Question

Prove that the following number is irrational: √5 - 2

Answer 1

√5 - 2
Let √5 - 2 be a rational number.
⇒ √5 - 2 = x
Squaring on both the sides, we get
`( √5 - 2 )^2 = x^2`

⇒ 5 + 4 - 2 x 2 x √5 = x²

⇒ 9 - x² = 4√5

⇒ √5 = `[ 9 - x^2 ]/4`

Here, x is a rational number.

⇒ x² is a rational number. 

⇒ 9 - x² is a rational number.

⇒ `[ 9 - x^2 ]/4` is also a rational number.

⇒ √2 = `[ 9 - x^2 ]/4` is a rational number

But √2 is an irrational number.

⇒ √5 = `[ 9 - x^2 ]/4` is an irrational number.

⇒ 9 - 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 √5 - 2 is a rational number is wrong.
∴ √5 - 2 is an irrational number.