The document outlines the various types of numbers, including natural numbers, whole numbers, integers, and rational numbers, emphasizing their properties and behaviors. It explains operations such as addition, subtraction, multiplication, and division in relation to rational numbers and highlights properties like closure, commutativity, and associativity. Additionally, it covers concepts like identity elements, inverses, and how to identify rational numbers within a specified range.

1. *
Numbers:- The systems of numbers expanded
with new needs.
*Natural numbers :- (Counting numbers)
1 , 2 ,3 ,4 , ………
*Whole numbers :-
0 , 1 , 2 , 3, 4 , …….

2. *
*Integers :- Positive & negative numbers & Zero
…., -2 , -1 , 0 , 1 , 2 , 3 , ……….
*Fractions :-
2
3
,
1
2
,
1
3
, etc.

3. *
*Need for rational numbers:
Do we have enough numbers to solve all simple
equations?
Lets us take 4𝑥 + 9 = 0
𝑥 =
−9
4
We need the number
−9
4
,which is neither a fraction
nor an integer, for solving the given equation.

4. *
*A number which can be written in the form
𝑝
𝑞
,
where p and q are integers and q ≠ 0 is called a
rational number.
Eg. 2 , 0 , -3 ,
2
3
,
−5
7
*Note:- Every natural number ,whole number
integer and fraction is also a rational number.

5. *Properties of Rational
numbers
How do rational numbers behave when they are
added , subtracted ,multiplied or divided with each
other?

6. *1) Closure Property:
Operation Example Rational
number?
Remarks
Addition 2 +
1
2
=
5
2
Yes
Rational numbers
are closed under
addition
Subtraction
5
8
-
3
4
=
5−6
8
=
−1
8
yes
R.Nos are closed
under subtraction
Multiplicatio
n
−4
5
X
5
8
=
−1
2
yes
R.Nos are closed
under
multiplication
Division
2
7
÷
5
3
=
2
7
X
3
5
=
6
35
3
5
÷ 0 is not defined
Not always
Note:- For any rational number a , a ÷ 0 is not defined .Rational numbers are not
closed under division.

7. *2) Commutativity:
Operation Example Remarks
Addition 1 +
1
2
=
1
2
+ 1 =
3
2
Addition is commutative for
Rational numbers
Subtraction
6
3
-
4
3
≠
4
3
-
6
3
Subtraction is not
commutative for Rational
numbers
multiplicatio
n -
7
3
x
6
5
=
6
5
x -
7
3
Multiplication is
commutative for Rational
numbers
Division -
5
4
÷
1
4
≠
1
4
÷ -
5
4
Division is not commutative
for
Rational numbers

8. *3) Associativity:
*Addition and Multiplication are associative for
rational numbers.
*For any three rational numbers a, b and c,
a + (b + c) = ( a + b) + c
Also, a x(b x c) = ( a x b) x c
Note: Subtraction and Division are not
associative for Rational numbers.

9. 5) The Role of
ONE:
*For any rational number ‘a’,
a + 0 = 0 + a = a
Zero is called the identity for the addition of Rational
numbers.
● For any rational number ‘a’,
a x 1 = 1 x a = a
One is the multiplicative identity for Rational numbers.

10. *
Additive Inverse and Multiplicative
Inverse
6) Negative of a number (Additive Inverse):
- 𝑎
𝑏
is the additive inverse of
𝑎
𝑏
since,
𝑎
𝑏
+ (-
𝑎
𝑏
) = 0
7) Reciprocal ( Multiplicative Inverse):
Eg:
2
3
x
3
2
= 1 Also, -
5
4
x -
4
5
= 1
𝑎
𝑏
and
𝑐
𝑑
are the reciprocals of each other if
𝑎
𝑏
x
𝑐
𝑑
= 1

11. *8) Distributivity of
Mutiplication over
Addition and
Subtraction:
For all rational numbers a , b and c
a( b + c) = ab + ac
a( b – c) = ab - ac

12. *Find the no./nos. in each case:
1)The rational number that does not have a reciprocal
Ans) Zero
2) The rational nos. that are equal to their reciprocals
Ans) 1 and -1
3) The rational number that is equal to its negative
Ans) Zero
4) The reciprocal of -5
Ans) -
1
5
5) The negative of -
1
4
Ans)
1
4

13. *Fill in the blanks:
1) The reciprocal of
1
𝑥
where x ≠ 0 is ---x------.
2) The product of two R.nos is always a –R.NOS----.
3) The reciprocal of a positive R.no is ----Possitive-
-----.

14. *Rational numbers
between two rational
numbers
Eg: Finding rational numbers between -2 and 0
-2 = -
2
1
= -
20
10
0 =
0
1
=
0
10
Ans) Rational nos. between -2 and 0 are:
-
19
10
, -
18
10
, ------------ , -
1
10

15. THANKYOU