Natural Numbers: Definition, Properties and Examples

# Natural Numbers: Definition, Properties and Examples

Natural Numbers

Natural numbers are a part of real numbers which includes positive integers. It is the set of whole numbers excluding 0. Natural numbers do not include negative numbers, fractions, decimals or rational numbers. They are just the counting numbers from 1 to infinity. We see many numbers around us such as to represent money, to tell time, to measure temperature, to count objects. In all these numbers, the counting numbers like 1 pencil, 2 pencils, 3 pencils, 4 pencils, etc. are natural numbers.

## What are Natural Numbers?

When we count the number of students in a class or the number of pages in a book, we naturally start counting as 1, 2, 3, 4, 5, 6, 7, 8 and so on. These numbers are called natural numbers.

Write a number as big as possible. How many digits does this number have—50, 100, or 200? If you write a 50-digit number and your friend writes a 60-digit number, the sum of these two will be another bigger number. Is it the largest natural number? Absolutely not. So, what is the largest natural number? Can you find it? No.

## Natural Numbers Definition

Natural numbers are defined as the group of all counting numbers is called a group of natural numbers. It is written as: 1, 2, 3, 4, 5, 6, 7, 8, 9, ……….

## Examples of Natural Numbers

Some examples of natural numbers are as follows:

34, 454, 3832, 19475, 595761, 5983012, 58377636, ……….

## Notation of Natural Numbers

Natural numbers are denoted by the symbol, N.

## Set of Natural Numbers

The set of natural numbers is the collection of all the counting numbers. The set of natural numbers is denoted by the capital letter N. It is written as:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, ………. }

The set of natural numbers can be written in three forms.

1. Description form: In description form, the set of natural numbers is written as:

N = {all counting numbers starting from 1}

2. Roaster form: In roaster form, the set of natural numbers is written as:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, …….}

3. Set builder form: In set builder form, the set of natural numbers is written as:

N = {x : x is a positive integer}

## Smallest and Largest Natural Numbers

The counting numbers start from 1. So, the smallest natural number is 1. What is the largest natural number? Can you find it?

Suppose I write a very large number, for example, 85423460873134086542145698869732190867231876598765432290876421578

Can you say this is the largest number? Of course, no. Because there are many numbers greater than this number. Hence, if you write a very large number, there are many numbers larger than that number.

Thus, no largest natural number exists.

## Natural Numbers from 1 to 100

The natural numbers from 1 to 100 are as follows:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100.

## Is Zero (0) a Natural Number?

No, zero (0) is not a natural number. Because natural numbers are counting numbers start from 1. We do not start counting from 0. Thus, 0 is not a natural number but it is a whole number.

## Even and Odd Natural Numbers

All the natural numbers divisible by 2 are called even natural numbers. The natural numbers divisible by 2 are: 2, 4, 6, 8, 10, 12, 14, 16, ……

Thus, the set of even natural numbers is written as:

{2, 4, 6, 8, 10, 12, 14, 16, ……}

All the natural numbers which are not divisible by 2 are called odd natural numbers. The natural numbers which are not divisible by 2 are: 1, 3, 5, 7, 9, 11, 13, 15, …..

Thus, the set of odd natural numbers is written as:

{1, 3, 5, 7, 9, 11, 13, 15, …..}

## Difference Between Natural Numbers and Whole Numbers

The natural numbers are the set of counting numbers starting from 1. They are written as: 1, 2, 3, 4, 5, 6, 7, 8, 9, …..

The whole numbers are the set of natural numbers including 0. They are written as: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …..

 Natural Numbers The set of natural numbers is denoted by N and written as: N = {1, 2, 3, 4, 5, 6, 7, 8, 9, ……} The set of whole numbers is denoted by W and written as: W = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …….} The smallest natural number is 1. The smallest whole number is 0. All the natural numbers are whole numbers. All the whole numbers are not natural numbers.

## Natural Numbers on Number Line

Let us represent natural numbers on a number line. First of all, draw a line and mark a point on it which represents the natural number 1. Mark the next point by leaving a gap of 1 cm or any suitable distance between the two points and mark it as 2. Let us take the distance between these two points as 1 unit. Now, mark a point to the right of the point marked ‘2’ at 1 unit distance from it and name it as 3. We can continue doing this till any natural number on the number line. The arrow head on the extreme right of the number line shows that this number line will continue till infinity.

## Properties of Natural Numbers

Closure Property

The sum and the product of two or more natural numbers are always natural numbers. But the difference and the quotient of two natural numbers are not always natural numbers. In general, a + b = c and a × b = c.

Closure property of addition: 4 + 8 = 12; 7 + 9 = 16; 30 + 15 = 45; 120 + 90 = 210, etc.

Thus, the natural numbers are closed under addition.

Closure property of multiplication: 3 × 5 = 15; 2 × 7 = 14; 9 × 8 = 72; 12 × 10 = 120, etc.

Thus, the natural numbers are closed under addition and multiplication.

The natural numbers are not closed under subtraction and division.

For example: 5 – 9 ≠ natural number; and 5 ÷ 10 ≠ natural number.

Commutative Property

The sum and the product of two natural numbers remains the same even if we change the order of the numbers. In general, a + b = b + a and a × b = b × a.

Commutative property of addition: 5 + 8 = 13 and 8 + 5 = 13. Thus, 5 + 8 = 8 + 5.

Commutative property of multiplication: 7 × 5 = 35 and 5 × 7 = 35. Thus, 7 × 5 = 5 × 7.

Thus, the natural number are commutative under addition and multiplication but they are not commutative under subtraction and division.

Associative Property

The sum and the product of three natural numbers remains the same even if we change the grouping of the numbers. In general, a + (b + c) = (a + b) + c

and a × (b × c) = (a × b) × c.

Associative property of addition: 5 + (8 + 4) = 5 + 12 = 17 and (5 + 8) + 4 = 13 + 4 = 17. Thus, 5 + (8 + 4) = (5 + 8) + 4.

Associative property of multiplication: 2 × (5 × 3) = 2 × 15 = 30 and (2 × 5) × 3 = 10 × 3 = 30. Thus, 2 × (5 × 3) = (2 × 5) × 3.

Thus, the natural number are associative under addition and multiplication but they are not associative under subtraction and division.

Distributive Property

According to the distributive property of multiplication over addition and subtraction, a × (b + c) = a × b + a × c and a × (b – c) = a × b – a × c.

Distributive property of multiplication over addition:

3 × (4 + 5) = 3 × 4 + 3 × 5

3 × 9 = 12 + 15

27 = 27

Distributive property of multiplication over subtraction:

5 × (4 – 2) = 5 × 4 – 5 × 2

5 × 2 = 20 – 10

10 = 10

## Solved Examples on Natural Numbers

Example 1: Fill in the blanks.

a. The smallest natural number is _____.

b. ______ is not a natural number but it is a whole number.

c. The natural numbers between 5 and 10 are _____________.

Solution:

a. The smallest natural number is 1.

b. 0 is not a natural number but it is a whole number.

c. The natural numbers between 5 and 10 are 6, 7, 8, 9.

Example 2: Identify the natural numbers in the following numbers.

12, -5, 5.2, 6.8, ¾, 102, -9, 523, 745, ½, -7/8, 95, 3/7, 3856, 10009

Solution: Natural numbers are:

12, 102, 523, 745, 95, 3856, 10009

Example 3: Which property holds in the following statement:

8 × (7 × 4) = (8 × 7) × 4

Solution: In the given statement, the associative property of multiplication holds true.

Example 4: Which property holds in the following statement:

12 + 9 = 9 + 12

Solution: In the given statement, the commutative property of addition holds true.

Related Topics:

What is an Addend in Maths

Minuend and subtrahend

Multiplicand and multiplier

Dividend, divisor, quotient and remainder

Natural numbers

Whole numbers

Properties of rational numbers

Are all integers rational numbers?

Find five rational numbers between 3/5 and 4/5