Question

Prove that `4 - 5sqrt(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.