Whole Numbers – Definition & Properties

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Natural numbers are commonly known as counting numbers. The set of natural numbers does not contain any number to represent ‘nothing’ or $0$. When $0$ is included in the set of natural numbers, it becomes a set of whole numbers. The set of whole numbers denoted by $W = \{0, 1, 2, 3, , 4, 5, 6, …, \}$.

Like natural numbers, whole numbers are also a part of a broader set of numbers called real numbers which also contain integers, rational numbers, and irrational numbers.

Let’s learn more about whole numbers in this article.

What Are Whole Numbers?

Whole numbers are the numbers starting from $0$, and moving on to $1$, $2$, $3$, …. The set of whole numbers is denoted by $W$. $W = \{0, 1, 2, 3, , 4, 5, …, \}$. The ‘three dots’ means that there are countless whole numbers with no limit or end.

Representing Set of Whole Numbers

The set of whole numbers can be represented in two ways.

• Statement Form: W = Set of numbers starting from $0$.
• Set Form: $W = \{0, 1, 2, 3, 4, 5, … \}$ 

Smallest and Largest Whole Numbers

As the whole numbers start from $0$, therefore, $0$ is the smallest whole number. The set of whole numbers contains countless (infinite) numbers, so there is no largest whole number.

If you pick any larger whole number, you can still find a whole number greater than that number.

For example, for the number $100000000$, $100000001 \left(100000000 + 1 \right)$ is still greater or for the number $99999999999$, $100000000000 \left(99999999999 + 1\right)$ is further greater.

Difference Between Whole Numbers and Natural Numbers

Whole numbers are the numbers like $0$, $1$, $2$, $3$, … and so on, whereas the set of natural numbers contains numbers starting from $1$ ($0$ removed from the set of whole numbers) and moving on to $2$, $3$, … Following are the differences between whole numbers and natural numbers.

Whole Numbers	Natural Numbers
The set of whole numbers is $W=\{0,1,2,3,…\}$	The set of natural numbers is $N= \{1,2,3,.. \}$
The smallest whole number is 0	The smallest natural number is 1
Each whole number is a natural number, except for zero	All natural numbers are whole numbers, but all whole numbers are not natural numbers

Whole Numbers on Number Line

The set of whole numbers can be shown on the number line. The line (in fact a ray) starts from the number $0$ and moves on to $1$, $2$, $3$, $4$, and so on on the right-hand side of $0$.

The whole number line (ray) has a starting point representing the number $0$ and the remaining numbers on the right-hand side of $0$ at constant intervals. The right side of the line is shown by an arrow which means it can be extended infinitely and can contain infinite (countless) numbers.

Properties of Whole Numbers

You can perform any of the following four basic operations on whole numbers. 

• Addition
• Subtraction
• Multiplication 
• Division

Each of these operations shows one or more of the following properties:

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property

Let’s understand these properties of whole numbers in detail.

Closure Property of Whole Numbers

The closure property states that if any two numbers from a set are operated by an arithmetic operation then their result also lies in the same set.

The closure property is exhibited by the operations addition and multiplication. The operations subtraction and division do not show exhibit closure property.

Closure Property of Addition of Whole Numbers

It states that when two whole numbers are added, then their sum is also a whole number.

Mathematically, it is represented as if $a, b \in W, \text {then  } a + b \in W$.

For example, $6$ and $13$ are whole numbers, then their sum $6 + 13 = 19$ is also a whole number.

Similarly, $23$ and $49$ are whole numbers, then their sum $23 + 49 = 72$ is also a whole number.

Closure Property of Multiplication of Whole Numbers

It states that when two whole numbers are multiplied, then their product is also a whole number.

Mathematically, it is represented as if $a, b \in W, \text {then  } a \times b \in W$.

For example, $8$ and $16$ are whole numbers, then their product $8 \times 16 = 128$ is also a whole number.

Similarly, $11$ and $35$ are whole numbers, then their product $11 \times 35 = 385$ is also a whole number.

Note: The operations subtraction and division do not show closure property.

Commutative Property of Whole Numbers

The commutative property deals with the ordering of numbers in an operation. It states that the result remains the same even if the order of numbers in the operation is changed or swapped.

The commutative property is exhibited by the operations addition and multiplication. The operations subtraction and division do not show exhibit commutative property.

Commutative Property of Addition of Whole Numbers

It states that for any two whole numbers their sum remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in W, \text {then  } a + b = b + a$.

For example, $8$ and $14$ are two whole numbers. $8 + 14 = 22$ and also $14 + 8 = 22$.

Similarly, for two whole numbers, $78$ and $57$, $78 + 57 = 135$ and $57 + 78 = 135$.

Commutative Property of Multiplication of Whole Numbers

It states that for any two whole numbers their product remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in W, \text {then  } a \times b = b \times a$.

For example, $9$ and $4$ are two whole numbers. $9 \times 4 = 36$ and also $4 \times 9 = 36$.

Similarly, for two whole numbers, $12$ and $15$, $12 \times 15 = 180$ and $15 \times 12 = 180$.

Associative Property of Whole Numbers

The associative property deals with the grouping of numbers in an operation. It states that the result remains the same even if the grouping of numbers is changed while performing the operation.

The associative property is exhibited by the operations addition and multiplication. The operations subtraction and division do not show exhibit associative property.

Associative Property of Addition of Whole Numbers

It states that the sum of any three whole numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in W, \text {then  } \left (a + b \right) + c = a + \left(b + c \right)$. 

For example, for three whole numbers $2$, $8$ and $6$, $\left( 2 + 8 \right) + 6 = 10 + 6 = 16$ and $2 + \left(8 + 6 \right) = 2 + 14 = 16$.

Similarly, for three whole numbers $15$, $12$ and $23$, $\left(15 + 12 \right) + 23 = 27 + 23 = 50 \text { and } 15 + \left(12 + 23 \right) = 15 + 35 = 50$.

Associative Property of Multiplication of Whole Numbers

It states that the product of any three whole numbers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in W, \text {then  } \left (a \times b \right) \times c = a \times \left(b \times c \right)$. 

For example, for three whole numbers $4$, $3$ and $9$, $\left( 4 \times 3 \right) \times 9 = 12 \times 9 = 108$ and $4 \times \left(3 \times 9 \right) = 4 \times 27 = 108$.

Similarly, for three whole numbers $15$, $10$ and $6$, $\left(15 \times 10 \right) \times 6 = 150 \times 6 = 900 \text { and } 15 \times \left(10 \times 6 \right) = 15 \times 60 = 900$. 

Distributive Property of Whole Numbers

The distributive property of whole numbers deals with the splitting of the distribution of whole numbers through addition and subtraction while performing the multiplication operation.

There are two forms of distributive property of whole numbers.

• Distributive property of multiplication over addition
• Distributive property of multiplication over subtraction

Distributive Property of Multiplication Over Addition of Whole Numbers

It states that for any three whole numbers the expression of the form $\left(a + b \right) \times c$ can be solved as $a \times b + a \times c$.

For example, $\left(5 + 3 \right) \times 9$ can be solved as $5 \times 9 + 3 \times 9 = 45 + 27 = 72$.

This also $\left(5 + 3 \right) \times 9$ on solving gives $8 \times 9 = 72$. 

Similarly, $\left(150 + 6 \right) \times 4$ can be solved as $150 \times 4 + 6 \times 4 = 600 + 24 = 624$.

This also $\left(150 + 6 \right) \times 4$ on solving gives $156 \times 4 = 624$. 

Distributive Property of Multiplication Over Subtraction of Whole Numbers

It states that for any three whole numbers the expression of the form $\left(a – b \right) \times c$ can be solved as $a \times b – a \times c$.

For example, $\left(14 – 6 \right) \times 8$ can be solved as $14 \times 8 – 6 \times 8 = 112 – 48 = 64$.

This also $\left(14 – 6 \right) \times 8$ on solving gives $8 \times 8 = 64$. 

Similarly, $\left(100 – 9 \right) \times 5$ can be solved as $100 \times 5 – 9 \times 5 = 500 – 45 = 455$.

This also $\left(100 – 9 \right) \times 5$ on solving gives $91 \times 5 = 455$. 

Note: The distributive property does not hold for division.

Conclusion

Whole numbers contain the same numbers as in natural numbers with one extra number $0$ and are represented as $W = \{0, 1, 2, 3, 4, … \}$. The whole numbers show the four main properties – closure property, commutative property, associative property, and distributive property of multiplication over addition and subtraction.

Practice Problems

State True or False

• $1$ is the smallest whole number.
• $0$ is the smallest whole number.
• Numbers in a set of whole numbers can be counted.
• Numbers in a set of whole numbers cannot be counted.
• The largest whole number is $99999999999$.
• There is no largest whole number.
• You get a set of whole numbers by including $0$ in the set of natural numbers.
• You get a set of whole numbers by removing $0$ from the set of natural numbers.

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties

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