Let x and y be rational numbers. Show that xy is a rational number.

Let x = $\dfrac{m}{n}$, n ≠ 0 and y = $\dfrac{p}{q}$, q ≠ 0, where m, n, p and q are integers. [By definition of rationals]

So,

$\Rightarrow x \times y = \dfrac{m}{n} \times \dfrac{p}{q} \\[1em] \Rightarrow xy = \dfrac{mp}{nq}$

Using properties of integers,

mp and nq are integers and since n, q are not equal to zero.

∴ nq ≠ 0.

∴ $\dfrac{mp}{nq}$ is a rational number.

Hence, proved that xy is also a rational number.