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Math Help - root three irrational

  1. #1
    Senior Member
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    root three irrational

    how do i prove that root three is irrational?
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  2. #2
    MHF Contributor
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    Hi

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

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

    p^2 = 3 q^2 which means that 3 divides p

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

    Let p=3k then p=9k and 9k=3q
    Then q=3k which means that 3 divides q
    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
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