On May 31, 12:59 am, "Glen Wheeler" <s...@gew75.com> wrote: > <chaja...@mail.com> wrote in message > > news:firstname.lastname@example.org... > > > > > > > > >> What is the distance between 0.99999... and one? > > >> Ciao > > >> Karl > > > The distance is a non-real number I call a dubious number. It is of > > indeterminate scale. It would take an infinity of them to equal one, > > as mentioned at the end of my analysis. No real number increment would > > be able to keep from surpassing one from zero in a finite number of > > increments. > > > While the distance it not a real number, the system of real numbers > > cannot allow 0.999... to be one as shown in my proof. In the Classic > > Proof, 10x - x requires an element of subtraction that decimal cannot > > hope to express in its own terms. > > > Bounds on the distance are not expressable in decimal notation. > > Decimal notation, as described, is unable to express a large class of > > real numbers. We know an infinity of numbers exist just after zero, > > but decimal can only ever hope to express some reasonbly small > > numbers. In between any smallest number decimal can express and zero > > lie an infinity of inexpressable real numbers. These real numbers > > would require only a finite mutliple to reach one. > > So you assert that R does not form a metric space? > > By the way, you are not using the word decimal appropriately. > > -- > Glen- Hide quoted text - > > - Show quoted text -
I see no limitations disallowing that interval between 0.999... and 1 to be a metric space. It is infinitely sub-dividable and can be understood to have its own metric. How this metric would relate to the real numbers would require its own study.
When I say decimal, I simply mean any number represent in our common base ten positional numbering system. If this seems an overuse of them word I would be interested in hearing a better simple description.