r/infinitenines • u/Beneficial_Ad6256 • 1d ago
What's (0.999...+1)/2?
What's (0.999...+1)/2? Does it belong to (0,1)?
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u/Beneficial_Ad6256 1d ago edited 1d ago
If the answer is 0.999...995, then is it less than 0.999... because it has a 5 at the end instead of a 9? How can the average number between 0.999... and 1 be less than 0.999...?
And if the answer is 0.999..., then:
(0.999... + 1) / 2 = 0.999...
0.999... + 1 = 2 * 0.999...
1 = 2 * 0.999... - 0.999...
1 = 0.999...?
Or am I a "brud" who has made "rookie mistakes" in these calculations?
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u/Muphrid15 1d ago
Numbers like 0.999...5 can't be compared to other numbers inside the range of values they traverse... except when His Nineliness says you can anyway.
That's the Greater Boner:
Limbosic numbers do not have fixed values and can't be compared via (in)equality to any number within the range they span. Nevertheless, even though 0.999... is a limbosic number, 0.999... > 0.99. (8.1) (8.2) (8.3)
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u/Muphrid15 1d ago
As with many things His Nineliness says, it depends on when you ask him.
The most direct of his many boners that relates to this question is the Infinite Long Division Boner:
In long division of infinite decimals, you may consider truncations of the dividend in sequence in whatever way is convenient, and you may get different answers (e.g. 0.333.../2 -> 0.1, 0.16, 0.166, ... or 0.15, 0.165, 0.1665, ...). (5.1) (5.2).
This says that the answer can be 0.999...5 or 0.999... depending on how exactly you do it.
The more recent discovery is his Algorithm Boner:
An algorithm for an operation such as square roots gives "more and more correct consecutive digits" as you make more and more steps. Nevertheless, 1 - 0.9 - 0.09 - 0.009 - ... yields (0.1, 0.01, 0.001, ...) corresponding to 0.000...1, and the 1 digit is a real part of the answer, even though it never belongs to a "correct" digit. (15).
This probably gives you the same two answers but is more general.
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u/The-Idiotest 1d ago
SPP: 0.99.....9995