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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

 Messages: [ Previous | Next ]
 Jim Burns Posts: 831 Registered: 9/26/15
Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Posted: Oct 5, 2017 9:01 AM

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?

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.

"infinitely many 9s following"? It might be the same as
what I mean (above). If it is, then we agree.

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
10/9/17 Jim Burns
10/7/17 Jim Burns