Question

Prove that `(5 - 2sqrt(3))` is an irrational number. It is given that `sqrt(3)` is an irrational number.

Answer 1

Let’s assume that `5 - 2sqrt(3)` is rational.

Therefore, it can be expressed in the form of `p/q` where p and q are integers and q ≠ 0.

Therefore, we write `5 - 2sqrt3 = p/q`.

or, `2sqrt3 = 5 - p/q`

or, `sqrt3 = (5q - p)/(2q)`

But `(5q - p)/(2q)` is a rational number as p and q are integers. This contradicts the fact that `sqrt(3)` is irrational, so our assumption is wrong.

Hence, `5 - 2sqrt3` is an irrational number.