RATIONAL NUMBER | REVISION NOTES | CLASS 8

 In our 7 class, we have studied about natural numbers, whole numbers, integers, and fractions. we have also studied various operations on the rational number. In this chapter, we shall study the properties of these operations on rational numbers 

Rational Numbers 

The numbers of the form , where  a and b are integers and   are called rational numbers.

for example : 

Positive Rationals 

A rational number is said to be positive if its numerator and denominator are either both positive or both negative. 

Negative Rationals

A rational number is said to be negative if its numerator and denominator are of opposite signs.

There Properties of Rational numbers : 

Property 1 

if   is a rational number and m is a nonzero integer then  

 

such rational numbers are called equivalent rational numbers.

Property 2 

if   is a rational number and m is common divisor of a and b , then 

Thus, we can write , 

"Standard form of a rational number "

A rational number   said to be in standard form is a and b are integers having no common divisor other than  1 and b is positive.

Property 3 

Let  and  be two rational numbers . then 

Comparison of rational numbers 

it is clear that : Every positive rational numbers is greater than 0 , and negtive rational number is less than 0 .

General method of comparing rational numbers 

Step 1  : Express each of the two given rational numbers with positive denominator 

Step 2  : Take the LCM of these positive denominators 

Step 3  : Express each rational number (obtained in step 1 ) with this LCM as the common denominator 

Step 4  : the number having the greater numerator is greater 

for example :

Which of the numbers 3/-4 or -5/6 is greater ?

Sol :  first we write each of the given numbers with positive denominator 

          our number 

           

          the other numbers = 

          LCM of 4 and 6 = 12 

          

         

          clearly -9 > -10 

          

       Hence , 

REPRESENTATION OF RATIONAL NUMBERS ON THE REAL NUMBERS 

In the last class we have learnt how to represent integers on the number line.

let us review it .

Draw any line .  Take a point O on it . Call it 0  (zero) . Set off equal distances on the right as well as on the left of O . such a distance is known as a unit length  Clearly. the points A, B,C, D and E Represent the integers -1,-2,-3,-4, and -5 respectively . 

thus , we may Represent any integer by a point on the number line. Clearly , every positive integer lies to the right of O and every negative integer lies to the left of O 

Similarly we can Represent rational numbers. 

Explain with Example  

Example :Represent 1/2 and -1/2 on the number line .

Solution : Draw a line . Take a point O on it . Let its represent 0 . set off unit length OA and OA` to  b Right and to the left of O respectively Then . A represents the integers 1 and A` represent integer -1 

                Now , divide OA into two equal parts let OP be first part out of these two parts then , The points P represent the rational number 1/2 

 

Again , Divide OA` into two equal parts . Let OP` be the first part out of these 2 parts. then the point p` represents the rational numbers -1/2 . 

ADDITION OF RATIONAL NUMBERS 

if two rational numbers are to be added, we should convert each of them into a rational number with positive denominator.

CASE 1

when given numbers have same denominator : In this case , we define 

For example : 

CASE 2

When denominators of given Numbers are unequal :

In the case we take the LCM of their denominators and express each of the given numbers with LCM as the common denominator . Now , we add these numbers as shown above 

For example 

Find the sum : -5/6 +4/9 

Solution : the denominators of the given rational numbers are 6 and 9 Respectively . LCM of 6 and 9 = (3 x 2 x 3 ) = 18

  and  

Short cut method 

Also visit on video to easily understand 

Properties of Addition of rational numbers 

Property 1 (closure property ) : 

The sum of two rational number is always a rational number . if a/b and c/d are any two rational numbers , then ( a/b + c/d ) is also rational number.

Property 2 ( Commutative Law) : 

Two rational Numbers can be added in any order. thus for any two rational number a/b and c/d we have 

Property 3 ( Associative Law ): 

while adding three rational numbers , they can be grouped in any order . thus for any three rational numbers a/b , c/d , and e/f , we have 

Property 4 (Existence of Addition identity ) : 

0 is a rational number such that the sum of any rational number and 0 is the rational number itself .

thus , 

Property 5 ( Existence of additive inverse ) :

for every rational number a/b , Thus 

-a/b is called the additive inverse of a/b

SUBTRACTION OF RATIONAL NUMBERS 

For rational number a/b and c/d , we define :

MULTIPLICATION OF RATIONAL NUMBERS 

For any Two rational a/b and c/d ,

Properties of multiplication of Rational Numbers 

Property 1 (Closure property ):

The sum of two rational number is always a rational number . if a/b and c/d are any two rational numbers , then ( a/b x c/d ) is also rational number.

Property 2 (Commutative property ):

Two rational Numbers can be added in any order. thus for any two rational number a/b and c/d we have 

Property 3 (Associative  property ):

while adding three rational numbers , they can be grouped in any order . thus for any three rational numbers a/b , c/d , and e/f , we have 

Property 4 (Existence of Multiplicative identity ):

For any rational number a/b , we have 

1 is called the multiplicative identity for rationals

Property 5 (Existence of Multiplicative inverse ):

Every nonzero rational number a/b  has its multiplicative inverse b/a Thus,

b/a is called the reciprocal a/b

clearly , Zero has no reciprocal.

Reciprocal of 1 is 1 and the reciprocal of (-1) is (-1)

Property 6  (Distributive Law of Multiplication Over addition ):

For any three rational numbers a/b , c/d and e/f , we have 

Property 7 (Multiplicative Property of 0 ):

Every rational number multiplied with 0 gives 0 .  thus for any rational number a/b, we have 

DIVISION OF RATIONAL NUMBERS 

If  and  are two rational numbers such that , we define , 

PROPERTIES OF DIVISION 

PROPERTY 1( Closure Property)

If  and  are two rational numbers such that  then  is also rational number. 

PROPERTY 2 ( Property of 1 ) 

For every rational number a/b we have 

PROPERTY 3 

For every non zero rational number a/b , we have 

More Important concept 

if x and y be two rational numbers such that x < y then  is a rational number between x and y . 

for example 

Find a rational number lying between  and

Solution : Required = 

                                 =

Hence , 5/12 is rational numbers