# Properties of Addition

## Trigonometry # Properties of Addition

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, the sign “+” is used. The numbers which are going to add are called “addends” and the result which we are going to obtain is called “sum”. The 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. The properties of addition are used in many algebraic equations in order to reduce the complex expressions into a simpler form. These properties are very helpful to the students as these properties obey all kinds of numbers. Here, we are going to discuss the important properties of addition with definitions and examples.

## 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
4. Additive Identity Property

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

### 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. 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.

### Distributive Property of Addition

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 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:

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, 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. The distributive property is very important as it has the combination of both the addition operation and the multiplication operation.

Also, read: Properties of Multiplication.

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

The identity property of the addition can be easily remembered by thinking it off by asking question and answer. 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 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. 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 are 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.

### Properties of Addition Examples

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 with the given number, the result should be zero.

Let assume that additive inverse be “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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## Frequently Asked Questions on Properties of Addition

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

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

### 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.

### 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.

### 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.

### 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