Prove that $\dfrac{1}{\sqrt{11}}$ is an irrational number.

Let $\dfrac{1}{\sqrt{11}}$ be a rational number, then

$\dfrac{1}{\sqrt{11}} = \dfrac{p}{q},$

where p, q are integers, q ≠ 0 and p, q have no common factors (except 1)

$\Rightarrow \dfrac{1}{11} = \dfrac{p^2}{q^2} \\[0.5em] \Rightarrow q^2 = 11p^2 \qquad \text{….(i)}$

As 11 divides 11p², so 11 divides q² but 11 is prime

$\Rightarrow 11 \text{ divides } q \qquad \text{(Theorem 1)}$

Let q = 11m, where m is an integer.

Substituting this value of q in (i), we get

$(11m)^2 = 11p^2 \\[0.5em] \Rightarrow 121m^2 = 11p^2 \\[0.5em] \Rightarrow 11m^2 = p^2 \\[0.5em]$

As 11 divides 11m², so 11 divides p² but 11 is prime

$\Rightarrow 11 \text{ divides } p \qquad \text{(Theorem 1)}$

Thus, p and q have a common factor 11. This contradicts that p and q have no common factors (except 1).

Hence, $\dfrac{1}{\sqrt{11}}$ is not a rational number. So, we conclude that $\dfrac{1}{\sqrt{11}}$ is an irrational number.