# Properties of Addition || Commutative, Associative, Distributive & Identity property of addition

Properties of addition is an important concept which is frequently used in advanced algebra and calculus.

The property may seem simple and straight – forward so may student may ignore its relevancy.

Here i will advise all the students to understand the underlying concept as your success in advanced Math depends on these simple concept only.

## What are Properties of addition?

There are basically six properties of addition

(a) Commutative Property
(b) Associative Property
(c) Distributive Property
(d) Identity Property
(f) Opposite of Sum Property

The word “commutative” means exchange or substitution.

The property says that the addition of two digits will remain same even after changing the order of the numbers.

This property is also called order property as change in the order of digits have no effect on addition result.

Where A & B can be any possible number.

Let us understand the concept with example:

Example 01
Let A = 4 and B = 9

According to the Commutative Property of Addition
⟹ A + B = B + A
⟹ 4 + 9 = 9 + 4
⟹ 13 = 13

You can see that both the addition yield same result.

Hence, the commutative property is verified.

Example 02
Use commutative property to express 10x in two different forms

10x can be written as 6x + 4x

Now using commutative property:
6x + 4x can also written as 4x + 6x

Hence 10x can be written as: (6x + 4x) and (4x + 6x)

The “Association” means “alliance” in English

The property says that the addition of three of more number will given same result regardless of how you group them in calculation

Where, A, B & C can be any possible number

Example 01
Let A = 5, B = 8, C = 10

A + (B + C) = (A + B) + C
⟹ 5 + (8 + 10) = (5 + 8) + 10
⟹ 5 + 18 = 13 + 10
⟹ 23 = 23

Hence even after regrouping you will get the same addition result

Example 02
Solve the expression: (7x + 6y) + 4x

⟹ (7x + 6y) + 4x

using associative property, the expression can be written as:

⟹ (7x + 4x) + 6y

⟹ 11x + 6y

The property says that adding two number and then multiplying it with third number will give the same results as multiplying the third number with the other two numbers and then adding it.

Where A, B & C can be any number possible

Example
A = 5, B = 3 and C = 1

Putting the values
⟹ A (B + C) = AB + AC
⟹ 5 (3 + 1) = 5.3 + 5.1
⟹ 5. 4 = 15 + 5
⟹ 20 = 20

This property is helpful in solving algebraic equation.
Given below are some examples for your understanding.

Example 01
Simplify the algebraic expression 6 ( x + y )

Using distributive property, the expression can be written as:

Note: You don’t have to remember the property, just remove the bracket and replace it with multiplication

Example 02
Rearrange the expression 7x + 14

The above expression can be written as:
⟹ 7x + 7 . 2

Now using distributive property;
⟹ 7 ( x + 2 )

Note: you don’t have to remember the property, just take the common number out of the bracket and you will get required expression.

It says that addition of any number with zero results in same number

A + 0 = A

Where, A can be any possible number

Zero is called additive identity because it does not change the identity of number while addition.

Example 01
A = 5

Putting the value in identity property;
A + 0 = A
5 + 0 = 5
Hence proved

It says that addition of any numbers with its inverse variant results in number 0

What is inverse variant?
The negative of any given number is its inverse variant.

Examples:
(a) Inverse variant of number 2 is -2
(b) Inverse variant of -9 is 9 [ -(-9) is 9 ]
(c) Inverse of 0 is 0

According to inverse property of addition:

Where A can be any number possible.

Let us understand the property with some examples.

Example 01
Prove inverse property of addition using number 9

Number given is 9
Inverse of number 9 is – 9

Adding both the number and inverse
⟹ 9 + ( – 9)
⟹ 9 – 9
⟹ 0

Showing the above calculation on number line

### Opposite of Sum Property

The opposite of sum of given numbers is equal to addition of opposite of individual number

According to the property

Where A & B can be any possible number.

Prove
Prove opposite sum property when A = 5 and B = 11

Sum of Number [A + B]
[ 5 + 11] = 16

Opposite of sum [A + B]
– [A + B] = -16 – – – eq(1)

Opposites of individual number
– [A] = – 5
– [B] = -11

Sum of individual opposites
[-A] + [-B] = [- 5] + [-11] = -16 – – -eq(2)

On comparing we can see that eq (1) = eq(2)

Hence, we proved the opposite sum property of addition
– [A + B] = [-A] + [-B]