> > 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.