Question

prove that
$\sqrt{2 \: is \: irrrationalno.}$
​

Answer 1

PROOF

Let us assume on the contrary that $\sqrt{2}$is a rational number. Then, there exist positive integers x and y such that

$\bold{\sqrt{2}=\frac{x}{y}}$ where x and y are co-prime numbers whose H.C.F is 1.

$\implies\bold{\sqrt{(2)²}=(\frac{x}{y})²}$

$\implies\bold{2=\frac{x²}{y²}}$

$\implies 2y² =x²$

$\implies 2|x²$

$[°.° 2|2x² \:and\: 2y² = x²]$

$\implies 2|x ......(i)$

$\implies x =2z (for\:some\:integer\:z)$

$\implies x² = 4z²$

$\implies 2y² =4z²$

$[°.° 2y² = x²]$

$\implies y² =2x²$

$\implies 2|y²$

$[°.° 2|2z²]$

$\implies 2|y ......(ii)$

From (i) and (ii), we obtain that 2 is a common factor of x and y. But, this contradicts the fact that x and y have no common factor other than 1. This means that our supposition is wrong.

Hence, $\sqrt{2}$ is an irrational number

PROVED