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