(i) No rational numbers are not always closed under division,

Since, \(\frac{a}{0}\) ∞ which is not a rational number

(ii) No rational numbers are not always commutative under division,

Let \(\frac{a}{b}\) and \(\frac{c}{d}\) be two rational numbers.

\(\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}\)

And

\(\frac{c}{d}\div\frac{a}{b}=\frac{bc}{ad}\)

Therefore,

\(\frac{a}{b}\div\frac{c}{d}\neq\frac{c}{d}\div\frac{a}{b}\)

Hence, rational numbers are not always commutative under division

(iii) No rational numbers are not always associative under division,

Let \(\frac{a}{b},\frac{c}{d}\) and \(\frac{e}{f}\) be two rational numbers.

\(\frac{a}{b}\div(\frac{c}{d}\div\frac{e}{f})= \frac{ade}{bcf}\)

And,

\((\frac{a}{b}\div\frac{c}{d})=\frac{e}{f}=\frac{adf}{bce}\)

Therefore,

\(\frac{a}{b}\div(\frac{c}{d}\div\frac{e}{f})\neq(\frac{a}{b}\div\frac{c}{d})\div\frac{e}{f}\)

Hence, rational numbers are not always associative under division.

(iv) No we cannot divide 1 by 0.

Since, \(\frac{a}{0}= ∞\) which is not defined.