Talk:0.999.../Archive 20

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Numbers and numerals

I just reverted a good faith but confused edit which wanted to say that 0.9recurring and 1 are "quantitatively" equal. It seems to me that this is a variety of the confusion of numbers with numerals; another edit recently basically claimed that 0.9recurring and 1 are different because they are different numerals, with a claim that fundamentally numbers are actually (misunderstanding of) numerals. I wonder if it would be helpful to add a paragraph pointing out this distinction? Imaginatorium (talk) 13:26, 10 September 2023 (UTC)

The difference between a number and a numeral is a basic concept that few people seem to grasp. The new math that was common in elementary schools in the sixties presented that concept to young students and it's a simple idea. Unfortunately, it no longer seems to be part of the curriculum.

I think a brief treatment of that difference would be an improvement to the article. Something like

A numeral is a symbol that represents a number, and for any given number there are multiple ways to express it symbolically. For example: 5, V, 2+3, 101(base 2), 4.9999.... are all representations of the same number.

I don't think it will prevent the weekly drive-by edit from someone insisting that 0.9999... is not really equal to one, but it might be helpful to the general reader. Mr. Swordfish (talk) 14:59, 10 September 2023 (UTC)

The problem "numeral vs. number" is a special case of "expression vs. object represented by the expression", and even "syntax vs. semantics". For example, $(x+y)^{2}=x^{2}+2xy+y^{2}$ is false as an equality of expressions and true as an equality of polynomials. The first paragraph was confusing on this by using the nonsensical "decimal number". I boldy tried to clarify this by replacing "denotes" by "is a notation for" (this emphasizes that this is a convention), and replacing "decimal number" by "number". Possibly one could add also a sentence like "In other words, 0.999... and 1 are two diferent numerals that represent the same number". D.Lazard (talk) 16:21, 10 September 2023 (UTC)

I think that the problem is that merely rewording the initial statement will have no effect. Of course it would be sufficient if readers were moderately mathematically sophisticated, but if they were this page would not exist. However verbosely you word it, the ordinary readers will just pass over; so I think a separate paragraph is essential, something like Swordfish's suggestion. Imaginatorium (talk) 16:51, 10 September 2023 (UTC)

Agree that a simple short paragraph would be an improvement. And perhaps trim some of the wall of text that comprises the rest of the article. I don't have specific edits in mind here, but support the basic idea of a simple treatment of number vs numeral. Mr. Swordfish (talk) 00:46, 11 September 2023 (UTC)

Agree that "numeral vs. number" is a special case of "expression vs. object represented by the expression", and even "syntax vs. semantics". But I don't see how that matters. An elephant is a special case of a mammal but that doesn't preclude an entire article about elephants.

Here, the stumbling block seems to be number vs numeral, and I don't see any reason to go into generalities about expression vs object or syntax vs semantics. Mr. Swordfish (talk) 00:44, 11 September 2023 (UTC)

How to prove

x=.999... 10x=9.999... 9x+x=9+x 9x=9 So 1=.999... JackJackRR (talk) 21:28, 4 October 2023 (UTC)

See the proof in the section 'Algebraic arguments'. Mindmatrix 12:59, 5 October 2023 (UTC)

Stillwell proof

I find starting off with the Stillwell proof in its current form quite counterproductive; the first, non-rigorous explanation is just as hand-wavey as the 10x - x = 9 proof, and far, far, more confusing (what does "no room" mean, informally?); and when the rigorous version is introduced, it's no more or less easy than the other rigorous proofs. I would expect the naive reader to leave this section of the article totally confused, and give up on reading the rest.

Update: I've now demoted the "elementary proof" section to "elementary demonstration", and removed the attempts at partially formalizing it that were making it confusing by smuggling in the concepts of least upper bound and limits without introducing them first. By avoiding premature formalization, I think this now flows much better into the start of the formal argument section. — The Anome (talk) 12:40, 7 October 2023 (UTC)

@D.Lazard: I see you've reverted my careful changes, in which I've tried to keep as much of the original structure as possible. The "informal proof" is neither informal, nor a proof; it implicitly pulls in things like limits, continuity, the idea of least upper bound, and so on, probably as a result of other editors attempting to tighten up the language. I think the best we can do with this part is to let it be fuzzy and to appeal to intutitions about the number line, and not attempt to improve on it by implicity pulling in more advanced concepts without explanation or discussion.

(Just a few examples of defects: the Archimedean property is pulled out of a hat; "0.999..." is not actually defined; without the idea of limits, the reader could argue "there's always a gap, it just gets smaller"; it's not intuitively obvious that two numbers without another number between them must be the same (consider, for example, the integers); and there are more...)

Then when the formal concepts are introduced as a lead-up to the rigorous proof, the reader has not been confused by their premature, and unexplained, introduction earlier. Can I suggest that we move to edit this point by point, in a way that can be justified? — The Anome (talk) 13:52, 7 October 2023 (UTC)

Where did you see an "informal proof"? The word "informal" does not appear in the article outside section § See also and your edits. One of your main changes consists of replacing "proof" a word that has a precise meaning by "demonstration", a word without real meaning. Another of your changes introduces a blatant mathematical error: you wrote "if 1 were greater than all of 0.9, 0.99, 0.999, etc.," when 1 is effectively greater than all these numbers. More generally, while the article is carefully written to distinguish between intuitive explanations and mathematical proofs, most your edits amount to confuse them.

However, I have just remarked that you may have been confused by the fact the the proof refered to by the heading § Elementary proof is not in the introductive paragraph of the section, but in subsection § Rigorous proof. I fixed this ambiguity. D.Lazard (talk) 14:48, 7 October 2023 (UTC)

Restructuring the sections like that helps a lot. It's still not elementary, as it still relies on the introduction of the idea of least upper bound and implicitly the notion of limits, and also the Archimedean property, which is not obvious at all, and are pulled out of thin air. Nor do even those suffice; there's a gaping hole in the assumption that two numbers without another "between them" must be the same; we know this is true for the reals, but this is not an elementary properly, see for example the integers, where 2 and 3 manage to be different without another integer between them.

Given all this, why not just introduce Stillwell's informal argument about "not enough space" (which is fine, because it gets the feels right) and then go directly for the Dedekind cut approach, which is both rigorous and explicit?

Oh, and just to nitpick your nitpick, when I wrote "if 1 were greater than all of 0.9, 0.99, 0.999, etc.," I had in mind the idea of the least upper bound of the infinite sequence ie. "if 1 were greater than (all of 0.9, 0.99, 0.999, etc.,)" (which it isn't); not "if 1 were greater than (every one of of 0.9, 0.99, 0.999, etc.,)" (which it is). I'm sorry if you didn't understand my careful wording; I should have been more careful. — The Anome (talk) 10:55, 8 October 2023 (UTC)

Terminating decimals

"Terminating decimal" is a technical term that must be linked in such an elementary article. Previously, it was linked to Repeating decimal. I agree that this is not a convenient target. I have created a redirect, and linked it to an anchor in the lead of Decimal. If this link is not correct, this is not a problem of 0.999... but a problem of the redirect page or of the target page. In any case, the link Terminating decimal must be kept. D.Lazard (talk) 15:09, 27 October 2023 (UTC)

Agree. Since this term may be unfamiliar to some of our readers a link is necessary. If the article that it the target of the link needs improvement that should be discussed/implemented there, not here. Mr. Swordfish (talk) 16:18, 27 October 2023 (UTC)

I am a native speaker of English, which is fundamentally the target of WP:en. "Terminating decimal" is not a technical term, at all, it is simply the participle adjective "terminating", which means "it stops", qualifying "decimal". If there really were an article "terminating decimal", a link would be unnecessary, IMO, but not confusing. The "repeating decimal" article is not very good, since the first paragraph tells us that a terminating decimal is not a "repeating decimal", then the second paragraph backtracks, and says that a "terminating decimal" is one where the repeating sequence is just zeros. It cannot help to link a self-explanatory term to this. Imaginatorium (talk) 19:09, 27 October 2023 (UTC)

@Imaginatorium: Adjectives need precise definitions in math texts and this is one instance. A decimal expansion is not an event in time or place in space so the English definition does not apply, and is in any case too imprecise. It is very unlikely you are going to get consensus in favor of your view.--Jasper Deng (talk) 19:27, 27 October 2023 (UTC)

It seems also that Imaginatorium did not notice that "terminating decimal" does not link anymore to Repeating decimal; the target is an anchor in Decimal . D.Lazard (talk) 19:38, 27 October 2023 (UTC)

One more proof

According to the formation rule, the reciprocal part of the number is 9 and the non-revolving part is zero. Accordingly (9-0)/9=1. Please add this.

Bera678 (talk) 19:14, 15 December 2023 (UTC)

This is not a proof. Moreover, for being added here, a proof requires to be published in a textbook, and you do not provide any source. D.Lazard (talk) 09:36, 16 December 2023 (UTC)

Maybe I didn't fully express what I meant. But I'm sure it's proof. Although this is based on personal research, we can find a reference. Bera678 (talk) 09:44, 16 December 2023 (UTC)

It certainly is not a proof. It is merely quoting a rule of thumb for obtaining the value, but the rule of thumb is valid only because there is a proof of it. JBW (talk) 18:30, 16 December 2023 (UTC)

Did you look at the formation rule in the 'in compressed form' section of the Repeating decimal article? If you looked you can see that our number is equal to 9/9 to 1. Moreover, this evidence may be more understandable to readers. Bera678 (talk) 12:26, 17 December 2023 (UTC)

Yes, that may be the most convincing "evidence" for many readers, but it is not proof, since it hinges on arithmetic algorithms that first should be proven to be valid for infinite decimals. Nø (talk) 13:27, 17 December 2023 (UTC)

OK Bera678 (talk) 13:51, 17 December 2023 (UTC)