Re: Military time (whoops!)

[David Auerbach auerbach@xxxxxxxx]

On Jun 17, at 3:34 PM, Carl Distefano wrote:
> > If that ellipsis is being used in the standard way in such
> > decimal contexts, then 2359.9999999999... =
> > 2359.999999999999999999... = 2359.9... = 0000 (and thus doesn't
> > belong to the day before). If all you meant by the ellipsis
> > was some unspecified by finite number of "9"'s, then what was
> > said is right.
>
> Gee, I had no idea that "0.9..." is the same as "1.0" or "1.0...".
> It seems off to me, or at least unconventional. I thought the
> convention was that rational numbers are not represented by
> repeating decimals (except 0's). For example, 5/4 = 1.250...
> (repeating 0), not 1.249... (repeating 9). I thought the repeating
> decimals were only used to represent irrational numbers:
> 1/3=0.333... OTOH, I suspect that 0.9... with 9's to infinity HAS
> to equal, or be defined to equal, 1 for the real number system to
> hold together. Yes?
Well, if were really going to quibble I'd point out the use/mention
errors above. (OK, I will; it should be "Gee, I had no idea that
0.9... is the same as 1.0 or 1.0...." or else "Gee, I had no idea
that "0.9..." names what "1.0" names ...".)
The convention you mention is, plausibly, the informal convention
concerning the *use* of the notation. The notation, which is to say
decimal notation for the reals*, behaves as you ended up saying. 
Place notation, in this case decimal, is a systematic way to produce
names for an infinite number of the reals (*but not all of them-- as
Georg Cantor famously showed). The notation is shorthand for a
polynomial expression; 12.23 is short for 1x10 + 2 x 1 + 2x 1/10 +
3x1/100.  (so, non-negative powers of ten to the left of, and
negative powers of 10 to the right of, the decimal point.) So that's
what the notation officially means; it turns out to be very very
handy that we can just stick with coefficients for our notation; it
saves space and permits efficient algorithms, like the technique for
multiplying that we all(?) learned in grade school. It would have
pleasant if each real was picked out by exactly one name. No such,
in either direction. Infinitely many reals go unnamed* and some
reals have many names. 1.2, 1.20, 1.200, etc.
Now a little legislated convention gets rid of all but finite strings
of zeros. But there's no simple convention to toss out the rest.
So yep, .9999... = 1. And it is so as a consequence of the
definition of the notation. (A relatively simple consequence, though
it does involve the sum of an infinite series.)
> In any event, what I meant was an arbitrarily large but finite
> number of 9's -- anything short of 0000.

I figured.


David Auerbach
Department of Philosophy & Religion
Box 8103
Raleigh, NC 27695-8103                  http://
slowfoodusa.org
auerbach@xxxxxxxx                  http://
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