Exercise-1.1, Class 9th, Maths, Chapter 1, NCERT

Q1. Is zero a rational number? Can you write it in the form $p\mathrm{/}q$, where $p$ and $q$ are integers and $q\mathrm{\ne }0$?

Solution.
Yes. Zero is a rational number because it can be written as a ratio of two integers with nonzero denominator. For example,

$$0=\frac{0}{1}=\frac{0}{2}=\frac{0}{-5},$$

etc., where the numerator $p=0$ and the denominator $q$ is any nonzero integer.

Answer: Yes; e.g. $0={\displaystyle \frac{0}{1}}$

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Q2. Find six rational numbers between $3$ and $4$.

Solution.
One easy way is to write $3$ and $4$ with a common denominator and pick fractions strictly between them. Using denominator $10$:

$$3=\frac{30}{10},4=\frac{40}{10}\mathrm{.}$$

Thus the fractions ${\displaystyle \frac{31}{10}},{\displaystyle \frac{32}{10}},{\displaystyle \frac{33}{10}},{\displaystyle \frac{34}{10}},{\displaystyle \frac{35}{10}},{\displaystyle \frac{36}{10}}$ all lie between $3$ and $4$.

(You can simplify some of them: $\frac{32}{10}=\frac{16}{5}$, $\frac{35}{10}=\frac{7}{2}$, etc.)

Answer (one possible list):

$$\frac{31}{10},\frac{32}{10},\frac{33}{10},\frac{34}{10},\frac{35}{10},\frac{36}{10}\mathrm{.}$$

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Q3. Find five rational numbers between ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{4}{5}}$.

Solution.
Write ${\displaystyle \frac{3}{5}}$ and ${\displaystyle \frac{4}{5}}$ with a common (larger) denominator. Using denominator $30$:

$$\frac{3}{5}=\frac{18}{30},\frac{4}{5}=\frac{24}{30}\mathrm{}$$

The fractions strictly between them are ${\displaystyle \frac{19}{30}},{\displaystyle \frac{20}{30}},{\displaystyle \frac{21}{30}},{\displaystyle \frac{22}{30}},{\displaystyle \frac{23}{30}}$. (Some can be simplified, e.g. $\frac{20}{30}=\frac{2}{3}$)

Answer (one possible list):

$$\frac{19}{30},\frac{20}{30},\frac{21}{30},\frac{22}{30},\frac{23}{30}\mathrm{.}$$

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Q4. State whether the following statements are true or false. Give reasons for your answers.

(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Solution.

(i) True. Whole numbers = $\{0,1,2,3,\dots \}$. Natural numbers (as used in the book) are $\{1,2,3,\dots \}$. Every natural number is in the set of whole numbers.

(ii) False. Integers include negative numbers (e.g. $-1,-2,\dots$), but whole numbers do not include negatives. For example, $-3$ is an integer but not a whole number.

(iii) False. Rational numbers include fractions like $\frac{1}{2},\frac{3}{4},$ etc., which are not whole numbers. For example, $\frac{1}{2}$ is rational but not a whole number.

Answers: (i) True. (ii) False. (iii) False.