About Properties of Whole Numbers
- What are Whole Numbers?
- What are the Different Properties of Whole Numbers?
- Commutative Property of Addition and Multiplication
- Associative Property of Addition and Multiplication
- Distributive Property of Multiplication over Addition
- Identity Property (for Addition and Multiplication)
- Rapid Recall
- Solved Examples
- Frequently Asked Questions
What are Whole Numbers?
The collection of positive integers or natural numbers, plus zero, are referred to as whole numbers. They can’t, however, be a negative number. Typically, the letter ‘W’ stands for the set of whole numbers. This set of numbers is represented as:
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,….}
What are the Different Properties of Whole Numbers?
The Properties of whole numbers are:
- Commutative property of addition and multiplication
- Associative property of addition and multiplication
- Distributive property of multiplication over addition
- Identity property
Commutative Property of Addition and Multiplication
The commutative property of whole numbers states that if two whole numbers are added or multiplied together, the outcome is unaffected by the order of the numbers. Two whole numbers can be multiplied or added in any order.
If A and B are two whole numbers, then;
- A + B = B + A
- A x B = B x A
Examples:
Commutative property of addition
4 + 6 = 6 + 4 = 10
7 + 5 = 5 + 7 = 12
Commutative property of multiplication
4 x 5 = 5 x 4 = 20
7 x 5 = 5 x 7 = 35
Associative Property of Addition and Multiplication
The regrouping of three whole numbers has no effect on their sum or product, according to the associative feature of addition and multiplication.
Let A, B and C are three whole numbers, then as per associative property,
A + (B + C) = (A + B) + C
A x (B x C) = (A x B) x C
We can understand the above two expressions, with the help of the examples.
Associative property of addition
(1 + 3) + 7 = 1 + (3 + 7) = 11
(4 + 7) + 8 = 4 + (7 + 8) = 19
Associative property of multiplication
(1 x 3) x 7 = 1 x (3 x 7) = 21
(4 x 7) x 8 = 4 x (7 x 8) = 224
Distributive Property of Multiplication over Addition
Multiplication is preferred to addition in this property. It indicates that if there are three whole numbers, A, B, and C, then;
A x (B + C) = A x B + A x C
Here is an example to prove the property.
3 × (8 + 2) = (3 × 8) + (3 × 2)
LHS = 3 × (8 + 2) = 3 × 10 = 30
RHS = (3 × 8) + (3 × 2) = 24 + 6 = 30
Hence, proved.
Identity Property (for Addition and Multiplication)
The identity property of whole numbers for addition and multiplication states that:
- W + 0 = W
- W x 1 = W
Where W is any whole number.
Let’s look at a few examples,
Identity for addition
11 + 0 = 11
51 + 0 = 51
120 + 0 = 120
Identity for multiplication
11 x 1 = 11
50 x 1 = 50
120 x 1 = 120
Therefore, 0 is the additive identity and 1 is the multiplicative identity of any whole number.
Rapid Recall
Properties of Whole Numbers:
Solved Examples
Example 1: Add the following expression using a suitable identity.
134 + 97 + 203
Solution:
134 + 97 + 203 = 134 + (97 + 203) [Use associative property]
= 134 + 300 [Apply PEMDAS rule]
= 434 [Add]
So, 134 + 97 + 203 = 434
Example 2: Use identities to simplify the following expression.
12 × 50 + 12 × 20
Solution:
12 × 50 + 12 × 20 = 12 × (50 + 20) [Use distributive property]
= 12 × 70 [Apply PEMDAS rule]
= 840 [Multiply]
Example 3: Mark was asked to calculate the total number of crayons if he were to distribute thirteen crayons to five children each. Ruth and twelve friends were given five crayons each and Ruth was asked to calculate the total number of crayons they all received. Mark says that the results will be different, and Ruth says that their results will be the same. Who is correct and which property of whole numbers is applied here?
Solution:
According to Mark’s calculation, the following expression is used to calculate the answer,
13 x 5 = 65
According to Ruth, the calculation she would have to do to get the result is,
5 x 13 = 65
So, both the calculations provide the same product. This means that Ruth was right.
13 x 5 = 5 x 13 = 65
In other words,
A x B = B x A [commutative property]
The commutative property was used in this scenario.
There are a total of four properties for whole numbers and they are:
- Commutative property for addition and multiplication
- Associative property for addition and multiplication
- Distributive property of multiplication over addition
- Identity for addition and multiplication
In a way, both the properties emphasise that the order of how the terms are multiplied or added does not affect the final answer. However, the associative property is considered for three terms whereas the commutative property is applied to two terms.
When adding whole numbers, zero is considered to be an identity.
One is considered as the identity when two whole numbers are multiplied.