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Math Help - Proof irrational

  1. #1
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    Proof irrational

    Can anyone give a complete proof step by step that the:

    sqrt(41)+5=irrational
    given that the sqrt(41) is irrational..

    Show that it is irrational by the fact that the
    sqrt(41) is irrational.
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  2. #2
    Senior Member roninpro's Avatar
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    How about by contradiction?

    What happens if \sqrt{41}+5 is irrational?
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  3. #3
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    by contradiction:
    I have the sqrt(41)+5=p/q
    but how does it help knowing the sqrt(41)=irrat.
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  4. #4
    Senior Member roninpro's Avatar
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    What if you subtract 5 from both sides?

    \sqrt{41}=\frac{p}{q}-5

    Anything wrong with this?
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  5. #5
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    still cant see it
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  6. #6
    Senior Member roninpro's Avatar
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    We have a rational \frac{p}{q} subtract another rational 5. What kind of number results?
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  7. #7
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    Rat - rat = rat
    I tried the prob before I submited it..
    I appreciate u trying to me to get to think but
    I posted it to get a complete proof..
    So anyone ELSE please prove
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  8. #8
    Senior Member roninpro's Avatar
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    You have it right there. First, you are given that \sqrt{41} is irrational. Then, you said that \sqrt{41}=\frac{p}{q}-5 is rational, since a rational subtract a rational is rational. Hence, \sqrt{41} is rational and irrational.

    Isn't this a problem?
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  9. #9
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    ok partner, i can see it..thanks for sticking it out
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  10. #10
    MHF Contributor undefined's Avatar
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    If it's any use to you, it's also possible to write

    \sqrt{41}=\frac{p}{q}-5=\frac{p-5q}{q}=\frac{p'}{q} \in \mathbb{Q}.
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