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olcott <polcott2@gmail.com> writes:0.999... means that is never reaches 1.0.
[...]It seems dead obvious that 0.999... is infinitesimally less than 1.0.Yes, it *seems* dead obvious. That doesn't make it true, and in fact it
isn't.
0.999... denotes a *limit*. In particular, it's the limit of the valueThat is how it has been misinterpreted yet it has always meant
as the number of 9s increases without bound. That's what the notation
"0.999..." *means*. (There are more precise notations for the sameI already know all that.
thing, such as "0.9̅" (that's a 9 with an overbar, or "vinculum") or
"0.(9)".
You have a sequence of numbers:Yet never reaching.
0.9
0.99
0.999
0.9999
0.99999
...
Each member of that sequence is strictly less than 1.0, but the *limit*
is exactly 1.0. The limit of a sequence doesn't have to be a member of
the sequence. The limit is, informally, the value that members of the
sequence approach arbitrarily closely.
<https://en.wikipedia.org/wiki/Limit_of_a_sequence>Infinitesimally less than 1.0 means one single geometric point
That we can say this in English yet not say this in conventionalThen I have good news for you. There are several such systems, for
number systems proves the need for another number system that can
say this.
example <https://en.wikipedia.org/wiki/Hyperreal_number>.
If your point is that you personally like hyperreals better than youThe Infinitesimal number system that I created.
like reals, that's fine, as long as you're clear which number system
you're using.
If you talk about things like "0.999..." withoutIt is already the case that 0.999...
qualification, everyone will assume you're talking about real numbers.
And if you're going to play with hyperreal numbers, or surreal numbers,--
or any of a number of other extensions to the real numbers, I suggest
that understanding the real numbers is a necessary prerequisite. That
includes understanding that no real number is either infinitesimal or
infinite.
Disclaimer: I'm not a mathematician. I welcome corrections.
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