why does ...999 apparently equal -1? is this notation valid?

Question

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?

Note that another solution of the equation $10x=x-9$ is $x=\infty$. — MJD, Oct 21 '20 at 18:26
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. — player3236, Oct 21 '20 at 18:27
Congratulations! You have just discovered $10$-adic numbers. You can read the Wikipedia article P-adic number for helpful information. — Somos, Oct 21 '20 at 18:27
Does this answer your question? Why does an argument similiar to 0.999...=1 show 999...=-1? but your question is a duplicate. — Somos, Oct 21 '20 at 18:28
Regarding p-adic numbers, it not only makes sense, the answer ($\ldots 999 = -1$) is correct; see Divergent series and p -adics — MJD, Oct 21 '20 at 18:28

score 3 · Answer 1 · answered Oct 21 '20 at 18:29

3

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

answered Oct 21 '20 at 18:29

GEdgar

111,679

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 – Lorem Ipsum1729 Oct 21 '20 at 18:30

why does ...999 apparently equal -1? is this notation valid?

1 Answers1