**Question 1: Define an irrational number.**

**Solution:**

A real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q â‰ 0. It is a non-terminating or non-repeating decimal. i.e. for example:

1.1000120010211…..

**Question 2: Explain, how irrational numbers differ from rational numbers?**

**Solution:**

An irrational number is a real number that cannot be expressed in the form of fractions i.e. p/q, where p and q are integers and q â‰ 0 i.e it cannot be expressed as a ratio of integers. It is a non-terminating or non-repeating decimal.

For example, âˆš2 is an irrational number

A rational number is a real number that can be expressed as a fraction and as a decimal i.e. it can be expressed as a ratio of integers. It is a terminating or repeating decimal.

For examples: 0.101 and 5/4 are rational numbers

**Question 3: Examine, whether the following numbers are rational or irrational:**

**(i) âˆš7**

**(ii) âˆš4**

**(iii) 2 + âˆš3**

**(iv) âˆš3 + âˆš2**

**(v) âˆš3 + âˆš5**

**(vi) (âˆš2 â€“ 2) ^{2}**

**(vii) (2 â€“ âˆš2)(2 + âˆš2)**

**(viii) (âˆš3 + âˆš2) ^{2}**

**(ix) âˆš5 â€“ 2**

**(x) âˆš23**

**(xi) âˆš225**

**(xii) 0.3796**

**(xiii) 7.478478â€¦â€¦**

**(xiv) 1.101001000100001â€¦â€¦**

**Solution:**

(i)âˆš7Given: âˆš7

Since, it is not a perfect square root,

Therefore, it is an irrational number.

(ii) âˆš4Given: âˆš4

Since, it is a perfect square root of 2.

Therefore, 2 can be expressed in the form of 2/1, thus it is a rational number.

(iii) 2 + âˆš3Given: 2 + âˆš3

Here, 2 is a rational number, and âˆš3 i is not a perfect square thus it is an irrational number.

Since, the sum of a rational and irrational number is always an irrational number.

Therefore, 2 + âˆš3 is an irrational number.

(iv) âˆš3 + âˆš2Given: âˆš3 + âˆš2

Here, âˆš3 is not a perfect square thus it is an irrational number.

Similarly, âˆš2 is not a perfect square, thus it is an irrational number.

Since, the sum of two irrational numbers is always an irrational number.

Therefore, âˆš3 + âˆš2 is an irrational number.

(v) âˆš3 + âˆš5Given: âˆš3 + âˆš5

Here, âˆš3 is not a perfect square thus it is an irrational number

Similarly, âˆš5 is not a perfect square thus it is an irrational number.

Since, the sum of two irrational numbers is always an irrational number.

Therefore, âˆš3 + âˆš5 is an irrational number.

(vi) (âˆš2 â€“ 2)^{2}Given: (âˆš2 â€“ 2)

^{2}(âˆš2 â€“ 2)

^{2}= 2 + 4 â€“ 4 âˆš2= 6 â€“ 4 âˆš2

Here, 6 is a rational number but 4âˆš2 is an irrational number.

Since, the sum of a rational and irrational number is always an irrational number.

Therefore, (âˆš2 â€“ 2)

^{2}is an irrational number.

(vii)(2 â€“ âˆš2)(2 + âˆš2)Given: (2 â€“ âˆš2)(2 + âˆš2)

(2 â€“ âˆš2)(2 + âˆš2) = ((2)

^{2}âˆ’ (âˆš2)^{2}) [As, (a + b)(a â€“ b) = a^{2}â€“ b^{2}]= 4 â€“ 2

= 2 or 2/1

Since, 2 is a rational number,

Therefore, (2 â€“ âˆš2)(2 + âˆš2) is a rational number.

(viii) (âˆš3 + âˆš2)^{2}Given: (âˆš3 + âˆš2)

^{2}(âˆš3 + âˆš2)

^{2}= (âˆš3)^{2}+ (âˆš2)^{2}+ 2âˆš3 x âˆš2 [ As, (a + b)^{2}= a^{2}â€“ 2ab + b2 ]= 3 + 2 + 2âˆš6

= 5 + 2âˆš6

Since, the sum of a rational and irrational number is always an irrational number.

Therefore, (âˆš3 + âˆš2)

^{2}is an irrational number.

(ix)âˆš5 â€“ 2Given: âˆš5 â€“ 2

Here, âˆš5 is an irrational number but 2 is a rational number.

Since, the difference between an irrational number and a rational number is an irrational number.

Therefore, âˆš5 â€“ 2 is an irrational number.

(x)âˆš23Given: âˆš23

âˆš23 = 4.795831352331â€¦

Since, the decimal expansion of âˆš23 is non-terminating and non-recurring

Therefore, âˆš23 is an irrational number.

(xi)âˆš225Given: âˆš225

âˆš225 = 15 or 15/1

Since, âˆš225 can be represented in the form of p/q and q â‰ 0.

Therefore, âˆš225 is a rational number

(xii) 0.3796Given: 0.3796

Since, the decimal expansion is terminating.

Therefore, 0.3796 is a rational number.

(xiii) 7.478478â€¦â€¦Given: 7.478478â€¦â€¦

Since, the decimal expansion is a non-terminating recurring decimal.

Therefore, 7.478478â€¦â€¦ is a rational number.

(xiv) 1.101001000100001â€¦â€¦Given: 1.101001000100001â€¦â€¦

Since, the decimal expansion is non-terminating and non-recurring.

Therefore, 1.101001000100001â€¦â€¦ is an irrational number

**Question 4: Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:**

**(i) âˆš4**

**(ii) 3âˆš18**

**(iii) âˆš1.44**

**(iv) âˆš9/27**

**(v) â€“ âˆš64**

**(vi) âˆš100**

**Solution:**

(i) âˆš4Given: âˆš4

Since, âˆš4 = 2 = 2/1, it can be written in the form of a/b.

Therefore, âˆš4 is a rational number.

The decimal representation of âˆš4 is 2.0

(ii) 3âˆš18Given: 3âˆš18

3âˆš18 = 9âˆš2

Since, the product of a rational and an irrational number is always an irrational number.

Therefore, 3âˆš18 is an irrational number.

(iii)âˆš1.44Given: âˆš1.44

Since, âˆš1.44 = 1.2, it is a terminating decimal.

Therefore, âˆš1.44 is a rational number.

The decimal representation of âˆš1.44 is 1.2

(iv)âˆš9/27Given: âˆš9/27

Since, âˆš9/27 = 1/âˆš3, as the quotient of a rational and an irrational number is an irrational number.

Therefore, âˆš9/27 is an irrational number.

(v)â€“ âˆš64Given: â€“ âˆš64

Since, â€“ âˆš64 = â€“ 8 or â€“ 8/1, as it can be written in the form of a/b.

Therefore, â€“ âˆš64 is a rational number.

The decimal representation of â€“ âˆš64 is â€“8.0

(vi)âˆš100Given: âˆš100

Since, âˆš100 = 10 = 10/1, as it can be written in the form of a/b.

Therefore, âˆš100 is a rational number.

The decimal representation of âˆš100 is 10.0

**Question 5: In the following equation, find which variables x, y, z etc. represent rational or irrational numbers:**

**(i) x ^{2} = 5**

**(ii) y2 = 9**

**(iii) z ^{2} = 0.04**

**(iv) u ^{2 }= 17/4**

**(v) v ^{2} = 3**

**(vi) w ^{2} = 27**

**(vii) t ^{2} = 0.4**

**Solution:**

(i) x^{2}= 5Given: x

^{2}= 5When we take square root on both sides, we get,

x = âˆš5

Since, âˆš5 is not a perfect square root,

Therefore, x is an irrational number.

(ii)y^{2}= 9Given: y

^{2}= 9When we take square root on both sides, we get,

y = 3

Since, 3 = 3/1, as it can be expressed in the form of a/b

Therefore, y is a rational number.

(iii)z^{2}= 0.04Given: z

^{2}= 0.04When we take square root on both sides, we get,

z = 0.2

Since, 0.2 = 2/10, as it can be expressed in the form of a/b and is a terminating decimal.

Therefore, z is a rational number.

(iv) u^{2}= 17/4Given: u

^{2}= 17/4When we take square root on both sides, we get,

u = âˆš17/2

Since, the quotient of an irrational and a rational number is irrational,

Therefore, u is an irrational number.

(v)v^{2}= 3Given: v

^{2}= 3When we take square root on both sides, we get,

v = âˆš3

Since, âˆš3 is not a perfect square root,

Therefore, v is an irrational number.

(vi) w^{2}= 27Given: w

^{2}= 27When we take square root on both sides, we get,

w = 3âˆš3

Since, the product of a rational and irrational is always an irrational number.

Therefore, w is an irrational number.

(vii)t^{2}= 0.4Given: t

^{2}= 0.4When we take square root on both sides, we get,

t = âˆš(4/10)

t = 2/âˆš10

Since, the quotient of a rational and an irrational number is always an irrational number.

Therefore, t is an irrational number.

**Question 6: Give an example of each, of two irrational numbers whose:**

**(i) Difference in a rational number**

**(ii) Difference in an irrational number**

**(iii) Sum in a rational number**

**(iv) Sum is an irrational number**

**(v) Product in a rational number**

**(vi) Product in an irrational number**

**(vii) Quotient in a rational number**

**(viii) Quotient in an irrational number**

**Solution:**

(i) Difference in a rational numberâˆš5 is an irrational number

Since, âˆš5 – âˆš5 = 0

Here, 0 is a rational number.

(ii)Difference in an irrational numberLet the two irrational number be 5âˆš3 and âˆš3

Since, (5âˆš3) – (âˆš3) = 4âˆš3

Here, 4âˆš3 is an irrational number.

(iii) Sum in a rational numberLet the two irrational numbers be âˆš5 and -âˆš5

Since, (âˆš5) + (-âˆš5) = 0

Here, 0 is a rational number.

(iv) Sum is an irrational numberLet the two irrational numbers be 4âˆš5 and âˆš5

Since, 4âˆš5 + âˆš5 = 5âˆš5

Here, 5âˆš5 is an irrational number.

(v)Product in a rational numberLet the two irrational numbers be 2âˆš2 and âˆš2

Since, 2âˆš2 Ã— âˆš2 = 2 Ã— 2 = 4

Here, 4 is a rational number.

(vi) Product in an irrational numberLet the two irrational numbers be âˆš2 and âˆš3

Since, âˆš2 Ã— âˆš3 = âˆš6

Here, âˆš6 is an irrational number.

(vii) Quotient in a rational numberLet the two irrational numbers be 2âˆš2 and âˆš2

Since, 2âˆš2 / âˆš2 = 2

Here, 2 is a rational number.

(viii)Quotient in an irrational numberLet the two irrational numbers be 2âˆš3 and 2âˆš2

Since, 2âˆš3 / 2âˆš2 = âˆš3/âˆš2

Here, âˆš3/âˆš2 is an irrational number.

**Question 7: Give two rational numbers lying between 0.232332333233332 and 0.212112111211112.**

**Solution:**

Let a = 0.212112111211112

Let b = 0.232332333233332

Here a<b as on the second decimal place

ahas digit 1 andbhas digit 3.If the second decimal place is considered as 2 then it lies between

aandb.Therefore, Let x = 0.22

and y = 0.22112211…

Thus, a < x < y < b

Hence, x and y are the rational numbers required.

**Question 8: Give two rational numbers lying between 0.515115111511115 and 0.5353353335**

**Solution:**

Let a = 0.515115111511115

Let b = 0.5353353335

Here a<b as on the second decimal place

ahas digit 1 andbhas digit 3.If the second decimal place is considered as 2 then it lies between

aandb.Therefore, Let x = 0.52

and y = 0.520520…

Thus, a < x < y < b

Hence, x and y are the rational numbers required.

**Question 9: Find one irrational number between 0.2101 and 0.2222… **

**Solution:**

Let a = 0.2101

and b = 0.2222…

Here a<b as on the second decimal place

ahas digit 1 andbhas digit 2.If the third decimal place is considered as 1 then it lies between

aandb.Therefore, Let x = 0.2110110011…

Thus, a < x < b

Hence, x is the irrational number required.

**Question 10: Find a rational number and also an irrational number lying between the numbers 0.3030030003… and 0.3010010001…**

**Solution:**

Let a = 0.3010010001…

and b = 0.3030030003…

Here a<b as on the third decimal place

ahas digit 1 andbhas digit 3.If the third decimal place is considered as 2 then it lies between

aandb.Therefore, Let x = 0.302

and y = 0.302002000200002…

Thus, a < x < y < b

Hence, x and y are the rational and irrational numbers required respectively.

**Question 11: Find two irrational numbers between 0.5 and 0.55.**

**Solution:**

Let a = 0.5

and b = 0.55

Here a<b as on the second decimal place a has digit 0 and b has digit 5.

If the second decimal place is considered between1 to 4 then it lies between a and b.

Therefore, Let x = 0.510510051000…

and y = 0.53053530…

Thus, a < x < y < b

Hence, x and y are the irrational numbers required.

**Question 12: Find two irrational numbers lying between 0.1 and 0.12.**

**Solution:**

Let a = 0.1

and b = 0.12

Here a<b as on the second decimal place a has digit 0 and b has digit 2.

If the second decimal place is considered 1 then it lies between a and b.

Therefore, Let x = 0.11011011000…

and y = 0.11100010100…

Thus, a < x < y < b

Hence, x and y are the irrational numbers required.

**Question 13: Prove that âˆš3 + âˆš5 is an irrational number.**

**Solution:**

Let âˆš3 + âˆš5 be a rational number equal to x.

Therefore, x = âˆš3 + âˆš5

x

^{2}= (âˆš3 + âˆš5)^{2}x

^{2}= (âˆš3)^{2}+ (âˆš5)^{2}+ 2 âˆš3 âˆš5= 3 + 5 + 2âˆš15

= 8 + 2âˆš15

x

^{2}– 8 = 2âˆš15(x

^{2}– 8)/2 = âˆš15Here, (x

^{2}– 8)/2 is a rational but âˆš15 is an irrational number.Therefore, âˆš3 + âˆš5 is an irrational number.