Prove that root two is an irrational number
Proof:-let √2 is a rational number
Then √2=p/q
(where p and q have not a common factor and q is not equal to zero)
Squaring both side LHS and RHS
(√2)^2=(p/q)^2
2=p^2/q^2
2q^2=p^2
p^2=2q^2
2 divide p^2
then 2 divide p
Now again let p=2k
Square each side LHS and RHS
p^2=4k^2 ....... 2
From 1 and 2
2q^2=4k^2
q^2=2k^2
2 divide q^2
Then 2 divide q
Since 2 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 √2 is not an rational number so √2 is an irrational number.
