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Math Help - Proof using sqrt(2) is irrational

  1. #1
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    Proof using sqrt(2) is irrational

    Prove that for any rationals x, y, with y not equal to 0, x+y.sqrt(2) is an irrational number.

    For this, I wrote:

    Suppose x+y.sqrt(2) is rational. Then, since x and y are rational, y.sqrt(2) is also rational, since the difference of two rationals is numbers is also rational.

    Then from there I proved by contradiction that sqrt(2) is irrational, and said that this implied that x+y.sqrt(2) is also irrational.

    Is this the correct method? Any help would be appreciated
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  2. #2
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    Re: Proof using sqrt(2) is irrational

    Quote Originally Posted by sakuraxkisu View Post
    Prove that\for any rationals x, y, with y not equal to 0, x+y.sqrt(2) is an irrational number.
    Suppose x+y.sqrt(2) is rational. Then, since x and y are rational, y.sqrt(2) is also rational, since the difference of two rationals is numbers is also rational.

    If each of r~\&~s is a rational number and (r+s)\sqrt2=t where t\in\mathbb{Q} then \sqrt2=\frac{t}{r+s}.

    What is wrong with wrong with that?
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  3. #3
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    Re: Proof using sqrt(2) is irrational

    i believe he means x + y√2, not (x + y)√2.

    since we know that y ≠ 0, we have, if x + y√2 = t, a rational number:

    √2 = (t - x)/y, which is rational.
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