2. OBJECTIVES:
• Students will be able to identify and differentiate different types
of numbers.
• Students will be able to know different properties of rational
numbers.
• Students will be able to represent the rational numbers on the
number line.
• Students will be able to find rational numbers between any two
rational numbers.
3. RATIONAL NUMBERS
• The numbers which can be expressed in the form of
𝑝
𝑞
, where 𝑝
and 𝑞 are integers and 𝑞 ≠ 0 are called as rational numbers.
• 𝑒. 𝑔.
2
3
, 5,
100
555
, 0, 2.4 etc.
4. TYPE OF NUMBERS:
• Natural Numbers: All counting numbers are called as natural
numbers.
e.g. 2, 35, 111, 1234 etc.
• Whole Numbers: Collection of all natural numbers with zero is
known as whole numbers.
e.g. 0, 2, 45, 234, 2947 etc.
• Integers: Collection of all natural numbers with its negative
numbers and zero is known as integers.
e.g. 56, -23, 0, -357, 6789 etc.
10. THE ROLE OF ZERO:
• Zero is called the identity for the addition of rational numbers.
It is the additive identity for integers and whole numbers as
well.
e.g. 0+2 = 2+0 = 2 𝑎 + 0 = 0 + 𝑎 = 𝑎
-5+0 = 0+(-5) = -5 where 𝑎 is any rational number
4
8
+ 0 = 0 +
4
8
=
4
8
THE ROLE OF ONE:
• 1 is the multiplicative identity for rational numbers, integers
and whole numbers.
e.g. 6 × 1 = 1 × 6 = 6 𝑎 × 1 = 1 × 𝑎 = 𝑎
−5
8
× 1 = 1 ×
−5
8
=
−5
8
where 𝑎 is any rational number
11. RECIPROCAL:
• A rational number
𝑐
𝑑
is called the reciprocal or multiplicative inverse of another
rational number
𝑎
𝑏
if
𝑎
𝑏
×
𝑐
𝑑
= 1 .
e.g.
7
9
×
9
7
= 1 and
−5
8
×
8
−5
= 1
We say that
9
7
is reciprocal of
7
9
and
8
−5
is reciprocal of
−5
8
.
NEGATIVE OF A NUMBER:
• We say that
−𝑎
𝑏
is the additive inverse of
𝑎
𝑏
and
𝑎
𝑏
is the additive inverse of
−𝑎
𝑏
as
𝑎
𝑏
+
−𝑎
𝑏
=
−𝑎
𝑏
+
𝑎
𝑏
= 0 .
2
5
+
−2
5
=
2 + −2
5
= 0
Here
−2
5
is the additive inverse of
2
5
.
14. • e.g. Find any ten rational number between
−5
8
and
7
10
.
Solution: We first convert
−5
8
and
7
10
to rational numbers with
same denominators.
−5
8
=
−5×5
8×5
=
−25
40
and
7
10
=
7×4
10×4
=
28
40
Thus we have
−24
40
,
−23
40
,
−22
40
,
−21
40
,
−20
40
, … . . ,
27
40
as the rational numbers
between
−5
8
and
7
10
.