Question

Prove that following number is irrational:

`4 + sqrt(2)`

Answer 1

Let us assume that `4+sqrt2` is rational. Then, there exist positive co-primes a and b such that

`4+sqrt2=a/b`

`sqrt2=a/b-4`

`sqrt2=(a-4b)/b`

`sqrt2` is a rational numbers which is a contradication.

Hence \[4 + \sqrt{2}\] is irrational.