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**Q1. Define an irrational number.**

**Solution:** An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers. It cannot be expressed as terminating or repeating decimal.

**Q2. Explain how an irrational number is differing from rational numbers?**

**Solution:** An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers. It cannot be expressed as terminating or repeating decimal.

For example, 0.10110100 is an irrational number

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

For examples,

0.10 and both are rational numbers

**Q3. Find, whether the following numbers are rational and irrational**

**(i) âˆš7**

**(ii) âˆš4**

**(iii) 2+âˆš3**

**(iv) âˆš3+âˆš2**

**(v) âˆš3+5**

**(vi) (âˆš2 âˆ’ 2) ^{2}**

**(vii) (2âˆ’âˆš2)(2+âˆš2)**

**(viii) (âˆš2+âˆš3) ^{2}**

**(ix) âˆš5 â€“ 2**

** **

**(xii) 0.3796**

**(xiii) 7.478478â€¦â€¦**

**(xiv) 1.101001000100001â€¦â€¦**

**Solution:**

(i) âˆš7is not a perfect square root so it is an Irrational number.

(ii) âˆš4 is a perfect square root so it is an rational number.

We have,

âˆš4 can be expressed in the form of

ab, so it is a rational number. The decimal

representation of âˆš9 is 3.0. 3 is a rational number.

(iii) 2+âˆš3

Here, 2 is a rational number and âˆš3 is an irrational number

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

(iv) âˆš3+âˆš2

âˆš3 is not a perfect square and it is an irrational number and âˆš2 is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so âˆš3+âˆš2 is an irrational number.

(v) âˆš3+âˆš5

âˆš3 is not a perfect square and it is an irrational number and âˆš5 is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so âˆš3+âˆš5 is an irrational number.

(vi) (âˆš2âˆ’2)^{2}

We have, (âˆš2âˆ’2)^{2}

= 2 + 4 - 4 âˆš2

= 6 + 4 âˆš2

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

The sum of a rational number and an irrational number is an irrational number, so (âˆš2+âˆš4)^{2} is an irrational number.

We have,

[Since, (a + b)(a - b) = a^{2} â€“ b^{2}]

Since, 2 is a rational number.

is a rational number.

We have,

[Since, (a+b)^{2} = a^{2} + 2ab + b^{2}

The sum of a rational number and an irrational number is an irrational number, so is an irrational number.

The difference of an irrational number and a rational number is an irrational number.

() is an irrational number.

= 4.795831352331â€¦

As decimal expansion of this number is non-terminating, non-recurring so it is an irrational number.

is rational number as it can be represented in p/q form.

(xii) 0.3796

0.3796, as decimal expansion of this number is terminating, so it is a rational number.

(xiii) 7.478478â€¦â€¦

7.478478 = 7.478, as decimal expansion of this number is non-terminating recurring so it is a rational number.

(xiv) 1.101001000100001â€¦â€¦

1.101001000100001â€¦â€¦, as decimal expansion of this number is non-terminating, non-recurring so it is an irrational number

**Q4. Identify the following as irrational numbers. Give the decimal representation of rational numbers:**

**Solution:**

(i) We have,

âˆš4 can be written in the form of

p/q. So, it is a rational number. Its decimal

representation is 2.0

(ii). We have,

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

is an irrational.

is an irrational number.

(iii) We have,

= 12/10

= 1.2

Every terminating decimal is a rational number, so 1.2 is a rational number.

Its decimal representation is 1.2.

We have,

Quotient of a rational and an irrational number is irrational numbers so

is an irrational number.

is an irrational number.

(v) We have,

= - 8

can be expressed in the form of a/b,

so

is a rational number.

Its decimal representation is - 8.0.

(vi) We have,

= 10 can be expressed in the form of a/b,

so is a rational number

Its decimal representation is 10.0.

**Q5. In the following equations, find which variables x, y and 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**

**Solution:**

(i) We have,

x^{2} = 5

Taking square root on both the sides, we get

x = âˆš5

âˆš5 is not a perfect square root, so it is an irrational number.

(ii) We have,

= y^{2} = 9

= 3

= 3/1 can be expressed in the form of a/b, so it a rational number.

(iii) We have,

z^{2} = 0.04

Taking square root on the both sides, we get

z = 0.2

2/10 can be expressed in the form of a/b, so it is a rational number.

(iv) We have,

u^{2} = 17/4

Taking square root on both sides, we get,

Quotient of an irrational and a rational number is irrational, so u is an Irrational number.

(v) We have,

v2 = 3

Taking square root on both sides, we get,

v = âˆš3

âˆš3 is not a perfect square root, so v is irrational number.

(vi) We have,

w^{2} = 27

Taking square root on both the sides, we get,

w = 3âˆš3

Product of a irrational and an irrational is an irrational number. So w is an irrational number.

(vii) We have,

t^{2} = 0. 4

Taking square root on both sides, we get,

Since, quotient of a rational and an Irrational number is irrational number. t^{2} = 0.4 is an irrational number.

**Q6. 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: ** is an irrational number.

Now,

0 is the rational number.

(ii) Let two irrational numbers are

is the rational number.

(iii) is an irrational number.

Now,

0 is the rational number.

(iv) Let two irrational numbers are

is the rational number.

(iv) Let two Irrational numbers are and âˆš 5

= 7 Ã— 5

= 35 is the rational number.

(v) Let two irrational numbers are

Now,

8 is the rational number.

(vi) Let two irrational numbers are

= 4 is the rational number

(vii) Let two irrational numbers are

Now, 3 is the rational number.

(viii) Let two irrational numbers are

Now is an rational number.

**Q7. Give two rational numbers lying between 0.232332333233332 and 0.212112111211112.**

**Solution:** Let a = 0.212112111211112

And, b = 0.232332333233332...

Clearly, a < b because in the second decimal place a has digit 1 and b has digit 3 If we consider rational numbers in which the second decimal place has the digit 2, then they will lie between a and b.

Let. x = 0.22

y = 0.22112211... Then a < x < y < b

Hence, x, and y are required rational numbers.

**Q8. Give two rational numbers lying between 0.515115111511115 and 0. 5353353335**

**Solution: **Let, a = 0.515115111511115...

And, b = 0.5353353335..

We observe that in the second decimal place a has digit 1 and b has digit 3, therefore, a < b.

So If we consider rational numbers

x = 0.52

y = 0.52062062...

We find that,

a < x < y < b

Hence x and y are required rational numbers.

**Q9. Find one irrational number between 0.2101 and 0.2222 ... **

**Solution:** Let, a = 0.2101 and,

b =0.2222...

We observe that in the second decimal place a has digit 1 and b has digit 2, therefore a < b in the third decimal place a has digit 0.

So, if we consider irrational numbers

x = 0.211011001100011....

We find that a < x < b

Hence x is required irrational number.

**Q10. 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...

We observe that in the third decimal place a has digit 1 and b has digit

3, therefore a < b in the third decimal place a has digit 1. So, if we

consider rational and irrational numbers

x = 0.302

y = 0.302002000200002.....

We find that a < x < b and, a < y < b.

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

**Q11. Find two irrational numbers between 0.5 and 0.55.**

**Solution:** Let a = 0.5 = 0.50 and b =0.55

We observe that in the second decimal place a has digit 0 and b has digit

5, therefore a < 0 so, if we consider irrational numbers

x =0.51051005100051...

y = 0.530535305353530...

We find that a < x < y < b

Hence x and y are required irrational numbers.

**Q12. Find two irrational numbers lying between 0.1 and 0.12.**

**Solution:** Let a = 0.1 =0.10

And b = 0.12

We observe that In the second decimal place a has digit 0 and b has digit 2.

Therefore, a < b.

So, if we consider irrational numbers

x = 0.1101101100011... y=0.111011110111110... We find that a < x < y < 0

Hence, x and y are required irrational numbers.

**Q13. Prove that is an irrational number.**

**Solution: **If possible, let be a rational number equal to x.

Now, is rational

is rational

Thus, we arrive at a contradiction.

Hence, is an irrational number.

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