Fill in the blanks:

(i) $-\dfrac{7}{8} + \dfrac{2}{11} = \dfrac{2}{11} + \left(-\dfrac{7}{8}\right)$ depicts ……… property of addition of rational numbers.

(ii) The multiplicative inverse of a rational number is also called its ………

(iii) The multiplicative inverse of a negative rational number is a ……… rational number.

(iv) The numbers ……… and ……… are their own reciprocals.

(v) $\left(-\dfrac{10}{11}\right) \div$ ……… = 1

(vi) $\left(-\dfrac{7}{9}\right) \times$ ……… = 1

(vii) Among $-\dfrac{5}{11}, \dfrac{-5}{13} \text{ and } \dfrac{-5}{7}$ the greatest rational number is ………

(i) Here the order of the two rational numbers has been changed but the sum remains the same.

If $\dfrac{a}{b} \text{ and } \dfrac{c}{d}$ are any two rational numbers, then $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{c}{d} + \dfrac{a}{b}$.

Hence, the given statement depicts the commutative property of addition of rational numbers.

(ii) For every non-zero rational number $\dfrac{a}{b}$, the rational number $\dfrac{b}{a}$ satisfies $\dfrac{a}{b} \times \dfrac{b}{a} = 1$

Hence, the multiplicative inverse of a rational number is also called its reciprocal.

(iii) The multiplicative inverse is obtained by interchanging the numerator and the denominator, so the sign of the rational number does not change. For example, the multiplicative inverse of $\dfrac{-2}{9} \text{ is } \dfrac{9}{-2} = \dfrac{-9}{2}$

Hence, the multiplicative inverse of a negative rational number is a negative rational number.

(iv) As 1 × 1 = 1 and (-1) × (-1) = 1, each of these numbers is its own reciprocal.

Hence, the numbers 1 and -1 are their own reciprocals.

(v) For every non-zero rational number $\dfrac{a}{b}$, we have $\dfrac{a}{b} \div \dfrac{a}{b} = 1$

$\left(\dfrac{-10}{11}\right) \div \left(\dfrac{-10}{11}\right) = \dfrac{-10}{11} \times \dfrac{11}{-10} = 1$

Hence, $\mathbf{\left(-\dfrac{10}{11}\right) \div \left(-\dfrac{10}{11}\right) = 1}$.

(vi) The product of a non-zero rational number and its reciprocal is 1, so the required number is the reciprocal of $-\dfrac{7}{9}$.

$\Rightarrow \text{Reciprocal of } \dfrac{-7}{9} = \dfrac{9}{-7} = \dfrac{-9}{7} \\[1em] \Rightarrow \dfrac{-7}{9} \times \dfrac{-9}{7} = 1$

Hence, $\mathbf{\left(-\dfrac{7}{9}\right) \times \left(\dfrac{-9}{7}\right) = 1}$.

(vii) All three rational numbers are negative and have the same numerator -5, so write them with a common denominator and compare.

$[\text{ LCM of } 11, 13 \text{ and } 7 = 1001] \\[1em] \Rightarrow \dfrac{-5}{11} = \dfrac{(-5) \times 91}{11 \times 91} = \dfrac{-455}{1001} \\[1em] \Rightarrow \dfrac{-5}{13} = \dfrac{(-5) \times 77}{13 \times 77} = \dfrac{-385}{1001} \\[1em] \Rightarrow \dfrac{-5}{7} = \dfrac{(-5) \times 143}{7 \times 143} = \dfrac{-715}{1001}$

As -715 < -455 < -385, so $\dfrac{-5}{7} \lt \dfrac{-5}{11} \lt \dfrac{-5}{13}$.

Hence, the greatest rational number is $\mathbf{\dfrac{-5}{13}}$.