Question

prove that √5 is irrational number

Answer 1

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
=> 5*q*q = p*p ------> 1
p*p is divisible by 5
p is divisible by 5
p = 5c [c is a positive integer] [squaring on both sides ]
p*p = 25c*c --------- > 2
sub p*p in 1
5*q*q = 25*c*c
q*q = 5*c*c
=> 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

Answer 2

let √5 is an rational number
√5= p \q
squraring both sides

(√5)² p2/q2

now cancel the root with square