On May 31, 9:07 pm, quasi <q...@null.set> wrote: > On Thu, 31 May 2007 19:13:25 -0700, chaja...@mail.com wrote: > >You cannot assert that students across the world are not taught > >that 0.999... is equal to one well before they are ever taught or told > >about limits. > > Limits are taught informally, very early. > > The formula for the area of a circle > > A=Pi*r^2 > > depends on the limit concept. This formula has been known for > thousands of years. But even then it was understood that the number Pi > exists as an exact number only in the limit. > > The concept of the sum of an infinite geometric progression, such as, > for example > > 1 + 1/2 + 1/4 + 1/8 + ... > > is taught early, possibly as early as elementary school, but certainly > it's taught to high school students. The concept of an infinite sum > _is_ explained to students by discussing limit concepts, at least > informally. > > By the standard limit-based definition of an infinite sum, > > 1 + 1/2 + 1/4 + 1/8 + ... equals 1, exactly > > Once again, this is old knowledge -- thousands of years old. > > If you won't allows limits to be regarded as real numbers, you lose a > lot of math and science. Your conservatism is too costly in terms of > all the wonderful (and useful) mathematical and scientific theories > that you won't be able to accept. > > In a prior reply, you made the analogy that someone might reject > negative numbers for lack of understanding. You're doing just that > with respect to the real numbers. You don't believe .999... = 1 so > you are effectively rejecting the standard definition of the reals. > > I suggest you table your bias so as to least learn the standard > foundations. Here are some topics you should master before exploring > nonstandard versions ... > > (1) Sets, logic, proofs at an elementary level > > (2) Limits, sequences and series at an elementary level. > > (3) Learn the standard axioms and definitions for > > the natural numbers > the integers > the reals > the complex numbers > > (4) Mathematical Logic, Model Theory, Set Theory > > If you can master the above, then perhaps you'll know what you're > talking about when you discuss nonstandard ideas. > > But even before you start, let me caution you about a serious error of > logic you've already made. > > You are not allowed to change an existing standard definition to suit > your own prejudices. Of course, if you can show a definition is > somehow _inconsistent_, that's different -- then you can reject the > definition, but still, you're not entitled to replace it with your > own. You _can_ make up _new_ terminology for your version of a > concept. > > Thus, you don't have the option of saying that .999... is not a real > number. It is a real number, by definition, and it equals 1, also by > definition. On the other hand, if you want to study a system for which > .999... is strictly less than 1, fine, you can do that, but don't call > your numbers "real numbers". Call them something else. > > quasi
I agree with most all you have presented here and I appreciate your input. What I'm trying to express is that people are taught 0.999... is equal to 1 in a way that leaves them having to reconcile what they intuitively believe 0.999... is with a value of 1 and it cannot be done as I attempt to show in my proof. This intuitive entity of 0.999... is what I attempt to show less than 1. It's all great and wonderful that mathematicians have fully owned the term 0.999... and assert it to be a limit and nothing else, yet they teach it out of this concept. You have helped make it very clear that the truth I wish to express must fully state a definition that leaves no room for misunderstanding, disowns other definitions I do not intend to bring to the table, and almost practically say I cannot even use the number 0.999... as part of my expression of this idea. Gee, I'll do be best, and my best I'll do.