1. SL. NO SUB-TOPIC SLIDE NO
1 INTRODUCTION 3
2 ADDITION OF RATIONAL NUMBERS
AND ITS PROPERTIES
5
3 SUBTRACTION OF RATIONAL
NUMBERS AND ITS PROPERTIES
8
4 MULTIPLICATION OF RATIONAL
NUMBERS AND ITS PROPERTIES
12
5 DIVISION OF RATIONAL NUMBERS
AND ITS PROPERTIES
18
6 RATIONAL NUMBERS BETWEEN TWO
RATIONAL NUMBERS
24
2. INTRODUCTION
One day two children of Real number, Whole number and Rational
number have a talk with each other;
Whole number: Do you know rational ,different operations like
Addition,Subtraction ,MULTIPLICATION and DIVISION can act on me
and they also satisfies different properties.
Rational number: But brother I do not have any knowledge that how
these operations act on me. Letβs ask our teacher about thisβ¦
Teacher- I will definitely clear your doubts but
before that answer my questionsβ¦
οΆ What is the additive identity for whole numbers ?
οΆ Does whole number obeys commutative law under Subtraction ?
οΆ What is the multiplicative inverse of 5 ?
οΆ Define distributive property of Multiplication over Addition ?
οΆ Say two whole numbers between 5 and 12 ?
οΆ Tell whether division obeys closure property or not ?
3. The numbers of the form
π
π
,Where π πππ π πππ ππππππππ πππ π β π πππ πππππ ππ ππππππππ ππππππ
Rational numbers include natural numbers, whole numbers, integer sand all positive and negative
fractions.
Example:
π
π
,
βπ
π
, π
π
π
,β¦ etc
Rational Numbers
As that of Whole numbers for rational numbers,also there are four operations they
are as following. Letβs discuss one after anotherβ¦
οΆ ADDITION
οΆ SUBTRACTION
οΆ MULTIPLICATION
οΆ DIVISION
4. ADDITION ON
RATIONAL NUMBERS
CASE-1
WHEN RATIONAL NUMBER
HAVING SAME DENOMINATOR
Σ Add numerators of the
rational numbers and then
divide by the common
denominator.
Ex- Add
π
π
and
π
π
Solution:
π
π
+
π
π
=
π+π
π
=
π
π
CASE-2
WHEN RATIONAL NUMBER DO
NOT HAVE SAME DENOMINATOR
Σ Take LCM of
denominators and then add
the rational numbers.
Example-Add
π
π
and
π
π
Solution:
π
π
+
π
π
LCM of denominators 4
and 3 is 12.
π
π
=
π Γ π
π Γ π
=
π
ππ
π
π
=
π Γ π
π Γ π
=
π
ππ
so
π
ππ
+
π
ππ
=
π+π
ππ
=
ππ
ππ
5. PROPERTIES OF ADDITION OF RATIONAL NUMBERS
οΆ Closure property:
Sum of any two rational number is always a rational number i.e if x and y are two rational numbers then x+y
will also became a rational number.
EXAMPLE :
1
2
+
3
4
=
5
4
Here
1
2
and
3
4
are both Rational number And sum
5
4
is also a rational number.
Hence Rational numbers are closed under Addition.
οΆ Commutative property:
The sum of two rational numbers does not depend on the order in which they are added.i.e π₯ + π¦ = π¦ +
LHS=
π
π
+
π
π
+
π
π
=
π+π
π
+
π
π
=
π
π
+
π
π
=
ππ+ππ
ππ
=
ππ
ππ
RHS=
1
2
+
2
3
+
5
4
=
1
2
+
8+15
12
=
1
2
+
23
12
=
6+23
12
=
29
12
6. οΆ Additive identity property:
When βoβ is added to any rational number the sum of the rational number is becomes
itself, If x is a rational number then π + π = π + π = π .
EXAMPLE: let π₯ =
5
4
LHS=
5
4
+ 0 =
5
4
RHS=0 +
5
4
=
5
4
As LHS = RHS so this property valids.
Zero is called the identity element in Addition .
οΆ Additive inverse property:
The negative of a rational number is itβs additive inverse.
The sum of a rational number and its additive inverse is always equal to Zero.
If x is a rational number then (-x) is the additive inverse of it and
π₯ + βπ₯ = 0
Let a rational number be
5
4
πππ ππ‘π πππππ‘ππ£π πππ£πππ π ππ
β5
4
Then
5
4
+
β5
4
=
5+ β5)
4
=
0
4
= 0
Negative of a rational number is called its additive inverse.
Try theseβ
(a) Simplify
β3
10
+
12
β10
+
14
10
(b) For x =
1
5
and y=
3
7
, verify that β(x+y) = (-x) + (-y)
(c) For x =
2
3
and y=
β5
6
and z=
7
9
, verify that (x+y)+ z = x + (y+z)
7. SUBTRACTION ON
RATIONAL NUMBERS
CASE-1
WHEN RATIONAL NUMBER
HAVING SAME DENOMINATOR
Σ Subtract numerators of
the rational numbers and
then divide by the common
denominator.
Ex- Add
π
π
and
π
π
Solution:
π
π
β
π
π
=
πβπ
π
=
βπ
π
CASE-2
WHEN RATIONAL NUMBER DO
NOT HAVE SAME DENOMINATOR
Σ Take LCM of
denominators and then
subtract the rational
numbers.
Example-Add
π
π
and
βπ
π
Solution:
π
π
β
βπ
π
)
LCM of denominators 4
and 3 is 12.
π
π
=
π Γ π
π Γ π
=
π
ππ
π
π
=
π Γ π
π Γ π
=
π
ππ
so
π
ππ
β
βπ
ππ
) =
π+π
ππ
=
ππ
ππ
8. PROPERTIES OF SUBTRACTION OF RATIONAL NUMBERS
οΆ EXAMPLE:
Difference of any two rational number is always a rational number i.e if x and y are two rational numbers then
x - y will also became a rational number.
VERIFICATION :
1
2
β
3
4
=
β1
4
Here
1
2
and
3
4
are both Rational numbers And their Difference
β1
4
is also
a rational number.
Hence Rational numbers are closed under Subtraction.
οΆ PROPERTY-1:
The difference of two rational numbers does not remain same if their order is changed i.e π₯ β π¦) β
LHS=
π
π
β
π
π
-
π
π
=
πβπ
π
β
π
π
=
βπ
π
+
π
π
=
βπ+ππ
ππ
=
ππ
ππ
RHS=
1
2
β
2
3
β
5
4
=
1
2
β
8β15
12
=
1
2
+
β7
12
=
6β7
12
=
β1
12
9. οΆ EXAMPLE ;
For all rational number π we have π β π = π πππ π β π = βπ
Show that,Ifx is a rational number then π β π β π β π .
VERIFICATION: let π₯ =
5
4
LHS=
5
4
β 0 =
5
4
But RHS=0 β
5
4
=
β5
4
As LHS β RHS so this property does not valid.
Identity element in Subtraction does not exist .
οΆ EXAMPLE:
Since identity of subtraction does not exist so inverse for Subtraction does
not arise.
Try theseeβ
(a) Simplify
β3
10
β
12
β10
β
14
10
(b) For x =
2
3
and y=
β5
6
and z=
7
9
, verify that (x-y)-z = x β(y-z)
10. MULTIPLICATION OF RATIONAL NUMBERS
Example: Find the product of
4
5
and
β10
3
Solution:
4
5
Γ
β10
3
=
4Γ β10)
5Γ3
=
β8
3
If
π
π
and
π
π
are two rational numbers
then
π
π
Γ
π
π
=
πΓπ
πΓπ
π·πππ πππ ππ πππππππππππ
π·πππ πππ ππ π«πππππππππππ
Solve the followingsβ¦
(a) Multiply
βπ
ππ
and
βπ
π
(b) Multiply 3
βπ
π
and 1
βπ
π
11. PROPERTIES OF MULTIPLICATION OF RATIONAL NUMBERS
οΆ CLOSURE PROPERTY:
Product of any two rational number is always a rational number
EXAMPLE:
1
2
Γ
3
4
=
3
8
Here
1
2
and
3
4
are both Rational number And product
3
8
are rational number
Hence Rational numbers are closed under multiplication.
οΆ COMMUTATIVE PROPERTY:
The product of two rational numbers remain same even if we change their
order . If π₯ and π¦ are two rational number then π₯ Γ π¦ = π¦ Γ π₯
Example: Verify that product of two rational number
β2
7
and
1
4
remains same
even if the order is changed .
LHS=
β2
7
Γ
1
4
=
β1
14
Hence LHS=RHS
RHS=
1
4
Γ
β2
7
=
β1
14
Commutative property holds TRUE in multiplication of rational number.
12. οΆ Associative Property:
The product of three rational numbers remains same even after
changing the groups.
If x , y, z are three rational numbers then π₯ Γ π¦) Γ π§ = π₯ Γ π¦ Γ π§
Example: Verify that product of three rational numbers
β3
4
,
5
7
and
β2
11
remains same
even after changing the groupings.
LHS=
β3
4
Γ
5
7
Γ
β2
11
=
β15
28
Γ
β2
11
=
15
154
RHS=
β3
4
Γ
5
7
Γ
β2
11
=
β3
4
Γ
β10
77
=
15
154
LHS=RHS
13. οΆ MULTIPLICATIVE IDENTITY:
When 1 is multiplied to any rational number then the product of the rational
number is equal to itself
If x is a rational number then π₯ Γ 1 = 1 Γ π₯ = π₯
Example: let π₯ =
5
4
LHS=
5
4
Γ 1 =
5
4
RHS=1 Γ
5
4
=
5
4
Hence LHS = RHS
1 is the identity element under the multiplication
οΆ PRODUCT OF A RATIONAL NUMBER AND ZERO IS ALWAYS
ZERO
If π₯ ππ πππ¦ πππ‘πππππ ππ’ππππ π‘βππ π₯ Γ 0 = 0 Γ π₯ = 0
EXAMPLE: The product of
β4
7
πππ 0
β4
7
Γ 0 =
β4Γ0
7Γ1
=
0
7
= 0
Again 0 Γ
β4
7
=
0Γβ4
1Γ7
=
0
7
= 0
15. οΆ Reciprocal of a rational number(Multiplicative Inverse)
Reciprocal of rational number means interchanging of numerator and
denominator in other words Rational number obtained after inverting the
given rational number is called Reciprocal of rational number
Example : The Reciprocal of
9
4
is
4
9
Note:
(1)Zero has no Reciprocal
(ii)Reciprocal of 1 is 1
(iii) if x is any non zero rational number,then its
reciprocal is denoted by π₯β1
which is equal to
1
π₯
16. DIVISION OF RATIONAL NUMBERS
EXAMPLE: Divide
7
3
ππ¦
5
3
Solution :
7
3
Γ·
5
3
=
7
3
Γ
5
3
β1
π₯ Γ· π¦ = π₯ Γ π¦β1
=
7
3
Γ
3
5
=
7
5
Dividing one rational number by
another except by zero, is the same
as the multiplication of the first by
the reciprocal of the second,
i.exΓ·y = xΓy-1
Divide followingsβ¦
(a) -10 by
1
5
(b)
1
13
by -2
17. PROPORTIES OF DIVISION OF RATIONAL NUMBER
Property: 1
Divisionof a rational number by another rational number except zero is always
rational number.
Example: (i)
5
3
β
4
3
=
5
3
Γ
3
β4
=
β5
4
(ii) 0 Γ·
β4
5
= 0
(iii)
β4
5
Γ· 0 =
β4
5
Γ
1
0
=
β4
0
π€βππβ ππ πππ‘ π πππ‘πππππ ππ’ππππ)
Property-2
When a nonzero rational number divided by itself we get always 1
Example :
3
4
Γ·
3
4
=
3
4
Γ
4
3
= 1
Property-3
When non zero rational number divided by 1 the quotient is
the same rational number
Example
β4
5
Γ· 1 =
β4
5
Γ
1
1
=
β4
5
18. PROPERTY-4
The division of two rational number does not remain same if
the order of the number are changed.
If π₯ πππ π¦ are two rational number then π₯ Γ· π¦ β π¦ Γ· π₯
Example:Verify that if π₯ =
2
3
and π¦ =
β4
5
then π₯ Γ· π¦ β π¦ Γ· π₯
β’ LHS=π₯ Γ· π¦
=
3
2
Γ·
β4
5
=
3
2
Γ
5
β4
=
β15
8
β’ RHS=π¦ Γ· π₯
=
β4
5
Γ·
3
2
=
β4
5
Γ
2
3
=
β8
15
LHSβ RHS
Hππππ πππ’ππ’π‘ππ‘πππ£π πππππππ‘π¦ ππππ πππ‘ βπππ π‘ππ’π πππ πππ£ππ πππ
19. PROPERTY-5
The division of three rational numbers does not remains same even after
changing the groups.
If x , y, z are three rational numbers then π₯ Γ· π¦) Γ· π§ β π₯ Γ· π¦ Γ· π§
Example:Verify that if π₯ =
2
3
, π¦ =
β4
9
,π§ =
5
6
then π₯ Γ· π¦) Γ· π§ β π₯ Γ· π¦ Γ· π§
β’ LHS= π₯ Γ· π¦) Γ· π§
=
2
3
Γ·
β4
9
Γ·
5
6
=
β3
2
Γ·
5
6
=
β3
2
Γ
6
5
=
β9
5
β’ RHS=π₯ Γ· π¦ Γ· π§)
=
2
3
Γ·
β4
9
Γ·
5
6
=
2
3
Γ·
β8
15
=
2
3
Γ
15
β8
=
β5
4
LHSβ RHS
Associative property does not holds true for division
22. RATIONALS BETWEEN TWO RATIONAL NUMBERS
Example: Find two rational number between
1
2
πππ
β1
2
Solution: Step-1 Find a rational number between
1
2
πππ
β1
2
1
2
Γ
1
2
+
β1
2
=
1
2
Γ 0 = 0
Step-2 Find a rational number between
1
2
πππ 0
1
2
Γ
1
2
+ 0 =
1
2
Γ
1
2
=
1
4
Hence 0,
1
4
are two rational number between
1
2
πππ
β1
2
If π πππ π πππ πππ ππππππππ πππππππ , ππππ
π+π
π
ππ π
ππππππππ ππππππ πππππππ π πππ π.
23. οΆIntroduction of Rational number
οΆOperation of rational number
οΆProperties of addition of rational number
οΆProperties of subtraction of rational number
οΆProperties of Multiplication of rational
number
οΆProperties of division of rational number
οΆInsert rational numbers between two given
rational number
KEY POINTS