Question

Prove that the following number is irrational:  3 - √2

Answer 1

3 - √2
Let  3 - √2 be a rational number.
⇒  3 - √2 = x
Squaring on both the sides, we get
(  3 - √2 )² = x²
⇒ 9 + 2 - 2 x  3 x √2 = x^2
⇒  11 - x² = 6√2
⇒ √2 = `[ 11 - x^2 ]/6`
Here, x is a rational number.
⇒ x² is a rational number. 

⇒ 11 - x² is a rational number.

⇒  `[ 11 - x^2 ]/6` is also a rational number.

⇒ `sqrt2 = [ 11 - x^2 ]/6`  is a rational number.

But √2 is an irrational number.

⇒  `[ 11 - x^2 ]/6 = sqrt2`  is an irrational number.

⇒ 11 - x² is an irrational number.

⇒ x² is an irrational number. 

⇒ x is an irrational number.

But we have assume that x is a rational number.

∴ we arrive at a contradiction.

So, our assumption that  3 - √2 is a rational number is wrong.
∴ 3 - √2 is an irrational number.