# Number Systems (Chapter 1, Exercise 1.1) – Class 9 Maths Solutions) NCERT

## Question 1: Is zero a Rational Number? Can you write it in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne0$?

Answer: Yes, zero $(0)$ is a Rational Number. The Number zero $(0)$ can be written in the form $\frac{p}{q}$.

Explanation:

To understand whether zero is a Rational Number or not, let us consider the following examples,

$\frac{0}{1}=0$ [here, $q=1$]

$\frac{0}{5}=0$ [here, $q=5]$

$\frac{0}{-5}=0$ [here, $q=-5$ ]

$\frac{0}{10}=0$ [here, $q=10$ ]

In all of the above examples, $p$ and $q$ are integers and the value of $q$ is not equal to zero $(q\ne 0)$ . Therefore, it can be concluded that zero $(0)$ is a Rational Number.

## Question 2: Find six rational numbers between 3 and 4?

Answer: The six Rational Numbers between $3$ and $4$ are: $\frac{31}{10},\, \frac{32}{10},\, \frac{33}{10},\, \frac{34}{10},\, \frac{35}{10},\, \frac{36}{10}$

Explanation:

Since, $3$ and $4$ are the two consecutive numbers, the rational numbers between them will be fractions of decimals.

So six decimals between $3$ and $4$ will be: $3.1,\,3.2,\,3.3,\,3.4,\,3.5,\,3.6$

No we can convert the above decimals into fractions as:

$3.1=\frac{3.1}{1}=\frac{3.1\times 10}{1\times 10}=\frac{31}{10}$

$3.2=\frac{3.2}{1}=\frac{3.2\times 10}{1\times 10}=\frac{32}{10}$

$3.3=\frac{3.3}{1}=\frac{3.3\times 10}{1\times 10}=\frac{33}{10}$

$3.4=\frac{3.4}{1}=\frac{3.4\times 10}{1\times 10}=\frac{34}{10}$

$3.5=\frac{3.5}{1}=\frac{3.5\times 10}{1\times 10}=\frac{35}{10}$

$3.6=\frac{3.6}{1}=\frac{3.6\times 10}{1\times 10}=\frac{36}{10}$

## Question 3: Find five rational numbers $\frac{3}{5}$ and $\frac{4}{5}$?

Answer: The five Rational Numbers between $\frac{3}{5}$ and $\frac{4}{5}$ are: $\frac{31}{50},\, \frac{13}{20},\, \frac{7}{10},\, \frac{18}{25}$ and $\frac{3}{4}$.

Explanation:

To find five rational numbers between $\frac{3}{5}$ and $\frac{4}{5}$, convert the fractions into decimals first.

$\frac{3}{5}=0.6$ and $\frac{4}{5}=0.8$

Now, we can find several Rational Numbers between $0.6$ and $0.8$. Such $5$ Rational Numbers will be,

$0.62,\, 0.65,\, 0.70,\, 0.72,\, 0.75$

Convert the above decimals into fractions,

$0.62=\frac{0.62}{1}=\frac{0.62\times 100}{1\times 100}=\frac{62}{100}=\frac{31}{50}$

$0.65=\frac{0.65}{1}=\frac{0.65\times 100}{1\times 100}=\frac{65}{100}=\frac{13}{20}$

$0.70=\frac{0.70}{1}=\frac{0.70\times 100}{1\times 100}=\frac{70}{100}=\frac{7}{10}$

$0.72=\frac{0.72}{1}=\frac{0.72\times 100}{1\times 100}=\frac{72}{100}=\frac{18}{25}$

$0.75=\frac{0.75}{1}=\frac{0.75\times 100}{1\times 100}=\frac{75}{100}=\frac{3}{4}$

## Question 4: State whether the following statements are true or false. Give reasons for your answers.

### i. Every Natural Number is a Whole Number.

Explanation:

The set of Natural Numbers are, $N = \{1, 2, 3, 4, 5, 6, 7, 8, 9 ……….\}$

The set of Whole Numbers are, $W=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ……….\}$ From above, it is clear that all of the Natural Numbers lie in the set of Whole Numbers and the Whole Numbers has one extra element $0$. Therefore, it can be said that every Natural Number is a Whole Number.

### ii. Every Integer is a Whole Number.

Explanation:

The Set of Integers are, $Z=\{……, -3, -2, -1, 0, 1, 2, 3, ……\}$

The set of Whole Numbers are, $W=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ……….\}$ From the above sets of Integers and Whole Numbers it is clear that the integers have both negative and positive numbers. However, the Whole numbers have only positive numbers along with $0$. Therefore, the negative integers are not the part of Whole Numbers. Hence, it may be concluded that every Integer is not a Whole Number.

### iii. Every Rational Number is a Whole Number.

Explanation:

The set of Whole Numbers is given by, $W=\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ……….\}$

We now that the Rational Numbers are the numbers that can be expressed in terms of $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$. Therefore, the Rational Numbers will also include fractions. Some rational numbers are:

$\frac{4}{2}=2,\, \frac{18}{3}=6,\, \frac{1}{2},\, \frac{2}{3},\, \frac{4}{7},\, \frac{-1}{5},\, \frac{2}{-5},\, \frac{3}{20}$ etc.

From above examples, it is clear that some of the Rational Numbers are not the Part of the Whole Numbers. Therefore, it can be concluded that Every Rational Number is Not a Whole Number.

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