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Topic: Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Replies: 42   Last Post: Oct 9, 2017 11:53 AM

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 netzweltler Posts: 473 From: Germany Registered: 8/6/10
Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Posted: Oct 5, 2017 9:50 AM

Am Donnerstag, 5. Oktober 2017 15:02:00 UTC+2 schrieb Jim Burns:
> On 10/4/2017 3:58 PM, netzweltler wrote:
> > Am Mittwoch, 4. Oktober 2017 18:27:18 UTC+2
> > schrieb Jim Burns:

> >> On 10/4/2017 4:19 AM, netzweltler wrote:
>
> >>> To me it looks like that we don't even agree, that there
> >>> are infinitely many 9s following.

> >>
> >> Maybe we agree, maybe we don't. We might be using the same
> >> words and meaning different things by them.
> >>
> >> I say there are infinitely many nines following the '.'
> >> What I mean by "infinitely many" here is that there is a
> >> map, one-to-one but not onto, from those after-dot decimal
> >> places to those after-dot decimal places. And '9' is in every
> >> place.
> >>
> >> I could say more, and I should, in order to say what 0.999...
> >> means, but that is what "infinitely many 9s following" means.
> >>
> >> What do you mean? That there is a '...' at the end?

> >
> > There is no end.

>
> I mean a '...' at the end of the description.
> When one writes
> 0.9, 0.99, 0.999, ...
> one puts '...' at the end of _that_ but what does it mean?

(1-(1/10)^n)n?N

> What I mean by "infinitely many 9s following" is broken down
> into concepts that we already share in order to explain what
> I mean -- which might not be the same as what you mean, even
> though we use the same words, "infinitely many 9s following".
>
> You raised this question. Do we agree? This is a question
>

> > Nothing follows after infinitely many 9s.
> > "infinitely many 9s following" replaces '...'.

>
> How do you say "infinitely many 9s following" without merely
> trading one thing that needs explaining for another thing that
> needs explaining?
>
> I'll give another example of what I'm talking about, taking
> some concept and expressing it using only more basic concepts.
> Suppose we want to say that a real function f: R -> R is
> continuous.
>
> Here's one way:
> A function f: R -> R is _continuous at b_ if
> for every eps > 0, there exists del > 0 such that
> for all x, abs(x - b) < del -> abs(f(x) - f(b)) < eps
>
> This might not be an obvious thing to mean by "continuous",
> but there are good reasons for it which can be explored.
> An important part of what mathematicians do is hash out
> definitions which refer to what we want our higher-level concepts
> to refer to, but which do it without calling on not-yet-defined
> concepts.
>
> (In my opinion, it is in the definitions that one most
> often see the brilliance of a mathematician. Good definitions
> are not at all trivial to devise.)
>
> This is what we do with the definition of the value of an
> infinite decimal expansion.
> (It is by that definition that 0.999... = 1)
> That definition uses addition and it uses infinite sets, but
> it does not use infinite repetitions of addition operations.
> We start, before our definition, with addition and infinite
> sets, so this is a good definition, one that does not merely
> trade one thing needing explanation for another thing needing
> explanation.
>
> To return to my question: what do you, netzweltler, mean by
> "infinitely many 9s following"? It might be the same as
> what I mean (above). If it is, then we agree.

I say there are infinitely many nines following the '.'
What I mean by "infinitely many" here is that there is a
map, one-to-one but not onto, from those after-dot decimal
places to those after-dot decimal places. And '9' is in every
place.

Date Subject Author
10/2/17 Guest
10/2/17 netzweltler
10/2/17 Jim Burns
10/3/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 bursejan@gmail.com
10/4/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 Jim Burns
10/4/17 netzweltler
10/4/17 Jim Burns
10/4/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 FromTheRafters
10/6/17 netzweltler
10/6/17 Jim Burns
10/7/17 FromTheRafters
10/8/17 FromTheRafters
10/8/17 netzweltler
10/8/17 Jim Burns
10/8/17 netzweltler
10/8/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/9/17 netzweltler
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