Chapter 1: Number System Exercise – 1.1
Question: 1
Is 0 a rational number? Can you write it in the form P/Q, where P and Q are integers and Q ≠ 0?
Solution:
Yes, 0 is a rational number and it can be written in P ÷ Q form provided that Q?
0 is an integer and it can be written various forms, for example
0 ÷ 2, 0 ÷ 100, 0 ÷ 95 etc.
Question: 2
Find five rational numbers between 1 and 2
Solution:
Given that to find out 5 rational numbers between 1 and 2
Rational number lying between 1 and 2
= 3/2
= 1 < 3/2 < 2
Rational number lying between 1 and 3/2
= 5/4
= 1 < 5/4 < 3/2
Rational number lying between 1 and 5/4
Rational number lying between 3/2 and 2
= 9/8
= 1 < 9/8 < 5/4
Rational number lying between 3/2 and 2
= 7/4
= 3/2 < 7/4 < 2
Rational number lying between 7/4 and 2
= 15/8
= 7/4 < 15/8 < 2
Therefore, 1 < 9/8 < 5/4 < 3/2 < 7/4 < 15/8 < 2
Question: 3
Find out 6 rational numbers between 3 and 4
Solution:
Given that to find out 6 rational numbers between 3 and 4
We have,
3 × 7/7 = 21/7 and
4 × 6/6 = 28/7
We know 21 < 22 < 23 < 24 < 25 < 26 < 27 < 28
21/7 < 22/7 < 23/7 < 24/7 < 25/7 < 26/7 < 27/7 < 28/7
3 < 22/7 < 23/7 < 24/7 < 25/7 < 26/7 < 27/7 < 4
Therefore, 6 rational numbers between 3 and 4 are
22/7, 23/7, 24/7, 25/7, 26/7, 27/7
Similarly to find 5 rational numbers between 3 and 4, multiply 3 and 4 respectively with 6/6 and in order to find 8 rational numbers between 3 and 4 multiply 3 and 4 respectively with 8/8 and so on.
Question: 4
Find 5 rational numbers between 3/5 and 4/5
Solution:
Given to find out the 5 rational numbers between 3/5 and 4/5
To find 5 rational numbers between 3/5 and 4/5, 3/5 and 4/5 with 6/6
We have,
3/5 × 6/6 = 18/30
4/5 × 6/6 = 24/30
We know 18 < 19 < 20 < 21 < 22 < 23 < 24
18/30 < 19/30 < 20/30 < 21/30 < 22/30 < 23/30 < 24/30
3/5 < 1930 < 20/30 < 21/30 < 22/30 < 23/30 < 4/5
Therefore, 5 rational numbers between 3/5 and 4/5 are 19/30, 20/30, 21/30, 22/30, 23/30
Question: 5
Answer whether the following statements are true or false? Give reasons in support of your answer.
(i) Every whole number is a rational number
(ii) Every integer is a rational number
(iii) Every rational number is an integer
(iv) Every natural number is a whole number
(v) Every integer is a whole number
(vi) Every rational number is a whole number
Solution:
(i) True. As whole numbers include and they can be represented
For example - 0/10, 1/1, 2/1, 3/1..... And so on.
(ii) True. As we know 1, 2, 3, 4 and so on, are integers and they can be represented in the form of 1/1, 2/1, 3/1, 4/1.
(iii) False. Numbers such as 3/2, 1/2, 3/5, 4/5 are rational numbers but they are not integers.
(iv) True. Whole numbers include all of the natural numbers.
(v) False. As we know whole numbers are a part of integers.
(vi) False. Integers include -1, -2, -3 and so on ....which is not whole number