Question

Prove that `2sqrt(3) - 1` is an irrational number.

Answer 1

Let us assume that \[2\sqrt{3} - 1\] is rational. Then, there exist positive co-primes a and b such that

`2sqrt3-1=a/b`

`2sqrt3=a/b+1`

`sqrt 3=(a/b+1)/2`

`sqrt3=(a+b)/(2b)`

This contradicts the fact that `sqrt3` is an irrational.

Hence `2sqrt3-1` is irrational.