Question

prove that root 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

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Answer 2

Let root five be irrational
Root five = a÷b
Here a and b are Co prime number
Root five square =(a) square ÷(b) square
B square =a square ÷ 5
::5 ÷ a square
So a square = c
B square =25c ÷ 5
B square ÷ 5 =c
So 5 divide b and a
So root 5 is rational but really root 5 is irrational
So are assumption is wrong
Root 5 is irrational number