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Math Help - Prove that -0 = 0

  1. #1
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    Prove that -0 = 0

    I know that this is probably trivial for all of you, but I'm having a little trouble figuring it out. This is what I wrote down, but I'm not too sure that it's correct:

    -0 = 0
    -0 + 0 = 0 + 0
    0 = 0
    Therefore -0 = 0

    Is this correct or am I missing something?
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  2. #2
    Member anonimnystefy's Avatar
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    Re: Prove that -0 = 0

    "0 is neither positive nor negative."

    from http://en.wikipedia.org/wiki/0_(number)
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Prove that -0 = 0

    Hmm, I don't think you've to prove that, 0 is neither negative neither positive.
    Why are you proving this? ...
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  4. #4
    Super Member TheChaz's Avatar
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    Re: Prove that -0 = 0

    -0 = 0
    Add 0 to both sides
    0 + (-0) = 0 + 0
    0 = 2(0)
    0 = 0
    QED

    edit... I guess this actually demonstrates, using ring axioms (ish), that
    "If -a = a, then a = 0"
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  5. #5
    Member anonimnystefy's Avatar
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    Re: Prove that -0 = 0

    ok prove that 2(0)=0
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  6. #6
    Super Member TheChaz's Avatar
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    Re: Prove that -0 = 0

    Quote Originally Posted by anonimnystefy View Post
    ok prove that 2(0)=0
    Surprised you couldn't look this one up on your own!
    PlanetMath: zero times an element is zero in a ring
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  7. #7
    Member anonimnystefy's Avatar
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    Re: Prove that -0 = 0

    but that proof is based on the fact that a-a=0,and a+0=a.
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  8. #8
    Super Member TheChaz's Avatar
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    Re: Prove that -0 = 0

    Quote Originally Posted by anonimnystefy View Post
    but that proof is based on the fact that a-a=0,and a+0=a.
    ... and??? Those are axioms. Is there some other punchline?
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  9. #9
    Member anonimnystefy's Avatar
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    Re: Prove that -0 = 0

    they could be axioms but you can prove them just by defining addition how you want,but let not go any deeper into this.
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  10. #10
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    Re: Prove that -0 = 0

    Quote Originally Posted by anonimnystefy View Post
    they could be axioms but you can prove them just by defining addition how you want,but let not go any deeper into this.
    Of course if you are allowed to redefine things the way you want, you can prove ANYTHING.

    Those ARE ring axioms, which means they are facts you can use whenever working with rings.

    Axioms are axioms for a reason. Generally, if they could be proven as theorems from the other axioms, then they wouldn't BE axioms.
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  11. #11
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    Re: Prove that -0 = 0

    Quote Originally Posted by TheChaz View Post
    edit... I guess this actually demonstrates, using ring axioms (ish), that
    "If -a = a, then a = 0"
    That isn't even true. A ring may certainly have a nonzero element that is equal to its own additive inverse. The simplest example probably being 1=-1 in \mathbb{Z}_2.
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  12. #12
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    Re: Prove that -0 = 0

    Quote Originally Posted by Alexrey View Post
    I know that this is probably trivial for all of you, but I'm having a little trouble figuring it out. This is what I wrote down, but I'm not too sure that it's correct:

    -0 = 0
    -0 + 0 = 0 + 0
    0 = 0
    Therefore -0 = 0

    Is this correct or am I missing something?
    In my view the first problem you have is that you assumed that -0 = 0 is true at the beginning of your proof.

    Another way to word this problem is:
    Prove that the additive inverse of zero is zero.
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  13. #13
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    Re: Prove that -0 = 0

    0+0=0

    That's the entire proof. Can you justify it from the group axioms? You should also try to prove that inverses are unique if you haven't done so already.
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  14. #14
    Super Member TheChaz's Avatar
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    Re: Prove that -0 = 0

    Quote Originally Posted by topspin1617 View Post
    That isn't even true. A ring may certainly have a nonzero element that is equal to its own additive inverse. The simplest example probably being 1=-1 in \mathbb{Z}_2.
    My edit/amendment was unto your point, not a claim of veracity. I proved the converse!
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  15. #15
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    Re: Prove that -0 = 0

    So my proof was wrong?
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