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January 11

0.999...

I wonder why the page on 0.999... is so biased and closed to opinion. I tried to edit it, and my edit was taken off and called "vandalism". I tried to initiate a discussion of the problem on the talk page and I was called out for it. The mathematical question is still open, and Wikipedia of all websites is not permitting of differing opinions. So my question is, why can't Wikipedia be more permitting of the opinion that 0.999... does not equal one. — Preceding unsigned comment added by 24.127.161.155 (talk) 15:24, 11 January 2021 (UTC)[reply]

The page is a correct as currently written; your contributions of edits backed by reliable sources are welcome, your opinion is irrelevant. --Jayron 32 15:29, 11 January 2021 (UTC)[reply]

There are some who disagree that 1 + 1 = 2 (in ordinary algebra). Do we permit their opinions any space outside the Talk pages? Hell, no. -- Jack of Oz ^{[pleasantries]} 22:10, 11 January 2021 (UTC)[reply]

I do not believe that reality exists, but they did not allow me to add that to the page Reality, which is terribly biased. --Lambiam 23:26, 11 January 2021 (UTC)[reply]

Wikipedia is based on published reliable sources like serious mathematical publications. They all agree that it's 1. It's some laypeople who disagree because they don't understand or don't like the mathematical definition of the notation "0.999...". It does have a precise definition and it does equal 1 with that definition. Mathematics is built on definitions. They aren't always intuitive to laypeople. PrimeHunter (talk) 00:01, 12 January 2021 (UTC)[reply]

I for 0.999... would love to know what the IP thinks real numbers are, and what it means for two of them to make the same Dedekind cut. JoelleJay (talk) 23:52, 11 January 2021 (UTC)[reply]

Wikipedia's interpretation of neutrality involves representing alternative viewpoints that can be supported by reliable sources (from the WP:UNDUE section). Check that you understand what is said in WP:FALSEBALANCE in particular, which addresses the kind of objection you make directly. It is not the case that all editors of maths articles are Platonists, but there is no professional-level disagreement between Platonists and anti-Platonists on what the standards are for provability or what the accepted formulation is of the real numbers. — Charles Stewart (talk) 06:41, 12 January 2021 (UTC)[reply]

Why? First of all: it is not an opinion. Saying "0.999.. is not 1" is not an opinion on real numbers. It's talking of something else, but not about real numbers and their notations. Which you are of course free to do, but possibly with different names and notations (and places, as this is an encyclopedia). pm a 16:53, 12 January 2021 (UTC)[reply]

It's not an unreasonable question if one has some uncertainty about why "0.999..." equals "1", and there are many approaches to explaining it. my favorite for simplicity is that X/9 = 0.XXX... for all single digits. Thus 1/9 = 0.111... 2/9 = 0.222... If you follow the pattern, 9/9 = 0.999... and scrupulously, 9/9 equals exactly 1. QED. There's no need for any other explanation. But insofar as there is some confusion, it has to do with the concept of a limit in mathematics, and at one time even honest to god mathematicians struggled with the notion of the fact that something that approaches but does not reach some definite value in any finite sense can be said to scrupulously equal that value in the infinite sense. It does, and the acceptance of the axiom that any limit equals the value it approaches is central to calculus. Without that axiom, things like differentials make no sense. Still, one doesn't even have to invoke limits. The decimal expansion of 9/9 is sufficient. --Jayron 32 17:23, 12 January 2021 (UTC)[reply]

It's a nice plausibility argument, but the weakness is that it relies on accepting (for example) the equation 1/9 = 0.111... in the first place. You can show your interlocutor the long division and that it keeps spitting out 1s, but it still takes another logical step to get from that to the fact that 0.111... represents exactly 1/9.

To really close the gap requires understanding what a real number is, and that brings in completed infinities and schools of mathematical foundational thought and things like that, which unfortunately are difficult to treat in an article pitched at the level of readers who would need the 0.999... article.

This is a recurring problem at that particular article. If you have ideas for addressing it, you might want to comment at its talk page. --Trovatore (talk) 18:21, 12 January 2021 (UTC)[reply]

True, but it's usually not the problem that the concept of a repeating decimal is the hangup, it's a problem that there are multiple ways to notate the same thing, and those multiple notations are arrived at from different directions. The bigger issue, however, is the problem with dealing with infinitesimals and limits, and that a limit really is scrupulously equal to the thing it approaches in the infinite. The hang up there is that people still think "infinite" is a synonym for "as much as I need to" or "until I have enough", and those are not the same things. Infinite is not "a lot". Those are different things. Also the problem that we can know about the end result of an infinite process (i.e. that an infinite process can actually stop) has been confusing people since Zeno. A mathematically rigorous difference between the continuous and the merely really finely divided, all of these distinctions have to be really internalized before one can understand the idea. --Jayron 32 18:31, 12 January 2021 (UTC)[reply]

@24.127.161.155: "The mathematical question" is not in fact "open". For mathematicians who accept the real numbers and the meaningfulness of infinite decimal expansions in the first place, there is no dispute that, interpreted as a real number, 0.999... is the same real number as 1. There is no unanswered mathematical question in this context.

However there are indeed some nuances. There are a few mathematicians who deny that the symbol 0.999... makes sense at all. And there are alternative ways of interpreting the symbol. These nuances are in fact treated in the article, though you have to scroll quite far down to find it. Take a look at the 0.999...#In alternative number systems section. Aside: I'm not sure that's the best possible section heading; ultrafinitism, for example, is not an "alternative number system" per se. --Trovatore (talk) 19:41, 12 January 2021 (UTC)[reply]

Yes, ultimately like in the question below about whether 1 is prime number it's matter of definition where the "..." means taking the limit to infinity. It's not all that different from 1 + 2 + 3 + 4 +.... = -1/12 and people claiming that this is wrong because the series diverges. The answer is then that the meaning of "..." in this case is not that of a limit of partial sums. Alternative definitions are possible because the definition of addition does not imply anything about infinite summations. It's just that "..." is conventionally chosen to the limit to infinity. Count Iblis (talk) 03:08, 15 January 2021 (UTC)[reply]