Prove that root of 5 is irrational?

Answer 1

Suppose we start with the opposite view. That is, the square root of 5 is rational. This means that it can be expressed as a ratio of two integers. Let the square root of 5 in its simplest terms be a/b so that a and b are integers with no common factors and b > 0.

sqrt(5) = a/b so 5 = a2/b2

that is, 5b2= a2

5 is a factor of the left hand side (LHS) so 5 must be a factor of the right hand side (RHS).

Since 5 is a prime and a is an integer, 5 must be a factor of a. That is, a = 5c for some integer c and since a and b are coprime b and c must also be coprime.

But now,

5 = (5c)2/b2so that 5 = 25c2/b2and so b2= 5c2. As before, this implies that 5 must be a factor of b. But that contradicts the supposition that a and b are coprime.

The contradiction implies that the original assumption was incorrect. That is, sqrt(5) cannot be rational.