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Homework Help: Prove that

  1. Nov 15, 2004 #1
    x=--x

    wtf =/

    (this isn't a homework problem but it was brought up today and I'm curious) :confused:
     
  2. jcsd
  3. Nov 15, 2004 #2

    NateTG

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    Well, what do you start with?
    If you start with
    [tex]-1 \times -1 = 1 [/tex]
    then
    [tex]--x = -1(-1(x))=(-1 \times -1) x= 1 x =x[/tex]
    The first equality is by definition, the second because multiplication is associative, the third because you know [itex]-1 \times -1 =1 [/tex] and the last because 1 is the multiplicative identity.

    To see that [tex]-1 \times -1 =1[/tex]:
    [tex]\frac{-1}{-1}=1=\frac{1}{1}[/tex]
    so
    [tex]\frac{-1}{1}=\frac{1}{-1}[/tex]
    but
    [tex]1=\frac{-1}{-1}=-1 \times \frac{1}{-1}=-1 \times -1[/tex]
     
  4. Nov 16, 2004 #3

    arildno

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    It's a bit easier than that.
    By definition of zero, x+0=x
    By definition of the additive inverse,
    x+(-x)=0

    --x, that is, (-(-x))
    fulfills therefore:
    (-x)+(-(-x)=0
    Add x on both sides:
    x+(-x)+(-(-x))=x, or, since x+(-x)=0, we get:
    (-(-x))=x

    That is the additive inverse to the additive inverse of x is x itself.
     
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