The document discusses rational numbers, defined as numbers in the form of p/q where q ≠ 0, covering properties such as closure, commutative, associative, and distributive properties for addition, subtraction, multiplication, and division of rational numbers. It explains multiplicative and additive identities, inverses, and how to represent rational numbers on a number line, including finding rational numbers between given values. Additionally, it includes practice questions to reinforce 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