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.