Question

How to prove weather root of 5 is irrational?

Answer 1

Answer:

So all 's prime factors are found among 's, and is an integer. This is a contradiction since 5 is not the square of an integer. This proof generalizes in a natural way to show that it is not possible for any integer that is not a square to have a rational square root

Answer 2

Answer:

let we using contary to supposing √5 is rational number

then their exit two uniqe integers

let √5=a/b (there coprime of a and b is 1 )

(√5)² = (a/b)²

5= a²/b²

5b²=a²

5divides b²

5divides b

a=5c for some integers c

(5c)²=(a)²

25c² =5b²

5c²=b²

5divides b²

5divides b

there is 5 is common factors of a and b so our contradiction is wrong

Hence √5 is an irrational number