Question

Prove that \[4 - 5\sqrt{2}\] is an irrational number.

Answer 1

Let us assume that \[4 - 5\sqrt{2}\]  is rational .Then, there exist positive co primes a and b such that

`4-5sqrt2=a/b`

`5sqrt2=a/b-4`

`sqrt2=(a/b-4)/5`

`sqrt5=(a-4b)/5b`

This contradicts the fact that `sqrt2`is an irrational

Hence `4-5sqrt2` is irrational