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Math Help - Proving Irrationals

  1. #1
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    Proving Irrationals

    I already posted this on another thread and then found this one! How good!

    Here's my problem...

    Assume we know that √2 is irrational,

    let x,y be any rational number such that y is not 0. Show that x + y√2 is irrational

    Any help would be awesome!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by lisa21a View Post
    I already posted this on another thread and then found this one! How good!

    Here's my problem...

    Assume we know that √2 is irrational,

    let x,y be any rational number such that y is not 0. Show that x + y√2 is irrational

    Any help would be awesome!
    well, assume it is rational..

    then \frac{p}{q} = x + y\sqrt{2} for p and q integers, q\neq 0.. solving for \sqrt{2} we have \sqrt{2} = \frac{\frac{p}{q} - x}{y}.. since the set rational numbers is a field (i.e. closed under addition and multiplication, and defined for non-zero divisor) the right hand side of the equality is still a rational which is a contradiction to the assumption that \sqrt{2} is irrational.
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