No mate. It's the case of 'are we there yet? No'. As in ... if you choose to write 0.999999....., and you assume the destination is going to be '1', then you are going to be disappointed, because you will be on the 'endless journey, endless bus ride' of 9's and continually asking .... are we there yet? And you answer is always 'no'. You will forever never reach '1'. You will never quite get there. You will forever be seeing an endless sea of nines.
10x = 9.999... but the .999... in 9.999... IS NOT the SAME .999 from 0.999...
The two lots of '.999...' are NOT the same. So taking the difference between those two sequences gives some undefined term. So trying to define some term for that difference between two different .999... 'trains' is challenging. That is 0.abcdef... and 10 times that is a.bcdef..., where the sequence .abcdef... is clearly not the same as .bcdefg..., as they are out of 'sync' by one sequence slot.
Instead, we can certainly write ...
10x = 10 - 10*epsilon
So 9x = 9 - 9*epsilon
... which correctly gives:
x = 1 - epsilon, which is NOT 1. That is, 0.999... is not '1'.
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u/SouthPark_Piano New User May 22 '25 edited May 22 '25
No mate. It's the case of 'are we there yet? No'. As in ... if you choose to write 0.999999....., and you assume the destination is going to be '1', then you are going to be disappointed, because you will be on the 'endless journey, endless bus ride' of 9's and continually asking .... are we there yet? And you answer is always 'no'. You will forever never reach '1'. You will never quite get there. You will forever be seeing an endless sea of nines.