On the subject of what "0.999..." _means_ (which we need to agree on before we can discuss what number it denotes), let me suggest that instead of thinking of it as the sum of an infinite series, one can think of it as an address on the number line. The two points of view can be shown to be equivalent, but pedagogically they're different. (Students don't need to know what an infinite series is or how limits are defined in order to appreciate that 3.1415... means a number that's between 3 and 4, and also between 3.1 and 3.2, etc.) Maybe someone who knows more of the history of mathematics than I do can comment on which of these two notions is closer to the meaning intended by the people who first used infinite decimals? Note that my use of the word "between" in the preceding paragraph is ambiguous vis-a-vis the handling of the endpoints. If 3.1415... means the unique number in all the intervals [3,4], [3.1,3.2], ..., then 0.999... means the unique number in all the intervals [0,1], [0.9,1.0], ..., namely 1. But one can argue that 3.1415... means the unique number in all the intervals [3,4), [3.1,3.2), ..., in which case 0.999... means the unique number in all the intervals [0,1), [0.9,1.0), ... --- but there is no such number! Jim Propp Jim On Wed, Jan 30, 2013 at 12:44 AM, James Propp <jamespropp@gmail.com> wrote:
Allan (and others),
A theory of formal infinite decimals in which 0.999... is not the same as 1.000... is described in the talk "Why 0.999... is greater than 1.000...", presented by my student Linda Zayas-Palmer at the ninth Gathering for Gardner (slides at http://jamespropp.org/g4g9-slides.pdf).
As long as you do arithmetic with just addition and multiplication (not subtraction or division), the distinction between 0.999... and 1.000... can be sustained. But once you bring subtraction into the picture, you basically have to identify the two expressions with one another. (Or, as I was taught to say forty or fifty years ago, these two _numerals_ represent the same _number_.)
There are connections with the Faltin-Metropolis-Ross-Rota construction of the real numbers as a wreath product, and a somewhat more arcane connection with the theory of chip-firing (aka the abelian sandpile model).
Jim Propp
On Tue, Jan 29, 2013 at 5:29 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I may be confused, but I know of no theories, however exotic, in which 0.999... has a well-defined meaning that is not 1. In the surreals, the sequence 0.9, 0.99, 0.999, ... has no single well-defined limit. It converges on the *set* of numbers within an infinitesimal distance of 1. I would love to hear of any theory in which it makes sense to say 0.999... != 1.
On Tue, Jan 29, 2013 at 3:46 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Mark, I respectfully could hardly disagree more strongly.
The question of why .99999... = 1 that was initially brought up by Eric Angelini was clearly about the ordinary real numbers.
Now, I have nothing against exotic theories of numbers. (In fact, I love the surreal numbers.) But when people want to know whether, and if so why, .99999... = 1, they are in almost all cases wanting to learn or be reminded of the conventional meaning of that statement.
And that requires understanding a) that an infinite decimal represents an infinite sequence, and b) that this sequence has a limit, and c) that having a limit has a very specific meaning that can be verified in this case.
I have no idea what vertiginous or any other kind of semi-circular reasoning the word "only" obscures. But let me remind you that the fundamental group of a semi-circle is trivial, as compared to that of a circle, which is the integers.
Of course, there's always this famous quote: << "When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean -- neither more nor less."
--Dan
On 2013-01-28, at 11:42 PM, Marc LeBrun wrote:
<< "Dan Asimov" <dasimov@earthlink.net> << The only way to understand an infinite decimal representation . . .
Hear hear! And here we go 'round the infinite decimal discussion, again...
Look, I've nothing 'gainst the Standard Theory of the Real Numbers (STotRN); an' I too'll testify t' y'all, it surely do come in plenty handy, of a time.
So (pacem, please, concerns of arithmetical apostasy) what bugs me is that darn word "only", and the vertiginous semi-circular reasoning it obscures.
Surely there are many ways to "understand" infinite decimal representations? . . .
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