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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: Does the number of nines increase?
Date: Wed, 26 Jun 2024 06:36:54 -0400
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Ross Finlayson has brought this to us :
> On 06/25/2024 04:17 PM, FromTheRafters wrote:
>> WM used his keyboard to write :
>>> Let the infinite sequence 0.999... be multiplied by 10. Does the
>>> number of nines grow?
>>
>> No, both sequences are infinite.
>>
>>> Corollary-question: Does the number of nines grow when in 0.999 the
>>> decimal point is shifted by one or more position?
>>
>> What do you think multiplying by ten does to a continued decimal
>> expansion representation?
>
> What does Simon Stevin say?
>
> The number of nines left of the radix grows, ....
>
> Though, one might aver it's the "count" of nines,
> it's also its number.
>
> Counting and numbering are two different things,
> though they're often conflated, not to be confused.
>
> Numbers "have" a number and "make" a count.

....000099.999...

There are countably many nines on either side of the radix point. This 
is a 'representation' of a number which has a countably infinite 
sequence of symbols left *and* right of the radix. Left of the radix 
point it has countably infinitely many zeroes and finitely many 
non-zero symbols. Right of the radix it has countably infinitely many 
symbols which in this case is all nines and possibly finitely many 
non-zero and non-nine leading symbols.

....000100.000...

Same number but no nines.