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Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 3:04 AM
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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.
--charlie
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