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Math Help - Basic Question about 2x = 0

  1. #1
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    Basic Question about 2x = 0

    If I have 2x = 0, can someone explain what's going on or what rule doesn't allow for these two cases to exist...or can they and why...

    Case 1:

    2x = 0

    (2x)/2 = 0/2

    x = 0

    O.K.

    ----------------------------------------------

    Case 2:

    2x = 0

    x + x = 0

    x = -x

    x = (-1)x

    x/x = ((-1)x)/x

    1 = -1

    ???

    ...so if anybody could clear this up, it would be great! It's funny that after all of these years of doing math that I could stumble on something so basic that REALLY confuses me Thanks!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Basic Question about 2x = 0

    Quote Originally Posted by Alexander4444 View Post
    x = (-1)x
    x/x = ((-1)x)/x
    1 = -1 ???
    You correctly proved that x=0 so, the step x/x = ((-1)x)/x is not valid because 1/x does not exist.
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  3. #3
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    Re: Basic Question about 2x = 0

    Case 1 :
    You can do so only if the field in which you are has characteristic not equal to 2(Group Theory).
    For example if:
    x^2=x in a field say F s.t xbelongs to F
    then, -x=(-x)^2=x^2 s.t xbelongs to F
    i.e. x=-x s.t xbelongs to F
    or 2x=0 s.t xbelongs to F
    This is a boolean Field. Here u cannot divide by a 2 which is equivalent to dividing by 0 in our number system.


    Case 2:

    2x = 0

    x + x = 0

    x = -x

    x = (-1)x

    x/x = ((-1)x)/x

    1 = -1

    ???

    For this one would say u cannot divide by a number whose multiplicative inverse is not defined eg the '0' here
    Also 1=-1 in a boolean field
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    Re: Basic Question about 2x = 0

    Thanks for the clarification. That was very insightful!
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  5. #5
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    Re: Basic Question about 2x = 0

    If you had not known in advance that a=0 and did the problem by saying that 2x= 0 gives x+ x= 0, x= -x, then the next step would be to say "If x is not 0 then -x/x= x/x, so -1= 1". Since that is obviously not true, you conclude that x= 0.
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    Re: Basic Question about 2x = 0

    That makes great sense.
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  7. #7
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    Re: Basic Question about 2x = 0

    in mod2 algebra truly -1=1
    x may or maynot be 0
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Re: Basic Question about 2x = 0

    Quote Originally Posted by Learner248 View Post
    in mod2 algebra truly -1=1 x may or maynot be 0
    That is right, but this thread is included in Pre-Algebra forum so I suppose the OP meant x\in\mathbb{R}.
    Last edited by FernandoRevilla; July 17th 2011 at 09:52 PM.
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  9. #9
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    Re: Basic Question about 2x = 0

    That is right, but this thread is included in Pre-Algebra forum so I suppose the OP meant x is in the set of the real numbers.
    Yes, I did. But still, bring on the advanced Much better to be thorough than to conform to the contextual limits of this portion of the forum. Unfortunately, I didn't know that this COULD go outside the scope of pre-algebra/algebra. Now I do.
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