Show that is 4√2 an irrational number.
Answer:
Let us assume that 4√2 is a rational number
where, b≠0 and a, b are integers
∵ a, b are integers ∴ 4b is also integer
is rational which cannot be possible
∵
which is an irrational number
∵ it is contradicting our assumption
∴ the assumption was wrong
Hence, 4√2 is an irrational number
Question 2.
Prove that 3 + √5 is an irrational number.
Answer:
Let us assume that 3 + √5 is a rational number
where, b≠0 and a, b are integers
∵ a, b are integers ∴ a – 3b is also integer
is rational which cannot be possible
∵
which is an irrational number
∵ it is contradicting our assumption ∴ the assumption was wrong
Hence, 3 + √5 is an irrational number
Question 3.
Represent the numbers √5 and √10 on a number line.
Answer:
By Pythagoras theorem,
(√5)2 = 22 + 12
⇒ (√5)2 = 4 + 1
First mark 0 and 2 on the number line. Then, draw a perpendicular of 1 unit from 2. And Join the top of perpendicular and 0. This line would be equal to √5. Now measure the line with compass and marc an arc on the number line with the same measurement. This point is √5.
Also,
By Pythagoras theorem,
(√10)2 = 32 + 12
⇒ (√10)2 = 9 + 1
First mark 0 and 3 on the number line. Then, draw a perpendicular of 1 unit from 3. And Join the top of perpendicular and 0. This line would be equal to √10. Now measure the line with compass and marc an arc on the number line with the same measurement. This point is √10.
Question 4.
Write any three rational numbers between the two numbers given below.
0.3 and -0.5
Answer:
0.3 and -0.5
To find a rational number x between two rational numbers 
and 
, we use
Therefore, to find rational number x (let) between
and 
Now if we find a rational number between 
and 
it will also be between 0.3 and -0.5 since lies between 0.3 and -0.5.
Therefore, to find rational number y (let) between and 
Now if we find a rational number between and 
it will also be between 0.3 and -0.5 since lies between 0.3 and -0.5.
Therefore, to find rational number z (let) between and 
Hence the numbers are -0.2, -0.1 and 0.1
Question 5.
Write any three rational numbers between the two numbers given below.
-2.3 and -2.33
Answer:
-2.3 and -2.33
To find a rational number x between two rational numbers and , we use
Therefore, to find rational number x (let) between 
and 
⇒ x = -2.315
Now if we find a rational number between 
and 
it will also be between -2.3 and -2.33 since -2.315 lies between -2.3 and -2.33
Therefore, to find rational number y (let) between 
and 
⇒ y = -2.3075
Now if we find a rational number between and it will also be between -2.3 and -2.33 since -2.315 lies between -2.3 and -2.33
Therefore, to find rational number z (let) between and 
⇒ z = -2.3225
Hence the numbers are -2.3225, -2.3075 and -2.315
Question 6.
Write any three rational numbers between the two numbers given below.
5.2 and 5.3
Answer:
5.2 and 5.3
To find a rational number x between two rational numbers and , we use
Therefore, to find rational number x (let) between 
and 
⇒ x = 5.25
Now if we find a rational number between 
and 
it will also be between 5.2 and 5.3 since 5.25 lies between 5.2 and 5.3
Therefore, to find rational number y (let) between and
⇒ y = 5.225
Now if we find a rational number between and 
it will also be between 5.2 and 5.3 since 5.25 lies between 5.2 and 5.3
Therefore, to find rational number z (let) between and
⇒ z = 5.275
Hence the numbers are 5.225, 5.25 and 5.275
Question 7.
Write any three rational numbers between the two numbers given below.
-4.5 and 4.6
Answer:
-4.5 and 4.6
To find a rational number x between two rational numbers and , we use
Therefore, to find rational number x (let) between 
and 
⇒ x = 0.05
Now if we find a rational number between 
and 
it will also be between -4.5 and 4.6 since 0.05 lies between -4.5 and 4.6
Therefore, to find rational number y (let) between and 
⇒ y = -2.225
Now if we find a rational number between 
and it will also be between -4.5 and 4.6 since 0.05 lies between -4.5 and 4.6
Therefore, to find rational number z (let) between and
⇒ z = 2.325
Hence the numbers are -2.225, 0.05and 2.325
