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Proof Question: Prove integer + 1/2 is not an integer

  1. Oct 25, 2005 #1
    I was in the middle of proving something when I reached a contradiction, that .5 + an integer = an integer. However, this cannot be true, and I'm curious if its acceptable to just say that by definition of integers .5 + an integer is not an integer, or do I have to prove it?
    Furthermore, if I have to prove it, how would I go about this? I would say let x and y be integers, so x + .5 = y, right?
    Since x and y are integers then x = x/1 and y = y/1, so x/1 + 1/2 = y/1.
    2x/2 + 1/2 = y/1
    so
    (2x + 1/2)/2 = y/1
    and then... If I said that 2x +1/2 was not a whole number so dividing it by two must give a fraction, and thus it can't be reduced to a whole number over 1... That doesn't sound like it works though becuase its just restating what I was trying to prove... Not to mention I'm not sure I can even say that a fraction divided by two doesn't give a whole number... Any ideas? Thanks.
     
  2. jcsd
  3. Oct 25, 2005 #2
    Are you saying that

    [tex]\frac{(2x+\frac{1}{2})}{2}=\frac{y}{1}[/tex]

    shows that [itex]\frac{1}{2}+n1=n2[/itex]?

    n is an arbitrary integer.

    (there's a problem with this post. unwanted spacing)
     
  4. Oct 25, 2005 #3
    There is a MUCH quicker way.
    supose 0.5+N=M where N and M are integers. Then 0.5=M-N. But if M and N are integers, M-N is an integer. But this implies 0.5 is an integer. This is a cointradiction. Done.
     
  5. Dec 17, 2008 #4
    How do you figure Setting x=x/1 forces x to be an integer?
     
  6. Apr 14, 2009 #5
    To show 1/2 is not an integer, use its definition:

    1/2 is the number that satisfies the equation 2x=1.

    Now, for any integer n, 2n is never 1 (why? because 2n is always even and 1 is odd, and no integer is both even and odd.) Hence, 1/2 cannot be an integer.

    Of course, to argue that no integer is both even and odd uses the quotient-remainder theorem, which in turn relies on the well-ordering principle of the positive integers.
     
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