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

(e) Inverse Property of Addition

(f) Opposite of Sum Property

**Commutative Property** **of Addition**

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)**

**Associative Property of Addition**

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

**Distributive Property of Addition**

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.

**Identity Property of Addition**

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**

**Inverse Property of Addition**

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]