Question

Prove that if x is irrational, then 1/x is irrational

Answer 1

Proof by contradiction.

Let assume that when $x$ is an irrational number, then $\dfrac{1}{x}$ is a rational number.

If $\dfrac{1}{x}$ is a rational number, then it can be expressed as a fraction $\dfrac{a}{b}$ where $a,b\in\mathbb{Z}$.

$\dfrac{1}{x}=\dfrac{a}{b} \Rightarrow x=\dfrac{b}{a}$

Since $a,b\in\mathbb{Z}$, the number $x=\dfrac{b}{a}$ is also a rational number. But this contardicts our initial assumption that $x$ is an irrational number. Therefore $\dfrac{1}{x}$ must be an irrational number.