Re: Does the number of nines increase?

On 6/26/2024 12:15 AM, WM wrote:

Le 26/06/2024 à 00:11, Jim Burns a écrit :

On 6/25/2024 4:18 PM, WM wrote:
>

Let the infinite sequence 0.999...
be multiplied by 10.
Does the number of nines grow?

>
Cardinalities which can grow by 1 are finite.
The number of nines in 0.999... is
larger than each finite cardinality.
It does not equal any finite cardinality.
It cannot grow by 1
>
tl;dr
No.

 Is the set of natural indices complete such that no natural number can be added?
 

Nuance:
There are _only_ positions in 0.999... which
are separated by some finite number,
even though there are infinitely.many of them.
>
The positions in 0.999... correspond to
numbers in well.ordered inductive ℕᴬ⤾⁺¹₀ᐣ⤓

 And they are fixed. Therefore your answer is correct:No. Therefore 9.999... has one 9 less after the decimal point than 0.999... .

NO! Think of: .(9) * 10 = 9.(9) = 10
There are still infinite nines... :^)
However, just keep in mind that .(9) represents 1 in base 10 therefore:
9.(9) is 10
So, .(9) * 10 = 1 * 10 = 10
and: .(9) * 9.(9) = 1 * 10 = 10
(.999...) * (9.999...) = 10
Got it?
:^)