Proof:-
Let √5 is a rational number
Then, √5=p/q
(where p and q have not common factor and q is not equal to zero)
Square each side LHS and RHS
5 =p^2/q^2
P^2=5q^2 ....... 1
5 divide p^2
Then 5 divide p
Now again let p=5k
Square each side LHS and RHS
p^2=25k^2 ....... 2
From 1 and 2
5q^2=25k^2
q^2=5k^2
5 divide q^2
Then 5 divide q
Since 5 divide both p and q means p and q have a common factor but we believe that p and q have not common factor so our recognition is failed.
Hense √5 is not an rational number so √5 is an irrational number.
Proved
