# Properties of Addition

Properties of addition are defined for the various conditions and rules of addition. These properties also indicate the closure property of the addition. In fact, like for addition, properties for subtraction, multiplication and division are also defined in Mathematics. But the for each operation the properties vary. There are basically four Maths properties defined for addition.

They are:

- Commutative Property of Addition
- Associative Property of Addition
- Additive Identity Property
- Distributive Property of Addition

## Commutative Property

According to this property, when two numbers or integers are added, the sum remains the same even if we change the order of numbers/integers. This property is also applicable in the case of multiplication. It can be represented as;

**A + B = B + A**

**Example: **10 + 5 = 5 + 10

15 = 15

In the above example, you can see, when we add the two numbers, 10 and 5 and we interchange the two numbers, the results remain the same as 15. Hence, addition follows commutative law.

## Associative Property

As per this property or law, when we add three numbers, the association of numbers in a different pattern does not change the result. We can represent this property as;

**A+(B+C) = (A+B)+C**

**Example: **(2 + 4) + 6 = 2 + (4 + 6)

6 + 6 = 2 + 10

12 = 12

As you can see from the above example, the left-hand side is equal to the right-hand side. Hence, the associative property is proved. This property is also applicable for multiplication.

## Distributive Property

This property is completely different from Commutative and Associative property. In this case, the sum of two numbers multiplied by the third number is equal to the sum when each of the two numbers is multiplied to the third number.

**A × (B + C) = A × B + A × C**

Here A is the monomial factor and (B+C) is the binomial factor.

**Example**: 2 × (3+5) = 2 × 3 + 2 × 5

2 × 8 = 6+10

16 = 16

In the above example, you can see, even we have distributed A (monomial factor) to each value of the binomial factor, B and C, the value remains the same on both sides.

## Additive Identity Property

This property states, for every number, there is a unique real number, which when added to the number gives the number itself. Zero is the unique real number, which is added to the number to generate the number itself. Hence, zero is called here the identity element of addition.

**A + 0 = A or 0 + A = A**

**Example: **9 + 0 = 9

### Some More Properties

**Property of Opposites: **In this case, if A is a real number then there exist a unique number -A such that;

**A + (-A) = 0 or (-A) + A = 0**

Since the result of the addition of two numbers is zero, therefore they both are called additive inverses.

**Sum of Opposite of Numbers:** Let the two numbers are A and B, then their opposites will be -A and -B. Then according to the property;

**-(A + B) = (-A) + (-B)**

**Example: **– (3+7) = (-3)+(-7)

-(10) = -3-7

-10 = -10

L.H.S = R.H.S