Question

Prove that √3 is irrational. You may use the fact that n2 is divisible by 3 only if n is divisible by 3.

Prove by contradiction that there is not a smallest positive rational number.

Answer 1

Let

Assume that there exists some integers such that a and b have no common factors. This means the fraction is in lowest terms.

Squaring on both sides, we get

-------(1)

(1) shows that a^2 is divisible by 3 and so a is divisible by 3

Therefore, a = 3p ---(A)

where p is some integer

Now substitute a = 3p in (1), we get

From this it can be said that b^2 is divisible by 3 and so b is divisible by 3

So, b = 3q ---(B)

From (A) and (B) we see that a and b have common factor 3 which is a contradiction.

Hence we cannot write  

So, √3 is irrational