NCERT Solutions for Class 9
Maths
Chapter 1 – Number System
Exercise 1.1
1. Do you think zero is a rational number? If it is, then can it be expressed in
p
the form of q , where p and q are integers and q 0 ? Describe it.
Ans: Remember that, according to the definition of rational number, a rational
p
number is a number that can be expressed in the form of , where p and q are
q
integers and q 0 .
0 0 0 0 0
Now, notice that zero can be represented as , , , , .....
1 2 3 4 5
0 0 0 0
Also, it can be expressed as , , , .....
−1 −2 −3 −4
p
Therefore, it is concluded from here that 0 can be expressed in the form of ,
q
where p and q are integers.
Hence, zero must be a rational number.
2. Write any 6 rational numbers between 3 and 4 .
Ans: It is known that there are infinitely many rational numbers between any two
numbers. Since we need to find 6 rational numbers between 3 and 4 , so multiply
and divide the numbers by 7 (or by any number greater than 6 )
Then it gives,
7 21
3 = 3 =
7 7
7 28
4 = 4 =
7 7
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Hence, 6 rational numbers found between 3 and 4 are , , , , , .
7 7 7 7 7 7
3 4
3. Write any five rational numbers between and .
5 5
Ans: It is known that there are infinitely many rational numbers between any two
numbers.
3 4
Since here we need to find five rational numbers between and , so multiply
5 5
and divide by 6 (or by any number greater than 5 ).
Then it gives,
3 3 6 18
= = ,
5 5 6 30
4 4 6 24
= = .
5 5 6 30
3 4 19 20 21 22 23
Hence, 5 rational numbers found between and are , , , , .
5 5 30 30 30 30 30
4. Verify all the statements given below and state whether they are true or
false. Show proper reasons for your answers.
i. Statement: Every natural number is a whole number.
Ans: Write the whole numbers and natural numbers in a separate manner.
It is known that the whole number series is 0,1,2,3,4,5..... . and
the natural number series is 1,2,3,4,5..... .
Therefore, it is concluded that all the natural numbers lie in the whole number
series as represented in the diagram given below.
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,Thus, it is concluded that every natural number is a whole number.
Hence, the given statement is true.
ii. Statement: Every integer is a whole number.
Ans: Write the integers and whole numbers in a separate manner.
It is known that integers are those rational numbers that can be expressed in the
p
form of , where q = 1 .
q
Now, the series of integers is like 0, 1, 2, 3, 4,... .
But the whole numbers are 0,1,2,3,4,... .
Therefore, it is seen that all the whole numbers lie within the integer numbers,
but the negative integers are not included in the whole number series.
Thus, it can be concluded from here that every integer is not a whole number.
Hence, the given statement is false.
iii. Statement: Every rational number is a whole number.
Ans: Write the rational numbers and whole numbers in a separate manner.
It is known that rational numbers are the numbers that can be expressed in the
p
form , where q 0 and the whole numbers are represented as 0,1,2,3,4,5,...
q
p
Now, notice that every whole number can be expressed in the form of as
q
0 1 2 3 4 5
, , , , , ,…
1 1 1 1 1 1
Thus, every whole number is a rational number, but all the rational numbers are
1 1 1 1
not whole numbers. For example, , , , ,... are not whole numbers.
2 3 4 5
Therefore, it is concluded from here that every rational number is not a whole
number.
Hence, the given statement is false.
Class IX Maths www.vedantu.com 3
, Exercise 1.2
1. Verify all the statements given below and state whether they are true or
false. Give proper reasons for your answers.
i. Every irrational number is a real number.
Ans: Write the irrational numbers and the real numbers in a separate manner.
● The irrational numbers are the numbers that cannot be represented in the form
p
, where p and q are integers and q 0.
q
For example, 2,3, .011011011... are all irrational numbers.
● The real number is the collection of both the rational numbers and irrational
numbers.
1
For example, 0, , 2 , ,... are all real numbers.
2
Thus, it is concluded that every irrational number is a real number.
Hence, the given statement is true.
ii. Every point on the number line is of the form m , where m is a natural
number.
Ans: Consider points on a number line to represent negative as well as positive
numbers.
Observe that, positive numbers on the number line can be expressed as
1, 1.1, 1.2, 1.3,... , but any negative number on the number line cannot be
expressed as −1, −1.1, −1.2, −1.3,... , because these are not real numbers.
Therefore, it is concluded from here that every number point on the number line
is not of the form = m , where m is a natural number.
Hence, the given statement is false.
iii. Every real number is an irrational number.
Ans: Write the irrational numbers and the real numbers in a separate manner.
Class IX Maths www.vedantu.com 4
Maths
Chapter 1 – Number System
Exercise 1.1
1. Do you think zero is a rational number? If it is, then can it be expressed in
p
the form of q , where p and q are integers and q 0 ? Describe it.
Ans: Remember that, according to the definition of rational number, a rational
p
number is a number that can be expressed in the form of , where p and q are
q
integers and q 0 .
0 0 0 0 0
Now, notice that zero can be represented as , , , , .....
1 2 3 4 5
0 0 0 0
Also, it can be expressed as , , , .....
−1 −2 −3 −4
p
Therefore, it is concluded from here that 0 can be expressed in the form of ,
q
where p and q are integers.
Hence, zero must be a rational number.
2. Write any 6 rational numbers between 3 and 4 .
Ans: It is known that there are infinitely many rational numbers between any two
numbers. Since we need to find 6 rational numbers between 3 and 4 , so multiply
and divide the numbers by 7 (or by any number greater than 6 )
Then it gives,
7 21
3 = 3 =
7 7
7 28
4 = 4 =
7 7
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Hence, 6 rational numbers found between 3 and 4 are , , , , , .
7 7 7 7 7 7
3 4
3. Write any five rational numbers between and .
5 5
Ans: It is known that there are infinitely many rational numbers between any two
numbers.
3 4
Since here we need to find five rational numbers between and , so multiply
5 5
and divide by 6 (or by any number greater than 5 ).
Then it gives,
3 3 6 18
= = ,
5 5 6 30
4 4 6 24
= = .
5 5 6 30
3 4 19 20 21 22 23
Hence, 5 rational numbers found between and are , , , , .
5 5 30 30 30 30 30
4. Verify all the statements given below and state whether they are true or
false. Show proper reasons for your answers.
i. Statement: Every natural number is a whole number.
Ans: Write the whole numbers and natural numbers in a separate manner.
It is known that the whole number series is 0,1,2,3,4,5..... . and
the natural number series is 1,2,3,4,5..... .
Therefore, it is concluded that all the natural numbers lie in the whole number
series as represented in the diagram given below.
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,Thus, it is concluded that every natural number is a whole number.
Hence, the given statement is true.
ii. Statement: Every integer is a whole number.
Ans: Write the integers and whole numbers in a separate manner.
It is known that integers are those rational numbers that can be expressed in the
p
form of , where q = 1 .
q
Now, the series of integers is like 0, 1, 2, 3, 4,... .
But the whole numbers are 0,1,2,3,4,... .
Therefore, it is seen that all the whole numbers lie within the integer numbers,
but the negative integers are not included in the whole number series.
Thus, it can be concluded from here that every integer is not a whole number.
Hence, the given statement is false.
iii. Statement: Every rational number is a whole number.
Ans: Write the rational numbers and whole numbers in a separate manner.
It is known that rational numbers are the numbers that can be expressed in the
p
form , where q 0 and the whole numbers are represented as 0,1,2,3,4,5,...
q
p
Now, notice that every whole number can be expressed in the form of as
q
0 1 2 3 4 5
, , , , , ,…
1 1 1 1 1 1
Thus, every whole number is a rational number, but all the rational numbers are
1 1 1 1
not whole numbers. For example, , , , ,... are not whole numbers.
2 3 4 5
Therefore, it is concluded from here that every rational number is not a whole
number.
Hence, the given statement is false.
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, Exercise 1.2
1. Verify all the statements given below and state whether they are true or
false. Give proper reasons for your answers.
i. Every irrational number is a real number.
Ans: Write the irrational numbers and the real numbers in a separate manner.
● The irrational numbers are the numbers that cannot be represented in the form
p
, where p and q are integers and q 0.
q
For example, 2,3, .011011011... are all irrational numbers.
● The real number is the collection of both the rational numbers and irrational
numbers.
1
For example, 0, , 2 , ,... are all real numbers.
2
Thus, it is concluded that every irrational number is a real number.
Hence, the given statement is true.
ii. Every point on the number line is of the form m , where m is a natural
number.
Ans: Consider points on a number line to represent negative as well as positive
numbers.
Observe that, positive numbers on the number line can be expressed as
1, 1.1, 1.2, 1.3,... , but any negative number on the number line cannot be
expressed as −1, −1.1, −1.2, −1.3,... , because these are not real numbers.
Therefore, it is concluded from here that every number point on the number line
is not of the form = m , where m is a natural number.
Hence, the given statement is false.
iii. Every real number is an irrational number.
Ans: Write the irrational numbers and the real numbers in a separate manner.
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