– 7) ÷ (– 14)

We observe that division is not commutative for integers.

##### Division of an integer by Zero

Any integer divided by zero is meaningless.

Example: 5 ÷ 0 = not defined

Zero divided by an integer other than zero is equal to zero.

Example: 0 ÷ 6 = 0

##### Division of an integer by 1

When we divide an integer by 1 it gives the same integer.

Example: (– 7) ÷ 1 = (– 7)

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

##### Some More Examples

Example: Evaluate [(– 8) + 4)] ÷ [(–5) + 1]

Solution: [(– 8) + 4)] ÷ [(–5) + 1]

= (-4) ÷ (-4)

= 1

Example: Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) when a = 8, b = – 2, c = 4.

Solution: L.H.S = a ÷ (b + c)

= 8 ÷ (-2 + 4)

= 8 ÷ 2 = 4

R.H.S = (a ÷ b) + (a ÷ c)

= [8 ÷ (-2)] + (8 ÷ 4)

= (-4) + 2

= -2

Here, L.H.S ≠ R.H.S

Hence verified

Example: (– 80) ÷ (4) is not same as 80 ÷ (–4). True/False

Solution: (– 80) ÷ (4) = -20

80 ÷ (–4) = -20

As (– 80) ÷ (4) = 80 ÷ (–4), so the above statement is false.