# KC Sinha Mathematics Solution Class 9 Chapter 1 Real numbers exercise 1.1

Page 1.24

### Exercise 1.1

Type 1

#### Question 1

State whether the following statements are true or false ?

(i) Every whole number is a natural number

Sol :  False, since 0 (zero) is a whole number but not a natural number.

(ii) Every integer is a rational number

Sol :  True, since every integer m may be written in the form so it is a rational number.

(iii) Every rational number is an integer

Sol :  False, since is a rational number but it is not an integer

(iv) If any rational number is an integer , then q=±1

Sol : False, since (integer) but q=2

#### Question 2

Write the following integers in the form of rational number

(i) 9

Sol :

(ii) -13

Sol :

(iii) 20

Sol :

#### Question 3

(i) Is a rational number, if p=0 ?

Sol :

Yes , because by definition of rational number which can be expressed in term form of p/q when q is non zero integer  and p is a integer .

(ii) Is a rational number , if q=0 ?

Sol :

No , According to definition of rational number which can be expressed in term form of p/q when q is non zero integer  and p is a integer .

#### Question 4

Fill up the blanks with the word terminating, non—terminating, repeating.

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.]

(i) On changing to a decimal , it will be __ decimal

Sol:

Terminating

(ii) On changing to a decimal, it will be __ decimal

Sol:

Terminating

(iii) On changing to a decimal, it will be __ decimal

Sol:

Non-Terminating

(iv) If denominator of a rational number has prime factors 2 and 5 only , the can be written in __ decimal form

Sol:

Terminating

#### Question 5

Without writing following rational numbers in decimal forms , state which will have terminating decimal expansion ?

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.]

(i)

Sol:

Here q=11 and it is not in the form of 2n5m, it is not a terminating decimal.

(ii)

Sol:

Here q=9

Can be written as 32 and it is not in the form of 2n5m, it is not a terminating decimal.

(iii)

Sol:

Here q=16

Can be written as 2and it is in the form of 2n5m, it is a terminating decimal.

(iv)

Sol :

Here q=30

Can be written as 2×3×5

Since 3 is also there and it is not in the form of 2n5m, it is not a terminating decimal.

#### Question 6

Without writing following rational numbers in decimal forms , state which will have non-terminating decimal expansion ?

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.]

(i)

Sol :

Here q=9

Can be written as 32 and it is not in the form of 2n5m, it is not a terminating decimal.

(ii)

Sol :

Here q=8

Can be written as 23 and it is in the form of 2n5m, it is a terminating decimal.

(iii)

Sol :

Here q=25

Can be written as 52 and it is in the form of 2n5m, it is a terminating decimal.

(iv)

Sol :

Here q=20

Can be written as 22×5 and it is in the form of 2n5m, it is a terminating decimal.

#### Question 7

State which of the following rational numbers represent terminating decimal expansion ?

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.]

(i)

Sol :

Denominator = 8 = 2×2×2

Can be written as terminating expansion [is in form 2n5m]

(ii)

Sol :

Denominator = 7

Can not be written as terminating expansion [not in form 2n5m]

(iii)

Sol :

Denominator = 40 = 2×2×2×5

Can be written as terminating expansion [is in form 2n5m]

(iv)

Sol :

Denominator = 130 = 2×5×13

Can not be written as terminating expansion because it is not in the form of 2n5m [as 13 is present in prime factorization]

(v)

Sol :

Denominator = 35 = 7×5

Can not be written as terminating expansion because it is not in the form of 2n5m [as 7 is present in prime factorization]

(vi)

Sol :

Denominator = 128 = 2×2×2×2×2×2×2

Can be written as terminating expansion [ is in the form of 2n5m]

(vii)

Sol :

Denominator = 138 = 2×3×23

Can not be written as terminating expansion because it is not in the form of 2n5m [as 3 and 23 is present in prime factorization]

(viii)

Sol :

Denominator = 50 = 2×5×5

Can be written as terminating expansion because it is in the form of 2n5m

#### Question 8

State which of the following rational numbers represent non-terminating decimal expansion ?

[Note:Let x = p/q be a rational number, such that the prime factorization of q is of the form 2n 5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.]

(i)

Sol :

Denominator = 7

Can not be written as terminating expansion [not in form 2n5m]

(ii)

Sol :

Denominator = 10 = 2×5

Can be written as terminating expansion [is in form 2n5m]

(iii)

Sol :

Denominator = 18 = 2×3×3

Can not be written as terminating expansion because it is not in the form of 2n5m [as 3 is present in prime factorization]

(iv)

Sol :

Denominator = 250 =2×5×5×5

Can be written as terminating expansion [is in form 2n5m]

(v)

Sol :

Denominator = 21 = 3×7

Can not be written as terminating expansion because it is not in the form of 2n5m [as 3 and 7 is present in prime factorization]

(vi)

Sol :

Denominator = 30 = 2×3×5

Can not be written as terminating expansion because it is not in the form of 2n5m [as 3 is present in prime factorization]

(vii)

Sol :

Denominator = 121 = 11×11

Can not be written as terminating expansion because it is not in the form of 2n5m [as 11 is present in prime factorization]

(viii)

Sol :

Denominator = 60 = 2×2×3×5

Can not be written as terminating expansion because it is not in the form of 2n5m [as 3 is present in prime factorization]

#### Question 9

Write the following in decimal form and state, what kind of decimal expansion each has ?

(i)

Sol :

Its denominator 100 = 22×52=(2×5)2

Since denominator has prime factors 2 and 5 only , so its decimal expansion is terminating

Now ,

(ii)

Sol :

Its denominator 3

Since its denomination is not in the form of 2m×5n , therefore its decimal expansion is non-terminating recuring

(iii)

Its denominator 11

Since its denomination is not in the form of 2m×5n , therefore its decimal expansion is non-terminating recuring

(iv)

Its denominator 8 = 23

Since denominator has prime factors 2 only , so its decimal expansion is terminating

=0.875

(v)

Its denominator 10 = 2×5

Since denominator has prime factors 2 and 5 only , so its decimal expansion is terminating

=0.3

(vi)

Its denominator 3

Since its denomination is not in the form of 2m×5n , therefore its decimal expansion is non-terminating recuring

#### Question 10

Write the following rational numbers in the decimal form:

(i)

Sol : 0.3125

(ii)

Sol : 3.3

(iii)

Sol : 0.27

(iv)

Sol :

(v)

Sol :

(vi)

Sol : 0.654

(vii)

Sol : 0.83

(viii)

Sol : 1.2692307

Type 2

#### Question 11

Show that the following numbers can be represented in the form where p and q are integers and q≠0:

(i)

Sol :

Let

Then , x=1.2727..   (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 27 , in which number of digits is 2 .

Multiply both sides of (i) by 10or 100

100x=100×(1.2727..)

100x=127.2727.. (ii)

Subtracting (i) from (ii) , we get

100x-x=127.2727.. – 1.2727..

99x=126

(ii) 0.3333…

Sol :

Let x=0.3.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 3 , in which number of digits is 1 .

Multiply both sides of (i) by 10or 10

10x=10×(0.3..)

10x=3.3.. (ii)

Subtracting (i) from (ii) , we get

10x-x=3.3.. – 0.3..

9x=3

(iii)

Sol :

Let x=0.6.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 6 , in which number of digits is 1 .

Multiply both sides of (i) by 10or 10

10x=10×(0.6..)

10x=6.6.. (ii)

Subtracting (i) from (ii) , we get

10x-x=6.6.. – 0.6..

9x=6

(iv) 0.2353532…

Sol :

Let x=0.235353.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 235353 , in which number of digits is 6 .

Multiply both sides of (i) by 10or 1000000

1000000x=1000000×(0.235353..)

1000000x=235353.235353.. (ii)

Subtracting (i) from (ii) , we get

1000000x-x=235353.235353.. – 0.235353..

99999x=235353

(v)

Sol :

Let x=3.142678.. (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 142678 , in which number of digits is 6 .

Multiply both sides of (i) by 10or 1000000

1000000x=1000000×(3.142678..)

1000000x=3142678.142678.. (ii)

Subtracting (i) from (ii) , we get

1000000x-x=3142678.142678.. – 3.142678..

99999x=3142675

Type 3

#### Question 12

Write the following in the form of , where p and q are integers and q≠0 :

(i) 0.25

Sol :

(ii) 0.54

Sol :

(iii)

Sol :

Let x=6.4646..   (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 46 , in which number of digits is 2 .

Multiply both sides of (i) by 10or 100

100x=100×(6.4646..)

100x=646.4646.. (ii)

Subtracting (i) from (ii) , we get

100x-x=646.4646.. – 6.4646..

99x=145

(iv)

Sol :

Let x=0.033..  (i)

Here, number of digits af‌ter decimal which are not in the repeating block, m=1 .

Now multiply both sides of (i) by 10m=101=10 we get

10x=10×0.03..

10x=0.3.. (ii)

Again, number of digits in repeating block, n=1

Multiplying both sides of (ii) by 10n=101=10 we get

100x=10×0.3..

100x=3.3.. (iii)

Subtracting (ii) from (iii) , we get

100x-10x=3.3.. – 0.3..

90x=3

(v)

Sol :

Let x=4.6732732..  (i)

Here, number of digits af‌ter decimal which are not in the repeating block, m=1 .

Now multiply both sides of (i) by 10m=101=10 we get

10x=10×4.6732732..

10x=46.732732.. (ii)

Again, number of digits in repeating block, n=3

Multiplying both sides of (ii) by 103=103=1000 we get

10000x=1000×46.732732

10000x=46732.732.. (iii)

Subtracting (ii) from (iii) , we get

10000x-10x=46732.732.. – 46.732732..

9990x=46686

(vi)

Sol :

Let x=4.2727..  (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 27 , in which number of digits is 2 .

Multiply both sides of (i) by 10or 100

100x=100×(4.2727..)

100x=427.27.. (ii)

Subtracting (i) from (ii) , we get

100x-x=427.27.. – 4.27..

99x=423

(vii)

Sol :

Let x=3.7..  (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 7 , in which number of digits is 1 .

Multiply both sides of (i) by 101 or 10

10x=10×(3.7..)

10x=37.7.. (ii)

Subtracting (i) from (ii) , we get

10x-x=37.7.. – 3.7..

9x=34

(viii)

Sol :

Let x=18.48..  (i)

In x , number of digits after decimal which are not in the repeating block of digits is zero as there is no such digit. Here, repeating block is 48 , in which number of digits is 2 .

Multiply both sides of (i) by 10or 100

100x=100×(18.48..)

100x=1848.48.. (ii)

Subtracting (i) from (ii) , we get

100x-x=1848.48.. – 18.48..

99x=1830

Type 4

#### Question 13

Find f‌ive rational numbers between 1 and 2.

Sol :

Here a=1 and b=2 and n=5

Now,

The required rational numbers will be

, , , ,

,   ,  , ,

,   ,  , ,

,   ,  , ,

,   ,  , ,

#### Question 14

Find five rational numbers between and

[Hint: ] Now write five rational numbers between ]

Sol :

Let and

Five rational numbers are

#### Question 15

Write one rational number between

Sol :

Here , and n=1

The required rational numbers will be

#### Question 16

Find three rational numbers between

Sol :

Here   , and n=3

Now,

The required rational numbers will be

, ,

, ,

,   ,

,   ,

,   ,

,   ,

,   ,

#### Question 17

Find three rational numbers between 0 and 0.2

Sol :

Average of 0 and 0.2

Average of 0.1 and 0

Average of 0.05 and 0.2

The required rational numbers are 0.1 , 0.05 , 0.12

Alternate method

Here a=0 , b=0.2 and n=3

Now,

The required rational numbers will be

, ,

, ,

, ,

⇒0.05 , 0.1 , 0.15

#### Question 18

Find two rational numbers between

Sol :

Here , and n=2

Now,

The required rational numbers will be

,

,

,

,

,

,

#### Question 19

Find one rational numbers between

Sol :

Here ,

The required rational numbers will be

Average

#### Question 20

Find two rational numbers between

Sol :

Here , and n=2

Now,

The required rational numbers will be

,

,

,

,

,

,

,

#### Question 21

Find two rational numbers between

Sol :

Here ,

Average

and another is

Average

The required rational numbers will be

,

Alternate Method

Here , and n=2

Now,

The required rational numbers will be

,

,

,

,

,

,

,

,

#### Question 22

Find two rational numbers between 0 and 0.1

Sol :

Here a=0 and b=0.1

The required rational numbers will be

Average of 0 and 0.1

Average of 0.05 and 0.1

Two rational numbers are

⇒0.05 , 0.07

#### Question 23

How many rational numbers can be written between 2.5 and 2.6 ? Write ten of these numbers

Sol :

⇒Infinitely many rational numbers can be written between 2.5 and 2.6

⇒ Ten rational numbers between 2.5 and 2.6 are 2.51 , 2.52 , 2.53 , 2.54 , 2.55 , 2.56 , 2.57  , 2.58 , 2.59 , 2.511 .

Type 5

#### Question 24

If x and y are rational numbers then, show that the following are also rational numbers :

Properties of rational numbers

⇒Sum of two rational numbers is rational number ..(i)

⇒Difference of two rational numbers is rational number ..(ii)

⇒Product of two rational numbers is rational number..(iii)

⇒Division of two rational numbers is rational number..(iv)

(i) x2-y2

Sol :

x2-y

can be written as (x+b)(x-b)

x+b is a rational number by (i)

x-b is a rational number by (ii)

So , x2-y2 is also a rational number

(ii) x-y

Sol :

x-y is a rational number by (ii)

(iii) , where y≠0

Sol :

, where y≠0 is a rational number by (iv)

(iv) x+y

Sol :

x+y is rational number by (i)

#### Question 25

If a is a rational number , then prove that an will be a rational number , where n is a rational number greater than 1 .

Sol :

Rational Number is a number that can be expressed in the form p/q, where q not equal to zero.

We know that product of two rational number is always a rational number.
Hence if a is a rational number then

a2 = a × a is a rational number.

a3 = a2× a is a rational number,

a4 = a3×a is a rational number,

……

……

an = an-1×a is a rational number.

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