# Basic properties of numbers

We have been performing the fundamental maths operations like addition, subtraction, multiplication, and division. There are certain properties that these operations follow. Also called as the basic properties of numbers.

There are four basic properties of real numbers:

1.  Commutative Property
2. Associative property
3. Identity Property
4. Distributive Property

#### COMMUTATIVE PROPERTY

When two or more numbers are added, then the order in which the addition of numbers takes place has no effect on the sum.

##### a + b = b + c

Other examples

•  5 + 6 = 6+5 =11
•  20 + 1 + 2 = 1 + 2 + 20 =23
##### Commutative Property for multiplication

When two or numbers are multiplied, then the product will always be the same, irrespective of the order of the multiplication.

##### a × b = b × a

Examples:

•  2 × 3 = 3 × 2 = 6
• 10 × 2 × 3 = 3 × 2 × 10 =30

#### ASSOCIATIVE PROPERTY

The term “Associative” means grouping or connecting.
When two or more numbers are added in groups, the sum will remain the same, no matter how we group them.

a + (b + c) = (a + b) + c

Examples

• 1 + (2 + 3) = (1+ 2) + 3 =6
• 4 + (3 + 2) + 6 = 4 + 3 + (2 + 6) =15
##### Associative Property for multiplication

When two or more numbers are multiplied then the product is always the same. The grouping while performing the multiplication has no effect on the answer. i .e.

##### ( a × b ) × c  = a × ( b × c )

Examples

• (2 × 3 ) × 5 = 2 × (3 × 5) = 30
• 1 × ( 5 × 3 ) = ( 1 × 5 ) × 3 = 15

#### IDENTITY PROPERTY

As the name suggests, “identity” means original value. When zero is added to a number, there is no change in the number’s value. That is, its identity remains the same. Thus, zero is the additive identity.

a + 0 = a
and 0 + 1 = a

Example,

• 0 + 11 = 11 + 0 = 11
• 0 + 30 = 30 + 0 =30
##### Identity property for multiplication

When a number is multiplied by “1” the answer is the number itself. Thus, 1 is the multiplicative identity.

a × 1 =1 × a  = a

Example

• 1 × 9 = 9 × 1 =9
• 123 × 1 = 1 × 123 = 123

#### DISTRIBUTIVE PROPERTY

When the numbers inside the brackets are added first and then multiplied by the number outside the brackets gives the same answer, when each number is multiplied with the number outside the bracket and then added.

a × ( b + c) = ( a × b) + (a × c )

Example

 2 ×( 1+ 5 ) = 2 × 6= 12 or 2 × ( 1 + 5)= (2×1)+(2×5)=2+10=12

##### To summarise, the basic properties of numbers can be given by-

It is to be noted that these properties are applicable to addition and multiplication only. These are not applicable to subtraction and division. Let us see why.

• For the commutative property of subtraction, it must follow
a – b = b – a
let us check this with an example.
5 – 2 = 3
but 2 – 5 = – 3
Thus, 5 – 2 ≠  2 – 5.
Hence subtraction does not follow the commutative property.

• For the commutative property of division, it must follow-
a ÷ b = b ÷ a
Let us check this with an example
1 ÷ 2 = 0.5
but 2 ÷ 1 = 2
Thus,  1 ÷ 2 ≠ 2 ÷ 1.
Hence, the division does not follow the commutative property.

• Associative property of subtraction is not true because it must follow-
(a – b ) – c = a- ( b – c )
Let us check whether it is true or not.
for associative ( 2 – 3 ) – 4 = 2 – (3 – 4)
⇒ ( 2 – 3 ) – 4 = -5
but 2 – (3 – 4) = 3
Thus ( 2 – 3 ) – 4  ≠  2 – (3 – 4).
Hence, subtraction does not follow the associative property.

• Associative property of division is also not true because it must follow-
a ÷ (b ÷ c) = (a ÷ b) ÷ c
Let us check whether it is true or not.
for associative 2 ÷ ( 8 ÷ 4) = ( 2 ÷ 8 ) ÷ 4
⇒ 2 ÷ ( 8 ÷ 4) = 1
but ( 2 ÷ 8 ) ÷ 4 = 0.0625
Thus  2 ÷ ( 8 ÷ 4) ≠ ( 2 ÷ 8 ) ÷ 4
Hence, the division does not follow the associative property.

Do not forget to check the other necessary topics.