Natural Numbers – Definition & Properties

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Mathematics starts with counting. The numbers that were developed as counting numbers are called natural numbers as they were used for counting naturally. The set of natural numbers denoted by $N = \{1, 2, 3, , 4, 5, 6, …, \}$.

Natural numbers are a part of a broader set of numbers called real numbers which also contains whole numbers, integers, rational numbers, and irrational numbers.

Natural numbers are significantly used in our day-to-day activities and speech. We see these numbers everywhere around us, for counting objects, representing or exchanging money, measuring the temperature, telling the time, etc.

For example, while counting objects, we say $12$ books, $20$ notebooks, $1$ instrument box, and so on. All these numbers are natural numbers.

What Are Natural Numbers?

Natural numbers are the numbers starting from $1$, and moving on to $2$, $3$, $4$, …. The set of natural numbers is denoted by $N$. $N = \{1, 2, 3, , 4, 5, 6, …, \}$. The ‘three dots’ means that there are countless natural numbers with no limit or end.

Representing Set of Natural Numbers

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

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

Smallest and Largest Natural Numbers

As the natural numbers start from $1$, therefore, $1$ is the smallest natural number. The set of natural numbers contains countless (infinite) numbers, so there is no largest natural number.

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

For example, for the number $1000000000$, $1000000001 \left(1000000000 + 1 \right)$ is still greater or for the number $999999999999$, $1000000000000 \left(999999999999 + 1\right)$ is further greater.

Is 0 a Natural Number?

As seen above, a set of natural numbers $N$ starts from $1$ ($1$ is the smallest natural number) and since $0 \lt 1$, therefore, $0$ does not belong to the set of natural numbers. So, $0$ is not a natural number. $0$ is the smallest whole number. 

Note: Whole numbers start from $0$.

Difference Between Natural Numbers and Whole Numbers

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

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

Natural Numbers on Number Line

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

The natural number line (ray) has a starting point representing the number $1$ and the remaining numbers on the right-hand side of $1$ 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 Natural Numbers

You can perform any of the following four basic operations on natural 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 natural numbers in detail.

Closure Property

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

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

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

For example, $4$ and $9$ are natural numbers, then their sum $4 + 9 = 13$ is also a natural number.

Similarly, $39$ and $52$ are natural numbers, then their sum $39 + 52 = 91$ is also a natural number.

Closure Property of Multiplication

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

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

For example, $7$ and $11$ are natural numbers, then their product $7 \times 11 = 77$ is also a natural number.

Similarly, $16$ and $49$ are natural numbers, then their product $16 \times 49 = 784$ is also a natural number.

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

Commutative Property

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

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

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

For example, $15$ and $7$ are two natural numbers. $15 + 7 = 22$ and also $7 + 15 = 22$.

Similarly, for two natural numbers, $43$ and $87$, $43 + 87 = 130$ and $87 + 43 = 130$.

Commutative Property of Multiplication

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

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

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

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

Associative Property

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 commutative property.

Associative Property of Addition

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

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

For example, for three natural numbers $7$, $9$ and $11$, $\left( 7 + 9 \right) + 11 = 16 + 11 = 27$ and $7 + \left(9 + 11 \right) = 7 + 20 = 27$.

Similarly, for three natural numbers $21$, $48$ and $108$, $\left(12 + 48 \right) + 108 = 60 + 108 = 168 \text { and } 12 + \left(48 + 108 \right) = 12 + 156 = 168$.

Associative Property of Multiplication

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

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

For example, for three natural numbers $2$, $4$ and $7$, $\left( 2 \times 4 \right) \times 7 = 8 \times 7 = 56$ and $2 \times \left(4 \times 7 \right) = 2 \times 28 = 56$.

Similarly, for three natural numbers $10$, $5$ and $12$, $\left(10 \times 5 \right) \times 12 = 50 \times 12 = 600 \text { and } 10 \times \left(5 \times 12 \right) = 10 \times 60 = 600$.

Distributive Property

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

There are two forms of distributive property of natural numbers.

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

Distributive Property of Multiplication Over Addition

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

For example, $\left(2 + 4 \right) \times 8$ can be solved as $2 \times 8 + 4 \times 8 = 16 + 32 = 48$.

This also $\left(2 + 4 \right) \times 8$ on solving gives $6 \times 8 = 48$. 

Similarly, $\left(100 + 2 \right) \times 8$ can be solved as $100 \times 8 + 2 \times 8 = 800 + 16 = 816$.

This also $\left(100 + 2 \right) \times 8$ on solving gives $102 \times 8 = 816$. 

Distributive Property of Multiplication Over Subtraction

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

For example, $\left(12 – 5 \right) \times 4$ can be solved as $12 \times 4 – 5 \times 4 = 48 – 20 = 28$.

This also $\left(12 – 5 \right) \times 4$ on solving gives $7 \times 4 = 28$. 

Similarly, $\left(100 – 3 \right) \times 6$ can be solved as $100 \times 6 – 3 \times 6 = 600 – 18 = 582$.

This also $\left(100 – 3 \right) \times 6$ on solving gives $97 \times 6 = 582$. 

Note: The distributive property does not hold for division.

Conclusion

Natural numbers also known as counting numbers are the numbers represented as $N = \{1, 2, 3, 4, … \}$. The natural 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 natural number.
• $0$ is the smallest natural number.
• Numbers in a set of natural numbers can be counted.
• Numbers in a set of natural numbers cannot be counted.
• The largest natural number is $99999999999$.
• There is no largest natural 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 $0$ in 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)
• Whole Numbers – Definition & Properties

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