Is 9 an irrational number?

No, because 9=±3 which are integers. So, 9 is not an irrational number.

No, π is approximately equal to 227. It is because π is an irrational number whereas 227 is a rational number.

Exercise 1A RS Aggarwal Class 9 Number Systems Best Solutions

Q: Is π a rational number?

No, π is not a rational number. It is an irrational number. The reason is that π is the ratio of circumference of a circle to the diameter of the circle. And this ratio always comes out to be an irrational number 3.1415926535... i.e. a non-terminating non-recurring decimal.

Q: What does every point on the number line represents?

Every point on the number represents a unique real number.

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 1A RS Aggarwal Class 9 of chapter 1 Number Systems contains 9 questions. The questions are based on the following topics:

Natural Numbers
Whole Numbers
Integers
Representation of integers on Number Line
Rational Numbers
Representation of Rational Numbers on Real Line
Finding Rational Numbers between two given Rational Numbers

Exercise 1A RS Aggarwal Class 9 Solutions

Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9

The first question of Exercise 1A RS Aggarwal Class 9 is about zero.

1. Is zero a rational number? Justify.

Answer

Yes, zero is a rational number.
We can write 0 in the form $\frac{p}{q}$ as follows:
$0 = < \frac{0}{1}$ , where $p = 0$ and $q = 1$ . Hence, justified.

The second question of Exercise 1A RS Aggarwal Class 9 is about representing rational numbers on the number line.

2. Represent each of the following rational numbers on the number line:

(i) $\frac{5}{7}$

Answer

$\frac{5}{7}$ is a proper rational number. It lies between 0 and 1. The steps to be followed in order to represent $\frac{5}{7}$ are

First Step: We draw a number line.

Second Step: Since $\frac{5}{7}$ lies between 0 and 1, we divide the distance between 0 and 1 into 7 equal parts. 7 is the denominator of the fraction.

Third Step: The numerator 5 of the fraction $\frac{5}{7}$ denotes the exact position of $\frac{5}{7}$ from 0.

(ii) $\frac{8}{3}$

Answer

Here, $\frac{8}{3}$ is an improper rational number, so first we convert it into a mixed rational number.
$\frac{8}{3} = 2 \frac{2}{3}$ .
∴ $\frac{8}{3}$ lies between 2 and 3.

(iii) $- \frac{23}{6}$

Answer

Here, we have $- \frac{23}{6} = - 3 \frac{5}{6}$ .
Clearly, it lies between −3 and −4.

(iv) 1.3

Answer

∵ 1.3 = $\frac{13}{10} = 1 \frac{3}{10}$
Clearly, it lies between 1 and 2.

(v) −2.4

Answer

∵ −2.4 = $- \frac{24}{10} = −2 \frac{4}{10}$ .
Clearly, it lies between −2 and −3.

The third question of Exercise 1A RS Aggarwal Class 9 is about finding a rational number between two rational numbers.

3. Find a rational number between

(i) $\frac{3}{8}$ and $\frac{2}{5}$

Answer

A rational number between $\frac{3}{8}$ and $\frac{2}{5}$ = $\frac{\frac{3}{8} + \frac{2}{5}}{2} = \frac{\frac{15 + 16}{40}}{2} = \frac{\frac{31}{40}}{2} = \frac{31}{40}$ .

(ii) 1.3 and 1.4

Answer

A rational number between 1.3 and 2.4 = $\frac{1.3 + 1.4}{2} = \frac{2.7}{2} = 1.35$ .

(iii) −1 and $\frac{1}{2}$

Answer

A rational number between −1 and $\frac{1}{2}$ is given by
$\frac{- 1 + \frac{1}{2}}{2} = \frac{\frac{- 1}{2}}{2} = \frac{- 1}{4}$ .

(iv) $- \frac{3}{4}$ and $- \frac{2}{5}$

Answer

A rational number between $- \frac{3}{4}$ and $- \frac{2}{5}$ is given by
$\frac{- \frac{3}{4} + (- \frac{2}{5})}{2} = \frac{- \frac{3}{4} - \frac{2}{5}}{2} = \frac{\frac{- 15 - 8}{20}}{2} = \frac{- 23}{40}$ .

(v) $\frac{1}{9}$ and $\frac{2}{9}$

Answer

A rational number between $\frac{1}{9}$ and $\frac{2}{9}$ is given by
$\frac{\frac{1}{9} + \frac{2}{9}}{2} = \frac{\frac{1 + 2}{9}}{2} = \frac{\frac{3}{9}}{2} = \frac{3}{18} = \frac{1}{6}$ .

The fourth question of Exercise 1A RS Aggarwal Class 9 is about finding three rational numbers between two rational numbers

4. Find three rational numbers lying between $\frac{3}{5}$ and $\frac{7}{8}$ .
How many rational numbers can be determined between these two numbers.

Answer

First Method

In this method, we add the two rational numbers and divide the result by 2 to get one rational number exactly in the middle of the two rational numbers.

First rational number between $\frac{3}{5}$ and $\frac{7}{8}$ = $\frac{\frac{3}{5} + \frac{7}{8}}{2} = \frac{\frac{24 + 35}{40}}{2} = \frac{59}{80}$

2nd rational number between $\frac{3}{5}$ and $\frac{7}{8}$ = $\frac{\frac{3}{5} + \frac{59}{80}}{2} = \frac{\frac{48 + 59}{80}}{2} = \frac{107}{160}$

3rd rational number between $\frac{3}{5}$ and $\frac{7}{8}$ = $\frac{\frac{59}{80} + \frac{7}{8}}{2} = \frac{\frac{59 + 70}{80}}{2} = \frac{129}{160}$

Second Method

The given rational numbers are $\frac{3}{5}$ and $\frac{7}{8}$ .
Firstly, we find the LCM of the denominators of the given rational numbers as we want to make the denominators same.

We can write $\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}$
and $\frac{7}{8} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}$ .
Now, we have $\frac{24}{40} < \frac{25}{40} < \frac{26}{40} < \frac{27}{40}$ ... $< \frac{34}{40} < \frac{35}{40}$ .
We can choose any three rational numbers from the above list of rational numbers.
Hence, the three rational numbers between $\frac{3}{5}$ and $\frac{7}{8}$ are
$\frac{25}{40}, \frac{26}{40}$ and $\frac{27}{40}$
After simplification: $\frac{5}{8}$ , $\frac{13}{20}$ and $\frac{27}{40}$

The fifth question of Exercise 1A RS Aggarwal Class 9 is about finding four rational numbers between two rational numbers.

5. Find four rational numbers between $\frac{3}{7}$ and $\frac{5}{7}$

Answer

The given rational numbers are $\frac{3}{7}$ and $\frac{5}{7}$ . Their denominators are equal and the difference between the numerators 3 and 5 is only two. So, by now only one rational number can be obtained easily. That rational number is $\frac{4}{7}$ . But we have to find four rational numbers. So, to increase the difference between numerators of the given numbers, we will do the following:
$\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}$ and
$\frac{5}{7} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21}$ .
Now, the gap between 9 and 15 is appropriate to find four rational numbers.
∴ $\frac{9}{21} < \frac{10}{21} < \frac{11}{21} < \frac{12}{21} < \frac{13}{21} < \frac{14}{21} < \frac{15}{21}$ .
Hence, four rational numbers between $\frac{3}{7}$ and $\frac{5}{7}$ are
$\frac{10}{21}, \frac{11}{21}, \frac{12}{21}, \frac{13}{21}$ .

The sixth question of Exercise 1A RS Aggarwal Class 9 is about finding six rational numbers between two given rational numbers.

6. Find six rational numbers between 2 and 3.

Answer

We know that 2 = $\frac{2}{1}$ and 3 = $\frac{3}{1}$ .
∵ The difference between the numerators is 1 and the denominator is also same on both. So, we have to multiply by 7 in the numerator and denominator of both 2 and 3 to create the appropriate difference between the numbers.
We can write
$\frac{2}{1} = \frac{2 \times 7}{1 \times 7} = \frac{14}{7}$ and
$\frac{3}{1} = \frac{3 \times 7}{1 \times 7} = \frac{21}{7}$ .
Now, the gap between the numerators 14 and 21 is appropriate to find six rational numbers easily.
∵ $\frac{14}{7} < \frac{15}{7} < \frac{16}{7} < \frac{17}{7} < \frac{18}{7} < \frac{19}{7} < \frac{20}{7} < \frac{21}{7}$ .
∴ The six rational numbers between 2 and 3 are
$\frac{15}{7}, \frac{16}{7}, \frac{17}{7}, \frac{18}{7}, \frac{19}{7}, \frac{20}{7}$ .

The seventh question of Exercise 1A RS Aggarwal Class 9 is about finding five rational numbers between two given rational numbers.

7. Find five rational numbers between $\frac{3}{5}$ and $\frac{2}{3}$ .

Answer

The given rational numbers are $\frac{3}{5}$ and $\frac{2}{3}$ .
The denominators are not equal. We will first make denominators equal and then we will try to find rational number by finding the difference between numerators of the given rational numbers.
∵ LCM of 3 and 5 = 15. We can write
$\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$ and $\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$ .
Since the gap between the numerators is 1, we cannot find a rational number directly.
So, We will create gap between the numerators by multiplying with a suitable number.
Multiplying $\frac{9}{15}$ by 6 (5+1) in both numerator and denominator. We get,
$\frac{9}{15} = \frac{9 \times 6}{15 \times 6} = \frac{54}{90}$ .
Multiplying $\frac{10}{15}$ by 6 in both numerator and denominator. We get,
$\frac{10}{15} = \frac{10 \times 6}{15 \times 6} = \frac{60}{90}$ .
Now, the denominators are same but the numerators have appropriate gap.
∵ $\frac{54}{90} < \frac{55}{90} < \frac{56}{90} < \frac{57}{90} < \frac{58}{90} < \frac{59}{90} < \frac{60}{90}$ .
After cancelling the common factors, we get
$\frac{54}{90} < \frac{11}{18} < \frac{28}{45} < \frac{19}{30} < \frac{29}{45} < \frac{59}{90} < \frac{60}{90}$ .
Hence the five rational numbers between $\frac{3}{5}$ and $\frac{2}{3}$ are
$\frac{11}{18}, \frac{28}{45}, \frac{19}{30}, \frac{29}{45}, \frac{59}{90}$ .

The eighth question of Exercise 1A RS Aggarwal Class 9 is about finding sixteen rational numbers between two given rational numbers.

8. Insert 16 rational numbers between 2.1 and 2.2.

Answer

We know that 2.1 = 2.100 and 2.2 = 2.200.
Clearly, there is a gap of 100 between the decimal part of each number. So, we can divide this gap into 16 parts roughly to get 16 rational numbers.
We can take a difference of 0.05 for the sake of simplicity.
We can write 2.1 = 2.100 < 2.105 < 2.110 < 2.115 < 2.120 < 2.125 < 2.130 < 2.135 < 2.140 < 2.145 < 2.150 < 2.155 < 2.160 < 2.165 < 2.170 < 2.175 < 2.180 < 2.185 < 2.190 < 2.195 < 2.200 = 2.2.
Hence, the sixteen rational numbers between 2.1 and 2.2 are as follows:
2.105, 2.115, 2.120, 2.125, 2.130, 2.135, 2.140, 2.145, 2.150, 2.155, 2.160, 2.165, 2.170, 2.175, 2.180, 2.185.

The ninth question of Exercise 1A RS Aggarwal Class 9 is a true/false question.

9. State whether the following statements are true or false. Give reasons for your answer.

(i) Every natural number is a whole number.

Answer

True, as all natural numbers: 1, 2, 3, 4, ...,∞ lie in the whole number set W = {0, 1, 2, 3, 4, ...,∞}

(ii) Every whole number is a natural number.

Answer

False, as 0 is a whole number but it is not a natural number.

(iii) Every integer is a whole number.

Answer

False, as negative numbers such as −1, −2, −3, ..., etc. are integers but not whole numbers.

(iv) Every integer is a rational number.

Answer

True, as all integers can be written in the form of $\frac{p}{q}$ . For example, let's take integer 2. We can write 2 = $\frac{2}{1}$ .

(v) Every rational number is an integer.

Answer

False as many rational numbers such as $\frac{1}{2}$ , $\frac{4}{3}$ , $\frac{−2}{7}$ ,..., etc. are not integers.

(vi) Every rational number is a whole number.

Answer

False, as many rational numbers $\frac{5}{7}$ , $\frac{25}{43}$ , $\frac{−73}{7}$ ,..., etc. are not whole numbers.

Frequently Asked Questions

Is $\sqrt{9}$ an irrational number?

No, because $\sqrt{9} = \pm 3$ which are integers. So, $\sqrt{9}$ is not an irrational number.

Is $π$ a rational number?

No, $π$ is not a rational number. It is an irrational number. The reason is that $π$ is the ratio of circumference of a circle to the diameter of the circle. And this ratio always comes out to be an irrational number 3.1415926535... i.e. a non-terminating non-recurring decimal.

Is $π = \frac{22}{7}$ ?

No, $π$ is approximately equal to $\frac{22}{7}$ . It is because $π$ is an irrational number whereas $\frac{22}{7}$ is a rational number.

What does every point on the number line represents?

Every point on the number represents a unique real number.

Exercise 1A RS Aggarwal Class 9 Solutions

1. Is zero a rational number? Justify.

Answer

2. Represent each of the following rational numbers on the number line:

(i) 57

Answer

(ii) 83

Answer

(iii) −236

Answer

(iv) 1.3

Answer

(v) −2.4

Answer

3. Find a rational number between

(i) 38 and 25

Answer

(ii) 1.3 and 1.4

Answer

(iii) −1 and 12

Answer

(iv) −34 and −25

Answer

(v) 19 and 29

Answer

4. Find three rational numbers lying between 35 and 78. How many rational numbers can be determined between these two numbers.

Answer

First Method

Second Method

5. Find four rational numbers between 37 and 57

Answer

6. Find six rational numbers between 2 and 3.

Answer

7. Find five rational numbers between 35 and 23.

Answer

8. Insert 16 rational numbers between 2.1 and 2.2.

Answer

9. State whether the following statements are true or false. Give reasons for your answer.

(i) Every natural number is a whole number.

Answer

(ii) Every whole number is a natural number.

Answer

(iii) Every integer is a whole number.

Answer

(iv) Every integer is a rational number.

Answer

(v) Every rational number is an integer.

Answer

(vi) Every rational number is a whole number.

Answer

Frequently Asked Questions

Is 9 an irrational number?

Is π a rational number?

Is π=227?

What does every point on the number line represents?

(i) $\frac{5}{7}$

(ii) $\frac{8}{3}$

(iii) $- \frac{23}{6}$

(i) $\frac{3}{8}$ and $\frac{2}{5}$

(iii) −1 and $\frac{1}{2}$

(iv) $- \frac{3}{4}$ and $- \frac{2}{5}$

(v) $\frac{1}{9}$ and $\frac{2}{9}$

4. Find three rational numbers lying between $\frac{3}{5}$ and $\frac{7}{8}$ .
How many rational numbers can be determined between these two numbers.

5. Find four rational numbers between $\frac{3}{7}$ and $\frac{5}{7}$

7. Find five rational numbers between $\frac{3}{5}$ and $\frac{2}{3}$ .

Is $\sqrt{9}$ an irrational number?

Is $π$ a rational number?

Is $π = \frac{22}{7}$ ?