(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.
