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Math Help - proving a formula

  1. #1
    Junior Member
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    proving a formula

    Hello!

    I want to prove the following:
    << k x >>= << k (1-x) >> where <<x>> is the distance to the nearest integer. x \in \[0,1 \] and k is an integer.

    Here is what I did so far:
    Since x \in \[0,1 \] I know there exists a c \in \{0, 1,..., 2^{n} -1 \} with x \in \left[\frac{c}{2^{n}}, \frac{c+1}{2^{n}} \right] , n \geq 1.. Thus I can conclude that x = \frac{s}{2^{n}} where c \leq s \leq c+1 (*)

    Now I got << k x >> = << k \frac{s}{2^{n}}>>

    I know because of (*) that \frac{c \cdot k}{2^{n}} \leq k \frac{s}{2^{n}} \leq \frac{c \cdot k}{2^{n}} + \frac{k}{2^{n}}

    I should now look at the cases that k is even and odd and greater or less than zero. But I donīt get to anything when I look at the last inequality to calculate  << k \frac{s}{2^{n}}>> The case k=0 is trivial.

    Plan is to look at k greater and less 0 and k even and odd for this case and then also do the same thing for the right hand side and compare those values and it should be the same result. Then I am done. Can anyone give me a hint on how to progress from here?

    Thanks.
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  2. #2
    Senior Member
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    Clarksville, ARk
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    Re: proving a formula

    Consider: kx + k(1-x)
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  3. #3
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    Re: proving a formula

    Hello SammyS! I donīt really get the point right now. What do you want to do with that term?
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  4. #4
    Senior Member
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    Re: proving a formula

    Let k be some integer, and x be some real number. Then kx is a real number.

    The max distance any real number is from the nearest integer is (1/2) .

    ∴ there is some integer M such that M-(1/2) ≤ kx ≤ M+(1/2).
    Case 1: M-(1/2) ≤ kx < M

    Then \ll kx\gg\,=M-kx  \,.

    M-(1/2) ≤ kx < M → -M < -kx ≤ (1/2) - M → k - M < k - kx ≤ (1/2) + k - M

    Since k - M is the closest integer to k - kx = k(1 - x), we have \ll k(1-x)\gg\,=k-kx-(k-M)=M-kx  \,.

    For Case 1: \ll kx\gg\,=\ll k(1-x)\gg  \,.

    Case 2: kx = M (This case should be easy.)

    Case 3: M < kx ≤ M+(1/2) (This case proceeds much like Case 1.
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  5. #5
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    Re: proving a formula

    Thanks for your answer. I guess I got it.
    In the 3rd case I get << kx>> = << k(1-x)>> = kx - M
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