Thread: root three irrational

how do i prove that root three is irrational?

Hi

One way is to suppose that exist p and q integers with no common prime factor such as

Then

which means that 3 divides p²

Therefore 3 divides p because
- if p=3k+1 then p² = 9k²+6k+1 = 3(3k²+2k)+1 cannot be divided by 3
- if p=3k+2 then p² = 9k²+12k+4 = 3(3k²+4k+1)+1 cannot be divided by 3

Let p=3k then p²=9k² and 9k²=3q²
Then q²=3k² which means that 3 divides q²
Therefore 3 divides q (same demonstration as per above with p)

3 divides both p and q, which is not possible because p and q have no common prime factor