Question

prove that under root 2 is irrational number

Answer 1

hey here is your ans.
hope it helps


if possible,let
$\sqrt{2}$
be rational and let it's simplest form be a/b
then , a and b are integer having no common factor other than 1 and b is not equal to 0
now , √2= a/b
2= a^2/ b^2 ( on squaring both side )
2b^2= a^2. -------------(I)
2 divides a^2 ( 2 divided 2b^2)
2divides a( 2 is prime and divides b^2= 2 divides b)
let a= 2c for some integer c
putting a = 2c in (I) we get
2b^2=4c^2
b^2= 2c^2
2 divides b^2 ( 2divides c^2)
2 divides b ( 2 is a prime. and divides b^2 = 2 divides b )
thus, 2 is common factor of a and b
but, this contradict this fact that a and b have no common factor other than 1
the contradiction arises by assuming that √2 is rational
hence, √2 is irrational