# Mathematics - Class 7 / Grade 7

- Division of Integers / Properties of Division of Integers
### Division of Integers / Properties of Division of Integers

Math - Class 7 – Division of Integers / Properties of Division of Integers - Key Points/Notes/Worksheets/Explanation/Lesson/Practice Questions Tags: Properties of Division Integers with Examples, Rule of Division of Integers for 7th Grade, Dividing integers, Practice Page and Free Worksheet PDF on Properties of Division of Integers for seventh class, Division is the inverse operation of multiplication, Practice Questions on Division of negative integer by positive integer, Division of a positive integer by a negative integer, Divide a negative integer by a negative integer, Closure property under Division, Commutative property of Division, Division of an integer by Zero, Division of an integer by 1, Steps to Divide integers### 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.

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