Decimal Inconsistency |
February 7th, 2015 |
math, ideas [html] |
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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