Rational numbers
m
where m, n are integers and n
n
≠ 0 is called a rational number. The set of rational numbers is denoted by Q. Thus,
Definition: Any number that can be expressed in the form
Q=
m
; m, n Z and n
n
0 .
Every terminating decimal or a recurring decimal is a rational number.
Properties of rational numbers: Let Q be the set of rational numbers and a, b be any
members of Q, then the following results hold:
1. a, b
2. a, b
3. a, b
Q
Q
Q
=>
=>
=>
4. a, b
Q, b ≠ 0 =>
a + b Q.
a – b Q.
ab Q.
a
Q.
b
Example 1: Insert one rational number between
5
4
and and arrange in ascending order.
7
9
Solution: The L.C.M. of 7 and 9 is 63.
5
7
5 9
7 9
Since 28 < 45,
45
63
4
9
4
9
4 7
9 7
28
63
5
.
7
4
5 5
A rational number between and =
9
7 7
4 73 5
ascending order are ,
, .
9 126 7
4
9
5
7
2
28 45
63
2
73
, and number in
126
Example 2: Insert three rational numbers between 3 and 3.5.
Solution: A rational number between 3 and 3.5 =
A rational number between 3 and 3.25 =
3 3.5
2
3 3.25
2
A rational number between 3 and 3.125 =
6.5
= 3.25
2
6.25
= 3.125
2
3 3.125
2
6.125
= 3.0625.
2
We note that 3 < 3.0625 < 3.125 < 3.25 < 3.5, therefore, three rational numbers between 3
and 3.5 are 3.0625, 3.125, 3.25.
Practice problems:
Problem 1: Insert a rational number between
2
3
and , and arrange them in ascending
9
8
order.
Problem 2: Insert two rational number between
order.
1
1
and , and arrange them in ascending
3
4
Problem 3: Insert two rational numbers between
ascending order.
1
1
and - and arrange them in
3
2