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I debated 9... != 1 claims for years now, but the discussion surfaced once again, this time I asked myself: what if I "change the direction" of the recurring digit, i.e. add 9s BEFORE the decimal point?

This means: 9 = 9 999= 900 + 90 + 9 ...999 = ? diverges?

First thing I tried was obviously the algebraic proof by just subracting equations:

(instead of)
0.9... = x |x10
9.9... = 10x
_____________ -

9x = 9


...999 = x
...9990 = 10x
(i feel like this step cheats, moving the decimal point to infinity)

10x = x-9 
x= -9/9 = -1 ??

...999 = -1

the anwser confuses me a lot. Is there another way of illustrating this problem? Moreover what would happen if I would subtract those two values? Am I simply breaking fundamental laws?

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    $\begingroup$ Note that another solution of the equation $10x=x-9$ is $x=\infty$. $\endgroup$
    – MJD
    Commented Oct 21, 2020 at 18:26
  • $\begingroup$ At first I would say it does not make sense to subtract a divergent series from another (...999 is divergent while 0.999...) is convergent, but then I realized that it could make sense, but in 10-adic numbers. $\endgroup$
    – player3236
    Commented Oct 21, 2020 at 18:27
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    $\begingroup$ Congratulations! You have just discovered $10$-adic numbers. You can read the Wikipedia article P-adic number for helpful information. $\endgroup$
    – Somos
    Commented Oct 21, 2020 at 18:27
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    $\begingroup$ Does this answer your question? Why does an argument similiar to 0.999...=1 show 999...=-1? but your question is a duplicate. $\endgroup$
    – Somos
    Commented Oct 21, 2020 at 18:28
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    $\begingroup$ Regarding p-adic numbers, it not only makes sense, the answer ($\ldots 999 = -1$) is correct; see Divergent series and p -adics $\endgroup$
    – MJD
    Commented Oct 21, 2020 at 18:28

1 Answer 1

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The series $9+90+900+9000+\cdots$ diverges (in the sense of the real numbers), so your calculations are invalid there.

There may be other "strange" metrics where this does make sense, and your argument does indeed show that the sum is $-1$. The so-called $10$-adic metric is an example of this.

See https://en.wikipedia.org/wiki/P-adic_number

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  • $\begingroup$ thanks for the explanations. this was just a quick thought I came across but apparently it makes sense. I will read up on the topic $\endgroup$
    – Lorem Ipsum1729
    Commented Oct 21, 2020 at 18:30

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