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Math Help - Sqrt of 4 is irrational

  1. #16
    MHF Contributor undefined's Avatar
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    @Mukilab: Your most recent post highlights the difference between "X is cheating" and "X feels like cheating." Maybe what I should have said is that my proof shouldn't feel like cheating. When there is a simple proof available, why use a more complex one?
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  2. #17
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    I agree with undefined. If you wanted a clever proof (ie, one that doesn't use the fact that the asquare root of 4 is known to be 2), you need to say so.
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  3. #18
    Senior Member Mukilab's Avatar
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    good point, thank you for the information.

    Would either be marked higher in an exam?
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  4. #19
    A Plied Mathematician
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    undefined @ 13: Well, some people might have a problem with 5=\pm 5, if they're using equality as being the same thing as identity. 5 is not identically equal to plus or minus 5. In effect, what I'm really doing here is a short-hand notation for (5=5)\vee(5=-5), where this equality does mean identity.
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  5. #20
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Ackbeet View Post
    undefined @ 13: Well, some people might have a problem with 5=\pm 5, if they're using equality as being the same thing as identity. 5 is not identically equal to plus or minus 5. In effect, what I'm really doing here is a short-hand notation for (5=5)\vee(5=-5), where this equality does mean identity.
    I think 5=\pm 5 universally stands for (5=5)\vee(5=-5). Reference (Wikipedia - Plus-minus sign - Subheading: Shorthand).

    Quote Originally Posted by Mukilab View Post
    good point, thank you for the information.

    Would either be marked higher in an exam?
    As long as you present a valid proof, you should get full marks. I think it would be unreasonable for a teacher/professor/grader to grade otherwise. The type of proof should only matter if the problem statement specifies a particular type of proof.
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  6. #21
    A Plied Mathematician
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    Fair point.
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  7. #22
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    \sqrt{4} = -2 is false.

    for all x \in \mathbb{R} , \sqrt{x^2} = |x| \ge 0
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  8. #23
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    What if I'm on a different branch of the square root function? You're correct, but that doesn't alter the fact that \sqrt{4}=\pm 2 is a true statement. If you think about the logical OR connective, with a boolean variable x, then it is the case that x\vee\text{false}=x.
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