When we say numbers, we may mean an integer or a fraction. Numbers are classified into whole numbers, natural number, integers, complex numbers etc. Integers are those which are the positive and negative numbers including zero on the number line. Where as the whole numbers are those which are positive integers including 0. What are counting numbers? Are the whole numbers and counting numbers same? Why do we call them as counting numbers? In this section let us see how the numbers are classified and why we call them as counting numbers.

What are counting numbers?
The numbers 1, 2, 3, 4, 5, . . . . . . . . . . . . .  are called counting numbers.
Whenever we count number of items like, number of pens, pencils, books, houses, etc, we start counting from 1 and then proceed with 2, 3, 4,............ Therefore, these numbers are called counting numbers of natural numbers.

Example 1: How many roses are there in the following picture?

Counting Numbers Examples
When we count we find that there are 10 Roses in the above picture.

Example 2: How many stars are there in the following figure.

Example of Counting Numbers
By counting from 1, we find that there are 6 stars in the above figure.
A number line is the one which consists of Real numbers. The real numbers include rational and irrational numbers.
The following diagram shows a real number line, where the natural numbers are highlighted in red.
Counting on Number Line
We see that the numbers on the number line are, positive, negative with zero in the middle.
The natural numbers are those which are the positive integers as shown on the number line. They also called as counting numbers.
Example 1: Number of members in a family.
Solution: Since the members of a family consists of minimum 1 and increases as we count the number of
members in the family, we can say that it is a natural number.

Example 2: Number of mobile phones in a mobile store.

Solution: A mobile store consists of different models of mobile phones, which is a natural number.
We can't express the number of mobile phones in fraction, since each mobile phone is a complete set.
Therefore the number of mobile phones in a mobile store is a natural number.

Example 3: Height of a place above sea level.

Solution: Since the height of a place above sea level can be expressed in decimal like 200.7 m, 315.7 m etc,
we cannot say that the height of the place is a natural number.
1. The counting numbers increase as we move from left to right.
(i.e ) For any two natural numbers a and b, such that if a lies to the left of b, then b > a or a < b.

2. Counting numbers are closed with respect to addition and multiplication.
For example, 2 + 5 = 7 $\epsilon$ N , and 2 x 5 = 10 $\epsilon$ N

3. Counting Numbers are commutative with respect to addition and multiplication.
For example 3 + 5 = 5 + 3 = 0
and 3 x 5 = 5 x 3 = 15

4. Counting Numbers are associative with respect to addition and multiplication.
For example, 1 + ( 3 + 5 ) = ( 1 + 3 ) + 5 = ( 1 + 5 ) + 3
1 + ( 3 + 5 ) = 1 + 8 = 9
( 1 + 3 ) + 5 = 4 + 5 = 9
( 1 + 5 ) + 3 = 6 + 3 = 9

Similarly for multiplication, we have
2 x ( 3 x 5 ) = ( 2 x 3 ) x 5 = ( 2 x 5 ) x 3
2 x ( 3 x 5 ) = 2 x 15 = 30
( 2 x 3 ) x 5 = 6 x 5 = 30
( 2 x 5 ) x 3 = 10 x 3 = 30

5. The inequality does not change when we add or multiply by a natural number on both sides of an inequality.
( i. e ) If a < b, then a + c < b + c where a, b, c are all natural numbers
For example, 4 < 9
=> 4 + 5 < 9 + 5
=> 9 < 14
if a < b, then ac < bc, where a, b and c are natural numbers.
For Example, 3 < 5
=> 3 x 2 < 5 x 2
=> 6 < 2