Whole numbers are a foundational concept in mathematics, encompassing all non-negative integers starting from zero. This set includes numbers like 0, 1, 2, 3, and so on, up to infinity. Unlike integers, whole numbers do not include negative numbers or fractions.
Examples of Whole Numbers are 0, 2, 34, 999, 345768, and so on. Basically, Natural Numbers and 0 together form Whole Numbers.
Let's learn about the definition, symbols, properties, and examples of whole numbers in detail along with some numerical examples and worksheets.
Whole Numbers Definition
Whole numbers are natural numbers starting with 0. The positive numbers 0, 1, 2, 3, 4, 5, 6, and (so on) constitute whole numbers.
It can be said that The whole number is a set of numbers without fractions, decimals, and negative numbers.
Whole Number Symbol
The symbol to represent whole numbers is the alphabet ‘W’ in capital letters.
The whole numbers list includes 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, to infinity.
W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,…}
Note -
- All whole numbers come under real numbers.
- All natural numbers are whole numbers but not vice-versa.
- All positive integers, including 0, are whole numbers.
Whole Numbers on Number Line
Whole numbers can easily be observed as the number line. They are represented as a collection of all the positive integers, along with 0.
The visual representation of whole numbers on the number line is given below :
.png)
Properties of Whole Numbers
A Whole Number has the following key properties:
- Closure Property
- Commutative Property
- Associative Property
- Distributive Property
| Property | Description(where W is a whole number) |
|---|
| Closure Property | x + y = W OR x × y = W |
| Commutative Property of Addition | x + y = y + x |
| Commutative Property of Multiplication | x × y = y × x |
| Additive Identity | x + 0 = x |
| Multiplicative Identity | x × 1 = x |
| Associative Property | x + (y + z) = (x + y) + z OR x × (y × z) = (x × y) × z |
| Distributive Property | x × (y + z) = (x × y) + (x × z) |
| Multiplication by Zero | a × 0 = 0 |
| Division by Zero | a/0 is undefined |
Let's discuss them in detail.
Closure Property
The sum and the product of two whole numbers will always be a whole number.
x + y = W
x × y = W
For example: Prove the closure property for 2 and 5.
2 is a whole number, and 5 is a whole number. To prove the closure property, add and multiply 2 and 5.
2 + 5 = 7 (Whole number).
2 × 5 = 10 (Whole number).
Commutative Property of Addition
In the commutative property of addition, the sum of any two whole numbers is the same. i.e., the order of addition doesn't matter. i.e.,
x + y = y + x
For Example: Prove the commutative property of addition for 5 and 8.
According to the commutative property of addition:
x + y = y + x
5 + 8 = 13
8 + 5 = 13
Therefore, 5 + 8 = 8 + 5
Commutative Property of Multiplication
The multiplication of any two whole numbers is the same. Any number can be multiplied in any order. i.e.,
x × y = y × x
For example: Prove the commutative property of multiplication for 9 and 0.
According to the commutative property of multiplication:
x + y = y + x
9 × 0 = 0
0 × 9 = 0
Therefore, 9 × 0 = 0 × 9
Additive Identity
In the additive property, When we add the value with zero, then the value of the integer remains unchanged. i.e.,
x + 0 = x
For example: Let's prove additive property for 7.
According to additive property
x + 0 = x
7 + 0 = 7
Hence, proved.
Multiplicative Identity
When we multiply a number by 1, then the value of the integer remains unchanged. i.e.,
x × 1 = x
For example: Prove multiplicative property for 13.
According to multiplicative property:
x × 1 = x
13 × 1 = 13
Hence, proved.
Associative Property
When adding and multiplying the number and grouped together in any order, the value of the result remains the same. i.e.,
x + (y + z) = (x + y) + z
and
x × (y × z) = (x × y) × z
For example: Prove the associative property of multiplication for the whole numbers 10, 2, and 5.
According to the associative property of multiplication:
x × (y × z) = (x × y) × z
10 × (2 × 5) = (10 × 2) × 5
10 × 10 = 20 × 5
100 = 100
Hence, Proved.
Distributive Property
When multiplying the number and distributing them in any order, the value of the result remains the same. i.e.,
x × (y + z) = (x × y) + (x × z)
For example: Prove the distributive property for 3, 6, and 8.
According to the distributive property:
x × (y + z) = (x × y) + (x × z)
3 × (6 + 8) = (3 × 6) + (3 × 8)
3 × (14) = 18 + 24
42 = 42
Hence, Proved.
Multiplication by Zero
Multiplication of the zero is a special multiplication as multiplying any number by zero yields the result zero. i.e.
a × 0 = 0
Example: Find 238 × 0.
= 238 × 0
we know that multiplying any number yield the result zero.
= 0
Division by Zero
We can not divide any number by zero, i.e.
a/0 is undefined
Division is the inverse operation of multiplication. But division by zero is undefined.
Read More :
Whole Numbers and Natural Numbers
A natural number is any whole number that is not zero. Furthermore, all natural numbers are whole numbers. Therefore, the set of natural numbers is a part of the set of whole numbers.
Difference between Whole Numbers and Natural Numbers
Let's discuss the difference between natural numbers and whole numbers.
Whole Numbers vs. Natural Numbers |
|---|
Natural Numbers | Whole Numbers |
|---|
| Smallest natural number is 1. | Smallest whole number is 0. |
| Set of natural numbers (N) is {1, 2, 3, ...}. | Set of whole numbers (W) is {0, 1, 2, 3, ...} |
| Every natural number is a whole number. | Every whole number is not a natural number. |
Image added below illustrates the difference between whole numbers and natural numbers.
.png)
Read More:
Examples on Whole Numbers
Let's solve some example questions on whole numbers.
Example 1: Are the numbers 100, 399, and 457 the whole numbers?
Solution:
Yes, the numbers 100, 399, 457 are the whole numbers.
Example 2: Solve the equation 15 × (10 + 5) using the distributive property.
Solution:
We know that distributive property are:
x × (y + z) = x × y + x × z
So, 15 × 10 + 15 × 5 = 150 + 75
= 225.
Example 3: Prove the associative property of multiplication for the whole numbers 1, 0, and 93.
Solution:
According to the associative property of multiplication:
x × (y × z) = (x × y) × z
1 × (0 × 93) = (1 × 0) × 93
1 × 0 = 0 × 93
0 = 0
Hence, Proved.
Example 4: Write down the number that does not belong to whole numbers:
4, 0, -99, 11.2, 45, 87.7, 53/4, 32.
Solution:
Out of the numbers mentioned above, it can easily be observed that 4, 0, 45, and 32 belong to whole numbers. Therefore, the numbers that do not belong to whole numbers are -99, 11.2, 87.7, and 53/4.
Example 5: Write 3 whole numbers occurring just before 10001.
Solution:
If the sequence of whole numbers are noticed, it can be observed that the whole numbers have a difference of 1 between any 2 numbers. Therefore, the whole numbers before 10001 will be: 10000, 9999, 9998.
Related Articles:
Whole Numbers Worksheet
You can download this worksheet from below with answers:
Conclusion
The set of natural numbers that includes zero is known as whole numbers: 0, 1, 2, 3, 4, and so on. In terms of whole numbers, they are non-negative integers, which means that they begin at zero and go indefinitely in a positive direction without containing fractions or decimals. In many mathematical operations, including counting, addition, subtraction, multiplication, and division, whole numbers are necessary. Understanding the characteristics and functions of whole numbers is essential in the teaching of mathematics and establishes the foundation for additional mathematical exploration.
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