how do i prove that root three is irrational?

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- Dec 13th 2008, 11:00 AMhmmmmroot three irrational
how do i prove that root three is irrational?

- Dec 13th 2008, 11:23 AMrunning-gag
Hi

One way is to suppose that exist p and q integers with no common prime factor such as $\displaystyle \sqrt{3} = \frac{p}{q}$

Then $\displaystyle 3 = \frac{p^2}{q^2}$

$\displaystyle p^2 = 3 q^2$ 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