## 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`.

`x R y → (x) R (y)``x R y → xz R yz``x R y → zx R zy``x = x0``x(x) = (x)``(xy) = x(yx)`

((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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