Prove that the reciprocal of an irrational number is irrational.

Let us consider, x as an irrational number.

Reciprocal of x is $\dfrac{1}{x}$.

Let us consider $\dfrac{1}{x}$ to be a non-zero rational number.

Then, $x × \dfrac{1}{x}$ will also be an irrational number as product of non-zero rational number and an irrational number is also an irrational number.

But $x × \dfrac{1}{x}$ = 1 is a rational number.

Hence, our supposition is wrong. So, $\dfrac{1}{x}$ is an irrational number.