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 50digit
number and your friend writes a 60digit 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.