The addition is an arithmetic operation in mathematics. There are many arithmetic operations in mathematics like addition, subtraction, multiplication and division. The addition is the arithmetic operation where the things are added together.

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To add the numbers, the symbol ‘+’ is used. The numbers to be added are called ‘addends’, and the outcome of the addition is called ‘sum’. The process of addition has two or more addends; these addends can be any digit number.

The addition process involves digit numbers like fractions, a positive integer, a negative integer, etc. This article will cover in detail the properties of addition with examples and their definitions.

**Properties of Addition**

The properties of addition are the ways we can add numbers. These properties help students to reduce complex equations into simpler equations. Various rules and conditions of addition are the properties of addition.

There are many arithmetic operations like subtraction, multiplication and division, which have properties like addition, but the properties can vary. There are four properties of addition:

- Commutative property

- Associative property

- Distribution property
- Additive identity property

**Commutative Property**

In this property of addition, the order of numbers written in an expression does not matter. The numbers can be rearranged anyway, and it still would not affect the answer. Like (a b) = (b a). For example,

Let us take x = 20 and y = 10,

20+10 = 10+20

30 = 30

The above example shows that when we add the two numbers, 10 and 20 and the order of the two numbers is rearranged, the results remain 30. Hence, addition follows the property of commutative law.

The user can easily remember this property of the word “commute”, which means switching between questions in two places.

**Associative Property**

In this rule, if three or more numbers are added to an equation, the order of arranged does not matter. The result is not affected by the reordering of the terms in the arithmetic equation. For example,

Let us take x = 12, y = 14 and z = 16

L.H.S =x+(y+z) = 12+(14+16)

= 42

R.H.S = (x+y)+z = (12+14)+16

=42

L.H.S = R.H.S

42 = 42

The above example states that the right-hand side is equal to the left-hand side. Hence, the property is proved. This property can be applied for multiplication as well. In this property, we use the parentheses to group the ‘addends’.

This forms the operations with another group of numbers. The associative property is easy to remember by the use of the word “associate”.

**Distributive Property**

The distributive rule of addition is slightly different from the commutative and associative properties of addition. It says when a number is being multiplied by the addition of two numbers, the output is the total sum of the products of two arithmetic terms, meaning the number outside of the sum of two digits will be multiplied individually.

For example,

Let us take x = 12, y = 13 and z = 15

L.H.S =x × (y z)= 12 × (13 15)

= 12 × 28

= 336

R.H.S = x × y x × z = 12 × 13+12 × 15

=156+180

=336

L.H.S = R.H.S

336 = 336

The above example states if we distribute x (monomial factor) to the values of the binomial factor, y and z, the result remains the same on both sides. The distributive property is an important property of addition because it comprises the multiplication and addition operation.

**Additive Identity Property**

Additive identity property states that for every number, there is a unique real number. When that unique real number is added to the number, the result is the number itself.

0 is a unique real number, which, when added to any number, generates the number itself. Hence, 0 is known as the identity element of addition.

10+0 = 10+(0)

0+10 = 10

**More Properties of Addition**

Property of Opposites: In this property, if x is a real number then there exist a unique number -x such that;

- x (-x) = 0 or (-x) x = 0

In this property of addition, the sum of two numbers is zero; therefore, both numbers are called additive inverses. This addition property is known as the inverse property.

In other words, the inverse property of addition defines that if any number (positive) is added to its opposite number (negative), the result should be zero. It should be noted that each real number has its unique additive inverse value.

For example, if x = 15

The inverse of 15 is -15. When these two numbers are added, the sum is zero, which means that

=15 (-15)

= 15 – 15

= 0

Hence, the inverse of 15 is -15.

Sum of Opposite of Numbers: If we take two numbers, x and y, then the opposites of those numbers will be -x and -y. According to the property;

- -(x+y) = (-x) (-y)

Let assume that, x = 15 and y = 13

On substituting the values in this property, we get

-(15+13) = (-15) (-13)

-(15+13) = -15 -13

-18 = -18

Hence, this proves the equality of this property.

Examples of properties of Addition

The examples below will help students to understand the properties of addition:

Example 1:

Prove:- (13+17) = (-13) (-17)

Proof:

-(20) = -13-17

-20 = -20

L.H.S = R.H.S

Example 2:

Identify the additive inverse of -7

Solution:

The given number is -7

As we know, according to the property of the additive inverse of numbers, when the inverse number is added to the given number, the result is zero.

Let assume that additive inverse be “y”

Therefore,

7+y = 0

On simplifying the above expression, we get

y = -7

**Conclusion**

To carefully navigate through the higher-level concepts of mathematics, the basic knowledge of numbers should be strong. This included the arithmetic operations of the mathematics containing addition, multiplication, subtraction and division.

Every operation has similar properties to follow, but the nature of properties can vary depending on the operation.

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