Question 1: Find two rational numbers between:

(i) (ii) (iii) (iv) and (v) and

Answer:

(i) The rational number between is

Now a rational number between is

Thus the required rational numbers are

(ii) The rational number between is

Now a rational number between is

Thus the required rational numbers are

(iii) The rational number between is

Now a rational number between is

Thus the required rational numbers are

(iv) The rational number between and is

Now a rational number between and

Thus the required rational numbers are

(v) The rational number between and is

Now a rational number between and

Thus the required rational numbers are

Question 2: Insert rational numbers between and

Answer:

We know that

Therefore, dividing each of these by would give us:

Which gives us the numbers between and .

Question 3: Insert numbers between and

Answer:

Insert numbers between and is equivalent to inserting numbers between and . We just multiplied the numerator and denominator by .

We know that:

Dividing by 110, we get the numbers

… … …

Question 4: Express the following rational numbers as decimals:

(i) (ii) (iii) (iv) (v) (vi) (viii) (ix) (x) (xi)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(viii)

(ix)

(x)

(xi)

Question 5: Express each of the following decimals in the form :

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

Answer:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

Question 6: Express each of the following decimals in the form :

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

Answer:

(i)

Let

Subtracting

(ii)

Let

Subtracting

(iii)

Let

Subtracting

(iv)

Let

Subtracting

(v)

Let

Subtracting

(vi)

Let

Subtracting

(vii)

Let

Subtracting

(viii)

Let

Subtracting

Question 7: Without performing long division, state if the following rational numbers are terminating decimals or non-terminating decimals:

(i) (ii) (iii) (iv) (v) (vi)

Answer:

(i) .

Since is of the form , hence the decimal expansion of is terminating.

(ii)

Since is not of the form , hence the decimal expansion of is a non-terminating repeating.

(iii)

Since is of the form , hence the decimal expansion of is terminating.

(iv)

Since is not of the form , hence the decimal expansion of is a non-terminating repeating.

(v)

Since is not of the form , hence the decimal expansion of is a non-terminating repeating.

(vi)

Since is of the form , hence the decimal expansion of is terminating.

Question 8: Write down the decimal expansion of the rational numbers by writing their denominators in the form of given m and n are non-negative integers.

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Answer:

(i) ( terminates are places decimals)

(ii) ( terminates are places decimals)

(iii) ( terminates are places decimals)

(iv) ( terminates are places decimals)

(v) ( terminates are places decimals)

(vi) ( terminates are places decimals)

(vii) ( terminates are places decimals)

(viii) ( terminates are places decimals)

(ix) ( terminates are places decimals)

Question 9: Show that the following numbers are irrational:

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi)

Answer:

(i)

Let us assume on the contrary that is a rational number. Then there exist positive integers and such that:

… … … … … (i)

for some integer

… … … … … (ii)

From (i) and (ii), we observe that and have at least as a common factor. But this contradicts the fact that and are co-primes. This means that our assumption is not correct.

Hence is an irrational number.

(ii)

Let us assume on the contrary that is a rational number. Then there exist positive integers and such that:

… … … … … (i)

for some integer

… … … … … (ii)

From (i) and (ii), we observe that and have at least as a common factor. But this contradicts the fact that and are co-primes. This means that our assumption is not correct.

Hence is an irrational number.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(iii)

Let us assume on the contrary that is a rational number. Then there exist positive integers and such that:

… … … … … (i)

for some integer

… … … … … (ii)

From (i) and (ii), we observe that and have at least as a common factor. But this contradicts the fact that and are co-primes. This means that our assumption is not correct.

Hence is an irrational number.

We know that the sum of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(iv)

From Problem 9(ii), we have proved that is an irrational number.

Therefore is an irrational number.

(v)

… … … … … (i)

for some integer

… … … … … (ii)

Hence is an irrational number.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Hence is an irrational number.

(vi)

… … … … … (i)

for some integer

… … … … … (ii)

Hence is an irrational number.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Hence is an irrational number.

(vii)

In the above problem 9(iii) we proved that is irrational.

We know that the sum of a non-zero rational number and an irrational number is an irrational number. Hence is an irrational number.

(viii)

In the above problem 9(iii) we proved that is irrational.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Hence is an irrational number.

(ix)

… … … … … (i)

for some integer

… … … … … (ii)

Hence is an irrational number.

We know that the sum of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(x)

In the problem 9(iii) we proved that is an irrational number.

We know that the sum of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(xi)

In the problem 9(iii) we proved that is an irrational number.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(xii)

In the problem 9(ix) we proved that is an irrational number.

We know that the product of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.

(xiii)

In the problem 9(ix) we proved that is an irrational number.

(xiv)

In the problem 9(ii) we proved that is an irrational number.

(xv)

Since and are rational number which implies means that is a rational number.

This contradicts the fact that is an irrational number. Hence our assumption is wrong. Therefore is irrational number.

Question 10: What can you say about the prime factorization of the denominators of the following rational number:

(i) (ii) (iii)

Answer:

(i)

This number is a rational terminating number. Therefore the denominator can be represented in the form of where are non-negative integers.

(ii)

This number is a rational non-terminating repeating number. Therefore the denominator can not be represented in the form of where are non-negative integers.

(iii)

This number is a rational terminating number. Therefore the denominator can be represented in the form of where are non-negative integers.

Question 11: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

(i) (ii) (iii) (iv) (vi)

Answer:

(i) . Therefore rational number.

(ii) . Therefore irrational number because we know, is an irrational number.

(iii) . Therefore rational number.

(iv) . Therefore, irrational number.

(v) . Therefore rational number.

(v) . Therefore rational number.

Question 12: In the following equations, find which variables etc. represent rational or irrational numbers:

(i) (ii) (iii) (iv) (v) (vi) (vii)

Answer:

(i)

. Therefore is an irrational number.

(ii)

. Therefore is a rational number.

(iii)

. Therefore is a rational number.

(iv)

. Therefore is an irrational number.

(v)

. Therefore is an irrational number.

(vi)

. Therefore is an irrational number.

(vii)

. Therefore is an irrational number.

Question 13: Find two irrational numbers between:

(i) (ii) ) (iii) (iv)

Answer:

(i)

The two numbers could be and

(ii) )

The two numbers could be and

(iii)

We have and

Therefore the number could be and

(iv)

The numbers could be and