 # Properties of Division- Closure, Commutative, Associative, Distributive | Basic Division Properties with Examples

Properties of Division definition is here. Check the formulae, various properties of division, and how they work on various problems. Know the basics regarding the division and also the division property of equality. Follow the various operators with examples and concepts. Get the expression form and also step by step procedure to solve the problems. Division rule follows many properties and those are important in solving various problems. Check the below sections to know the complete details regarding properties of division, formulae, rules, examples, etc.

## Properties of Division

Of the four basic arithmetic operations, the division is the one. In the operation of division, we distribute or share the number or a group of things into equal parts. Division operation defines the fair result of sharing. It is the inverse property of multiplication. Division operation has five properties which are discussed in the below sections. The division is defined as the most complicated part of the arithmetic functions. But will be easy if you have a clear idea of all the methods, concepts, rules, and formulae along with a clear understanding of its usage.

### Representation of Division Operator

The notation of the division operator is a short horizontal line with 2 dots one above the line and the other below the line.

Notation:

The division is represented with the notation “÷”

### Basic Terms Used in Division

Various parts involving in the division rule have a special name.

Dividend – Dividend is the term that is being divided.

Divisor – The term which is being divided by the dividend is called the divisor

Quotient – The term quotient is defined as the result that is obtained in the division process

Remainder – The term remainder is defined as the leftover portion after the division process

### Rules of Division

• The first division rule is when the number is divided by zero, then the result is always 0. For suppose, 0 ÷ 4 = 0, i.e., 0 chocolates are shared among 4 pupils and each one gets 0 chocolates.
• No number can be divided with zero, the result gives the undefined value. For suppose, 4 ÷ 0. You have 4 chocolates but no pupil to distribute it, hence you cannot divide it by 0.
• On dividing the number with 1, the result is the same number with which you are dividing. For suppose, 4 ÷ 1 = 4. 4 chocolates divided among one pupil.
• If you divide the number by 2, it means that you are halving the number. For suppose, 4 ÷ 2 = 2. 4 chocolates dividing among two pupils, each gets 2 chocolates.
• On dividing the same number, the result value will always be one. For suppose, 4 ÷ 4 = 1. If 4 chocolates are divided among 4 pupils, then each gets one chocolate.
• The dividend rule must be applied in a proper way because if we interchange the numbers, the result value changes. For suppose, 20 ÷ 4 = 5 and 4 ÷ 20 = 0.2. Hence, the division rule must be applied in the correct order.
• The fractions like ¼, ½, ¾ are known as the division sums. ¼ is nothing but 1 ÷ 4, i.e., 1 chocolate is divided among 4 pupils.

### Division Properties

There are various properties of a division operation. They are explained in detail by considering few examples and they are as under

#### Closure Property

In general closure property states that, the resultant value will be always an integer. But when it comes to the division operation, the resultant value of the division need not be an integer value always. Hence, division fundamental operation does not follow closure property. i.e., a ÷ b is not an integer always. Therefore a ÷ b does not follow closure property.

Example: 7 ÷ 3 is not an integer

If we divide 7 with 3, then the resultant value is 2.33 which is not an integer. Thus, it is proved that closure property is not applicable for division operation.

#### Commutative Property

In general commutative property states that, even after swapping or shifting of numbers, the resultant value will be the same. When it comes to division operation, it gives the different resultant value when the operators are shifted or swapped. Hence, division operation does not follow the commutative property. i.e., a ÷ b ≠ b ÷ a. Therefore, a ÷ b does not follow the commutative property.

Example: 10 ÷ 5 ≠ 5 ÷ 10

If we divide 7 with 3, the resultant value is 2.33. If we divide 3 with 7, the resultant value is 0.42. Therefore, both the values are not equal. Thus, it is proved that commutative property is not applicable for division operation.

#### Associative Property

In general associative property states that, even if the parentheses of the expression are rearranged, the resultant will not be changed. When it comes to the division operation, it gives a different value when the parentheses are rearranged. Hence, division operation does not follow the associative property. i.e., a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c. Therefore, a ÷ (b ÷ c) does not follow the associative property.

Example: (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2)

If we solve (16 ÷ 4) ÷ 2, the resultant value is 2 and if we solve 16 ÷ (4 ÷ 2), the resultant value is 8. Therefore, both the values are not the same. Thus, it is proved that associative property is not applicable for division operation.

#### Distributive Property

In general distributive property states that, the resultant value is the same, even if the sum of two or more addends are multiplied or each addend multiplied separately, and then the products to be added together. When it comes to division operation, it gives different results when the addends are multiplied separately. Hence, division operation does not follow the distributive property. Therefore a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c).

Example: 12 ÷ (4+ 2) ≠ (12 ÷ 4) + (12 ÷ 2)

If we solve the equation 12 ÷ (4+ 2), we get the resultant value as 2 and if we solve the equation (12 ÷ 4) + (12 ÷ 2), we get the resultant value as 9. Therefore, both the values are not similar. Thus, it is proved that commutative property is not applicable for division operation.

#### Division by 1

Any number that is divided by 1 gives the resultant value as the same number.

Example:

5 ÷ 1 = 5

### Example Problems on Division Properties

Problem 1:

There are 80 chocolates. Each packet must be packed with 5 chocolates. How many packets do we need in total?

Solution:

Total number of chocolates = 80

Toffees that are to be packed in 1 packet = 5

Packets needed to pack 80 toffees = 80 ÷ 5

= 16

Therefore, we require 16 packets to pack 80 chocolates

Problem 2:

There are 100 donuts. They are equally packed in 10 packets. How many donuts are there in each box?

Solution:

Total number of donuts = 100

Total number of packets = 10

Number of donuts in each packet = 100 ÷ 10

= 10

Therefore, there are 10 donuts in each box

Problem 3:

50 bottles are placed in 5 equal trays. Find the number of bottles in each tray?

Solution:

Total no of bottles = 50

No of trays = 5

Number of bottles in each tray = 50 ÷ 5

= 10

Therefore, there are 10 bottles in each tray

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