Fill in the blanks:

(i) The product of two positive rational numbers is always …………… .

(ii) The product of two negative rational numbers is always …………… .

(iii) If two rational numbers have opposite signs then their product is always …………… .

(iv) The reciprocal of a positive rational number is …………… and the reciprocal of a negative rational number is …………… .

(v) Rational number 0 has …………… reciprocal.

(vi) The product of a non-zero rational number and its reciprocal is …………… .

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

(viii) If $m$ is reciprocal of $n$, then the reciprocal of $n$ is …………… .

(i) The product of two positive rational numbers is always positive.

(ii) The product of two negative rational numbers is always positive.

(iii) If two rational numbers have opposite signs then their product is always negative.

(iv) The reciprocal of a positive rational number is positive and the reciprocal of a negative rational number is negative.

(v) Rational number 0 has no reciprocal.

(vi) The product of a non-zero rational number and its reciprocal is 1.

(vii) The numbers 1 and -1 are their own reciprocal.

(viii) If m is reciprocal of n, then the reciprocal of n is m.

Explanation

(i) Let 2 positive rational numbers be $\dfrac{a}{b}$ and $\dfrac{c}{d}$.

Hence,

$\dfrac{a}{b} \times \dfrac{c}{d}\\[1em] = \dfrac{a \times c}{b \times d}\\[1em] = \dfrac{ac}{bd}$

$\dfrac{ac}{bd}$ is also positive rational number.

(ii) Let 2 negative rational numbers be -$\dfrac{a}{b}$ and -$\dfrac{c}{d}$.

Hence,

$-\dfrac{a}{b} \times -\dfrac{c}{d}\\[1em] = \dfrac{-a \times -c}{b \times d}\\[1em] = \dfrac{ac}{bd}$

$\dfrac{ac}{bd}$ is positive rational number.

(iii) Let 2 rational numbers be $\dfrac{a}{b}$ and -$\dfrac{c}{d}$.

Hence,

$\dfrac{a}{b} \times -\dfrac{c}{d}\\[1em] = \dfrac{a \times -c}{b \times d}\\[1em] = \dfrac{-ac}{bd}$

-$\dfrac{ac}{bd}$ is negative rational number.

(iv) Let the positive rational number be $\dfrac{a}{b}$.

Reciprocal of $\dfrac{a}{b} = \dfrac{b}{a}$

$\dfrac{b}{a}$ is a positive rational number.

Let the negative rational number be -$\dfrac{a}{b}$.

Reciprocal of -$\dfrac{a}{b} = -\dfrac{b}{a}$

-$\dfrac{b}{a}$ is a negative rational number.

(v) Reciprocal of $\dfrac{0}{1} = \dfrac{1}{0}$

$\dfrac{1}{0}$ is not defined.

(vi) Let the positive rational number be $\dfrac{a}{b}$.

Reciprocal of $\dfrac{a}{b} = \dfrac{b}{a}$

$\dfrac{a}{b} \times \dfrac{b}{a} =\dfrac{a \times b}{b \times a} =\dfrac{ab}{ab} =1$

(vii) Reciprocal of $\dfrac{1}{1} = \dfrac{1}{1} = 1$.

Reciprocal of $\dfrac{-1}{1} = \dfrac{1}{-1} = -1$.

(viii) If reciprocal of $\dfrac{m}{1} = \dfrac{n}{1}$

Reciprocal of $\dfrac{n}{1} = \dfrac{m}{1}$