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Math Help - Prove that -0 = 0

  1. #16
    MHF Contributor Siron's Avatar
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    Re: Prove that -0 = 0

    I think you're step 2 is incorrect, if you're working with ring axioms, you can say in general for a number a\in R (where R is a ring), there exists a neutral element e (and for the addition: 0) wherefore: a+e=a=e+a, that means in this case (for the addition): -0+0=-0.

    I would prove it this way:
    0+0=0 (this is an axiom of the existence of a neutral element in a commutative group in a ring)
    \Leftrightarrow 0=0-0 (putting 0 to the other side)
    \Leftrightarrow 0=-0+0 (in a ring the addition is commutative)
    Because of the existence of the neutral element 0 for the addition, like I stated earlier: -0+0=-0 therefore:
    0=-0+0 \Leftrightarrow 0=-0
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  2. #17
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    Re: Prove that -0 = 0

    Quote Originally Posted by Alexrey View Post
    So my proof was wrong?
    Yes. You started with -0=0, which is what you're trying to prove in the first place, and you end up with the true statement 0=0. This isn't a valid form of proof unless you can show that each step is reversible, so we have to start from the bottom and see if that's true.

    0=0. So far so good...

    -0+0=0+0. This is true because -0=0, but unfortunately that's what you're trying to prove, so you can't use this step.

    I still say that any proof over one line is unnecessarily long. 0+0=0 is all you need. Think about what it says. If you add 0 to itself you get 0, and isn't 0 the additive identity in a ring?
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  3. #18
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    Re: Prove that -0 = 0

    Quote Originally Posted by Alexrey View Post
    So my proof was wrong?
    Well... yeah. You started off by assuming what you want to prove. You can't prove a statement P by starting like "assume P is true".

    All you need to do is say 0+0=0. By definition, this means -0=0.
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  4. #19
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    Re: Prove that -0 = 0

    How does 0+0=0 imply -0=0? I would really appreciate it if that statement could be explained further.
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  5. #20
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    Re: Prove that -0 = 0

    What is, say, -7? It's 7 reflected across 0 on the number line, it's -1 times 7, but more fundamentally, it's the solution to 7+x=0. The solution to -2+x=0 is 2 and so by definition, -(-2)=2. The solution to 0+x=0 is 0, so by definition, -0=0.
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  6. #21
    Super Member TheChaz's Avatar
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    Re: Prove that -0 = 0

    Quote Originally Posted by Alexrey View Post
    How does 0+0=0 imply -0=0? I would really appreciate it if that statement could be explained further.
    a + b = 0 implies that b is the (additive) inverse of a.
    That b is the inverse of a means, exactly, that b = -a.
    So if
    a + a = 0
    Then a (the one on the right, if you will) = -a (i.e. the inverse of a, the one on the left)
    Replace a with 0.
    If you don't understand, read it again. Otherwise, it just looks like the old "dumb troll" routine.
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  7. #22
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    Re: Prove that -0 = 0

    Awesome, thanks a lot guys.
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