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)
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. Hence is an irrational number.
(vi)
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. 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)
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.
(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.
We know that the sum 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.
We know that the sum 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.
We know that the product of a non-zero rational number and an irrational number is an irrational number. Therefore 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.
(xiv)
In the problem 9(ii) 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.
We know that the sum of a non-zero rational number and an irrational number is an irrational number. Therefore is an irrational number.
(xv)
Let us assume on the contrary that is a rational number. Then there exist positive integers
and
such that:
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