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?

Answer 1

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

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

(ii) No

$$\begin{gathered} \frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc} \\ \mathrm{Also}, \\ \frac{c}{d}÷\frac{a}{b}=\frac{c}{d}\times \frac{b}{a}=\frac{cb}{da}\mathrm{Thus},\frac{a}{b}÷\frac{c}{d}\ne \frac{c}{d}÷\frac{a}{b} \end{gathered}$$

Therefore, division is not commutative.

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

$\frac{a}{b}÷\left(\frac{c}{d}÷\frac{e}{f}\right)\ne \left(\frac{a}{b}÷\frac{c}{d}\right)÷\frac{e}{f}$

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