Question

Show that \[5 - 2\sqrt{3}\] is an irrational number.

Answer 1

Let us assume that `5-2sqrt3` is rational .Then, there exist positive co primes a and b such that

\[5 - 2\sqrt{3}\] `=a/b`

`2sqrt3=a/b-5`

`sqrt3=((a/b)-5)/2`

`sqrt3= (a-5b)/(2b)`

This contradicts the fact that `sqrt3` is an irrational