## Decimal Inconsistency |
February 7th, 2015 |

ideas, math |

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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