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NNTP-Posting-Date: Mon, 01 Jul 2024 20:18:22 +0000
Subject: Re: hot to write out this summation:
Newsgroups: sci.math
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From: Mike Terry <news.dead.person.stones@darjeeling.plus.com>
Date: Mon, 1 Jul 2024 21:18:22 +0100
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On 01/07/2024 17:45, Chris M. Thomasson wrote:
> On 6/30/2024 3:44 PM, Mike Terry wrote:
>> On 30/06/2024 22:38, Chris M. Thomasson wrote:
>>> r[0] = .01
>>> r[1] = .0011
>>> r[2] = .000111
>>> r[3] = .00001111
>>> r[4] = .0000011111
>>> ...
>>>
>>> Now, think of a possible formula. Something like this shit:
>>>
>>> ;^) lol.
>>>
>>> r[0] = .01
>>> r[1] = .01 * 10^(-1) + .1 * 10^(-3) = .0011
>>> r[2] = .0011 * 10^(-1) + .1 * 10^(-5) = .000111
>>> r[3] = .000111 * 10^(-1) + .1 * 10^(-7) = .00001111
>>> r[4] = .00001111 * 10^(-1) + .1 * 10^(-9) = .0000011111
>>> r[5] = .0000011111 * 10^(-1) + .1 * 10^(-11) = .000000111111
>>> ...
>>>
>>> Taken to infinity, 
>>
>> .. to Infinity And Beyond !!  :)
>>
>>> it would be: .(0)(1) or something? How to properly write it:
>>>
>>> .0...1...
>>>
>>> ?
>>
>> I'd go for .(0)(1)  specifying a point '.' then w (first infinite ordinal) '0's then w '1's.
> 
> I agree. .(0)(1) the first part is an infinity of zeros and the last part is an infinity of ones.
> 
> 
> 
>>
>> If you settle for .0...1... you will have to clarify how .022332...23550... should be interpreted.
> 
> Yeah. It's more confusing that just .(0)(1)
> 
> 
>> A harder question is what the above actually means, beyond just a way of describing (well-ordered) 
>> infinite strings.
> 
> This might be missing your main point, but when zooming into some of my work, or even adjusting a 
> parameter that alters the field, there are some big time differences between say:
> 
> .000111
> 
> and
> 
> .00001111
> 
> The differences show up in the final render result.
> 
> Not quite sure if that even relevant or not...
> 
> Humm.... Just thinking out loud here. Sorry! ;^o
> 
> 
>> E.g. you didn't put quotes around anything, making it look like you were describing regular real 
>> numbers, but regular real numbers have decimal notation with digits (after the decimal point) just 
>> at positions 1,2,...n,... with n < w.  So a "string" like .(0)(1) does not represent a real 
>> number. And "taken to infinity" /as a sequence of real numbers/ your sequence obviously converges 
>> to the real number 0.
> 
> Well, humm... Each iterate of .(0)(1) is a real number as in:
> 
> r[0] = .01
> r[1] = .0011
> r[2] = .000111
> r[3] = .00001111
> ...
> 
> Each one is a real number, and each index is an unsigned integer? Fair enough? Mapping is fun. :^)

Sure - we have a properly defined mapping from N to R, aka a "sequence of real numbers".  That 
sequence "taken to infinity" converges to 0.  Lol, here I assigned my own interpretation of "taken 
to infinity", deciding it to mean "what is the limit of the sequence?".  Hmmm, now we need to 
interpret what I meant by the "limit" of the sequence - but mathematicians already understand what 
that means.  But if they didn't, we'd have to give it some definition otherwise we'd still be in the 
dark...

The interpretation I decided on was natural for a mathematician given a sequence of real numbers. 
You would maybe like to come up with some other interpretation of "taken to infinity", which better 
recognises the pattern apparent in the base-10 representation of the real numbers.  That's a 
reasonable goal, but it becomes your job to explain your alternative interpretation so we can be on 
the same page.  I suppose my point is that it is easier to Say Words than it is to come up with a 
coherent and useful mathematical system.  Just Saying Words is playing around, and that can be fun 
and may ultimately lead to new mathematics so who would want to discourage that?

But some people [WM comes to mind] put too much stock in Words, believing that because they've 
framed a sentence using Words they must have said something with a definite meaning!  Your phrase 
"taken To infinity" is rather like that - what does it actually mean?  In contrast, "the /limit/ of 
the sequence of real numbers", has a well understood meaning, but probably not the one you were 
aiming for.


Mike.

> 
> 
>> More generally you might search for a notation to represent "strings" of digits indexed by any 
>> ordinal, rather than just the simple ordinals w and 2*w in your example.  As a starter, what might 
>> the following designate:
> 
> Hummm... For some reason this is making me think of L-Systems.
> 
> 
>>   .((1)(2))

    0 . 111111111...  22222222... 111111111...  22222222... 111111111...  22222222...
         111111111...  22222222... 111111111...  22222222... 111111111...  22222222...
         ...

>>   .((12))

    0 . 12121212... 12121212... 12121212... 12121212... 12121212... 12121212... 12121212...
         12121212... 12121212... 12121212... 12121212... 12121212... 12121212... 12121212...
          ...

    [hey, they're just infinite "strings", no suggestion by me of any other meaning...]

>> and what might be an example of a string with w^w digits?
>>
>> ..or why stick to ordinals?  Be brave!  To Infinity and Beyond !!
> 
> Thanks for your input Mike!  Thanks.
>