The document explains rational numbers, which are numbers in the form of p/q where q ≠ 0. It details properties of rational numbers such as closure, commutativity, associativity, and identities for addition and multiplication, along with how to find rational numbers between two values on a number line. Additionally, it includes practice questions to solidify understanding of these concepts.

1. RATIONAL NUMBERS

2. Rational numbers are numbers in the form
of
𝑝
𝑞
where, 𝑞 ≠ 0.
For e.g.:
3
4
,
−9
8
,
−3
5
,
6
4
RATIONAL NUMBERS

3. DIFFERENCE BETWEEN FRACTION &
RATIONAL NUMBER
Rational Numbers
Fractions
RATIONAL NUMBERS = POSITIVE FRACTIONS + NEGETIVE FRACTIONS

4. NUMBER SYSTEM

5. PROPERTIES OF RATIONAL NUMBERS

6. CLOSURE PROPERTY
 Closure property means that if we divide/ add/ subtract/ multiply a rational
number the answer should always be a rational number if one of the case
becomes wrong then the whole law is wrong.

7. 2
3
+
4
5
=
10 + 12
15
=
22
15
Addition
Since by adding two rational numbers answer is also rational number .
therefore , addition of rational numbers is closed.

8. 2
3
−
4
5
=
10 − 12
15
=
−2
15
Subtraction
Since by subtracting two rational numbers answer is also rational number.
Therefore, subtraction of rational number is closed.

9. 2
4
×
−8
9
=
−4
9
Multiplication
Since by multiplying two rational numbers answer is also rational number.
Therefore, multiplication of rational number is closed.

10. 2
5
÷
3
10
=
2
5
×
10
3
=
4
3
7
1
÷
0
7
=
7
1
×
7
0
= 𝑛 ⋅ 𝑑
Division of rational number is not closed.

11. COMMUTATIVE PROPERTY
 Commutative property means, that by interchanging
the position of rational numbers the answer should
be same.
𝑝
𝑤
+
𝑞
𝑛
=
𝑞
𝑛
+
𝑝
𝑤

12. 1
2
+
6
2
=
7
2
6
2
+
1
2
−
7
2
Addition
Since by interchanging the position of rational numbers the answer is same.
Therefore, addition of rational numbers is commutative.

13. −4
9
−
4
9
=
−4
9
−
4
9
=
−8
9
4
9
−
−4
9
=
4
9
+
4
9
=
8
9
Subtraction
Since, by interchanging the position of rational
numbers the answer is not same. Therefore,
subtraction of rational numbers is not
commutative.

14. 2
4
×
3
−8
=
3
−16
3
−8
×
2
4
=
3
−16
Multiplication
Since by interchanging the position of rational numbers the answer is
same. Therefore, multiplication of rational numbers is commutative.

15. −2
9
÷
3
−11
=
−2
9
×
−11
3
=
22
27
3
−11
÷
−2
9
=
3
−11
𝑥
9
−2
=
27
22
Division
Since by interchanging the position of rational numbers the answer is not
same. Therefore, division of rational numbers is not commutative.

16. ASSOCIATIVE PROPERTY
 Associative property means that by interchanging the
position of three rational numbers the answer should
be same.
𝑝
𝑞
+
𝑛
𝑚
+
𝑥
𝑦
=
𝑥
𝑦
+
𝑛
𝑚
+
𝑝
𝑞

17. 9
2
+
6
2
+
1
2
=
9
2
+
7
2
=
16
2
1
2
+
6
2
+
9
2
=
15
2
+
1
2
=
16
2
Addition
Since by interchanging the position of rational numbers the answer is same.
Therefore, addition of rational numbers is associative.

18. −𝟏
𝟑
−
4
3
−
𝟐
𝟑
=
−𝟏
𝟑
−
𝟐
𝟑
=
−𝟑
𝟑
2
3
−
4
3
−
−1
3
=
𝟐
𝟑
−
4
3
+
1
3
=
2
3
−
5
3
=
−3
3
Subtraction
Since by interchanging the position of rational numbers the answer is same.
Therefore, subtraction of rational numbers is associative.

19. 2
4
×
4
9
×
12
4
=
2
4
×
4
3
=
2
3
12
4
×
4
9
×
2
4
=
12
4
×
2
9
=
4
6
=
2
3
Multiplication
Since by interchanging the position of rational numbers the answer is
same. Therefore, multiplication of rational numbers is associative.

20. 1
4
÷
6
9
÷
3
1
=
1
4
÷
6
9
×
1
3
=
1
4
÷
3
9
=
1
4
×
9
3
=
3
4
3
1
÷
6
9
÷
1
4
=
3
1
÷
6
9
×
4
1
=
3
1
÷
8
3
=
3
1
×
3
8
=
9
8
Division
Since by interchanging the position of rational numbers the answer is not
same. Therefore, division of rational numbers is not associative.

21. DISTRIBUTIVE PROPERTY
 This property is only of multiplication and is
taken over by addition or subtraction.

22. Multiplication over addition
3
4
×
5
8
+
3
4
×
11
8
=
3
4
5
8
+
11
8
=
3
4
×
16
8
=
3
2

23. Multiplication over subtraction
3
4
×
5
8
−
3
4
×
11
8
=
3
4
5
8
−
11
8
=
3
4
5
8
−
11
8
=
3
4
×
−6
8
=
−9
16

24. MULTIPLICATIVE IDENTITY
• Multiplicative identity means that if we will multiply any number
by a number the answer will always be same.
• 1 is the multiplicative identity for rational numbers. As it will not
change the identity of rational number or any integer.
• For example:
a.
1
2
× 1 =
1
2
b.
5
6
× 1 =
5
6

25. ADDITIVE IDENTITY
• Additive identity means that if we add any number by a number
the answer will always be same.
• Zero is called the identity for the addition of rational numbers. It
is the additive identity for integers and whole numbers as well.
• For example:
a. 0 + −9 = −9
b. 0 +
9
2
=
9
2

26. RECIPROCAL OF RATIONAL
NUMBERS
• Reciprocal of rational numbers means upside down.
• NUMBER -2 RECIPROCAL –½

27. MULTIPLICATIVE INVERSE
• Upside down
• Number -4 Multiplicative inverse = -¼
• Number -0 Multiplicative = 1/0 = n.d ( does not exist )

28. ADDITIVE INVERSE
• Opposite in sign
• Number =3 additive inverse = (-3 )
• Number = -5 additive inverse = 5

29. REPRESENTATION OF RATIONAL
NUMBERS ON NUMBER LINE
0
-1
-2
-4 -3
-5 5
4
3
2
1

32. HOW TO FIND RATIONAL NUMVERS
BETWEEN NUMBER LINE?
•Take out the equivalent fraction of the rational
numbers and write the rational numbers lying
between them.

33. a. −3 & − 4
−
36
12
& −
48
12
−37/12 , −38/12 , −39 /12 , −40/12 , −41
TO FIND RATIONAL NOS. IN BETWEEN RATIONAL NUMBERS.

34. 3
4
&
4
5
15
20
&
16
20
150
200
&
160
200
151/200 , 152/200 , 153 /200 , 154 /200 ,
155/200 , 156/200 , 157 /200

35. PRACTICE TIME

36. Q1. Is
0
7
a rational number?
YES NO

37. YOU GOT IT!

38. Additive identity of 56 is:

39. YOU GOT IT!

40. Reciprocal of 6/7 is:
-7/6
7/6
-6/7
6/-7

41. YOU GOT IT!

42. Which of the following is the Multiplicative
identity for rational numbers?
1
None of these
0
-1

43. YOU GOT IT!

44. Which of the following lies between 0 and -1?
0
4/3
-2/3
-3

45. YOU GOT IT!

46. Which of the following is additive inverse of 7/29?
29/7
7/29
-7/29
-29/7

47. YOU GOT IT!

48. Which of the following is multiplicative inverse of 15/31?
31/15
15/31
-15/31
-31/15

49. YOU GOT IT!

50. Which of the properties indicate give operation:
a + b + c = c + b + a
Associative Property
Closure Property
Commutative property
Distributive property

51. YOU GOT IT!

52. NEED PRACTICE