# Concise Rational and Irrational Numbers ICSE Class-9 Selina Solutions

**Concise Rational and Irrational** Numbers ICSE Class-9 Mathematics Selina Solutions Chapter-1 . We provide step by step Solutions of Exercise / lesson-1 **Rational and Irrational** Numbers for ICSE Class-9** Concise** Selina Mathematics by RK Bansal. Our Solutions contain all type Questions with Exe-1 A, Exe-1 B and Exe-1 C , to develop skill and confidence. Visit official Website **CISCE** for detail information about ICSE Board Class-9.

**Concise Rational and Irrational** Numbers ICSE Class-9 Mathematics Selina Solutions Chapter-1

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Note:- Before viewing Solution of Chapter -1 **Rational and Irrational Numbers Class-9** of Concise Selina Solutions. Read the Chapter Carefully then solve all example given in Exercise-1.1, Exercise-1.2, Exercise-1.3. For more practice on The **Rational and Irrational Numbers** Visit ML Aggarwal **Rational and Irrational** Numbers Chapter Also.

**Exercise – 1(A) Rational and Irrational Numbers for ICSE Class-9 Maths Concise Selina Solutions **

#### Question 1

Is zero a rational number ? Can it be written in the form , where p and q are integers and q≠0 ?

**Answer**

Yes, zero is a rational number.

As it can be written in the form of , where p and q are integers and q≠0 ?

⇒ 0 =

#### Question 2

**Are the following statement true or false ? Give reason for your answer.**

- Every whole number is a natural number.
- Every whole number is a rational number.
- Every integer is a rational number.
- Every rational number is a whole number.

**Answer**

i) False, zero is a whole number but not a natural number.

ii) True, Every whole can be written in the formof, where p and q are integers and q≠0.

iii) True, Every integer can be written in the form of , where p and q are integers and q≠0.

iv) False.

**Example : ** is a rational number, but not a whole number.

#### Question 3

Arrange in ascending order of their magnitudes.

Also, find the difference between the largest and smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

**Answer**

#### Question 4

Arrange in descending order of their magnitudes.

Also, find the sum of the lowest and largest of these fractions. Express the result obtained as a decimal fraction correct to two decimal places.

**Answer**

#### Question 5.1

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

**Answer**

Given number is 716

Since 16 = 2 x 2 x 2 x 2 = 2^{4} = 2^{4} x 5^{0}

I.e. 16 can be expressed as 2^{m} x 5^{n}

∴ is convertible into the terminating decimal.

#### Question 5.2

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

**Answer**

Given number is

Since 125 = 5 x 5 x 5 = 5^{3 = }2^{0} x 5^{3}

I.e. 125 can be expressed as 2^{m} x 5^{n}

∴ is convertible into the terminating decimal.

#### Question 5.3

Without doing any actual division, find which of the following rational numbers have terminating decimal representation :

**Answer**

Given number is

Since 14 = 2 x 7 = 2^{1} x 7^{1}

I.e. 14 cannot be expressed as 2^{m} x 5^{n}

∴ is not convertible into the terminating decimal.

#### Question 5.4

**Answer**

Given number is

Since, 45 = 3 x 3 x 5 = 3^{2} x 5^{1}

I.e. 45 cannot be expressed as 2^{m} x 5^{n}

∴ is not convertible into the terminating decimal.

#### Question 5.5

**Answer**

Given number is

Since 50 = 2 x 5 x 5 = 2^{1} x 5^{2}

I.e. 50 can be expressed as 2^{m} x 5^{n}

∴ is convertible into the terminating decimal.

#### Question 5.6

**Answer**

Given number is

Since 40 = 2 x 2 x 2 x 5 = 2^{3} x 5^{1}

I.e. 40 can be expressed as 2^{m} x 5^{n}

∴ is convertible into the terminating decimal.

#### Question 5.7

**Answer**

Given number is

Since 75 = 3 x 5 x 5 = 3^{1} x 5^{2}

i.e. 75 cannot be expressed as 2^{m} x 5^{n}

∴ is not convertible into the terminating decimal.

#### Question 5.8

**Answer**

Given number is

Since 250 = 2 x 5 x 5 x 5 = 2^{1} x 5^{3}

i.e. 250 can be expressed as 2^{m} x 5^{n.1}

∴ is convertible into the terminating decimal.

End of Exe-1 A **Concise Rational and Irrational Numbers** for ICSE Class-9

** ICSE Class-9 Mathematics Concise Selina Solutions Exercise – 1(B)**** Rational and Irrational Numbers**

#### Question 1.1

**State, whether the following numbers is rational or not : **( 2 + √2 )^{2}

**Answer**

( 2 + √2 )^{2 } = 2^{2} + 2 ( 2 ) ( √2 ) + ( √2 )^{2}

= 4 + 4√2 + 2

= 6 + 4√2

**Irrational number.**

#### Question 1.2

**State, whether the following numbers is rational or not : **( 3 – √3 )^{2}

**Answer**

( 3 – √3 )^{2 } = 3^{2} + 2 ( 3 ) ( √3 ) + ( √3 )^{2}

= 9 – 6√3 + 3

= 12 – 6√3 = 6 ( 2 – √3 )

**Irrational number.**

#### Question 1.3

**State, whether the following numbers is rational or not : **( 5 + √5 )( 5 – √5 )

**Answer**

( 5 + √5 )( 5 – √5 ) = ( 5 )^{2} – ( √5 )^{2}

= 25 – 5 = 20

**Rational Number**

#### Question 1.4

**State, whether the following numbers is rational or not : **( √3 – √2 )^{2}

**Answer**

( √3 – √2 )^{2 } = ( √3 )^{2} – 2 ( √3 )( √2 ) + ( √2 )^{2}

= 3 – 2√6 + 2

= 5 – 2√6

**Irrational Number**

#### Question 1.5

**State, whether the following numbers is rational or not :
**

**Answer**

#### Question 1.6

**State, whether the following number is rational or not :**

**Answer**

#### Question 2.1

**Find the square of :**

**Answer**

#### Question 2.2

**Find the square of : **√3 + √2

**Answer**

( √3 + √2 )^{2} = ( √3 )^{2} + 2( √3 )( √2 ) + ( √2 )^{2}

= 3 + 2√6 + 2

= 5 + 2√6

#### Question 2.3

**Find the square of : **√5 – 2

**Answer**

( √5 – 2 )^{2} = ( √5 )^{2} – 2( √5 )( 2 ) + ( 2 )^{2}

= 5 – 4√5 + 4

= 9 – 4√5

#### Question 2.4

**Find the square of :** 3 + 2√5

**Answer**

( 3 + 2√5 )^{2} = 3^{2} + 2( 3 )( 2√5) + ( 2√5 )^{2}

= 9 + 12√5 + 20

= 29 + 12√5

#### Question 3.1

**State, in each case, whether true or false :****
**√2 + √3 = √5

##### Options

- True
- False

**Answer**

False

#### Question 3.2

**State, in each case, whether true or false :****
**2√4 + 2 = 6

##### Options

- True
- False

**Answer**

2√4 + 2 = 2 x 2 + 2 = 4 + 2 = 6 which is **True.**

#### Question 3.3

**State, in each case, whether true or false :**** **

3√7 – 2√7 = √7

##### Options

- True
- False

**Answer**

3√7 – 2√7 = √7 –** True.**

#### Question 3.4

**State, in each case, whether true or false :****
**27 ia an irrational number.

##### Options

- True
- False

**Answer**

False Because 27=0.285714¯ which is recurring and non-terminating and hence it is rational.

#### Question 3.5

**State, in each case, whether true or false :****
** is a rational number.

##### Options

- True
- False

**Answer**

True, because 0.45¯ which is recurring and non-terminating

#### Question 3.6

**State, in each case, whether true or false :**** **

All rational numbers are real numbers.

##### Options

- True
- False

**Answer**

True

#### Question 3.7

**state, in each case, whether true or false :****
**All real numbers are rational numbers.

##### Options

- True
- False

**Answer**

False

#### Question 3.8

**State, in each case, whether true or false :**** **

Some real numbers are rational numbers.

##### Options

- True
- False

**Answer**

True

#### Question 4.1

**Answer**

Given Universal set is

#### Question 4.2

**Answer**

#### Question 4.3

Given universal set =

`{ -6, -5 3/4, -sqrt4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, sqrt8, 3.01, π, 8.47 }`

**From the given set, find :** set of integers

**Answer**

Given Universal set is

`{ – 6, -5 3/4, -sqrt4, -3/5, -3/8, 0, 4/5, 1, 1 2/3, sqrt8, 3.01, π, 8.47 }`

We need to find the set of integers.

Set of integers consists of zero, the natural numbers and their additive inverses.

The set of integers is Z.

Z = `{ …… -3, -2, -1, 0, 1, 2, 3,……. }`

Here the set of integers is U ∩ Z = `{ -6, √4, 0, 1 }`

#### Question 4.4

**Answer**

We need to find the set of non-negative integers.

Set of non-negative integers consists of zero and the natural numbers.

The set of non-negative integers is Z^{+} and Z^{+ }= { 0, 1, 2, 3,…… }

Here the set of integers is U ∩ Z^{+ }= {0, 1}

#### Question 5

Use method of contradiction to show that √3 and √5 are irrational numbers.

**Answer**

#### Question 6.1

**Prove that the following number is irrational: **√3 + √2

**Answer**

3 + √2

Let √3 + √2 be a rational number.

⇒ √3 + √2 = x

Squaring on both the sides, we get

( √3 + √2 )^{2} = x^{2
}⇒ 3 + 2 + 2 x √3 x √2 = x^{2}

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √3 + √2 is a rational number is wrong.

∴ √3 + √2 is an irrational number.

#### Question 6.2

**Prove that the following number is irrational: **3 – √2

**Answer**

3 – √2

Let 3 – √2 be a rational number.

⇒ 3 – √2 = x

Squaring on both the sides, we get

( 3 – √2 )^{2} = x^{2
}⇒ 9 + 2 – 2 x 3 x √2 = x^{2}

^{⇒ } 11 – x^{2} = 6√2

⇒ 11 – x^{2} is an irrational number.

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that 3 – √2 is a rational number is wrong.

∴ 3 – √2 is an irrational number.

#### Question 6.3

**Prove that the following number is irrational: **√5 – 2

**Answer**

√5 – 2

Let √5 – 2 be a rational number.

⇒ √5 – 2 = x

Squaring on both the sides, we get

(√5-2)2=x2

⇒ 5 + 4 – 2 x 2 x √5 = x^{2}

⇒ 9 – x^{2 }= 4√5

⇒ √5 = 9-x24

Here, x is a rational number.

⇒ 9 – x^{2} is an irrational number.

⇒ x^{2} is an irrational number.

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that √5 – 2 is a rational number is wrong.

∴ √5 – 2 is an irrational number.

Question 7

Write a pair of irrational numbers whose sum is irrational.

**Answer**

√3 + 5 and √5 – 3 are irrational numbers whose sum is irrational.

( √3 + 5 ) + ( √5 – 3 ) = √3 + √5 + 5 – 3 = √3 + √5 + 2 which is irrational.

#### Question 8

Write a pair of irrational numbers whose sum is rational.

**Answer**

√3 + 5 and 4 – √3 are two irrational numbers whose sum is rational.

( √3 + 5 ) + ( 4 – √3 ) = √3 + 5+ 4 – √3 = 9

#### Question 9

Write a pair of irrational numbers whose difference is irrational.

**Answer**

√3 + 2 and √2 – 3 are two irrational numbers whose difference is irrational.

( √3 + 2 ) – ( √2 – 3 ) = √3 – √2 + 2 + 3 = √3 – √2 + 5 which is irrational.

#### Question 10

Write a pair of irrational numbers whose difference is rational.

**Answer**

√5 – 3 and √5 + 3 are irrational numbers whose difference is rational.

( √5 – 3 ) – ( √5 + 3 ) = √5 – 3 – √5 – 3 = -6 which is rational.

#### Question 11

Write a pair of irrational numbers whose product is irrational.

**Answer**

Consider two irrational numbers ( 5 + √2 ) and ( √5 – 2 )

Thus, the product, ( 5 + √2 ) x ( √5 – 2 ) = 5√5 – 10 + √10 – 2√2 is irrational.

#### Question 12

Write a pair of irrational numbers whose product is rational.

**Answer**

Consider √2 as an irrational number.

√2×√2= √4= 2 which is a rational number.

#### Question 13.1

**Write in ascending order: **3√5 and 4√3

** Answer**

** **

#### Question 13.2

**Write in ascending order : **253 and323

** Answer**

** **

#### Question 13.3

**Write in ascending order : **6√5, 7√3, and 8√2

** Answer**

6√5 = 62×5=180

7√3 = 72×3=147

8√2 = 82×2=128

and 128 < 147 < 180

∴ √128 < √147 < √180

⇒ 8√2 < 7√3 < 6√5

#### Question 13.4

**Write in ascending order : **6√5, 7√3 and 8√2

** Answer**

65=62×5=180

73=72×3=147

82=82×2=128

and 128 < 147 < 180

∴ 128<147<180

⇒ 82<73<65

#### Question 14.1

** Write in descending order: **

** Answer**

#### Question 14.2

**Write in descending order: **7√3 and 3√7

** Answer**

#### Question 15.1

** Answer**

#### Question 15.2

** Compare : **

** Answer**

#### Question 16.

Insert two irrational numbers between 5 and 6.

** Answer**

#### Question 17.

Insert five irrational numbers between 2√5 and 3√3.

**Answer**

We know that 2√5 = 4×5 = √20 and 3√3 = √20

Thus, We have, √20 < √21 < √22 < √23 < √24 < √25 < √26 < √27.

So any five irrational numbers between 2√5 and 3√3 are :

√21, √22, √23, √24, and √26.

#### Question 18.

Write two rational numbers between √2 and √3.

**Answer**

#### Question 19.

Write three rational numbers between √3 and √5.

**Answer**

#### Question 20.1

**Answer**

#### Question 20.2

**Answer**

#### Question 20.3

**Simplify :** ( 3 + √2 )( 4 + √7 )

**Answer**

( 3 + √2 )( 4 + √7 )

= 3 x 4 + 3 x √7 + 4 x √2 + √2 x √7

= 12 + 3√7 + 4√2 + √14

#### Question 20.4

**Simplify :** (√3 – √2 )^{2}

**Answer**

(√3 – √2 )^{2}

= ( √3 )^{2} + ( √2 )^{2} – 2 x √3 x √2

= 3 + 2 – 2√6

= 5 – 2√6

End of Exe-1 B **Concise Rational and Irrational Numbers** for ICSE Class-9

### Selina Solutions of Exercise – 1(C)** Rational and Irrational Numbers Concise Maths for ICSE Class-9**

#### Question 1.1

**State, with reason, of the following is surd or not : **√180

**Answer**

√180 = 2×2×5×3×3 = 6√5 which is irrational.

∴ √180 is a surds.

#### Question 1.2

**State, with reason, of the following is surd or not :
**

**Answer**

#### Question 1.3

**Answer**

#### Question 1.4

**Answer**

#### Question 1.5

**State, with reason, of the following is surd or not : **

**Answer**

#### Question 1.6

**State, with reason, of the following is surd or not : **

…..

**Answer**

#### Question 1.7

**State, with reason, of the following is surd or not : **√π

**Answer**

√π not a surds as π is irrational.

#### Question 1.8

**State, with reason, of the following is surd or not : **

**Answer**

is not a surds because 3 + √2 is irrational.

#### Question 2.1

**Write the lowest rationalising factor of :** 5√2

**Answer**

5√2 x 5√2 = 5 x 2 = 10 which is rational.

∴ lowest rationalizing factor is √2

#### Question 2.2.

**Write the lowest rationalising factor of : **√24

**Answer**

#### Question 2.3

**Write the lowest rationalising factor of : **√5 – 3

**Answer**

( √5 – 3 )( √5 + 3 ) = ( √5 )^{2} – (3)^{2} = 5 – 9 = -4

∴ lowest rationalizing factor is ( √5 + 3 )

#### Question 2.4

**Write the lowest rationalising factor of : **7 – √7

**Answer**

7 – √7

( 7 – √7 )( 7 + √7 ) = 49 – 7 = 42

Therefore, lowest rationalizing factor is ( 7 + √7 ).

#### Question 2.5

**Write the lowest rationalising factor of : **√18 – √50

**Answer**

∴ lowest rationalizing factor is √2

#### Question 2.6

**Write the lowest rationalising factor of : **√5 – √2

**Answer**

√5 – √2

( √5 – √2 )( √5 + √2 ) = ( √5 )^{2} – ( √2 )^{2} = 3

Therefore lowest rationalizing factor is √5 + √2

#### Question 2.7

**Write the lowest rationalising factor of : **√13 + 3

**Answer**

( √13 + 3 )( √13 – 3 ) = ( √13 )^{2} – 3^{2} = 13 – 9 = 4

Its lowest rationalizing factor is √13 – 3.

#### Question 2.8

**Write the lowest rationalising factor of : **15 – 3√2

**Answer**

#### Question 2.9

**Write the lowest rationalising factor of : **3√2 + 2√3

**Answer**

3√2 + 2√3

= ( 3√2 + 2√3 )( 3√2 – 2√3 )

= ( 3√2)^{2} – (2√3)^{2}

= 9 x 2 – 4 x 3

= 18 – 12

= 6

its lowest rationalizing factor is 3√2 – 2√3.

#### Question 3.1

**Rationalise the denominators of :**

**Answer**

3√5×√5√5=3√5/5

#### Question 3.2

**Rationalise the denominators of : **

**Answer**

#### Question 3.3

**Rationalise the denominators of : **

**Answer**

#### Question 3.4

**Rationalise the denominators of :** 3/√5+√2

**Answer**

#### Question 3.5

**Rationalise the denominators of : **

**Answer**

#### Question 3.6

**Rationalise the denominators of : **

**Answer**

#### Question 3.7

**Rationalise the denominators of : **

**Answer**

#### Question 3.8

**Rationalise the denominators of : **

**Answer**

#### Question 3.9

**Rationalise the denominators of : …………………**

**Answer**

#### Question 4.1

**Find the values of ‘a’ and ‘b’ in each of the following :
**………

**Answer**

#### Question 4.2

**Find the values of ‘a’ and ‘b’ in each of the following:…..**

**Answer**

#### Question 4.3

**Find the values of ‘a’ and ‘b’ in each of the following: **

**Answer**

#### Question 4.4

**Find the values of ‘a’ and ‘b’ in each of the following**

**Answer**

#### Question 5.1

**Simplify :**

**Answer**

#### Question 5.2

**Simplify :
**…………….

**Answer**

#### Question 6.1

**f x =**…………………………**find **

**Answer**

#### Question 6.2

**f x =**…………………**; find : **y^{2}

**Answer**

#### Question 6.3

**If x =**…………………**; find : **xy

**Answer**

#### Question 6.4

**If x =**………………….**; find :**

x^{2} + y^{2} + xy.

**Answer**

x^{2} + y^{2} + xy

= 161 – 72√5 + 161 +72√5 + 1

= 322 + 1 = 323

#### Question 7.1

If m = ………… find m^{2}

**Answer**

#### Question 7.2

If m = ……………….., find n^{2}

**Answer**

#### Question 8.1

If x = 2√3 + 2√2 , find : …..

**Answer**

#### Question 8.2

If x = 2√3 + 2√2 , find : ……..

**Answer**

#### Question 8.3

If x = 2√3 + 2√2 , find :…..

**Answer**

#### Question 9

If x = 1 – √2, find the value of ……

**Answer**

#### Question 10

If x = 5 – 2√6, find x2+1×2

**Answer**

#### Question 11

**Show that : ……………..**

**Answer**

#### Question 12

**Rationalise the denominator of : **

**Answer**

#### Question 13.1

**If √2 = 1.4 and √3 = 1.7, find the value of :…………..**

**Answer**

#### Question 13.2

**If √2 = 1.4 and √3 = 1.7, find the value of :** ………

**Answer**

#### Question 13.3

**Simplify** :

**Answer**

#### Question 14

**Evaluate :**

**Answer**

#### Question 15

If ; find the value of x^{2} – y^{2}.

**Answer**

#### Question 16

**Simplify : **

**Answer**

— End of Exe-1 C Concise **Rational and Irrational Numbers **Solutions :–

Return to **– Concise Selina Maths Solutions for ICSE Class -9**

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