Can 99.999...8 be described as:

i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))

?

i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
....

;^)

On Monday, October 10, 2022 at 2:14:42 PM UTC-7, Chris M. Thomasson wrote:
> Can 99.999...8 be described as:
>
> i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
>
> ?
>
> i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
> i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
> i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
> i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
> ...
>
> ;^)

Euh, how do you write that?

Please feel free [to refer] to exactly how I just wrote that.

The, how we say, .998, ....

On Monday, October 10, 2022 at 8:24:10 PM UTC-7, Ross A. Finlayson wrote:
> On Monday, October 10, 2022 at 2:14:42 PM UTC-7, Chris M. Thomasson wrote:
> > Can 99.999...8 be described as:
> >
> > i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
> >
> > ?
> >
> > i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
> > i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
> > i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
> > i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
> > ...
> >
> > ;^)
> Euh, how do you write that?
>
> Please feel free [to refer] to exactly how I just wrote that.
>
> The, how we say, .998, ....

"What you were just telling me, Ross, was ...".

"The .998 you made a notation and some implicit ellipses
for the modular, same as .999... about .000...".

On Monday, October 10, 2022 at 11:29:49 PM UTC-4, Ross A. Finlayson wrote:
> On Monday, October 10, 2022 at 8:24:10 PM UTC-7, Ross A. Finlayson wrote:
> > On Monday, October 10, 2022 at 2:14:42 PM UTC-7, Chris M. Thomasson wrote:
> > > Can 99.999...8 be described as:
> > >
> > > i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
> > >
> > > ?
> > >
> > > i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
> > > i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
> > > i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
> > > i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
> > > ...
> > >
> > > ;^)
> > Euh, how do you write that?
> >
> > Please feel free [to refer] to exactly how I just wrote that.
> >
> > The, how we say, .998, ....
> "What you were just telling me, Ross, was ...".
>
> "The .998 you made a notation and some implicit ellipses
> for the modular, same as .999... about .000...".

Let x = 999...98 .
One thing I can tell you is that the square of x is:
999...96000...04
On the usage of the decimal point: just put it in four places over rather than two I think.
The decimal point is optional. It's mechanics are well defined, but this particular usage may be ambiguous. It is for me and my new method even though the repeating decimal of old has no trouble with it. Call it an aleph mark and all will be well.... well, maybe I'll have to rethink this detail. It's a matter of heads or tails, and the decimal point should be introduced on tails first, I used to think. You see, your representation will require an infinite value if your decimal point is taken from the least significant digit. Taken from the most significant digit you really are using an aleph mark; you are up at infinity. Certainly it works, and it works about the same, so what's with all this clap-trap? Structural analysis of numerical formats exposes that these augmentations to the natural value deserve sensible and logical treatment. The functional aspect of the decimal point as a unital indicator exists as a secondary form of unity, and so my fog of uncertainty is lifted, yet the fact that it works well the other way around troubles me. Our numbers are two-sided... but for the modulo-one form.

On 10/10/2022 8:24 PM, Ross A. Finlayson wrote:
> On Monday, October 10, 2022 at 2:14:42 PM UTC-7, Chris M. Thomasson wrote:
>> Can 99.999...8 be described as:
>>
>> i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
>>
>> ?
>>
>> i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
>> i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
>> i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
>> i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
>> ...
>>
>> ;^)
>
> Euh, how do you write that?
>
> Please feel free [to refer] to exactly how I just wrote that.
>
> The, how we say, .998, ....

When I see 999... I think of endless 9's, so:
_____________________
i[0] = 9
i[1] = 99
i[2] = 999
i[3] = 9999
i[4] = 99999
_____________________
on and on... Notice iteration i[2], call that a.

I do not necessarily automatically think of a fixed three 9 prefix wrt:
_____________________
i[0] = 999
i[1] = 9999
i[2] = 99999
i[3] = 999999
i[4] = 9999999
_____________________
Keep in mind that a = i[0] here. Starting conditions and all... :^)

On 10/10/2022 2:14 PM, Chris M. Thomasson wrote:
> Can 99.999...8 be described as:
>
> i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
>
> ?
>
> i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
> i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
> i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
> i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
> ...
>
> ;^)

Humm... Is the limit of this 100? What about that pesky ever postfixed
8? This will never exactly equal 100, right? Or am I thinking to
"step-by-step", so to speak?

On Tuesday, October 11, 2022 at 4:55:50 PM UTC-4, Chris M. Thomasson wrote:
> On 10/10/2022 2:14 PM, Chris M. Thomasson wrote:
> > Can 99.999...8 be described as:
> >
> > i[n] = (((floor((1 / 9)*10^(n+4))*9)-1)/10^(n+2))
> >
> > ?
> >
> > i[0] = (((floor((1 / 9)*10^(0+4))*9)-1)/10^(0+2)) = 99.98
> > i[1] = (((floor((1 / 9)*10^(1+4))*9)-1)/10^(1+2)) = 99.998
> > i[2] = (((floor((1 / 9)*10^(2+4))*9)-1)/10^(2+2)) = 99.9998
> > i[3] = (((floor((1 / 9)*10^(3+4))*9)-1)/10^(3+2)) = 99.99998
> > ...
> >
> > ;^)
> Humm... Is the limit of this 100? What about that pesky ever postfixed
> 8? This will never exactly equal 100, right? Or am I thinking to
> "step-by-step", so to speak?

I think it's a good question. It does possibly advance the movement, too.
Pretty sure the serious answer or at least some critical ground will be through the highly respected Dedekind cut.
Yet you are varying a value, so the ordinary sense of the thing is not quite as it was on say the square root of two.
Another angle will be convergence. Pretty sure you do have convergence to 100 here.
Either way digit chasing is the theme.
You could rewrite this as 100.0 - 2.0 x 10 ^ -m
and in the limit as m approaches infinity that second term is rapidly headed for zero.

What I think is interesting about the value is that the discrepancy to 99.999... is felt by altering the tail.
It somewhat challenges the infinite resolution ideal that some people think of the real value as having.
In some regards you've made way for the gray value.

The discrepancy between the engineer's or physicist's real value and the mathematician's real value is more easily exposed on a less controversial form, say back at 99.98. To what degree this value implies 99.98000...: this would be the mathematician's version of the real value. The engineer and physicist know full well that the next digit is not well defined; past the eight, that is. The distinction is not desirable. As to who has it wrong: this is a question about the continuum. To state that the mathematician's version of the continuum lacks physical correspondence is an accurate statement. Their sense of perfection is not accurate nor is their naming then of the 'real' number. That this brings us back to digit chasing all over again is appropriate. That the rational value could be dragged into a theoretical quagmire sounds distant, but it is not at all distant.

Worst of all mathematics was to be the pure one. Perfection not only of value, but in theory too. That the true nature of the continuum got snuffed out along the way exposes blatant disregard for the geometer, who has now turned machinist, or at least the state of the art has gone that way. A serious geometer would now do their finest works on polished sheet metal; simply dropping into the groove for perfect transcription, in a work room held at constant temperature. Claims of going any finer than a ten thousandth of an inch should be scrutinized I would think. Such fine measurements are the first form of uncertainty. That mathematics has divorced itself from any observer or observation at all grants it freedom to wander wherever it likes. As to how then names like 'natural' and 'real' are entering into those works... cause for great skepticism.