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Math Help - the largest integer number

  1. #1
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    the largest integer number

     For \ x \in R , let [x] be the largest integer less than or equal to x . prove that :  [x] +[x+\frac{1}{2}] = [2x] , \forall x \in R
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    Moo
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    Hello,

    x=[x]+{x}, where 0<{x}<1 (the decimal part)

    If {x}<1/2, then [x+1/2]=[x] and [2x]=[2[x]+2{x}]=[2[x]]=2[x].
    So if {x}<1/2, we indeed have [x]+[x+1/2]=2[x]=[2x]


    and if {x}>1/2, [x+1/2]=[x]+1 and [2x]=[2[x]+2{x}]. But 1<2{x}<2. So [2x]=[2[x]]+1=2[x]+1.
    So if {x}>1/2, we indeed have [x]+[x+1/2]=2[x]+1=[2x]
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    MHF Contributor Bruno J.'s Avatar
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    Mais c'est Léonard et le disciple!



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    Moo
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    Ouais, bien vu !
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