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
`(1)/(x)` is a rational number.

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

`(1)/(x)=(a)/(b) \Rightarrow x=(b)/(a)`

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