What is an Addition Operation?
The process of adding objects or numbers together is called addition. The symbol, ‘+’ is used to represent the addition of two or more numbers. The numbers we add are referred to as ‘addends’ and the result is called the ‘sum’.
For example: 2 + 3 = 5.
Here, the numbers 2 and 3 are the addends, and 5 is the sum.
When referring to an addition operation, two or more single or multi-digit numbers can be used as addends.
What are the Properties of Addition?
There are four fundamental properties of addition:
- Commutative Property
- Associative Property
- Distributive Property
- Additive Identity or Identity Property of Addition
Let us further explore each of these properties.
Commutative Property of Addition
According to this property, when two numbers are added, the sum remains the same even if we change the order of numbers being added. It can be represented as:
A + B = B + A
Let us take A = 9 and B = 4
9 + 4 = 4 +9
13 = 13
Associative Property of Addition
According to the associative property of addition, when we add three numbers, the association of numbers in a different pattern does not change the result. This implies that when adding three or more numbers, the total or the sum is always the same, even when the grouping of addends is changed.
We can represent this property as;
A+ (B + C) = (A + B) + C
Consider this example,
A = 7, B = 6 and C = 5
So, 7 + (6 + 5) = (7 + 6) + 5
18 = 18
Distributive Property of Addition
In this case, the sum of two numbers multiplied by the third number is equal to the sum of the products of each addend and the third number. This property can be represented as:
A × (B + C) = A × B + A × C
Let us take A = 1, B = 2 and C = 5
L.H.S = A × (B + C)= 1 × (2 + 5)
= 1 × 7
= 7
R.H.S = A × B + A × C = 1 × 2 + 1 × 5
= 2+ 5
= 7
Hence, L.H.S = R.H.S
In the above example, we observe that even though we have distributed A to each addend of (B + C), the value remains the same on both sides. The distributive property shows a combination of both an addition operation and a multiplication operation.
Identity Property of Addition
This property states that for every number, there is a unique real number, which when added to the number gives the number itself. The number zero is this unique real number. Hence, zero here is called the additive identity of any given number.
- A + 0 = A or 0 + A = A
Example: 8 + 0 = 0 + 8 = 8
[Note: There is also the concept of additive inverse when referring to addition. An additive inverse is a number (the inverse of a number, with a different sign) when added to the original number results in zero as the result.]
Rapid Recall
Solved Examples
Example 1: Find 27 + 3 using the properties of addition.
Solution:
Write 27 as sum of 20 and 7
27 + 3
= (20 + 7) + 3
= 20 + (7 + 3) [Associative Property of Addition]
= 20 + 10 [Add]
= 30 [Add]
Hence, 27 + 3 = 30.
Example 2: Find the additive inverse of – 18.
Solution:
The given number is – 18
According to the additive inverse of numbers, when the inverse of a number is added to the given number, the result should be zero.
Let’s assume that the additive inverse of -18 is x, then,
– 18 + x = 0
– 18 + x + 18 = 0 + 18 [Add 18 on both sides]
x = 18 [Simplify]
So, the additive inverse of – 18 is 18.
Example 3: Simplify 6(2 + 3) using the properties of addition.
Solution:
6(2 + 3) [Write the expression]
= 6.2 + 6.3 [Distributive Property of Addition]
= 12 + 18 [Multiply]
= 30 [Add]
Hence, 6(2 + 3) = 30.
There are four fundamental characteristics of addition.
- Property of Identity
- Commutative property
- Associative property
- Distributive property
In many mathematical situations, the properties of addition can be used to simplify complex expressions into simpler chunks to make calculations easier.
The commutative property of addition implies that even if the order of the addends changes during the addition process, the sum stays the same.
Additive identity of 7 is 0. Since, 7 + 0 = 7.