### Decimal Inconsistency

February 7th, 2015
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

```
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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