NOTES
I.1
FOR
RATIONAL NUMBERS
Rational numbers
A rational number is a number that can be expressed as the quotient or fraction
Integers, p and q, with denominator q  0. Since q can be equal
p
p
q of two
q to 1 So, every Integer is a
rational number.
e.g.
2 8
8
, ,  3,
3 7
81
Real
Numbers
Rational
Numbers
Irrational
Numbers
Integers
0.75
–3
Rational
Numbers
349
12
5
–24
0
–10
3 4 32
–2 5
3.2125
3
– 24
–1.7
5
–4 10
3
7.66...

2.1515...
1.2
Properties of Rational Numbers
1.2.1 Closure Property
Addition of two rational number will result in a rational number.
e.g.
7 3 7  3 10
 

8 8
8
8
2 3 2  7  3  3 14  9 5
 


3 7
21
21
21
•
Subtraction of two rational number will result in a rational number.
e.g.
•
2 3 2  5  3  7 10  21 11
 


7 5
35
35
35
4 1 4  3  1 5 12  5 7
 


10 6
30
30
30
Product of two rational numbers is again a rational number.
e.g.
2  7  14
  
3  11  33
7  3  21 7
  

9  2  18 6
Page 1
•
p
p
When you divide a rational number a b by a rational number q (where q  0 ) then a
rational number is obtained.
e.g.
3 9 3 8 8
   
7 8 7 9 21
2 7 2 6 12
 
 
5 6 5 7 35
Important Note:Division of a rational number by zero is not defined. So, rational number are not closed under division.
1.
2.
3.
Add the following rational numbers
(a)
7 2 1
, ,
3 7 2
(b)
7 4 3
, ,
11 8 2
(c)
2 1 11
, ,
3 5 2
(d)
2 9
, ,7
4 3
Subtract the following rational number.
(a)
7
2
from
3
9
(c)
4
2
from
9
7
(b)
4
1
from
9
11
Multiply the following rational numbers
(a)
2 4 9
, ,
7 3 11
(b)
1 7
,
2 4
(c)
4 9
,
9 8
(d)
2 2 7
, ,
7 5 4
Page 2
4.
Divide the following rational number.
(a)
3
2
by
9
7
(b)
1
25
by
5
6
(c)
11
2
by
37
5
(d)
91
13
by
7
2
Solution Div.
1.
2.
3.
4.
(a)
131
92
(b)
29
11
(c)
179
30
(d)
39
4
(a)
19
9
(b)
35
99
(c)
46
63
(a)
24
77
(b)
7
8
(c)
1
2
(d)
1
5
(a)
6
7
(b)
6
125
(c)
55
74
(d) 2
1.2.2 Commutative Property.
a p p a
 All rational numbers obey Commutative law under addition b  q  q  b
e.g.
5  1  15  7 8
  

7 3
21
21
1 5 7  15 8
 

3 7
21
21
5  1  1 5
  

7  3 3 7
 Two given rational numbers do not obey Commutative law under subtraction.
Page 3
a p p a
  
b q q b
e.g.
3 4 21  20
1
 

5 7
35
35
4 3 20  21 1
 

7 5
35
35
3 4 4 3
  
5 7 7 5
Two given rational numbers obey Commutative law under multiplication.
So,
a p p a
  
b q q b
e.g.
2 3 6 2



9 7 63 21
3 2 6 2
 

7 9 63 21
2 3 3 2



9 7
7 9
 Two given rational numbers do not obey Commutative law under division.
So,
a p p a
  
b q q b
e.g.
3  1  3
6
      
5  2 5
5
1 3 1 5 5
 
 
2 5 2 3 6
So,
3  1  1 3
  

5  2 2 5
1.2.3 Associativity Property.
 Irrespective of the order in which they are grouped, sum of three or more rational numbers
remain the same.
a c p a c p
      
b d q b d q
 1 3   2  1  3  2  
e.g.             
 2 7   4  2  7  4 
Page 4
 7  6   1  1  3  1  

         
 14   2  2  7  2  
13  1  1  6  7 
   

14  2  2  14 
13  7 1  1 
  
14
2  14 
6 7 1

14
14
3 3

7 7
So, Addition of Rational numbers is Associative
The difference of three of more rational numbers is not the same if the order in which they are
grouped is changed.
a c p a c p
       
 b d q b  d q
 2 3  5 2  3 5 
   
e.g.   
 7 14  2 7  14 2 
 4  3  5 2  3  35 

   

 14  2 7  14 
1 5 2  32 
  

14 2 7  14 
1  35 4  32

14
14
34 36

14
14
So, subtraction of Rational numbers is not Associative.
 Irrespective of the order in which they are grouped, multiplication of three or more rational
number remains the same
a c p a c p
      
 b d q b  d q
 7 7   1  7  4  1  
e.g.             
 3 5   11  3  5  11  
28  1  7  4 
    
15  11  3  55 
Page 5
28 28

165 165
So,Multiplication of Rational numbers is Associative
 The division of three or more rational numbers is not the same if the order in which they are
divided is changed.
a c  p a c p
      
b d q b d q
2 4 7 2 4 7 
e.g.         
 3 5  11 3  5 11 
2 5 7 2 4 7 
      
 3 4  11 3  5 11 
10 7 2 44

 
12 11 3 35
10 11 2 35
  
12 7 3 44
55 35

42 66
So,the division of rational number is not associative.
DIY.
1.
Rearrange and add.
(a)
3  2  4 9
  
11  7  11 7
(b)
3 6  7   10 
  

5 9  5   9 
Page 6
2.
Rearrange and Multiply
(a)
21 5 22 16
 

16 11 40 7
2 3 9 14
  
9 7 8 10
Solutions Diy.
(b)
1.
(a)
18
11
(b)
56
45
2.
(a)
3
4
(b)
3
20
1.4
Distributive property of multiplication over addition of rational numbers.
For rational numbers
p
a c
, and
q
b d
a c p a c a p


   
b  d q  b d b q
e.g. Consider 3 rational numbers
2 4 2
, &
3 7 5
2  4 2  2 4  2  2
   
  
3 7 5 3 7  3  5
2  20  14  8  4 

 

3  35  21  15 
2 34 8  4 


 
3 35 21  15 
68 40  28

105
105
68 68

105 105
1.4
Distributive property of multiplication over subtraction of rational numbers:For three rational numbers
p
a c
, &
q
b d
a c p a c a p


   
b  d q  b d b q
Page 7
2 4 3 2 4 2 3
      
3 7 2 3 7 3 2
2  8  21  8

1

3  14  21
2 13 8  21


3 14
21
13 13

21
21
1.5
Additive Inverse of a rational number:For any non-zero rational number
a
a
, there exists a unique rational number
Such that
b
b
a
a
a  a 
    0.
and
are additive inverse. Each one is called the additive inverse of the
b
b
b  b 
other.
e.g.
4
7
Additive inverse of
4
 4  4
will be =    
 7  7
7
4 4
 0
7 7
Important Note:a
a a
where b is not equal to zero, 0  
b
b b
Therefore, zero is the identity element for addition in rational numbers.
Additive Identity: For any rational number
e.g. 0 
1.6
7 7

9 9
Multiplicative inverse of a rational number for any non-zero rational number
a
, there exists
s
b
b
a b
a
b
such that   1 .
and
are multiplicative inverses.
a
b a
b
a
Each one is called the multiplicative inverse of the other.
a unique rational number
e.g.
Page 8
4
7
Multiplicative inverse of
4 7

7
4
4  7 
  1
7  4 
Important Note:a
a a
, where b is not equal to D, 1  Therefore, one
b
b b
is the identity element for multiplication in rational number.
Multiplicative identity – For any rational number
e.g.
DIY.
1.
2
2
 
81
81
Write the Additive inverse of the following
(a)
7
8
(c) 7
2.
1
2
(b)
3
41
(d)
2
7
Is 0.3 the multiplicative Inverse of 2
4
?
3
Solution Diy.
1.
2.
(a)
7
8
(b)
3
41
(c)
15
2
(d)
2
7
Yes
Page 9
1.7
Finding rational numbers between any two given rational number.
There are only definite number of numbers existing between 2 numbers when we talk about whole
numbers or Integers. But in the case of rational number this is not true.
There are infinite rational numbers present between any two given rational numbers.
3
1
and ?
5
4
Ans. First convert their denominators into a common denominator by taking LCM.
LCM = 20
e.g. Find out how many rational number exists between
1 5
5
 
4 5 20
3 4 12
 
5 4 20
So, now the rational numbers are
5 6 7 8 9 10 11 12
,
,
,
,
,
,
,
20 20 20 20 20 20 20 20
If, we again make the denominator common. For e.g. 200.
1 50 50


4 50 200
3 40 120


5 40 200
Now, the rational numbers between them are,
50 51 52
118 119 120
,
,
, ,
,
,
200 200 200
200 200 200
We can further take the common denominator 2000 and repeat the above steps.
If we go on increasing the divisions between two rational numbers, we can accommodate infinite
number of rational numbers.
DIY.
1.
Find any 5 rational numbers between
2.
Find any two rational numbers
Page 10
2 6
&
3 7
1
1
and
2
2
Solution Diy.
1.
4 3 2 1 1
,
,
, ,
and many more.
10 10 10 10 10
2.
15 16 17
, ,
21 21 21
Solved Examples.
1.
 3 2   4 10   1  
  
Simplify:- 
   
3  2 2
 7 3 5
Ans.
 3 2   4 10   1 5 
  

   
3  2 2
 7 3 5
2.

3  2 4  10) 1 5


73
53
22

2 8 5
7 3 4

2  12  8  28  5  21
84

24  224  105
84

353
84
Simplify:-
1 6
9
 11  13 13





10 15 20  10  15 20
We first need to rearrange and group rational numbers with common denominator for easier
calculation.
1  11  6 13
9
13





10  10  15 15 20 20

1 11 6 13 9 13





10 10 15 15 20 20

12 19 22


10 15 20

6 19 11


5 15 10
Page 11
3.

6  6  19  2  11 3
30

36  38  33
30

31
30
Rearrange the following and find the sum
2 2
5 11
 4

7 6
6
7
Ans. We will group in such a manner that common denominator are solved first and then the whole
expression.
2 2
5 11
 4

7 6
6
7
4.

2 11 2 5
   4
7
7 6 6

2  11 2  5

4
7
6

9  3 
 4
7  6 

9 1
 4
7 2

9  2  1 7  4  14
14

18  7  96
14

74  7
14

67
14
Find the Multiplicative inverse of:(a)
3
4
(b)
5
11
(c)
7
2
(d)
9
34
Page 12
Ans. 1.
3 a

4 b
Multiplicative Inverse =
2.
5
a

11 b
Multiplicative Inverse =
3.
b 2

a 7
9 a

34 b
Multiplicative Inverse =
5.
b 11

a
5
7 a

2 b
Multiplicative Inverse =
4.
b
4
4


a 3 3
b 34
34

or
a 9
9
Find the Additive Inverse of:81
4
(a)
(b)
91
99
6
7
(c)
(d)
11
22
Ans.
1.
81 a

91 b
Additive Inverse =
2.
4 a

99 b
Additive Inverse =
3.
a 4

b 99
6
a

11 b
Additive Inverse =
4.
a
 81  81
 

b
 91  91
a 6
6


b
11 11
7 a

22 b
Additive Inverse =
a
 7  7
   
b
 22  22
Page 13
6.
Apply distributive property in the below Expression.
 3 6   10 6 
   

 7 5  7 5
Ans.
7.
6
6
is common between the expression that are being subtracted, so we will take
out of the
5
5
brackets

6  3 10 
 

5 7
7 

6  13 


5  7 

78
35
Write any 10 rational numbers between 2 to 3.
200
300
and 3 as
100
100
The rational number between them are
Ans. Let us write 2 as
201 202 203
298 299
,
,
, ,
,
100 100 100
100 100
You can choose any 10 rational number from above rational numbers.
8.
Add the additive inverse of
Ans. Additive inverse of
7
12
with the multiplicative inverse of
?
9
13
7 7

9 9
12 13

13
12
Now, we need to Add both the terms
Multiplicative inverse of
7  13 


9  12 

7  4  13  3
36

28  39
36

Page 14
67
36
9.
Arrange the following rational numbers in ascending order
2 7 5 1 7
, , , ,
3 9 12 2 4
Ans. First you need to make the denominator same for all. So, we will find the LCM of 3, 9, 12, 2, 4
which is = 36. Now, make the denominator 36 for all:2  12 24

3  12
36
7  4 28

9  4 36
5  3 15

12  3 36
1 18 18

2  18 36
7  9 63

49
36
Now, we can simply arrange these in ascending order:63 24 15 18 28
,
,
,
,
36 36 36 36 36

10.
7 2 5 1 7
,
, , ,
4 3 12 2 9
Arrange the following rational numbers in descending order:-
1 7 2 4 9
,
, ,
,
5 25 3 15 5
Ans. First you need to make the denominator same for all by taking the LCM of 5, 2, 5, 3, 15, 5 that is
75. Make all the denominators as 75:-
1 15 15

5  15 75
7  3 21

25  3 75
2  25 50

3  25 75
4  5 20

15  5
75
9  15 135

5  15 75
Now arranging these in descending order:135 50 15 20 21
,
,
,
,
75 75 75 75 75

11.
9 2 1 4 7
, , ,
,
5 3 5 15 25
State the following
(a) Specify rational numbers that are equal to its additive inverse.
(b) Specify rational number that do not have a multiplicative inverse.
(c) Specify rational number that have same multiplicative inverse.
Page 15
Ans. (a) The rational number 0 is equal to its additive inverse and it is the only one.
(b) Rational number 0 when multiplied by any rational number wont give 1. So, 0 does not have a
multiplicative inverse.
(c) Reciprocal of 1 is 1 and -1 is also -1. So, 1 and -1 are the only two rational numbers having
same multiplicative inverse.
4
1
the multiplicative inverse of 1 ?
5
4
12.
Is
Ans.
4 a

5 b
Multiplicative inverse =
b 

a 4
1
5
1 
4
4
5 4
  1
4 5
No, 1
13.
Is
4
1
is not the multiplicative inverse of
.
5
4
9
1
is the additive inverse of 1 ?
8
8
Ans. Yes,
9
1
is the additive inverse of 1
8
8
1
9
1 
8
8
9  9 
 0
8  8 
14.
Is 0.13 is the multiplicative inverse of
Ans. Yes 0.13 is the multiplicative inverse of 7
0.13 
13
100
13 100

1
100 13
Page 16
7
9
100

13
13
7
9
13
9
?
13