**Question:**

(i) Are rational numbers always closed under division?

(ii) Are rational numbers always commutative under division?

(iii) Are rational numbers always associative under division?

(iv) Can we divide 1 by 0?

**Solution:**

(i) No, rational numbers are not closed under division in general.

$\frac{a}{0}=\infty$; it is not a rational number.

(ii) No

$\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}=\frac{a d}{b c}$

Also,

$\frac{c}{d} \div \frac{a}{b}=\frac{c}{d} \times \frac{b}{a}=\frac{c b}{d a}$ Thus, $\frac{a}{b} \div \frac{c}{d} \neq \frac{c}{d} \div \frac{a}{b}$

Therefore, division is not commutative.

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

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

(iv) No, we cannot divide 1 by 0 . The answer will be $\infty$, which is not defined.