> Jesse I'm glad you brought this up. It is imperative people understand > I am using 0.999... to be literally the series. I do this for a very > important reason.
Right. The reason is that you want 0.999... to be distinct from 1, but it is clear that if 0.999... has the usual meaning, it is equal to 1. So you change the definition.
But you did not change the definition in any coherent manner. What does it mean for 0.999... to be equal to a series rather than its sum? Does 0.999... represent a syntactic object now? In any case, a series is not a real number and so congrats! You win! 0.999... does not equal 1!
Of course, everyone else will continue to use 0.999... to represent the sum of the series, as before, and ignore this breakthrough, but don't let that discourage you. Truth is on your side.
> I do not claim the limit of 0.999... is not one as > the number of positions approaches infinity. People are taught that > 0.999... is equal to 1 years before they are ever taught about limits. > I am saying that the full sum of the series would never equal 1, it is > less, and only the limit is equal to one. We try to teach students in > early math that the series itself ~equals~ one, rather than has a > limit of it. It is only this equality I refute, not the limit. -- Jesse F. Hughes Quincy (age 3 1/2, looking at a picture): Are these people Canadians? Me: Uh, no, they're Australian Aborigines. Quincy: Do they fight Canadians?