# Whole Numbers – Definition, Properties and Examples

Last Updated : 10 Apr, 2024

Whole numbers are a set of numbers that include all natural numbers and zero. They are a collection of all the positive numbers from zero to infinity.

Let’s learn about the symbols, properties, and examples of whole numbers in detail.

## What are Whole Numbers?

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.

## 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).

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

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.

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.

## 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 :

## Natural Number and Whole Number

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.

## 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,

## Conclusion of Whole Number

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.

## Whole Numbers 1 to 100 – FAQs

### What are Whole Numbers? Give examples.

The group of natural number including the number zero is called whole number. It is represented by the symbol ‘W’.

Example of whole number are, 0, 11, 23, 45, 25, etc.

### Can Whole Numbers be Negative?

No, a whole number can never be negative as the set of whole numbers “W” is represented as:

W = {0, 1, 2, 3, …}

Therefore, whole numbers do not contain negative numbers.

### Are all Whole Numbers Real Numbers?

Yes, all whole numbers are real number. i.e. real number include whole number in themselves. But the opposite is not true i.e. all real number are not whole number.

### What is the Smallest Whole Number?

As we know that whole number starts from 0 and goes to infinity. Thus, the smallest whole number is 0.

### Is 0 a whole number?

Yes, 0 (zero) is a whole number as a whole number includes zero with natural numbers. Thus zero is the first whole number and the set of the whole number starts from zero.

### How many Whole Numbers are between 32 and 53?

The whole number between 32 and 59 are, 19 which include 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, and 52.

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