Exercise 1C RS Aggarwal Class 9 Number Systems Best Solutions

Exercise 1A Exercise 1B Exercise 1C Exercise 1D Exercise 1E Exercise 1F Exercise 1G Exercise 2A Exercise 2B Exercise 2C Exercise 2D Exercise 3A Exercise 3B Exercise 3C Exercise 3D Exercise 3E Exercise 3F

Exercise 1C RS Aggarwal Class 9 contains a total of 14 questions. The questions are based on the following topics:

Irrational Numbers
Properties of Irrational Numbers

Exercise 1C RS Aggarwal Class 9 Solutions

Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13 Q.14

The first question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

1. What are irrational numbers? How do they differ from rational numbers? Give examples.

Answer

Irrational numbers : A number which can neither be expressed as a terminating decimal nor as a repeating decimal is called an irrational number. In other words, the numbers which cannot be expressed in the form $\frac{p}{q}$ are called irrational numbers.

For example $\sqrt{2}, \sqrt{3},$ 1.01001000100001..., 0.34034340343434..., 9.404004000400004..., 187.2112111211112..., etc. are irrational numbers.
And $\frac{2}{3}, \frac{3}{4}, \frac{5}{6}$ ..., etc. are rational numbers.

The second question of Exercise 1C RS Aggarwal Class 9 is based on classification of real numbers

2. Classify the following numbers as rational or irrational. Give reasons to support your answer.

(i) $\sqrt{\frac{3}{81}}$

Answer

We have $\sqrt{\frac{3}{81}}$ = $\sqrt{\frac{3}{3 \times 3 \times 3 \times 3}}$ = $\sqrt{\frac{1}{3 \times 3 \times 3}}$ = $\frac{1}{3 \sqrt{3}}$ = $\frac{rational number}{irrational number}$ = $irrational number$

Clearly, $\sqrt{\frac{3}{81}}$ is an irrational number.
Reason: A rational number divided by an irrational number is irrational.

(ii) $\sqrt{361}$

Answer

We have $\sqrt{361} = 19$ = a rational number.

Reason: $\sqrt{361}$ is a perfect square root.

(iii) $\sqrt{21}$

Answer

$∵ \sqrt{21}$ is not a perfect square. It is an irrational number.

(iv) $\sqrt{1.44}$

Answer

∵ $\sqrt{1.44}$ = $\sqrt{\frac{144}{100}}$ = $\sqrt{\frac{12^{2}}{10^{2}}}$ = $\frac{12}{10}$ is a rational number. ⁣

(v) $\frac{2}{3} \sqrt{6}$

Answer

Multiplication of a rational number by an irrational number is an irrational number.
$\frac{2}{3}$ is a rational number, and $\sqrt{6}$ is an irrational number. So, $\frac{2}{3} \sqrt{6}$ is an irrational number.

(vi) $4.1276$

Answer

4.1276 is a terminating decimal expansion. So, it is a rational number.

(vii) $\frac{22}{7}$

Answer

∵ $\frac{22}{7}$ is of the form $\frac{p}{q}$ . Hence, it is a rational number.

(viii) $1.232332333 . . .$

Answer

∵ 1.232332333... is a non-terminating and non-recurring decimal expansion. It is an irrational number.

(ix) $3.040040004 . . .$

Answer

∵ 3.040040004... is a non-terminating and non-recurring decimal expansion. It is an irrational number.

(x) $2.356565656 . . .$

Answer

∵ 2.356565656... = 2.356, here the block of digits 56 is repeating. So, it's a non-terminating recurring decimal expansion. Hence, it is a rational number.

(xi) $6.834834 . . .$

Answer

∵ 6.834834... = 6.834, here the block of digits 834 is repeating. So, it's a non-terminating recurring decimal expansion. Hence, 6.834834... is a rational number.

The third question of Exercise 1C RS Aggarwal Class 9 is based on rational and irrational numbers.

3. Let $x$ be a rational number and $y$ be an irrational number. Is $x + y$ necessarily an irrational number? Give an example in support of your answer.

Answer

Yes, $x + y$ is necessarily an irrational number if x is a rational number and y is an irrational number.

For example, 3 is a rational number, and 1.04004000400004... is an irrational number. After adding, we get <br>
3 + 1.04004000400004... = 4.04004000400004...
Clearly, the result is an irrational number, as it is a non-terminating and non-recurring decimal expansion.
Hence, a rational number and an irrational number added together give an irrational number.

The fourth question of Exercise 1C RS Aggarwal Class 9 is about rational and irrational numbers.

4. Let $a$ be a rational number and $b$ be an irrational number. Is $a b$ necessarily an irrational number? Justify your answer with an example.

Answer

Yes, $a b$ is necessarily an irrational number.

For example, 2 is a rational number, and 1.01011011101111... is an irrational number.
Multiplying 2 by 1.01011011101111... , we get
2 × 1.01011011101111... = 2.02022022202222...
Clearly, the product obtained above is a non-terminating and non-recurring decimal expansion.
Hence, the product of a rational and an irrational number is always an irrational number.

The fifth question of Exercise 1B RS Aggarwal Class 9 is about irrational numbers.

5. Is the product of two irrationals always irrational? Justify your answer

Answer

No, the product of two irrationals is not always irrational.
For example, $2 \sqrt{2}$ and $\sqrt{2}$ are irrational numbers. But $2 \sqrt{2}$ × $\sqrt{2}$ = 2 × 2 = 4, which is a rational number.

The sixth question of Exercise 1C RS Aggarwal Class 9 is based on rational and irrational numbers.

6. Give an example of two irrational numbers whose

(i) difference is an irrational number.

Answer

Two irrational numbers whose difference is an irrational number are $\sqrt{2}$ and $\sqrt{3}$ as $\sqrt{2}$ − $\sqrt{3}$ is an irrational number.

The following examples can also be taken.

$- \sqrt{3}$ and $\sqrt{3}$
$2 - \sqrt{3}$ and $2 + \sqrt{3}$
$\sqrt{5} - 7$ and $- \sqrt{5} + 7$

Note: Similarly you can choose infinitely many pairs whose difference is an irrational number.

(ii) difference is a rational number.

Answer

The two irrational numbers whose difference is a rational number are $2 + \sqrt{3}$ and $5 + \sqrt{3}$ .
$(2 + \sqrt{3}) - (5 + \sqrt{3})$ = −3 (a rational number)

Numerous irrational number pairs that could also be the solution are shown here.

$(4 + \sqrt{7}) and (5 + \sqrt{7})$
$(- 7 + \sqrt{6}) and (2 + \sqrt{6})$
$(11 + \sqrt{3}) and (21 + \sqrt{3})$
$(1 - \sqrt{13}) and (5 - \sqrt{13})$
$(8 - \sqrt{2}) and (−5 - \sqrt{2})$

Likewise, you can create a lot of these pairs. All you have to do is maintain the irrational number's sign.

(iii) sum is an irrational number.

Answer

Two irrational numbers whose sum is an irrational number are $(5 + \sqrt{2})$ and $(\sqrt{3} - 5)$ .

$(5 + \sqrt{2})$ + $(\sqrt{3} - 5)$ = $\sqrt{2} + \sqrt{3}$ (an irrational number).

The following lists additional irrational number pairs whose sum is also irrational.

$(3 + \sqrt{7}) and (6 + \sqrt{2})$
$(8 - \sqrt{3}) and (−8 + \sqrt{2})$
$(32 + \sqrt{13}) and (−4 + \sqrt{3})$

You must choose the pairs so that the numbers containing roots do not vanish.

(iv) sum is a rational number.

Answer

Two irrational numbers whose sum is a rational number are $(3 + \sqrt{2})$ and $(3 - \sqrt{2})$ .

$(3 + \sqrt{2})$ + $(3 - \sqrt{2})$ = 6 (a rational number).

The following lists additional irrational number pairs whose sum is also irrational.

$(4 + \sqrt{7}) and (4 - \sqrt{7})$
$(2 + \sqrt{6}) and (2 - \sqrt{6})$
$(11 + \sqrt{3}) and (11 - \sqrt{3})$
$(1 - \sqrt{13}) and (1 + \sqrt{13})$
$(8 - \sqrt{2}) and (8 + \sqrt{2})$

Likewise, you can create a lot of these pairs. All you have to do is take the conjugate pairs.

(v) product is an irrational number.

Answer

Two irrational numbers whose product is a rational number are $(2 + \sqrt{2})$ and $(3 - \sqrt{2})$ .

$(2 + \sqrt{2})$ × $(3 - \sqrt{2})$
= $2 (3 - \sqrt{2}) + \sqrt{2} (3 - \sqrt{2})$ .
= $6 - 2 \sqrt{2} + 3 \sqrt{2} - 2$ .
= $4 + \sqrt{2}$ (an irrational number).

The following lists additional irrational number pairs whose product is also irrational.

$(2 + \sqrt{7}) and (4 - \sqrt{7})$
$(8 + \sqrt{6}) and (2 - \sqrt{6})$
$(15 + \sqrt{3}) and (11 - \sqrt{3})$
$(7 - \sqrt{13}) and (1 + \sqrt{13})$
$(9 - \sqrt{2}) and (8 + \sqrt{2})$

Likewise, you can create a lot of these pairs. Since the product of conjugate pairs is a rational number, you should not take the conjugate pairs.

(vi) product is a rational number.

Answer

Two irrational numbers whose product is a rational number are $(3 + \sqrt{2})$ and $(3 - \sqrt{2})$ .

$(3 + \sqrt{2})$ × $(3 - \sqrt{2})$
= $3^{2} - {(\sqrt{2})}^{2}$ ... [∵ $(a + b) (a - b)$ = $a^{2} - b^{2}$ ]
= $9 - 2$
= 7 (a rational number)

The following lists additional irrational number pairs whose product is also irrational.

$(2 + \sqrt{7}) and (2 - \sqrt{7})$
$(8 + \sqrt{6}) and (8 - \sqrt{6})$
$(12 + \sqrt{3}) and (12 - \sqrt{3})$
$(7 - \sqrt{13}) and (7 + \sqrt{13})$
$(8 - \sqrt{2}) and (8 + \sqrt{2})$

Likewise, you can create a lot of these pairs. Only conjugate pairs must be taken.

(vii) quotient is an irrational number.

Answer

Two irrational numbers whose quotient is an irrational number are $\sqrt{18}$ and $\sqrt{3}$ .

$\sqrt{18} \div \sqrt{3}$ = $\frac{\sqrt{18}}{\sqrt{3}}$ = $\sqrt{\frac{18}{3}}$ = $\sqrt{6}$ (an irrational number).

The following lists additional irrational number pairs whose quotient is also irrational.

$\sqrt{28}$ and $\sqrt{2}$
$\sqrt{27}$ and $\sqrt{9}$
$\sqrt{50}$ and $\sqrt{5}$
$\sqrt{12}$ and $\sqrt{2}$

(viii) quotient is a rational number.

Answer

Two irrational numbers whose quotient is a rational number are $\sqrt{27}$ and $\sqrt{3}$ .

$\sqrt{27} \div \sqrt{3}$ = $\frac{\sqrt{27}}{\sqrt{3}}$ = $\sqrt{\frac{27}{3}}$ = $\sqrt{9}$ = $3$ (a rational number)

Other irrational number pairs with rational quotients are listed below.

$\sqrt{18}$ and $\sqrt{2}$
$\sqrt{45}$ and $\sqrt{5}$
$\sqrt{50}$ and $\sqrt{2}$
$\sqrt{12}$ and $\sqrt{3}$

The seventh question of Exercise 1C RS Aggarwal Class 9 is based on rational and irrational numbers.

7. Examine whether the following numbers are rational or irrational.

(i) $3 + \sqrt{3}$

Answer

$3 + \sqrt{3}$ is an irrational number, as 3 is rational, $\sqrt{3}$ is irrational and sum of a rational and an irrational is always an irrational number.

(ii) $\sqrt{7} - 2$

Answer

$\sqrt{7} - 2$ is an irrational number, as $\sqrt{7}$ is irrational, 2 is rational, and the difference between an irrational number and a rational number is always irrational.

(iii) $\sqrt[3]{5} \times \sqrt[3]{25}$

Answer

We have
$\sqrt[3]{5} \times \sqrt[3]{25}$ = $\sqrt[3]{5 \times 25}$ = $\sqrt[3]{5 \times 5 \times 5}$ = 5 (a rational number).
Hence, $\sqrt[3]{5} \times \sqrt[3]{25}$ is a rational number.

(iv) $\sqrt{7} \times \sqrt{343}$

Answer

We have
$\sqrt{7} \times \sqrt{343}$ = $\sqrt{7 \times 343}$ = $\sqrt{7 \times 7 \times 7 \times 7}$ = 7 (a rational number).
Hence, $\sqrt{7} \times \sqrt{343}$ is a rational number.

(v) $\sqrt{\frac{13}{117}}$

Answer

We have
$\sqrt{\frac{13}{117}}$ = $\sqrt{\frac{13}{13 \times 9}}$ = $\sqrt{\frac{1}{9}}$ = $\frac{1}{\sqrt{9}}$ = $\frac{1}{3}$ (a rational number).
Hence, $\sqrt{\frac{13}{117}}$ is a rational number.

(vi) $\sqrt{8} \times \sqrt{2}$

Answer

We have
$\sqrt{8} \times \sqrt{2}$ = $\sqrt{8 \times 2}$ = $\sqrt{2 \times 2 \times 2 \times 2}$ = $4$ (a rational number).
Hence, $\sqrt{8} \times \sqrt{2}$ is a rational number.

The eighth question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

8. Insert a rational and an irrational number between 2 and 2.5.

Answer

A rational number between 2 and 2.5 = $\frac{2 + 2.5}{2}$ = 2.25.

An irrational number between 2 and 2.5 = $\sqrt{2 \times 2.5}$ = $\sqrt{5}$ .

The ninth question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

9. How many irrational numbers lie between $\sqrt{2}$ and $\sqrt{3}$ ? Find any three irrational numbers lying between $\sqrt{2}$ and $\sqrt{3}$ .

Answer

There are infinitely many irrational numbers that lie between $\sqrt{2}$ and $\sqrt{3}$ .

The decimal form of $\sqrt{2}$ = 1.414...

The decimal form of $\sqrt{3}$ = 1.732...

Now, we know the decimal expansion of both $\sqrt{2}$ and $\sqrt{3}$ .
$\sqrt{2}$ = 1.414...
$\sqrt{3}$ = 1.732...
Three irrational numbers between $\sqrt{2}$ and $\sqrt{3}$ are given below.
(i) 1.5050050005...
(ii) 1.6161161116...
(iii) 1.707707770...

Many other irrational numbers between $\sqrt{2}$ and $\sqrt{3}$ can be obtained simply by finding a number between 1.41 and 1.73. Some such numbers are 1.42, 1.43, 1.44, 1.45, 1.46, 1.47, 1.48, 1.49, 1.50,... etc. We can form irrational numbers using these numbers in the beginning as given below.
● 1.42020020002...
● 1.43030030003...
● 1.44040040004...
● 1.45050050005...
● 1.46060060006...
● 1.47070070007...
While forming these numbers, keep in mind that no block of digits should repeat and the digits should form a progressive pattern.

The tenth question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

10. Find two rational and two irrational numbers between 0.5 and 0.55.

Answer

The given numbers are 0.5 and 0.55.
We can write 0.5 = 0.50.
Now, we can see a clear difference between 0.50 and 0.55.
Two rational numbers between 0.5 and 0.55 are 0.51 and 0.52.

Two irrational numbers between 0.50 and 0.55 are
(i) 0.5151151115...
(ii) 0.545545554...

Similarly, many irrational numbers can be obtained just by first choosing a number between 0.50 and 0.55.
(i) 0.52020020002...
(ii) 0.53053005300053...
(iii) 0.54054005400054...
(iv) 0.53253225322253...

The 11th question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

11. Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$ .

Answer

First, we convert the given rational numbers into decimal form.

$\frac{5}{7}$ = 0.714285714285... = 0.714285 .

$\frac{9}{11}$ = 0.818181... = 0.81 .

Clearly, we can find many numbers between 0.71 and 0.81. Such numbers could be 0.72, 0.73, 0.74, 0.75, 0.76, ... 0.80. Corresponding to each of these numbers, we can form the following irrational numbers.

0.717117111711117...
0.7272272227...
0.7373373337...
0.7474474447...
0.757757775...
0.7677677767777...
0.8080080008...
etc...

The 12th question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

12. Find two rational numbers of the form $\frac{p}{q}$ between the numbers 0.2121121112... and 0.2020020002... .

Answer

The given numbers 0.2121121112... and 0.2020020002... are irrational numbers.
We can write many rational numbers between these numbers by finding numbers between them.

0.2020020002... < 0.203 < 0.204 < 0.205 < 0.206 < 0.207 < . . . < 0.211 < 0.2121121112...

Clearly, the two rational numbers between 0.2020020002... and 0.2121121112... are given below.

(i) 0.204 = $\frac{204}{1000}$ = $\frac{51}{250}$ .

(ii) 0.206 = $\frac{206}{1000}$ = $\frac{103}{500}$ .

Similarly, you can find other rational numbers between these numbers.

The 13th question of Exercise 1C RS Aggarwal Class 9 is about irrational numbers.

13. Find two irrational numbers between 0.16 and 0.17.

Answer

We can write
0.16 = 0.160 and 0.17 = 0.170.
We have,

0.160 < 0.161 < 0.162 < 0.163 < 0.164 < . . . < 0.170.

With the aid of the numbers above, we can form the following irrational numbers.

0.16116111611116...
0.168668666866668...
0.1636336333633336...
0.16416441644416...
etc...

The 14th question of Exercise 1C RS Aggarwal Class 9 is about rational and irrational numbers.

14. State, in each case, whether the given statement is true or false.

(i) The sum of two rational numbers is rational.

Answer

True, as we can infer from the following.

2 (a rational) + 3 (a rational) = 5 (a rational)

$\frac{4}{7} (a rational) + \frac{6}{7} (a rational) = \frac{10}{7} (a rational)$

\frac{3}{4} (a rational) + \frac{1}{3} (a rational) = \frac{13}{12} (a rational)

1.121212...(a rational) + 2.333333...(a rational) = 3.454545...(a rational)

(ii) The sum of two irrational numbers is irrational.

Answer

True, as we can infer from the following.

\sqrt{2}

\sqrt{3}

is an irrational number.

$\sqrt{2}$ (an irrational number) + $2 \sqrt{2}$ (an irrational number) = $3 \sqrt{2}$ (an irrational number)

(iii) The product of two rational numbers is rational.

Answer

True , as we can infer from the following.

2(a rational) × 2(a rational) = 4(a rational)
4(a rational) × (−5)(a rational) = −20(a rational)

$\frac{2}{3} (a rational) \times \frac{3}{5} (a rational) = \frac{2}{5} (a rational)$

(iv) The product of two irrational numbers is irrational.

Answer

False, as we can infer from the following.

$\sqrt{2}$ (an irrational) × $\sqrt{18}$ (an irrational) = $\sqrt{2 \times 18}$ = $\sqrt{36}$ = 6 (a rational).

$\sqrt{3}$ (an irrational) × $\sqrt{12}$ (an irrational) = $\sqrt{3 \times 12}$ = $\sqrt{36}$ = 6 (a rational).

$\sqrt{7}$ (an irrational) × $\sqrt{28}$ (an irrational) = $\sqrt{7 \times 28}$ = $\sqrt{196}$ = 14 (a rational).

(v) The sum of a rational number and an irrational number is irrational.

Answer

True, as we can infer from the following.

2 is rational and $\sqrt{3}$ is an irrational. $2 + \sqrt{3}$ is an irrational number.

5 is rational and $\sqrt{3}$ is an irrational. $2 + \sqrt{3}$ is an irrational number.

(vi) The product of a nonzero rational number and an irrational number is a rational number.

Answer

False, as we can infer from the following.

2 (a rational) × $\sqrt{3}$ (an irrational) = $2 \sqrt{3}$ (an irrational). A single example is sufficient to validate the statement.

(vii) Every real number is rational.

Answer

False, as all irrational numbers are real numbers but are not rationals. For example
$\sqrt{2}$ is a real number but not a rational.

(viii) Every real number is either rational or irrational.

Answer

True , as the set of real numbers is the result of combining the sets of rational and irrational numbers.

(ix) $π$ is irrational and $\frac{22}{7}$ is rational.

Answer

True, as $π = \frac{Circumference of a circle}{Diameter of the circle}$ is an irrational number and its value is approximately equal to $\frac{22}{7}$ .

In other words, the value of $π$ is not exactly equal to $\frac{22}{7}$ .

Exercise 1C RS Aggarwal Class 9 Solutions

1. What are irrational numbers? How do they differ from rational numbers? Give examples.

Answer

2. Classify the following numbers as rational or irrational. Give reasons to support your answer.

(i) 381

Answer

Clearly, 381 is an irrational number. Reason: A rational number divided by an irrational number is irrational.

(ii) 361

Answer

(iii) 21

Answer

(iv) 1.44

Answer

(v) 236

Answer

(vi) 4.1276

Answer

(vii) 227

Answer

(viii) 1.232332333 . . .

Answer

(ix) 3.040040004 . . .

Answer

(x) 2.356565656 . . .

Answer

(xi) 6.834834 . . .

Answer

3. Let x be a rational number and y be an irrational number. Is x+y necessarily an irrational number? Give an example in support of your answer.

Answer

4. Let a be a rational number and b be an irrational number. Is ab necessarily an irrational number? Justify your answer with an example.

Answer

5. Is the product of two irrationals always irrational? Justify your answer

Answer

6. Give an example of two irrational numbers whose

(i) difference is an irrational number.

Answer

(ii) difference is a rational number.

Answer

(iii) sum is an irrational number.

Answer

(iv) sum is a rational number.

Answer

(v) product is an irrational number.

Answer

(vi) product is a rational number.

Answer

(vii) quotient is an irrational number.

Answer

(viii) quotient is a rational number.

Answer

7. Examine whether the following numbers are rational or irrational.

(i) 3+3

Answer

(ii) 7−2

Answer

(iii) 53×253

Answer

(iv) 7×343

Answer

(v) 13117

Answer

(vi) 8×2

Answer

8. Insert a rational and an irrational number between 2 and 2.5.

Answer

9. How many irrational numbers lie between 2 and 3? Find any three irrational numbers lying between 2 and 3.

Answer

10. Find two rational and two irrational numbers between 0.5 and 0.55.

Answer

11. Find three different irrational numbers between the rational numbers 57 and 911.

Answer

12. Find two rational numbers of the form pq between the numbers 0.2121121112... and 0.2020020002... .

Answer

13. Find two irrational numbers between 0.16 and 0.17.

Answer

14. State, in each case, whether the given statement is true or false.

(i) The sum of two rational numbers is rational.

Answer

(ii) The sum of two irrational numbers is irrational.

Answer

(i) $\sqrt{\frac{3}{81}}$

Clearly, $\sqrt{\frac{3}{81}}$ is an irrational number.
Reason: A rational number divided by an irrational number is irrational.

(ii) $\sqrt{361}$

(iii) $\sqrt{21}$

(iv) $\sqrt{1.44}$

(v) $\frac{2}{3} \sqrt{6}$

(vi) $4.1276$

(vii) $\frac{22}{7}$

(viii) $1.232332333 . . .$

(ix) $3.040040004 . . .$

(x) $2.356565656 . . .$

(xi) $6.834834 . . .$

3. Let $x$ be a rational number and $y$ be an irrational number. Is $x + y$ necessarily an irrational number? Give an example in support of your answer.

4. Let $a$ be a rational number and $b$ be an irrational number. Is $a b$ necessarily an irrational number? Justify your answer with an example.

(i) $3 + \sqrt{3}$

(ii) $\sqrt{7} - 2$

(iii) $\sqrt[3]{5} \times \sqrt[3]{25}$

(iv) $\sqrt{7} \times \sqrt{343}$

(v) $\sqrt{\frac{13}{117}}$

(vi) $\sqrt{8} \times \sqrt{2}$

9. How many irrational numbers lie between $\sqrt{2}$ and $\sqrt{3}$ ? Find any three irrational numbers lying between $\sqrt{2}$ and $\sqrt{3}$ .

11. Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$ .

12. Find two rational numbers of the form $\frac{p}{q}$ between the numbers 0.2121121112... and 0.2020020002... .

(ix) $π$ is irrational and $\frac{22}{7}$ is rational.