In article <email@example.com>, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
> firstname.lastname@example.org wrote: > > > > > > 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. > > Charlie, > > I'm almost certain that you would be interested in "Is 0.999... = 1?", > Fred Richman, _Mathematics Magazine_ 72 (1999), 404-408. Based on what you > said above, I think that your ideas are quite similar to Richman's. His > article is also available on the web at > > <http://www.math.fau.edu/Richman/HTML/999.htm>.
This article defines define Dedekind cuts in an obscure way. Why not say a Dedekind cut is a partition of Q = X U Y s.t. x in X && y in Y -> x < y.
But that is secondary. When folks argue that 0.999... != 1, they never tell me what _they_ mean when they write `0.999...' It is only fair that they tell me first what they mean. Otherwise they are writing meaningless prose. I want a an opportunity to slang their thesis.