Properties of addition are different for different addition operations. It is necessary to know addition properties because we use every addition operation in our daily life activities. The numbers used for addition operations are called addends. All these properties of addition are used to reduce complex algebraic equations. We will also have properties for subtraction, multiplication, and division in our mathematics. Learn all the properties of addition in this article along with solved examples and practice questions.
The main properties of addition are
1. Commutative property
2. Associative Property
3. Distributive Property
4. Additive Identity Property
Check out all the properties with examples and explanation below.
Also, check
1. Commutative Property of Addition
When two numbers or integers are added the total remains the same even when the values of the numbers or integers changed it is considered as commutative property of addition
For example, let us consider A and B as two numbers or integers, then according to the commutative property of addition A+B = B+A.
Example:
Let us assume A = 15 and B = 10 then commutative property of addition becomes,
Firsly, add A and B: A + B = 15 + 10 = 25
Then, add B and A: B + A = 10 + 15 = 25.
Now, A + B is equal to the B + A.
Here adding 15 and 10 & and adding 10 and 15 by interchanging the values gives the same result i.e 25.
2. Associative Property of Addition
When three numbers or integers are added the result remains the same even when the grouping or associating of the numbers or integers changed is considered as Associative Property of Addition.
For example, let us assume A, B, and C as three numbers or integers. According to Associative Property of Addition A + (B + C) = (A + B) + C. The associative property of addition is also applicable for multiplication. We use parenthesis to group the addends.
Example of Associative Property of Addition:
Let us assume A = 2, B = 3, and C = 4 then Associative property of addition becomes,
Firsly, add A with the B + C : A + (B + C) = 2 + (3 + 4) = 2 + 7 = 9.
Then, add (A + B) with C : (A + B) + C = (2 + 3) + 4 = 5 + 4 = 9.
Now, A + (B + C) is equal to the (A + B) + C
Here adding 2, 3, and 4 & and adding 4, 3, and 2 by interchanging the values gives the same result i.e 9.
3. Distributive Property of Addition
Distributive Property of Addition says that When the addition of two numbers or integers is multiplied with the third number the result remains the same when the addition of each of the two numbers or integers is multiplied by the third number or integer.
For example, let us assume A, B, and C as three numbers or integers. According to Distributive Property of Addition, A × (B + C) = (A × B) + (A × C). The distributive property is the combination of both the multiplication operation and the addition operation.
Do Refer:
Example:
Let us assume A = 4, B = 5, and C = 6 then Distributive Property of Addition becomes,
Firsly, add (B + C) and multiply with A : A × (B + C) = 4 × (5 + 6) = 4 × 11 = 44.
Then, multiply (A × B) and Add it with the multiplication of (A × C) : (A × B) + (A × C) = (4 × 5) + (4 × 6) = 20 + 24 = 44.
Now, A × (B + C) is equal to the (A × B) + (A × C).
4. Additive Identity Property of Addition
Additive Identity Property of Addition says that when any number gives the same number after adding it with another number i.e, zero. When you add a number with zero, then you will get the same number as the output. Therefore, the number zero is known as the identity element of addition.
For example, let us take a number A. According to Additive Identity Property of Addition, A + 0 = A or 0 + A = A.
Example:
Let us assume A = 3, then Additive Identity Property of Addition becomes,
Firsly, add A + 0 = A
Then, add 0 + A = A
Therefore, A + 0 = A or 0 + A = A.
Some More Properties of Addition
Property of Opposites:
According to the Property of Opposites, if you add a number with its negative number, then you will get zero. As the addition of two numbers became zero, they are called additive inverses. This property is known as the inverse property of addition. Or if any number is added to its opposite number, the result becomes zero. Every real number will have its unique additive inverse value.
For example, let us take a number B. According to inverse property of addition, A + (-A) = 0 or (-A) + A = 0
Example:
Let us assume A = 8, then inverse property of addition becomes,
8 + (-8) = 8 – 8 = 0.
Hence Proved.
Sum of Opposite of Numbers:
Let us take two numbers C and D, then their opposites number will become -C and -D. The property becomes -(C + D) = (-C) + (-D)
Example:
Let us assume A = 6, B = 4 then the property to prove its equality becomes,
-(6 + 4) = (-6) + (-4)
-(6 + 4) = -6 -4
-10 = -10
Hence, the equality of this property is proved.
Properties of Addition Examples
Check out the below examples to understand the complete concepts of Properties of Addition.
Example 1:
Prove -(5 + 2) = (-5) + (-2)
Solution:
Given that -(5 + 2) = (-5) + (-2)
-(7) = -5 -2
-7 = -7
L.H.S = R.H.S
Example 2:
Prove 2, 3, and 4 obeys the Distributive Property of Addition.
Solution:
According to the Distributive Property of Addition, A × (B + C) = A × B + A × C
Let A = 2, B = 3, and C = 4.
2 × (3 + 4) = 2 × 3 + 2 × 4
2 × (7) = 6 + 8
14 = 14.
Hence the given numbers obey the Distributive Property of Addition.
Different Practice Questions on Properties of Addition
1. Use properties of addition: – (6+1) = _____.
(i) -7
(ii) -5
(iiii) -12
(iv) -9
Answer: (i) -7
Simplification: Given that – (6+1).
– (6+1) = -6 -1
– (6+1) = -7
Therefore, the answer is -7.
2. Which property describes the following problem? 5 + 4 = 4 + 5
(i) Identity Property
(ii) Commutative Property
(iii) Associative Property
(iv) Math Property
Answer: (ii) Commutative Property
Simplification: Given that 5 + 4 = 4 + 5.
9 = 9
According to the Commutative Property of addition A + B = B + A.
Therefore, the answer is the Commutative Property of addition.
3. Which property uses parentheses?
(i) identity
(ii) commutative
(iii) associative
(iv) all of the properties
Answer: (iv) all of the properties
4. Which property describes this problem? 9 + 0 = 9
(i) Commutative Property of Addition
(ii) Identity Property of Addition
(iii) Associative Property of Addition
(iv) This problem isn’t any of the properties.
Answer: (ii) Identity Property of Addition
Simplification: Given that 9 + 0 = 9
According to the Additive Identity Property of Addition A + 0 = 0 + A = A.
Therefore, the answer is the Additive Identity Property of Addition.
5. The following problem is an example of which property? (6 + 5) + 3 = 6 + (5 + 3)
(i) Identity Property of Addition
(ii) Commutative Property of Addition
(iii) This problem isn’t any of the given properties.
Answer: (iii) This problem isn’t any of the properties.
Simplification: Given that (6 + 5) + 3 = 6 + (5 + 3)
According to the Associative Property of Addition (A + B) + C = A + (B + C).
Therefore, the answer is the (iii) This problem isn’t any of the given properties.
6. What number fills in the blank?
2 × ____ = (2 × 3) + (2 × 7).
(i) 10
(ii) 8
(iii) 6
(iv) 7
Answer: (i) 10
Simplification: Given that 2 × ____ = (2 × 3) + (2 × 7).
According to Distributive Property of Addition, A × (B + C) = (A × B) + (A × C).
A = 2, B = 3, and C = 7.
2 × ____ = (2 × 3) + (2 × 7)
2 × ____ = 6 + 14
2 × ____ = 20.
Therefore, the answer is (i) 10.
Frequently Asked Questions on Properties of Addition
1. Mention the different properties of addition?
Different properties of addition are
Commutative property
Associative property
Identity Property
Distributive property
2. What is the additive identity of 8?
The additive identity says that A + 0 = 0 + A = A.
Here the A = 8.
8 + 0 = 0 + 8 = 8.
3. Explain which property uses addition and multiplication operations at the same time?
The distributive property of addition uses both addition and multiplication operations at the same time. The distributive property of addition is A × (B + C) = A × B + A × C.
4. What does the commutative property of addition explains to us?
The commutative property of addition explains that the sum becomes the same even if the order of addends is changed in the process of addition.