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Math Help - Proving irrational

  1. #1
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    Proving irrational

    Hi,

    I have been reading a lot on here about proving that a number is irrational, which has helped. But I have not seen anything when it is simply variables that are either said to be rational or irrational. Here is my question:

    Prove that, if x and y are rational, x is not equal to 0, and z is irrational, y + xz is irrational, and that y +x/z is irrational.

    I am not sure at all on how to approach, and unfortunately, I have been assigned this problem without it being addressed in lecture or the text. =/

    Thank you
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  2. #2
    Moo
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    Hello !
    Quote Originally Posted by Sinuyen View Post
    Hi,

    I have been reading a lot on here about proving that a number is irrational, which has helped. But I have not seen anything when it is simply variables that are either said to be rational or irrational. Here is my question:

    Prove that, if x and y are rational, x is not equal to 0, and z is irrational, y + xz is irrational, and that y +x/z is irrational.

    I am not sure at all on how to approach, and unfortunately, I have been assigned this problem without it being addressed in lecture or the text. =/

    Thank you
    Here is my go...
    Let x=\frac{x'}{x''} and y=\frac{y'}{y''} (x',x'',y',y'' are nonzero integers if there are, you can simply see if the results are rational or not)

    See for y+xz. Suppose it is a rational, that is to say y+xz=\frac pq

    xz=\frac pq-\frac{y'}{y''}=\frac{py''-qy'}{qy''}

    z=\frac{py''-qy'}{qy''} \cdot \frac{x''}{x'}=\frac{px''y''-qy'x''}{qx'y''}

    Is the RHS a rational ?
    Is z a rational ?


    Same reasoning for the second one
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  3. #3
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    I understand all the steps your present and it makes sense. All I have left to show is that the RHS is rational, but how do i do that? Gah! The reasoning behind number theory confuses me. >.< But the whole idea behind the proof is to present a contradiction no? So by saying that z is rational when in the beginning it was irrational correct?

    The thing is I do not know how to say this mess of variables that represent integers will always come out to a rational number. Do I have to repeat the process by setting it equal to some rational expression r/s where r and s are non-zero integers?

    Sorry for my incompetence in the area.

    Edit:
    Cool, after looking at it long enough, it makes sense. Since z is equal to a ratio of integers, then that is enough to prove it is rational. But since in the beginning it was said that z is irrational, this is a contradiction. Therefore, y + xz is irrational. Sweet! Thanks Moo.
    Last edited by Sinuyen; September 25th 2008 at 12:10 PM.
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  4. #4
    Moo
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    Quote Originally Posted by Sinuyen View Post
    I understand all the steps your present and it makes sense. All I have left to show is that the RHS is rational, but how do i do that? Gah! The reasoning behind number theory confuses me. >.< But the whole idea behind the proof is to present a contradiction no? So by saying that z is rational when in the beginning it was irrational correct?
    Exactly, it was a proof by contradiction, but I didn't say that lol

    The thing is I do not know how to say this mess of variables that represent integers will always come out to a rational number. Do I have to repeat the process by setting it equal to some rational expression r/s where r and s are non-zero integers?

    Sorry for my incompetence in the area.
    It is not a matter at all.

    Here is the key : a rational number is in the form \frac ab where a and b are integers.

    Are the ugly stuff integers ? It doesn't matter what they represent, all you have to know is whether their sum & product are integers ! Do you kind of get it or do you need further explanation ?
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  5. #5
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    LOL it came to me just now and then I saw your post, so I can see that it is as i thought. Thank you very much.
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