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
⇒ 3 + 2 + 2 x √3 x √2 = x^2
⇒ x² - 5 = 2√6
⇒ √6 = `[ x^2 - 5 ]/2`
Here, x is a rational number.
⇒ x² is a rational number. 

⇒ x² - 5 is a rational number.

⇒  `[ x^2 - 5]/2`  is also a rational number.

But √6 is an irrational number.

⇒  `[ x^2 - 5]/2`  is also a irrational number.

⇒ x² - 5 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.