 ## 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. The four basic properties of addition are:

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

Let us learn these properties of addition one by one.

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, that 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. This property is easily remembered using the word “commute”. It means switching between two places.

As per this property or law, when we add three numbers, the association of numbers in a different pattern does not change the result. It means that when the addition of three or more numbers, the total/sum will be the same, even when the grouping of addends are changed. 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) =  2 + (4 + 6)

= 12

R.H.S = (A+B)+C = (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 multiplicationIn this property, the parenthesis is used to group the addends. It forms the operations with a group of numbers. The associative property can be easily remembered using the word “associate”,  which means that associate with a certain group of people.

This property is completely different from the 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 by the third number.

• A × (B + C) = A × B + A × 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 × (B + C)=  2 × (3+5)

= 2 × 8

= 16

R.H.S = A × B + A × C =  2 × 3 + 2 × 5

=6+ 10

=16

L.H.S = R.H.S

16 = 16

In the above example, you can see, that even though we have distributed A (monomial factor) to each value of the binomial factor, B and C, the value remains the same on both sides. The distributive property is very important as it has the combination of both the addition operation and the multiplication operation.

This property states, that 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

The identity property of the addition can be easily remembered by thinking it off by asking questions and answers. It means that we have to think about which number should be added to the given number so that the value of the original number cannot be changed. If you think that, the answer should be definitely zero. Hence, the identity element of the addition operation is zero.

### Some More Properties of Addition

Property of Opposites: In this case, if A is a real number then there exists 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. This property is called the inverse property of addition. In other words, the inverse property of addition defines that if any number is added to its opposite number, the sum should be zero. It is noted that every real number has its unique additive inverse value.

For example, assume that A = 5

The inverse of 5 is -5. When these two numbers are added together, it results in zero. It means that

=5 + (-5)

= 5-5

= 0

Hence, the inverse of 5 in the addition operation is -5.

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

• -(A + B) = (-A) + (-B)

Let assume that, A = 5 and B = 3

Now, substitute the values in the property to prove its equality, hence it becomes

-(5+3) = (-5) + (-3)

-(5+3)  = -5 -3

-8 = -8

Hence, the equality of this property is proved.

Go through the below examples to understand the properties of addition:

Example 1:

Prove:- (3+7) = (-3)+(-7)

Proof:

-(10) = -3-7

-10 = -10

L.H.S = R.H.S

Example 2:

Identity the additive inverse of -9

Solution:

The given number is -9

We know that, according to the additive inverse of numbers, when the inverse number is added to the given number, the result should be zero.

Let’s assume that the additive inverse is “x”

Therefore,

9 +x = 0

By simplifying the above expression, we get

x = -9

Therefore, the additive inverse of 9 is -9.

### Properties of Addition Practice Questions

1. Simplify 5(2+3) using properties of addition.
2. Fill in the blank: 5 + ____ = 0.
3.  Use properties of addition: – (7+2) = _____.

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Q1

### What are the four basic properties of addition?

The four basic properties of addition are:
Commutative property
Identity Property
Associative property
Distributive property

Q2

### Why do we use the properties of addition

The properties of addition are applied in many mathematical problems to reduce the complex expression into a simple expression.

Q3

### What does the commutative property of addition tell us?

The commutative property of addition tells that the sum remains the same even if the order of addends are changed in the addition process.

Q4

### What is the additive identity of 7?

The additive identity of 7 is 0. Because zero is the only additive element, which does not change the value of the original number. It means that 7 + 0 = 7.

Q5

### Which property uses both addition and multiplication operations?

The property that uses both the addition and multiplication operation is the distributive property. (i.e.,) A × (B + C) = A × B + A × C