Prove that root 5 is irrational.
let root 5 be rational
then it must in the form of p/q [q is not equal to 0][p and q are co-prime]
root 5=p/q
=> root 5 * q = p
squaring on both sides
=> 5qq = pp ------> 1
pp is divisible by 5
p is divisible by 5
p = 5c [c is a positive integer] [squaring on both sides ]
pp = 25cc --------- > 2
sub pp in 1
5qq = 25cc
qq = 5cc
=> q is divisble by 5
thus q and p have a common factor 5
there is a contradiction
as our assumsion p &q are co prime but it has a common factor
so √5 is an irrational