Rational and Irrational Numbers — Exercise 1.1

Insert a rational number between $\dfrac{2}{9}$ and $\dfrac{3}{8}$ arrange in descending order.

Answer

The L.C.M of 9 and 8 is 72.

$\dfrac{2}{9} = \dfrac{2 \times 8}{9 \times 8} = \dfrac{16}{72} \\[0.5em] \dfrac{3}{8} = \dfrac{3 \times 9}{8 \times 9} = \dfrac{27}{72} \\[0.5em] \text{Since } 16 \lt 27, \dfrac{2}{9} \lt \dfrac{3}{8}$

A rational number between $\dfrac{2}{9}$ and $\dfrac{3}{8}$

$= \dfrac{\dfrac{2}{9} + \dfrac{3}{8}}{2} \\[0.5em] = \dfrac{\dfrac{16 + 27}{72}}{2} \\[0.5em] = \bold{\dfrac{43}{144}} \\[0.5em]$

Numbers in descending order are:

$\bold{\dfrac{3}{8}}, \bold{\dfrac{43}{144}}, \bold{\dfrac{2}{9}}$

Insert two rational numbers between $\dfrac{1}{3}$ and $\dfrac{1}{4}$ and arrange in ascending order.

Answer

The L.C.M of 3 and 4 is 12.

$\dfrac{1}{3} = \dfrac{1 \times 4}{3 \times 4} = \dfrac{4}{12} \\[0.5em] \dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12} \\[0.5em] \text{Since } 3 \lt 4, \dfrac{1}{4} \lt \dfrac{1}{3}$

A rational number between $\dfrac{1}{4}$ and $\dfrac{1}{3}$

$= \dfrac{\dfrac{1}{4} + \dfrac{1}{3}}{2} \\[0.5em] = \dfrac{\dfrac{4 + 3}{12}}{2} \\[0.5em] = \bold{\dfrac{7}{24}} \\[0.5em]$

A rational number between $\dfrac{1}{4}$ and $\dfrac{7}{24}$

$= \dfrac{\dfrac{1}{4} + \dfrac{7}{24}}{2} \\[0.5em] = \dfrac{\dfrac{6 + 7}{24}}{2} \\[0.5em] = \bold{\dfrac{13}{48}} \\[0.5em]$

Numbers in ascending order are:

$\bold{\dfrac{1}{4}}, \bold{\dfrac{13}{48}}, \bold{\dfrac{7}{24}}, \bold{\dfrac{1}{3}}$

Insert two rational numbers between $-\dfrac{1}{3}$ and $-\dfrac{1}{2}$ and arrange in ascending order.

Answer

The L.C.M of 2 and 3 is 6.

$-\dfrac{1}{3} = -\dfrac{1 \times 2}{3 \times 2} = -\dfrac{2}{6} \\[0.5em] -\dfrac{1}{2} = -\dfrac{1 \times 3}{2 \times 3} = -\dfrac{3}{6} \\[0.5em] \text{Since } -3 \lt -2, -\dfrac{1}{2} \lt -\dfrac{1}{3}$

A rational number between -$\dfrac{1}{2}$ and -$\dfrac{1}{3}$

$= \dfrac{-\dfrac{1}{2} + \Big(-\dfrac{1}{3}\Big)}{2} \\[0.5em] = \dfrac{\dfrac{-3 + (-2)}{6}}{2} \\[0.5em] = \dfrac{\dfrac{-3 -2}{6}}{2} \\[0.5em] = \bold{-\dfrac{5}{12}} \\[0.5em]$

A rational number between -$\dfrac{1}{2}$ and -$\dfrac{5}{12}$

$= \dfrac{-\dfrac{1}{2} + \Big(-\dfrac{5}{12}\Big)}{2} \\[0.5em] = \dfrac{\dfrac{-6 + (-5)}{12}}{2} \\[0.5em] = \dfrac{\dfrac{-6 -5}{12}}{2} \\[0.5em] = \bold{-\dfrac{11}{24}} \\[0.5em]$

Numbers in ascending order are:

$\bold{-\dfrac{1}{2}}, \bold{-\dfrac{11}{24}}, \bold{-\dfrac{5}{12}}, \bold{-\dfrac{1}{3}}$

Insert three rational numbers between $\dfrac{1}{3}$ and $\dfrac{4}{5}$, and arrange in descending order.

Answer

The L.C.M of 3 and 5 is 15.

$\dfrac{1}{3} = \dfrac{1 \times 5}{3 \times 5} = \dfrac{5}{15} \\[0.5em] \dfrac{4}{5} = \dfrac{4 \times 3}{5 \times 3} = \dfrac{12}{15} \\[0.5em] \text{Since } 5 \lt 12, \dfrac{1}{3} \lt \dfrac{4}{5}$

A rational number between $\dfrac{1}{3}$ and $\dfrac{4}{5}$

$= \dfrac{\dfrac{1}{3} + \dfrac{4}{5}}{2} \\[0.5em] = \dfrac{\dfrac{5 + 12}{15}}{2} \\[0.5em] = \bold{\dfrac{17}{30}} \\[0.5em]$

A rational number between $\dfrac{1}{3}$ and $\dfrac{17}{30}$

$= \dfrac{\dfrac{1}{3} + \dfrac{17}{30}}{2} \\[0.5em] = \dfrac{\dfrac{10 + 17}{30}}{2} \\[0.5em] = \bold{\dfrac{27}{60}} \\[0.5em]$

A rational number between $\dfrac{17}{30}$ and $\dfrac{4}{5}$

$= \dfrac{\dfrac{17}{30} + \dfrac{4}{5}}{2} \\[0.5em] = \dfrac{\dfrac{17 + 24}{30}}{2} \\[0.5em] = \bold{\dfrac{41}{60}} \\[0.5em]$

Numbers in descending order are:

$\bold{\dfrac{4}{5}}, \bold{\dfrac{41}{60}}, \bold{\dfrac{17}{30}}, \bold{\dfrac{27}{60}}, \bold{\dfrac{1}{3}}$

Using concept of decimals, insert

(i) three rational numbers between 3 and 3.5

(ii) four rational numbers between $\dfrac{1}{4}$ and $\dfrac{2}{5}$.

(iii) five rational numbers between $1\dfrac{1}{2}$ and $1\dfrac{3}{4}$.

Answer

(i) We want three rational numbers between 3 and 3.5.

We know that,

Terminating decimals are rational numbers.

Let us take 3.1, 3.2, 3.3

Thus, we have :

⇒ 3 < 3.1 < 3.2 < 3.3 <3.5

Hence, three rational numbers between 3 and 3.5 are 3.1, 3.2 and 3.3.

(ii) Converting fraction in decimal form,

$\dfrac{1}{4}$ = 0.25

$\dfrac{2}{5}$ = 0.40

We know that,

Terminating decimals are rational numbers.

Let us take 0.28, 0.30, 0.32, 0.34

Thus, we have :

⇒ 0.25 < 0.28 < 0.30 < 0.32 < 0.34 < 0.40

Hence, four rational numbers between $\dfrac{1}{4} \text{ and } \dfrac{2}{5}$ = 0.28, 0.30, 0.32 and 0.34

(iii) We want five rational numbers between $1\dfrac{1}{2}$ and $1\dfrac{3}{4}$.

Numbers : $1\dfrac{1}{2} = \dfrac{3}{2} = 1.50$ and $1\dfrac{3}{4} = \dfrac{7}{4} = 1.75$

We know that,

Terminating decimals are rational numbers.

Let us take 1.61, 1.62, 1.63, 1.64, 1.65

Thus, we have :

⇒ 1.50 < 1.61 < 1.62 < 1.63 <1.64 < 1.65 <1.75

Hence, five rational numbers between $1\dfrac{1}{2} \text{ and } 1\dfrac{3}{4}$ = 1.61, 1.62, 1.63, 1.64 and 1.65.

Find six rational numbers between 3 and 4.

Answer

The numbers 3 and 4 can be written as $\dfrac{3}{1}$ and $\dfrac{4}{1}$.

Since we want to find six rational numbers between the given numbers, multiplying the numerator and denominator of the above numbers by 6 + 1 i.e. by 7, we get $\dfrac{21}{7}$ and $\dfrac{28}{7}$, which are equivalent to the given numbers.

$\text{As } 21 \lt 22 \lt 23 \lt 24 \lt 25 \lt 26 \lt 27 \lt 28, \\[0.7em] \dfrac{21}{7} \lt \dfrac{22}{7} \lt \dfrac{23}{7} \lt \dfrac{24}{7} \lt \dfrac{25}{7} \lt \dfrac{26}{7} \lt \dfrac{27}{7} \lt \dfrac{28}{7}$

Therefore, six rational numbers between 3 and 4 are:

$\bold{\dfrac{22}{7}}, \bold{\dfrac{23}{7}}, \bold{\dfrac{24}{7}}, \bold{\dfrac{25}{7}}, \bold{\dfrac{26}{7}}, \bold{\dfrac{27}{7}}$

Find five rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$.

Answer

Since we want to find five rational numbers between the given numbers, multiplying the numerator and denominator of the above numbers by 5 + 1 i.e. by 6, we get $\dfrac{18}{30}$ and $\dfrac{24}{30}$, which are equivalent to the given numbers.

$\text{As } 18 \lt 19 \lt 20 \lt 21 \lt 22 \lt 23 \lt 24, \\[0.7em] \dfrac{18}{30} \lt \dfrac{19}{30} \lt \dfrac{20}{30} \lt \dfrac{21}{30} \lt \dfrac{22}{30} \lt \dfrac{23}{30} \lt \dfrac{24}{30} \\[0.7em] \Rightarrow \dfrac{3}{5} \lt \dfrac{19}{30} \lt \dfrac{2}{3} \lt \dfrac{7}{10} \lt \dfrac{11}{15} \lt \dfrac{23}{30} \lt \dfrac{4}{5}$

Therefore, six rational numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$ are:

$\bold{\dfrac{19}{30}}, \bold{\dfrac{2}{3}}, \bold{\dfrac{7}{10}}, \bold{\dfrac{11}{15}}, \bold{\dfrac{23}{30}}$

Find ten rational numbers between $-\dfrac{2}{5}$ and $\dfrac{1}{7}$.

Answer

Writing the given numbers with same denominator 35 (L.C.M. of 5 and 7), we get:

$-\dfrac{2}{5} = -\dfrac{14}{35} \\[0.5em] \dfrac{1}{7} = \dfrac{5}{35}$

$\text{As } -14 \lt -13 \lt -12 \lt -11 \lt -10 \lt -9 \lt -8 \lt -7 \lt 0 \lt 1 \lt 2 \lt 5, \\[0.7em] -\dfrac{14}{35} \lt -\dfrac{13}{35} \lt -\dfrac{12}{35} \lt -\dfrac{11}{35} \lt -\dfrac{10}{35} \lt -\dfrac{9}{35} \lt -\dfrac{8}{35} \lt -\dfrac{7}{35} \lt 0 \lt \dfrac{1}{35} \lt \dfrac{2}{35} \lt \dfrac{5}{35} \\[0.7em] \Rightarrow -\dfrac{2}{5} \lt -\dfrac{13}{35} \lt -\dfrac{12}{35} \lt -\dfrac{11}{35} \lt -\dfrac{2}{7} \lt -\dfrac{9}{35} \lt -\dfrac{8}{35} \lt -\dfrac{1}{5} \lt 0 \lt \dfrac{1}{35} \lt \dfrac{2}{35} \lt \dfrac{5}{35}$

Therefore, ten rational numbers between $-\dfrac{2}{5}$ and $\dfrac{1}{7}$ are:

$\bold{-\dfrac{13}{35}}, \bold{-\dfrac{12}{35}}, \bold{-\dfrac{11}{35}}, \bold{-\dfrac{2}{7}}, \bold{-\dfrac{9}{35}}, \\[0.5em] \bold{-\dfrac{8}{35}}, \bold{-\dfrac{1}{5}}, 0, \bold{\dfrac{1}{35}}, \bold{\dfrac{2}{35}}$

Find six rational numbers between $\dfrac{1}{2}$ and $\dfrac{2}{3}$.

Answer

Writing the given numbers with same denominator 6 (L.C.M. of 2 and 3), we get:

$\dfrac{1}{2} = \dfrac{3}{6} \\[0.5em] \dfrac{2}{3} = \dfrac{4}{6}$

Since we want to find six rational numbers between the given numbers, multiplying the numerator and denominator of the above numbers by 6 + 1 i.e. by 7, we get $\dfrac{21}{42}$ and $\dfrac{28}{42}$, which are equivalent to the given numbers.

$\text{As } 21 \lt 22 \lt 23 \lt 24 \lt 25 \lt 26 \lt 27 \lt 28, \\[0.7em] \dfrac{21}{42} \lt \dfrac{22}{42} \lt \dfrac{23}{42} \lt \dfrac{24}{42} \lt \dfrac{25}{42} \lt \dfrac{26}{42} \lt \dfrac{27}{42} \lt \dfrac{28}{42} \\[0.7em] \Rightarrow \dfrac{1}{2} \lt \dfrac{11}{21} \lt \dfrac{23}{42} \lt \dfrac{4}{7} \lt \dfrac{25}{42} \lt \dfrac{13}{21} \lt \dfrac{9}{14} \lt \dfrac{2}{3}$

Therefore, six rational numbers between $\dfrac{1}{2}$ and $\dfrac{2}{3}$ are:

$\bold{\dfrac{11}{21}}, \bold{\dfrac{23}{42}}, \bold{\dfrac{4}{7}}, \bold{\dfrac{25}{42}}, \bold{\dfrac{13}{21}}, \bold{\dfrac{9}{14}}$