Exams Academic Natural Numbers: Definition, Number Line, Example, Properties

Written By Abhishek_verma
Last Modified 26-01-2023

Natural Numbers: Definition, Number Line, Example, Properties

Natural Numbers are commonly known as counting numbers. All the numbers starting from 1 to infinity are known as natural numbers. In other words, natural numbers are all the positive integers from 1 to infinity.

Natural Numbers are part of Real Numbers that include only the positive integers, i.e., 1, 2, 3, 4, etc., excluding zero, fractions, decimals and negative numbers. This article will know more about natural number definition, examples, properties, and more.

What are Natural Numbers?

Natural Numbers are positive integers and include numbers from 1 till infinity(∞). These are the numbers which we usually count in our day-to day-life. Natural numbers will not include fractions, decimals, negative numbers and zero. The set of Natural Numbers is represented by the letter ‘N‘.
N = {1,2,3,4,5,6,7,8,9,10…….}

Natural Numbers Vs Whole Numbers

Students may have questions like, Is Every Natural Number a Whole Number?, Is zero a Whole Number, etc and to understand this we need to know the difference between Natural Numbers and Whole Numbers.

Natural Numbers are numbers starting from 1 to infinity. Whereas, Whole Numbers start from 0 and go till infinity.
Natural Numbers = {1,2,3,4,5,………}
Whole Numbers = {0,1,2,3,4,……..}

Therefore, every natural number is a Whole number but every whole number is not a natural number. Because Zero is a whole number and not a natural number.

Check the difference between Whole Numbers & Natural Numbers from below:

Natural Numbers	Whole Numbers
The natural numbers is N={1,2,3,…}	The 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.	Each whole number is a natural number, except zero.

Representation of Natural Numbers on Number Line

The natural numbers can be represented on the number line as follow:

Set of Natural Numbers

The set of natural numbers will be from 1 to infinity. Check the representation from below:

Statement:

N = Set of all numbers starting from 1.
Where N stands for the set of natural numbers.

In Roster Form it is represented as:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …………………}

In Set Builder Form it is represented as:
N = {x : x is an integer starting from 1}

Properties of Natural Numbers

The operations of addition, subtraction, multiplication and division, on natural numbers, lead to four main properties of natural numbers such as:

Closure Property
Associative Property
Commutative Property
Distributive Property

Closure Property

The addition and multiplication of two or more natural numbers will always yield a natural number. However, in the case of subtraction and division, natural numbers do not obey the closure property. This means that subtracting or dividing two natural numbers might not yield a natural number.

– Closure Property of Addition: a+b=c this implies, 1+2 = 3 or 10+5 = 15. This shows that the addition of two natural numbers is always a natural number.
– Closure Property of Multiplication: a×b=c this implies, 3×2 = 6 or 10x 4 = 40. This shows that, the product of natural numbers is always a natural number.

Associative Property

The associative property is true in the case of the addition and multiplication of natural numbers. That is the sum or product of any three natural numbers remains the same even if the grouping of numbers is changed. On the other hand, the associative property does not hold good for subtraction and division of natural numbers.

– For Addition: a + ( b + c ) = ( a + b ) + c => 2+ (3+ 5) = (2+3) + 5 = 10
– For Multiplication: a × ( b × c ) = ( a × b ) × c => 2x(3×4) = (2×3)x4 = 24
– For Subtraction: a – ( b – c ) ≠ ( a – b ) – c => 3 – (5 – 1) = – 1 and ( 3 – 5 ) – 1 = – 3.
– For Division: a ÷ ( b ÷ c ) ≠ ( a ÷ b ) ÷ c => 2 ÷( 9 ÷ 3 ) = 0.666 and ( 2 ÷ 9 ) ÷ 3 = 0.074.

Commutative Property

In the case of commutative property, the addition and multiplication of natural numbers show the commutative property. Whereas, subtraction and division of natural numbers do not show the commutative property.

For Addition and Multiplication: x + y = y + x and a × b = b × a
For Subtraction and Division: x – y ≠ y – x and x ÷ y ≠ y ÷ x

Distributive Property

In the case of distributive property, the Multiplication of natural numbers is always distributive over addition. Also, the Multiplication of natural numbers is also distributive over subtraction.

Distributive property of multiplication over addition: a × (b + c) = ab + ac
Distributive property of multiplication over subtraction: a × (b – c) = ab – ac

Operations With Natural Numbers

Let’s have an overview of the operations and properties of natural numbers from below:

Operation	Closure Property	Associative Property	Commutative Property
Addition	yes	yes	yes
Subtraction	no	no	no
Multiplication	yes	yes	yes
Division	no	no	no

Solved Examples

Let’s go through some solved examples on natural numbers from below:

Question 01: Sort the natural numbers from the following list: 13, 12, 0.5, 2/3, 55, 1003, 10745, -56, -20
Solution: From the above-given numbers, the following are natural numbers: 13, 12, 55, 1003 and 10745.

Question 02: What are the first 10 natural numbers?
Solutions: The first 10 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Question 03: Is Zero a natural number?
Solution: No, Zero is not a natural number.

Study Materials On Embibe

Make use of the following study materials of Embibe which will definitely help you in your exams:

NCERT Solutions	NCERT Books
Class 8 Mock Test Series	Class 8 Practice Questions
Class 9 Mock Test Series	Class 9 Practice Questions
Class 10 Mock Test Series	Class 10 Practice Questions
JEE Main Mock Tests (Class 11-12 PCM)	JEE Main Practice Questions (Class 11-12 PCM)
NEET Mock Tests (Class 11-12 PCB)	NEET Practice Questions (Class 11-12 PCB)