August 10, 2024

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

## Various types of numbers :

According to the properties and how they are represented in the number line, there are various types of numbers . The numbers are classified into different types, like natural number, integers,  whole numbers   etc.

# 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 $\mathbb{N}$,       $\mathbb{N}=\left&space;\{&space;1,2,3,4,5,............&space;\right&space;\}$

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, $\small&space;\mathbb{N}$ is closed under addition and multiplication .  If a and b are any two natural numbers, then $\small&space;(a+b)$ and $\small&space;ab$ is also a natural number. e.g. 7+2=9    and  $\small&space;7\times&space;2=14$ 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 $\small&space;\mathbb{N}$  is not closed under subtraction.

Similarly $\small&space;\mathbb{N}$  is also not closed under division.

(ii) Commutative property :  Addition  and multiplication of two natural numbers is   commutative. If $\small&space;a$ and $\small&space;b$ are any two natural numbers, then,  $\small&space;a&space;+&space;b&space;=&space;b&space;+&space;a$  and    $\small&space;ab=ba$

Subtraction  and division of two natural numbers is not commutative.

If a and b are any two natural numbers,  then $\small&space;(a&space;-&space;b)&space;\neq&space;(b&space;-&space;a)$    and   $a÷b$$\small&space;\neq$$b÷a$.

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 $\small&space;a&space;+&space;(b&space;+&space;c)&space;=&space;(a&space;+&space;b)&space;+&space;c$   and $\small&space;a&space;\times&space;(b&space;\times&space;c)&space;=&space;(a&space;\times&space;b)&space;\times&space;c$.

e.g.   (a)   $\small&space;2&space;+&space;(5&space;+&space;1)&space;=&space;2&space;+&space;(6)&space;=&space;8$   and  $\small&space;(2&space;+&space;5)&space;+&space;1&space;=&space;(7)&space;+&space;1&space;=&space;8$.    Hence, $\small&space;2&space;+&space;(5&space;+&space;1)&space;=&space;(2&space;+&space;5)&space;+&space;1$
(b) $\small&space;2\times&space;(5\times&space;3)=30$   and  $\small&space;(2\times&space;5)\times&space;3=30$. Hence  $\small&space;2\times&space;(5\times&space;3)=(2\times&space;5)\times&space;3$

Subtraction  and division of natural numbers  is not associative .

It means for any natural number a , b and c    $a-(b-c)\neq&space;(a-b)-c$  and $a÷\left(b÷c\right)$$\small&space;\neq$$\left(a÷b\right)÷c$

(iv)Identity element :  The additive identity  of a natural numbers  is zero and multiplicative identity of natrural numbers is 1.  If a is any natural number, then $\small&space;a+0=&space;0&space;+&space;a&space;=&space;a$   and  $\small&space;a\times&space;1&space;=&space;1&space;\times&space;a&space;=&space;a.$

(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 $\mathbb{Z}$,

$\mathbb{Z}=\left&space;\{&space;..........-4,-3,-2,-1,0,1,2,3,4,.....&space;\right&space;\}$

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 $\frac{p}{q},$ where $p$ and $q$ are integers  and $q\neq&space;0$, are known as rational numbers. The collection of rational numbers is denoted  by $\mathbb{Q}$ and is written as

$\mathbb{Q}=\left&space;\{&space;\frac{p}{q}&space;;p,&space;q&space;\:&space;are&space;\:&space;integers,\:&space;q\neq&space;0&space;\right&space;\}$

Thus, $\frac{1}{2},\:&space;\frac{2003}{79},&space;\:&space;\frac{7}{5}$  , 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

$p=\frac{p}{1},&space;\:&space;\;$ where $p$  is either any natual no. or  whole number or integer.

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