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Math Help - proof rational + irrational

  1. #1
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    proof rational + irrational

    How do I prove a rational +an irrational number = an irrational number?
    thanks
    Barry
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tinhats View Post
    How do I prove a rational +an irrational number = an irrational number?
    thanks
    Barry
    It is quite simple - let n \in \mathbb{R} \setminus \mathbb{Q}, p,q,r,s \in \mathbb{Z}. Then, if n+p/q=r/s we can quite easily manipulate the equality to get n=x/y, x,y \in \mathbb{Z}, a contradiction. I shall, however, leave the manipulation up to you.
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  3. #3
    Moo
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    Hello,
    Quote Originally Posted by tinhats View Post
    How do I prove a rational +an irrational number = an irrational number?
    thanks
    Barry
    Here is a similar thing : http://www.mathhelpforum.com/math-he...rrational.html .

    Put x=1, that is to say x'=x'' and you're done...
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  4. #4
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    here's another example of 2 similar proofs:
    http://www.mathhelpforum.com/math-he...bers-even.html

    Halfway down the page is a proof that an uneven * even number is an even number.
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  5. #5
    Moo
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    Quote Originally Posted by bmp05 View Post
    here's another example of 2 similar proofs:
    http://www.mathhelpforum.com/math-he...bers-even.html

    Halfway down the page is a proof that an uneven * even number is an even number.
    Hmm but yours is dealing with even/uneven, while we're talking about rational/irrational here ?
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    Yes, well, I feel that any retort would probably be irrational at this stage.
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  7. #7
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by Moo View Post
    Hmm but yours is dealing with even/uneven, while we're talking about rational/irrational here ?
    They may not be the same but they are similar.

    In general:

    Let G be a group and H be a subgroup. Then for any x,y\in G, x\in H and y\notin H \implies xy\notin H.

    Proof is simple: Suppose x (and therefore x^{-1})\,\in\,H. If xy\in H, then y=x^{-1}(xy)\in H. Therefore by contrapositivity, y\notin H\ \implies\ xy\notin H.

    In bmp05’s original example, G is the addibive group of the reals and H is the additive group of the rationals; hence rational + irrational = irrational. In the odd/even example, G is the additive group of the integers and H is the subgroup of the even integers – \therefore even + odd = odd.

    Further examples:

    (i) Even permutation odd permutation = odd permutation. (Take G to be a symmetric group, H to be the corresponding alternating group.)

    (ii) Rotation reflection = reflection. (Take G to be a dihedral group, H to be the subgroup of rotations.)
    Last edited by TheAbstractionist; May 25th 2009 at 07:53 AM.
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