Natural Numbers

Natural numbers are a basic part of the number system. They include all the positive integers starting from 1 and go on till infinity. These numbers are used mainly for counting. They do not include zero, negative numbers, decimals, or fractions.

 

Table of Contents

• Natural Numbers Definition
• Natural Numbers and Whole Numbers
• Natural Numbers and Whole Numbers Set Representation
• Is 0 a Natural Number?
• Representing Natural Numbers on a Number Line
• Set of Natural Numbers
• Natural Numbers Symbol
• Smallest Natural Number
• Natural Numbers from 1 to 100
• Natural Numbers Examples
• Properties of Natural Numbers
• Properties and Operations on Natural Numbers
• Conclusion
• FAQs on Natural Numbers

 

Natural Numbers Definition

Natural numbers are all positive whole numbers starting from 1. These numbers are used for counting objects or quantities.
Example: 1, 2, 3, 4, 5, 6, ...

Set of natural numbers:
N = {1, 2, 3, 4, 5, 6, ...}

If someone asks, what is natural numbers? The answer is: numbers starting from 1 and used for counting are called natural numbers.

 

Natural Numbers and Whole Numbers

Natural numbers and whole numbers are closely related.

• Natural numbers start from 1 and go on: {1, 2, 3, 4, 5, ...}

• Whole numbers include 0 and all natural numbers: {0, 1, 2, 3, 4, ...}
  
  

So, all natural numbers are whole numbers, but 0 is only a whole number, not a natural number.

 

Natural Numbers and Whole Numbers Set Representation

Set representation helps show the difference between natural numbers and whole numbers.

Set of natural numbers: N = {1, 2, 3, 4, 5, 6, ...}
Set of whole numbers: W = {0, 1, 2, 3, 4, 5, ...}

In a Venn diagram, natural numbers are inside the whole numbers set, but zero stands alone in the whole numbers.

 

Is ‘0’ a Natural Number?

No, 0 is not a natural number.

Natural numbers are the numbers we use for counting, like 1, 2, 3, 4, and so on. We never start counting things from 0. That’s why natural numbers always begin from 1 and go up to infinity.

So, the set of natural numbers is:
{1, 2, 3, 4, 5, ...}

But 0 is still a very important number in maths. It is part of whole numbers, not natural numbers.

The set of whole numbers is:
{0, 1, 2, 3, 4, ...}

Simple Difference:

• Natural numbers: Start from 1

• Whole numbers: Start from 0

Representing Natural Numbers on a Number Line

Natural numbers can be easily represented on a number line. A number line is a straight horizontal line with numbers placed at equal intervals.

To represent natural numbers:

1. Draw a horizontal line.

2. Mark a point and label it as 0.

3. Starting from 1, place equal divisions to the right of 0.

4. Label each point with consecutive numbers: 1, 2, 3, 4, 5, and so on.

Natural numbers start from 1 and go towards the right infinitely. So, only the numbers to the right of 0 are considered.

Example:

• Numbers like 1, 2, 3, 4, etc., are all natural numbers.

• This number line clearly shows that natural numbers do not include 0 or any numbers to the left of 0.

Important note:

• 0 is the starting point for whole numbers.

• Natural numbers start from 1 and move endlessly to the right.

 

Set of Natural Numbers 

The set of natural numbers is a collection of all positive counting numbers starting from 1 and going on infinitely. These numbers are used to count objects, represent quantities, and perform basic arithmetic operations.

Definition:

A set is a well-defined group or collection of objects, in this case, numbers.
The set of natural numbers is written using the letter N.

Standard Form:

The natural numbers set is represented as:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}

• It starts from 1

• It goes on without any end (infinite set)

• It does not include 0, fractions, decimals, or negative numbers

Roster Form:

In roster form, we list the elements of the set:
N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}

Here, the dots (…) show that the list continues forever.

Set-Builder Form:

In set-builder form, we define the rule for the set:
N = {x : x is a positive integer}
Or
N = {x : x is a whole number greater tha

 

Natural Numbers Symbol

The symbol used to represent the set of natural numbers is the capital letter N.

Standard Symbol:

N = {1, 2, 3, 4, 5, 6, 7, …}

This symbol stands for the set of all positive integers starting from 1 and going on infinitely. It is one of the most commonly used symbols in mathematics when referring to counting numbers.

Variations in Notation:

In higher mathematics, different forms of the symbol N are used depending on the context:

• N - Represents the set of natural numbers

• N⁺ or N{0} - Sometimes used to emphasize that zero is excluded

• N₀ or N ∪ {0} - Used when zero is included in the set

But in standard school-level mathematics:

• N = {1, 2, 3, 4, 5, ...}

 

Smallest Natural Number

The smallest natural number is 1.

Natural numbers are the numbers used for counting. They begin from 1 and continue to infinity.
So, the list of natural numbers starts like this:
1, 2, 3, 4, 5, 6, …

Out of all these, 1 is the first and therefore the smallest.

 

Natural Numbers from 1 to 100

List of natural numbers from 1 to 100:

Natural Numbers Examples

Here are some examples of natural numbers:
5, 17, 29, 100, 2021

These numbers are all positive integers and are used for counting.
Some real-life examples:

• 4 pencils

• 10 rupees

• 6 mangoes

• 25 students

 

Properties of Natural Numbers

Natural numbers follow several important mathematical properties that help us perform basic operations like addition and multiplication. These properties include:

Closure Property

The closure property states that when two natural numbers are added or multiplied, the result is also a natural number.

• Addition: 3 + 4 = 7 (7 is a natural number)

• Multiplication: 5 × 6 = 30 (30 is a natural number)
  
  

But this does not work for:

• Subtraction: 4 - 7 = -3 (-3 is not a natural number)

• Division: 5 ÷ 2 = 2.5 (2.5 is not a natural number)
  
  

Commutative Property

The commutative property says that changing the order of numbers does not change the result in addition and multiplication.

• Addition: 2 + 5 = 5 + 2 = 7

• Multiplication: 3 × 4 = 4 × 3 = 12
  
  

It does not work for:

• Subtraction: 6 - 2 ≠ 2 - 6

• Division: 8 ÷ 4 ≠ 4 ÷ 8
  
  

Associative Property

The associative property states that the way in which numbers are grouped in addition or multiplication does not change the result.

• Addition: (2 + 3) + 4 = 2 + (3 + 4) = 9

• Multiplication: (2 × 3) × 4 = 2 × (3 × 4) = 24
  
  

This property does not apply to:

• Subtraction: (5 - 2) - 1 ≠ 5 - (2 - 1)

• Division: (12 ÷ 3) ÷ 2 ≠ 12 ÷ (3 ÷ 2)
  
  

Distributive Property

The distributive property connects multiplication with addition or subtraction. It is used to simplify expressions.

• Multiplication over addition:
  2 × (3 + 4) = 2 × 3 + 2 × 4 = 6 + 8 = 14

• Multiplication over subtraction:
  3 × (5 - 2) = 3 × 5 - 3 × 2 = 15 - 6 = 9

 

Properties and Operations on Natural Numbers

• Natural numbers are closed under addition and multiplication

• They are not closed under subtraction or division

• Addition and multiplication are both commutative and associative

• Multiplication is distributive over addition and subtraction
  
  

Conclusion

Natural numbers are fundamental in mathematics as they form the basis of counting and basic operations. They are positive integers starting from 1 and continue infinitely. Understanding the difference between whole number and natural numbers, their properties, and examples helps build strong foundational knowledge in math. Natural numbers play a vital role in real-life situations, problem-solving, and higher-level math concepts.

 

Related Links

Prime Numbers - Understand what prime numbers are, how to identify them, and explore their significance in mathematics with clear examples and explanations.

Rational Numbers - Learn about rational numbers, how they differ from other number types, and see real-life examples that illustrate their properties and usage.

Co-prime Numbers - Learn what co-prime numbers are, how they differ from prime numbers, and how to identify pairs of numbers that share no common factor other than 1.

 

Frequently Asked Questions on Natural numbers 

1. Is 0 a natural number?

Ans: No, 0 is not a natural number.
Natural numbers start from 1 and are used for counting. The set is:
{1, 2, 3, 4, 5, ...}
Since we do not count using 0, it is not considered a natural number. However, 0 is a whole number.

 

2. What are natural numbers?

Ans: Natural numbers are positive integers starting from 1 and going up to infinity.
They do not include 0, negative numbers, fractions, or decimals.
Set of natural numbers:
{1, 2, 3, 4, 5, ...}
They are mainly used for counting things in real life.

 

3. Why is 23 not a natural number?

Ans: Actually, 23 is a natural number.
It is a positive whole number greater than 0, and it belongs to the set of natural numbers.
So, the question is incorrect,23 is a natural number.

 

4. How many even natural numbers are there from 1 to 100?

Ans:There are 50 even natural numbers between 1 and 100.
Even natural numbers are divisible by 2.
List: 2, 4, 6, ..., 100
This is an arithmetic sequence with the first term 2, last term 100, and a common difference of 2.

Number of terms = ((100 - 2) ÷ 2) + 1 = 49 + 1 = 50

 

5. What is the difference between a natural number and a whole number?

Ans:

• Natural numbers: Start from 1 and include only positive counting numbers.
  Example: {1, 2, 3, 4, 5, ...}

• Whole numbers: Start from 0 and include all natural numbers.
  Example: {0, 1, 2, 3, 4, 5, ...}

 

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