Question

Show that `sqrt(2) - sqrt(5)` is an irrational number.

Answer 1

To prove that `(sqrt(2) - sqrt(5))` is an irrational number, we will use the contradiction method.

Let, if possible, `sqrt(2) - sqrt(5)` = x, where x is any rational number (Clearly x ≠ 0).

So, `sqrt(2) = x + sqrt(5)`

⇒ `2 = (x + sqrt(5))^2`

⇒ `2 = x^2 + 5 + 2sqrt(5)x`

⇒ `-x^2 - 3 = 2sqrt(5)x`

⇒ `(-x^2 - 3)/(2x) = sqrt(5)`   ...(1)

`sqrt(5)` is an irrational number, as the square root of any prime number is always an irrational number.

In equation (1), LHS is a rational number while RHS is an irrational number but an irrational number cannot be equal to a rational number.

So, our assumption is wrong.

Thus, `(sqrt(2) - sqrt(5))` is an irrational number.