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Division of Integers

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Division of integers:

Division is the inverse operation of multiplication.

Since 3 × 5 = 15
So 15 ÷ 5 = 3 and 15 ÷ 3 = 5
Similarly, 4 × 3 = 12 gives 12 ÷ 4 = 3 and 12 ÷ 3 = 4
We can say for each multiplication statement of whole numbers there are two division statements.

1) Dividing any negative integer by a positive integer:

Observe the following example:
(– 12) ÷ 2 = (– 6)
(– 20) ÷ 5 = (– 4)
(– 32) ÷ 4 = (– 8)
(– 45) ÷ 5 = (– 9)

We observe that when we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient.

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

2) Dividing any positive integer by a negative integer:

We also observe that:
72 ÷ (– 8) = –9
50 ÷ (– 10) = – 5
72 ÷ (– 9) = – 8
50 ÷ (– 5) = – 10

So we can say that when we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.

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

3) Dividing any negative integer by a negative integer:

Lastly, we observe that
(–12) ÷ (– 6) = 2;
(–20) ÷ (– 4) = 5;
(–32) ÷ (– 8) = 4;
(– 45) ÷ (–9) = 5
So, we can say that when we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).

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

Rules of the division of integers:

  • We cannot divide any number by zero.
  • The quotient of two positive integers is a positive number.
  • The quotient of two negative integers is a positive number.
  • The quotient of a positive integer and a negative integer is always a negative number.

Example

Solve: 7 ÷ (- 2)

`7/(-2)`

`= (7 xx 1)/((-1) xx 2)`

`= 7 xx 1/(-1) xx 1/2`

`= 7/1 xx (-1) xx 1/2`

`= ((7) xx (-1))/2`

`= (-7)/2`

Example

Solve: `(- 13)/(- 2)`

`(-13)/(-2)`

`= ((-1) xx 13)/((-1) xx 2)`

`= (-1)/(-1) xx 13 xx 1/2`

`= (- 1) xx (-1)/1 xx 13/2`

`= 1 xx 13/2`

`= 13/2`

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Shaalaa.com | Division Involving Negative Integers

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Division Involving Negative Integers [00:07:42]
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