• Posts
  • RSS
  • ◂◂RSS
  • Contact

  • Decimal Inconsistency

    February 7th, 2015
    ideas, math
    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.

    Comment via: google plus, facebook

    Recent posts on blogs I like:

    Moral aesthetics

    “Doing good” differs by subculture The post Moral aesthetics appeared first on Otherwise.

    via Otherwise September 29, 2022

    Futurist prediction methods and accuracy

    I've been reading a lot of predictions from people who are looking to understand what problems humanity will face 10-50 years out (and sometimes longer) in order to work in areas that will be instrumental for the future and wondering how accurate thes…

    via Posts on September 12, 2022

    On the Beach

    I really like going in the water and this beach is a great place for building sand castles and boogie boarding. I also like trying to float on top of big waves. I'm not very good at it. I only float on the flat waves.

    via Anna Wise's Blog Posts July 12, 2022

    more     (via openring)


  • Posts
  • RSS
  • ◂◂RSS
  • Contact