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  • Decimal Inconsistency

    February 7th, 2015
    ideas, math  [html]
    Ben Orlin and I were playing around with a weird kind of infinite repeating decimal. He wrote up a blog post, there was some discussion, but now I think there's a contradiction in what I thought were reasonable axioms.

    First a summary. This is all based around the idea that you can have "0.9̅3" or "0.9̅4," and in fact "0.9̅3 < 0.9̅4". The first one is 0.999...3 while the latter is 0.999...4. Or "first you have nines forever, and then either a three or a four". Since 3 < 4, we should have 0.9̅3 < 0.9̅4. If this is confusing Ben's post goes into more detail.

    (I'll note here that this isn't normal math. You can't add these, subtract them, multiply, etc. Normally 0.9̅ is exactly 1 and 0.9̅3 is meaningless. We're playing with some things that are kind of like numbers, but not entirely.)

    Here are some properties it seems like these numbers should have, where x and y are infinite decimals and R is any of >, =, or <. To simplify writing in text we're writing 0.x̅y as (x)y.

    1. x R y → (x) R (y)
    2. x R y → xz R yz
    3. x R y → zx R zy
    4. x = x0
    5. x(x) = (x)
    6. (xy) = x(yx)
    Here's the contradiction, which my coworker Shawn figured out:
      ((x)x) = (x)(x(x))   by #6
             = (x)((x))    by #5
             = ((x))       by #5
    
    so
    
        (x)x = (x)         by #1
             = (x)0        by #4
    
    so
    
           x = 0           by #3
    
    which is a contradiction.
    
    This seems right to me, but all of the axioms also seem reasonable. I'm not sure what you would drop to make this more reasonable.

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