We can tell that 0.11111111...*9=0.999999999... And that 1/9=0.11111111111... Therefore 1/9*9=0.999999999...However, we know that 1/9⋅9=9/9=1 Note: I'm taking in account that ... are the other rational digits left. What am I making wrong? What is misunderstood? Thanks for the help in clearing this problem.

Braylon Lester 2022-07-21 Answered
Is 9 ( 1 / 9 ) = 0.99999999... statement correct?
We can tell that
0.11111111... 9 = 0.999999999...
And that
1 9 = 0.11111111111...
Therefore
1 9 9 = 0.999999999...
However, we know that
1 9 9 = 9 9 = 1
Note: I'm taking in account that ... are the other rational digits left.
What am I making wrong? What is misunderstood?
Thanks for the help in clearing this problem.
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Answers (2)

nuramaaji2000fh
Answered 2022-07-22 Author has 18 answers
Nothing. What you wrote is correct, and you just proved that 0. 9 ¯ = 1
You can do the same thing with 1 3 and 3 for example:
1 = 3 3 = 3 1 3 = 3 ( 0.333333 ) = 0.999999 = 0. 9 ¯
Remark
That holds only for infinite periodic decimals. You cannot, for example, state that 0.999999999999999999999999999999999 = 1
That is not true!

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Cristofer Graves
Answered 2022-07-23 Author has 6 answers
A more typical way to prove is to subtract 0.1 0. 9 ¯ from 0. 9 ¯
0. 9 ¯ 0.0 9 ¯ = 0.9
But X 0.1 X = 0.9 X = 0.9 proves X = 1

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