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**Q.1. Define an irrational number.**

**Proof: **An irrational number is a real number that cannot be reduced to any ratio between an integer p and a natural number q.

If the decimal representation of an irrational number is non-terminating and non-repeating, then it is called irrational number. For example

**Q.2. Explain, how irrational numbers differ from rational numbers?**

**Proof: **

Every rational number must have either terminating or non-terminating but irrational number must have non- terminating and non-repeating decimal representation.

A rational number is a number that can be written as simple fraction (ratio) and denominator is not equal to zero while an irrational is a number that cannot be written as a ratio.

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

**(i) √7**

**(ii) √4**

**(iii) 2 + √3**

**(iv)√3 + √2**

**(v) √3 + √5**

**(vi) (√ 2 – 2) ^{2}**

**(vii) (2−√2) (2+√2) **

**(viii) (√2 + √3–√) ^{2}**

**(ix) √5−2**

**(x) √23**

**(xi) √225**

**(xii) 0.3796**

**(xiii) 7.478478**

**(xiv) 1.101001000100001**

**Proof: **(i) Let

Therefore,

It is non-terminating and non-repeating

Hence is an irrational number

(ii) Let

Therefore,

It is terminating.

Henceis a rational number.

(iii) Let be the rational

Squaring on both sides

Since, x is rational

is rational

is rational

is rational

is rational

But,is irrational

So, we arrive at a contradiction.

Hence is an irrational number

(iv) Let be the rational number

Squaring on both sides, we get

Since, x is a rational number

is rational number

is rational number

is rational number

is rational number

But is an irrational number

So, we arrive at contradiction

Hence is an irrational number

(v) Let is an irrational number

Squaring on both sides, we get

Now, x is rational number

is rational number

is rational number

is rational number

is rational number

But

is an irrational number

So, we arrive at a contradiction

Hence is an irrational number

(vi) Let be a rational number.

Since, x is rational number,

⇒ x – 6 is a rational number

⇒is a rational number

⇒is a rational number

But we know that is an irrational number, which is a contradiction

So is an irrational number

(vii) Let

Using the formula

⇒is a rational number

⇒is a rational number

But we know that is an irrational number

So, we arrive at a contradiction

So is an irrational number.

(ix) Let x = √5−2 be the rational number

Squaring on both sides, we get

Now, x is rational

x^{2} is rational.

So, x^{2}−29 is rational

But, √5 is irrational. So we arrive at contradiction

Hence x = √5−2 is an irrational number

(x) Let

It is non-terminating or non-repeating

Hence is an irrational number

(xi) Let

Hence is a rational number

(xii) Given x= 0.3796.

It is terminating

Hence it is a rational number

(xiii) Given number

It is repeating

Hence it is a rational number

(xiv) Given number is

It is non-terminating or non-repeating

Hence it is an irrational number

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

(i) (√4)

(ii) 3√18

(iii) √1.44

(iv) √927

(v) −√64

(vi) √100

**Proof: **(i) Given number is x =

x = 2, which is a rational number

(ii) Given number is

So it is an irrational number

(iii) Given number is

Now we have to check whether it is rational or irrational

So it is a rational

(iv) Given that

Now we have to check whether it is rational or irrational

So it is an irrational number

(v) Given that

Now we have to check whether it is rational or irrational

Since,

So it is a rational number

(vi) Given that

Now we have to check whether it is rational or irrational

Since,

So it is rational number

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

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

**(ii) y ^{2} = 9**

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

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

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

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

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

**Proof: **(i) Given that

Now we have to find the value of x

So it x is an irrational number

(ii) Given that

Now we have to find the value of y

So y is a rational number

(iii) Given that

Now we have to find the value of z

So it is rational number

(iv) Given that

Now we have to find the value of u

So it is an irrational number

(v) Given that

Now we have to find the value of v

So it is an irrational number

(vi) Given that

Now we have to find the value of w

So it is an irrational number

(vii) Given that

Now we have to find the value of t

So it is an irrational number

**Q.6. Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.**

**Proof: **Let a = 0.232332333233332

b = 0.212112111211112

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place of a and b have the same digit but digit in the second place of their decimal representation are distinct. And the number a has 3 and b has 1. So a > b.

Hence two rational numbers are lying between 0.232332333233332... and 0.212112111211112...

**Q.7. Give two rational numbers lying between 0.515115111511115...0.5353353335...**

**Proof: **Let a = 0.515115111511115... and b = 0.535335333533335...

Here the decimal representation of a and b are non-terminating and non-repeating. So we observe that in first decimal place a and b have the same digit but digit in the second place of their decimal representation are distinct. And the number a has 1 and b has 3. So a < b.

Hence two rational numbers are lying between 0.515115111511115.. and 0.535335333533335...

**Q.8. Find one irrational number between 0.2101 and 0.222... = 0.2¯.**

**Proof: **Let

Here a and b are rational numbers .Since a has terminating and b has repeating decimal. We observe that in second decimal place a has 1 and b has 2. So a < b.

Hence one irrational number is lying between 0.2101 and 0.2222...

**Q.9. Find a rational number and also an irrational number lying between the numbers 0.3030030003 ... and 0.3010010001 ...**

**Proof: **Let

Here decimal representation of a and b are non-terminating and non-repeating. So a and b are irrational numbers. We observe that in first two decimal place of a and b have the same digit but digit in the third place of their decimal representation is distinct.

Therefore, a > b.

Hence one rational number is lying between 0.3030030003... and 0.3010010001...

And irrational number is lying between and

**Q.10. Find three different irrational numbers between the rational numbers and .**

**Proof: **Let and

Here we observe that in the first decimal x has digit 7 and y has 8. So x < y. In the second decimal place x has digit 1. So, if we considering irrational numbers

a = 0.72072007200072..

b = 0.73073007300073..

c = 0.74074007400074....

We find that

x<a<b<c<y

Hence are required irrational numbers.

**Q.11. ****Give an example of each, of two irrational numbers whose:**

**(i) difference is a rational number.**

**(ii) difference is an irrational number.**

**(iii) sum is a rational number.**

**(iv) sum is an irrational number.**

**(v) product is an rational number.**

**(vi) product is an irrational number.**

**(vii) quotient is a rational number.**

**(viii) quotient is an irrational number.**

**Proof: **(i) Let

And, so

Therefore,and are two irrational numbers and their difference is a rational number

(ii) Let are two irrational numbers and their difference is an irrational number

Because is an irrational number

(iii) Letare two irrational numbers and their sum is a rational number

That is

(iv) Letare two irrational numbers and their sum is an irrational number

That is

(v) Let are two irrational numbers and their product is a rational number

That is

(vi) Letare two irrational numbers and their product is an irrational number

That is

(vii) Let are two irrational numbers and their quotient is a rational number

That is

(viii) Letare two irrational numbers and their quotient is an irrational number

That is

**Q.12. Find two irrational numbers between 0.5 and 0.55.**

**Proof: **Let

a = 0.5

b = 0.55

Here a and b are rational number. So we observe that in first decimal place a and b have same digit .So a < b.

Hence two irrational numbers areand

lying between 0.5 and 0.55.

**Q.13. Find two irrational numbers lying between 0.1 and 0.12.**

**Proof: **Let

a = 0.1

b = 0.12

Here a and b are rational number. So we observe that in first decimal place a and b have same digit. So a < b.

Hence two irrational numbers areandlying between 0.1 and 0.12.

**Q.14. Prove that √3 + √5 is an irrational number.**

**Proof: **Given that √3 + √5 is an irrational number

Now we have to prove √3 + √5 is an irrational number

Let x = √3 + √5 is a rational

Squaring on both sides

Now x is rational

is rational

is rational

is rational

But,is an irrational

Thus we arrive at contradiction thatis a rational which is wrong.

Henceis an irrational

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