December 9, 2022

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

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, 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 \mathbb{N}

,       \mathbb{N}=\left \{ 1,2,3,4,5,............ \right \}

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

Similarly \small \mathbb{N}  is also not closed under division.

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

Subtraction  and division of two natural numbers is not commutative. 

If a and b are any two natural numbers,  then \small (a - b) \neq (b - a)    and   a÷b\small \neqb÷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 a + (b + c) = (a + b) + c   and \small a \times (b \times c) = (a \times b) \times c.

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

Subtraction  and division of natural numbers  is not associative . 

It means for any natural number a , b and c    a-(b-c)\neq (a-b)-c  and a÷(b÷c)\small \neq(a÷b)÷c

(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 \small a+0= 0 + a = a   and  \small a\times 1 = 1 \times a = 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 \{ ..........-4,-3,-2,-1,0,1,2,3,4,..... \right \}

 

                                                   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 0, are known as rational numbers. The collection of rational numbers is denoted  by \mathbb{Q} and is written as

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

 Thus, \frac{1}{2},\: \frac{2003}{79}, \: \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}, \: \; where p  is either any natual no. or  whole number or integer.

 

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