** Various types of numbers: natural numbers, whole numbers, integers and rational numbers**

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**Various types of numbers** :

According to the properties and how they are represented in the number line, the numbers are classified into different types.

**Natural numbers**

The natural numbers are those numbers that are used for counting and ordering. The set of natural numbers is often denoted by the symbol ,

The least natural number is 1 and there are infinitely many natural numbers. They are located at the right side of the number line (after 0)

**Properties of natural numbers: **

**(i)** **closure property: **The sum and multiplication of any two natural numbers is always a natural number. This is called “Closure property of addition and multiplication” of natural numbers. Thus, is closed under addition and multiplication . If a and b are any two natural numbers, then and is also a natural number. e.g. 7+2=9 and is also a natural number.

The difference between any two natural numbers need not be a natural number.

Example : 3 – 5 = -2 is a not natural number. Hence is not closed under subtraction.

Similarly is also not closed under division.

**(ii) Commutative property : **Addition and multiplication of two natural numbers is commutative. If and are any two natural numbers, then, and .

Subtraction and division of two natural numbers is not commutative.

If a and b are any two natural numbers, then and $a\xf7b$$b\xf7a$.

e.g. (i) 5 – 3 = 2 and 3 – 5 = -2 . Hence 5 – 3 ≠ 3 – 5

(ii)2 ÷ 1 = 2 and 1 ÷ 2 = 1.5 . Hence 2 ÷ 1 ≠ 1 ÷ 2

Therefore, Commutative property is not true for subtraction and addition.

**(iii) Associative property : **Addition and multiplication of natural numbers is associative.

If a, b and c are any three natural numbers, then and .

e.g. (a) and . Hence,

(b) and . Hence

Subtraction and division of natural numbers is not associative .

It means for any natural number a , b and c and $a\xf7(b\xf7c)$$(a\xf7b)\xf7c$

**(iv)Identity element : ** The additive identity of a natural numbers is zero and multiplicative identity of narural numbers is 1. If a is any natural number, then and

**(v)Distributive Property: ** Multiplication of natural numbers is distributive over addition and subtraction . If a, b and c are any three natural numbers, then a x (b + c) = ab + ac and a x (b – c) = ab – ac.

## **Whole numbers**

All natural numbers together with ‘0’ are called whole numbers. The set of Whole numbers is denoted by *W* and written as

*W=*{0,1,2,3,4,5,……………………}

**Integers **

It includes all natural numbers , 0 and negative of natural numbers . It is denoted by *,*

* representation of integers on number line*

Negative integers are on the left side of 0

Positive integers are on the right side of the zero

0 is neither +ve nor -ve.

#### Rational numbers

The numbers of the form where and are integers and , are known as rational numbers. The collection of rational numbers is denoted by and is written as

Thus, , etc. are all rational numbers.

Rational numbers include natural numbers , whole numbers and integers since every natural numbers , whole numbers and integers can be written as

where is either any natual no. or whole number or integer.