**Properties of addition** define in what ways we can add the given integers. “**Addition**” is one of the basic arithmetic operations in Mathematics. The addition is the process of adding things together. To add the numbers together, a sign “+” is used. The numbers which are going to add is called “addends” and the result which we are going to obtain is called “sum”. Addition process involves two or more addends which can be any digit number. Addends can be any numbers such as positive integer, a negative integer, fractions and so on.

Also read:Â Properties of Multiplication

## What are the four 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 for each operation, the properties might vary. There are basically four Maths properties defined for addition.

They are:

Let us learn these properties of addition one by one.

### Commutative Property of Addition

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: **

Let us take A = 10 and B = 5

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 of Addition

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: **

Let us take A = 2, B = 4 and C = 6

L.H.S =A+(B+C) =Â 2 + (4 + 6)

= 12

R.H.S =Â (A+B)+C =Â (2 + 4) + 6

=12

L.H.S = R.H.S

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 of Addition

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 Ã— (B + C) = A Ã— B + A Ã— C**

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

**Example**:

Let us take A = 2, B = 3 and C = 5

L.H.S =A Ã— (B + C)=Â 2 Ã— (3+5)

=Â 2 Ã— 8

= 16

R.H.S = A Ã— B + A Ã— C =Â 2 Ã— 3 + 2 Ã— 5

=6+ 10

=16

L.H.S = R.H.S

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 of Addition

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 (or)

0 + 9 = 9

### Some More Properties of Addition

**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 1: Prove:Â **– (3+7) = (-3)+(-7)

-(10) = -3-7

-10 = -10

L.H.S = R.H.S

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