Does this mean that because the difference or “lateness” gets smaller tending to zero each time a single identical digit is added to 0.A and 0.9 respectively, then 0.A… = 0.9...?
(Whereas the difference we get when we do this to say 0.8 and 0.9 gets larger each time so we can’t say 0.8… = 0.9...)
No I believe you are reaching a different concept. It is true that the difference squashes towards 0 but that would be different line of thinking. In a contex where infinidesimal are allowed (ie non-real) we might associate the series to different amounts and indeed find that they differ by a “minuscule amount”. But as we normally operate on reals we only get a “real precision” result. For example if you had to say whether 3⁄4, 1 and 5⁄4 name which integers probalby your best bet would be that all of them name the same integer 1, if you are only restricted to integer precision. In the same way you might have 1 and 1-epsilon to be differnt numbers when infinidesimal accuracy is allowed but a real + anything infinidesimal is going to be the same real regardless of the infinidesimal (1 and 1-epsilon are the same real in real precision)
What I was actually going fo is that, for any r < 1 you can ask how many terms you need to get up to that level and both series will give a finite answer. Ie to get to the same “depth” as 0.999999… gets with 6 digits you might need a bit less with 0.AAAAA… .It’s a “horizontal” difference instead of a “vertical” one. However there is no number that one of the series could reach but the other does not (and the number that both series fails to reach is 1, it might be helpful to remember that an suprenum is the smallest upper limit). if one series reaches a sum with 10 terms and other reaches the same sum in 10000 terms it’s equally good, we are only interested what happens “eventually” or after all terms have been accounted for. The way we have come up what the repeating digit sign means refers to limits and it’s pretty guaranteed to produce reals.
I think I see your first point.
0.A{base11} = 10⁄11
0.9 = 9⁄10
0.A − 0.9 = 0.0_09...
0.AA = 10⁄11 + 10⁄121
0.99 = 9⁄10 + 9⁄100
0.AA − 0.99 = 0.00_1735537190082644628099...
Does this mean that because the difference or “lateness” gets smaller tending to zero each time a single identical digit is added to 0.A and 0.9 respectively, then 0.A… = 0.9...?
(Whereas the difference we get when we do this to say 0.8 and 0.9 gets larger each time so we can’t say 0.8… = 0.9...)
No I believe you are reaching a different concept. It is true that the difference squashes towards 0 but that would be different line of thinking. In a contex where infinidesimal are allowed (ie non-real) we might associate the series to different amounts and indeed find that they differ by a “minuscule amount”. But as we normally operate on reals we only get a “real precision” result. For example if you had to say whether 3⁄4, 1 and 5⁄4 name which integers probalby your best bet would be that all of them name the same integer 1, if you are only restricted to integer precision. In the same way you might have 1 and 1-epsilon to be differnt numbers when infinidesimal accuracy is allowed but a real + anything infinidesimal is going to be the same real regardless of the infinidesimal (1 and 1-epsilon are the same real in real precision)
What I was actually going fo is that, for any r < 1 you can ask how many terms you need to get up to that level and both series will give a finite answer. Ie to get to the same “depth” as 0.999999… gets with 6 digits you might need a bit less with 0.AAAAA… .It’s a “horizontal” difference instead of a “vertical” one. However there is no number that one of the series could reach but the other does not (and the number that both series fails to reach is 1, it might be helpful to remember that an suprenum is the smallest upper limit). if one series reaches a sum with 10 terms and other reaches the same sum in 10000 terms it’s equally good, we are only interested what happens “eventually” or after all terms have been accounted for. The way we have come up what the repeating digit sign means refers to limits and it’s pretty guaranteed to produce reals.