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class="ext-discussiontools-init-sidebar-meta">5 comments</span> <ul id="toc-Terminating_decimals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-One_more_proof" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#One_more_proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>One more proof</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">7 comments</span> <ul id="toc-One_more_proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-FAR_notice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#FAR_notice"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>FAR notice</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">20 comments</span> <ul id="toc-FAR_notice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Root_of_student_confusion?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Root_of_student_confusion?"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Root of student confusion?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">1 comment</span> <ul id="toc-Root_of_student_confusion?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_proof_supported_by_Stillwell?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementary_proof_supported_by_Stillwell?"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Elementary proof supported by Stillwell?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">8 comments</span> <ul id="toc-Elementary_proof_supported_by_Stillwell?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_"proof"?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementary_"proof"?"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Elementary "proof"?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">23 comments</span> <ul id="toc-Elementary_"proof"?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-efn?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#efn?"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>efn?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">8 comments</span> <ul id="toc-efn?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-lede_that_gets_to_the_point_without_jargon." class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#lede_that_gets_to_the_point_without_jargon."> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>lede that gets to the point without jargon.</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">19 comments</span> <ul id="toc-lede_that_gets_to_the_point_without_jargon.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Two_representations_in_every_positional_numeral_system_with_one_terminating?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Two_representations_in_every_positional_numeral_system_with_one_terminating?"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Two representations in every positional numeral system with one terminating?</span> </div> </a> <span class="ext-discussiontools-init-sidebar-meta">22 comments</span> <ul id="toc-Two_representations_in_every_positional_numeral_system_with_one_terminating?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sourcing_question" class="vector-toc-list-item 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src="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/40px-Replacement_filing_cabinet.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/60px-Replacement_filing_cabinet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/2/2a/Replacement_filing_cabinet.svg/80px-Replacement_filing_cabinet.svg.png 2x" data-file-width="200" data-file-height="200"/></span></span></td><td class="mbox-text" style="text-align:center">This is an <b><a href="/wiki/Help:Archiving_a_talk_page" title="Help:Archiving a talk page">archive</a></b> of past discussions about <a href="/wiki/0.999..." title="0.999...">0.999...</a>. <b>Do not edit the contents of this page.</b> If you wish to start a new discussion or revive an old one, please do so on the <a href="/wiki/Talk:0.999..." title="Talk:0.999...">current talk page</a>.</td></tr></tbody></table> <table style="text-align:center;margin:0 auto 0.5em;max-width:35em;width:100%;width:-moz-available;width:-webkit-fill-available;width:stretch"><tbody><tr><td><a href="/wiki/Talk:0.999.../Archive_15" title="Talk:0.999.../Archive 15">Archive 15</a></td><td>←</td><td><a href="/wiki/Talk:0.999.../Archive_18" title="Talk:0.999.../Archive 18">Archive 18</a></td><td><a href="/wiki/Talk:0.999.../Archive_19" title="Talk:0.999.../Archive 19">Archive 19</a></td><td><span style="font-size:115%;"><a class="mw-selflink selflink">Archive 20</a></span></td></tr></tbody></table> <meta property="mw:PageProp/toc"/> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Numbers_and_numerals" data-mw-thread-id="h-Numbers_and_numerals-20230910132600"><span data-mw-comment-start="" id="h-Numbers_and_numerals-20230910132600"></span>Numbers and numerals<span data-mw-comment-end="h-Numbers_and_numerals-20230910132600"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Imaginatorium-20230910132600-Numbers_and_numerals"></span>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 <b>numbers</b> with <b>numerals</b>; 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? <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> (<a href="/wiki/User_talk:Imaginatorium" title="User talk:Imaginatorium">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Imaginatorium-20230910132600-Numbers_and_numerals" class="ext-discussiontools-init-timestamplink">13:26, 10 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Imaginatorium-20230910132600-Numbers_and_numerals"></span><span data-mw-comment-end="c-Imaginatorium-20230910132600-Numbers_and_numerals"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Mr_swordfish-20230910145900-Imaginatorium-20230910132600"></span>The difference between a number and a numeral is a basic concept that few people seem to grasp. The <a href="/wiki/New_math" class="mw-redirect" title="New math">new math</a> 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.</dd> <dd>I think a brief treatment of that difference would be an improvement to the article. Something like <dl><dd>A <i>numeral</i> 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.</dd></dl></dd> <dd>I don't think it will prevent the weekly drive-by edit from someone insisting that 0.9999... is not <i>really equal</i> to one, but it might be helpful to the general reader. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20230910145900-Imaginatorium-20230910132600" class="ext-discussiontools-init-timestamplink">14:59, 10 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20230910145900-Imaginatorium-20230910132600"></span><span data-mw-comment-end="c-Mr_swordfish-20230910145900-Imaginatorium-20230910132600"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20230910162100-Mr_swordfish-20230910145900"></span>The problem "numeral vs. number" is a special case of "expression vs. object represented by the expression", and even "syntax vs. semantics". For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0111b619a713abcef925f71a14b68d6d011741cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.215ex; height:3.176ex;" alt="{\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}}"/></span> 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". <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20230910162100-Mr_swordfish-20230910145900" class="ext-discussiontools-init-timestamplink">16:21, 10 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20230910162100-Mr_swordfish-20230910145900"></span><span data-mw-comment-end="c-D.Lazard-20230910162100-Mr_swordfish-20230910145900"></span> <dl><dd><span data-mw-comment-start="" id="c-Imaginatorium-20230910165100-D.Lazard-20230910162100"></span>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. <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> (<a href="/wiki/User_talk:Imaginatorium" title="User talk:Imaginatorium">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Imaginatorium-20230910165100-D.Lazard-20230910162100" class="ext-discussiontools-init-timestamplink">16:51, 10 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Imaginatorium-20230910165100-D.Lazard-20230910162100"></span><span data-mw-comment-end="c-Imaginatorium-20230910165100-D.Lazard-20230910162100"></span> <dl><dd><span data-mw-comment-start="" id="c-Mr_swordfish-20230911004600-Imaginatorium-20230910165100"></span>Agree that a simple short paragraph would be an improvement. And perhaps trim some of the <i>wall of text</i> 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. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20230911004600-Imaginatorium-20230910165100" class="ext-discussiontools-init-timestamplink">00:46, 11 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20230911004600-Imaginatorium-20230910165100"></span><span data-mw-comment-end="c-Mr_swordfish-20230911004600-Imaginatorium-20230910165100"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Mr_swordfish-20230911004400-D.Lazard-20230910162100"></span>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 <a href="/wiki/Elephant" title="Elephant">elephant</a> is a special case of a <a href="/wiki/Mammal" title="Mammal">mammal</a> but that doesn't preclude an entire article about elephants.</dd> <dd>Here, the stumbling block seems to be <i>number vs numeral</i>, and I don't see any reason to go into generalities about <i>expression vs object</i> or <i>syntax vs semantics</i>. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20230911004400-D.Lazard-20230910162100" class="ext-discussiontools-init-timestamplink">00:44, 11 September 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20230911004400-D.Lazard-20230910162100"></span><span data-mw-comment-end="c-Mr_swordfish-20230911004400-D.Lazard-20230910162100"></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="How_to_prove" data-mw-thread-id="h-How_to_prove-20231004212800"><span data-mw-comment-start="" id="h-How_to_prove-20231004212800"></span>How to prove<span data-mw-comment-end="h-How_to_prove-20231004212800"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-JackJackRR-20231004212800-How_to_prove"></span>x=.999... 10x=9.999... 9x+x=9+x 9x=9 So 1=.999... <a href="/w/index.php?title=User:JackJackRR&action=edit&redlink=1" class="new" title="User:JackJackRR (page does not exist)">JackJackRR</a> (<a href="/w/index.php?title=User_talk:JackJackRR&action=edit&redlink=1" class="new" title="User talk:JackJackRR (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-JackJackRR-20231004212800-How_to_prove" class="ext-discussiontools-init-timestamplink">21:28, 4 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-JackJackRR-20231004212800-How_to_prove"></span><span data-mw-comment-end="c-JackJackRR-20231004212800-How_to_prove"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Mindmatrix-20231005125900-JackJackRR-20231004212800"></span>See the proof in the section 'Algebraic arguments'. <a href="/wiki/User:Mindmatrix" title="User:Mindmatrix"><span style="color: #8b4513;">Mind</span></a><a href="/wiki/User_talk:Mindmatrix" title="User talk:Mindmatrix"><span style="color: #ee8811;">matrix</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mindmatrix-20231005125900-JackJackRR-20231004212800" class="ext-discussiontools-init-timestamplink">12:59, 5 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mindmatrix-20231005125900-JackJackRR-20231004212800"></span><span data-mw-comment-end="c-Mindmatrix-20231005125900-JackJackRR-20231004212800"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Stillwell_proof" data-mw-thread-id="h-Stillwell_proof-20231007124000"><span data-mw-comment-start="" id="h-Stillwell_proof-20231007124000"></span>Stillwell proof<span data-mw-comment-end="h-Stillwell_proof-20231007124000"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-The_Anome-20231007124000-Stillwell_proof"></span>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. </p><p><i>Update:</i> 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. — <a href="/wiki/User:The_Anome" title="User:The Anome">The Anome</a> (<a href="/wiki/User_talk:The_Anome" title="User talk:The Anome">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-The_Anome-20231007124000-Stillwell_proof" class="ext-discussiontools-init-timestamplink">12:40, 7 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-The_Anome-20231007124000-Stillwell_proof"></span><span data-mw-comment-end="c-The_Anome-20231007124000-Stillwell_proof"></span> </p> <dl><dd><span class="template-ping"><span data-mw-comment-start="" id="c-The_Anome-20231007135200-The_Anome-20231007124000"></span>@<a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a>:</span> 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 <i>let</i> 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.<p>(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...)</p><p>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? — <a href="/wiki/User:The_Anome" title="User:The Anome">The Anome</a> (<a href="/wiki/User_talk:The_Anome" title="User talk:The Anome">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-The_Anome-20231007135200-The_Anome-20231007124000" class="ext-discussiontools-init-timestamplink">13:52, 7 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-The_Anome-20231007135200-The_Anome-20231007124000"></span><span data-mw-comment-end="c-The_Anome-20231007135200-The_Anome-20231007124000"></span> </p><dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20231007144800-The_Anome-20231007135200"></span>Where did you see an "informal proof"? The word "informal" does not appear in the article outside section <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ See also</a> 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.</dd> <dd>However, I have just remarked that you may have been confused by the fact the the proof refered to by the heading <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Elementary proof</a> is not in the introductive paragraph of the section, but in subsection <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Rigorous proof</a>. I fixed this ambiguity. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20231007144800-The_Anome-20231007135200" class="ext-discussiontools-init-timestamplink">14:48, 7 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20231007144800-The_Anome-20231007135200"></span><span data-mw-comment-end="c-D.Lazard-20231007144800-The_Anome-20231007135200"></span></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><span data-mw-comment-start="" id="c-The_Anome-20231008105500-D.Lazard-20231007144800"></span>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.<p>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? </p><p> 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 <i>my</i> careful wording; I should have been more careful. — <a href="/wiki/User:The_Anome" title="User:The Anome">The Anome</a> (<a href="/wiki/User_talk:The_Anome" title="User talk:The Anome">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-The_Anome-20231008105500-D.Lazard-20231007144800" class="ext-discussiontools-init-timestamplink">10:55, 8 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-The_Anome-20231008105500-D.Lazard-20231007144800"></span><span data-mw-comment-end="c-The_Anome-20231008105500-D.Lazard-20231007144800"></span></p></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Terminating_decimals" data-mw-thread-id="h-Terminating_decimals-20231027150900"><span data-mw-comment-start="" id="h-Terminating_decimals-20231027150900"></span>Terminating decimals<span data-mw-comment-end="h-Terminating_decimals-20231027150900"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-D.Lazard-20231027150900-Terminating_decimals"></span>"Terminating decimal" is a technical term that must be linked in such an elementary article. Previously, it was linked to <a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a>. I agree that this is not a convenient target. I have created a redirect, and linked it to an anchor in the lead of <a href="/wiki/Decimal" title="Decimal">Decimal</a>. If this link is not correct, this is not a problem of <a href="/wiki/0.999..." title="0.999...">0.999...</a> but a problem of the redirect page or of the target page. In any case, the link <a href="/wiki/Terminating_decimal" class="mw-redirect" title="Terminating decimal">Terminating decimal</a> must be kept. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20231027150900-Terminating_decimals" class="ext-discussiontools-init-timestamplink">15:09, 27 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20231027150900-Terminating_decimals"></span><span data-mw-comment-end="c-D.Lazard-20231027150900-Terminating_decimals"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Mr_swordfish-20231027161800-D.Lazard-20231027150900"></span>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. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20231027161800-D.Lazard-20231027150900" class="ext-discussiontools-init-timestamplink">16:18, 27 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20231027161800-D.Lazard-20231027150900"></span><span data-mw-comment-end="c-Mr_swordfish-20231027161800-D.Lazard-20231027150900"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Imaginatorium-20231027190900-Mr_swordfish-20231027161800"></span>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. <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> (<a href="/wiki/User_talk:Imaginatorium" title="User talk:Imaginatorium">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Imaginatorium-20231027190900-Mr_swordfish-20231027161800" class="ext-discussiontools-init-timestamplink">19:09, 27 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Imaginatorium-20231027190900-Mr_swordfish-20231027161800"></span><span data-mw-comment-end="c-Imaginatorium-20231027190900-Mr_swordfish-20231027161800"></span> <dl><dd><span class="template-ping"><span data-mw-comment-start="" id="c-Jasper_Deng-20231027192700-Imaginatorium-20231027190900"></span>@<a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a>:</span> 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.--<a href="/wiki/User:Jasper_Deng" title="User:Jasper Deng">Jasper Deng</a> <a href="/wiki/User_talk:Jasper_Deng" title="User talk:Jasper Deng">(talk)</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Jasper_Deng-20231027192700-Imaginatorium-20231027190900" class="ext-discussiontools-init-timestamplink">19:27, 27 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Jasper_Deng-20231027192700-Imaginatorium-20231027190900"></span><span data-mw-comment-end="c-Jasper_Deng-20231027192700-Imaginatorium-20231027190900"></span></dd> <dd><span data-mw-comment-start="" id="c-D.Lazard-20231027193800-Imaginatorium-20231027190900"></span>It seems also that <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> did not notice that "<a href="/wiki/Terminating_decimal" class="mw-redirect" title="Terminating decimal">terminating decimal</a>" does not link anymore to <a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a>; the target is an anchor in <a href="/wiki/Decimal" title="Decimal">Decimal</a> . <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20231027193800-Imaginatorium-20231027190900" class="ext-discussiontools-init-timestamplink">19:38, 27 October 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20231027193800-Imaginatorium-20231027190900"></span><span data-mw-comment-end="c-D.Lazard-20231027193800-Imaginatorium-20231027190900"></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="One_more_proof" data-mw-thread-id="h-One_more_proof-20231215191400"><span data-mw-comment-start="" id="h-One_more_proof-20231215191400"></span>One more proof<span data-mw-comment-end="h-One_more_proof-20231215191400"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Bera678-20231215191400-One_more_proof"></span>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. </p><p><a href="/wiki/User:Bera678" title="User:Bera678">Bera678</a> (<a href="/wiki/User_talk:Bera678" title="User talk:Bera678">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Bera678-20231215191400-One_more_proof" class="ext-discussiontools-init-timestamplink">19:14, 15 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bera678-20231215191400-One_more_proof"></span><span data-mw-comment-end="c-Bera678-20231215191400-One_more_proof"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20231216093600-Bera678-20231215191400"></span>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. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20231216093600-Bera678-20231215191400" class="ext-discussiontools-init-timestamplink">09:36, 16 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20231216093600-Bera678-20231215191400"></span><span data-mw-comment-end="c-D.Lazard-20231216093600-Bera678-20231215191400"></span> <dl><dd><span data-mw-comment-start="" id="c-Bera678-20231216094400-D.Lazard-20231216093600"></span>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. <a href="/wiki/User:Bera678" title="User:Bera678">Bera678</a> (<a href="/wiki/User_talk:Bera678" title="User talk:Bera678">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Bera678-20231216094400-D.Lazard-20231216093600" class="ext-discussiontools-init-timestamplink">09:44, 16 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bera678-20231216094400-D.Lazard-20231216093600"></span><span data-mw-comment-end="c-Bera678-20231216094400-D.Lazard-20231216093600"></span> <dl><dd><span data-mw-comment-start="" id="c-JBW-20231216183000-Bera678-20231216094400"></span>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. <a href="/wiki/User:JBW" title="User:JBW">JBW</a> (<a href="/wiki/User_talk:JBW" title="User talk:JBW">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-JBW-20231216183000-Bera678-20231216094400" class="ext-discussiontools-init-timestamplink">18:30, 16 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-JBW-20231216183000-Bera678-20231216094400"></span><span data-mw-comment-end="c-JBW-20231216183000-Bera678-20231216094400"></span> <dl><dd><span data-mw-comment-start="" id="c-Bera678-20231217122600-JBW-20231216183000"></span>Did you look at the formation rule in the 'in compressed form' section of the <a href="/wiki/Repeating_decimal" title="Repeating decimal">Repeating decimal</a> 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. <a href="/wiki/User:Bera678" title="User:Bera678">Bera678</a> (<a href="/wiki/User_talk:Bera678" title="User talk:Bera678">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Bera678-20231217122600-JBW-20231216183000" class="ext-discussiontools-init-timestamplink">12:26, 17 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bera678-20231217122600-JBW-20231216183000"></span><span data-mw-comment-end="c-Bera678-20231217122600-JBW-20231216183000"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20231217132700-Bera678-20231217122600"></span>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. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20231217132700-Bera678-20231217122600" class="ext-discussiontools-init-timestamplink">13:27, 17 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20231217132700-Bera678-20231217122600"></span><span data-mw-comment-end="c-Nø-20231217132700-Bera678-20231217122600"></span> <dl><dd><span data-mw-comment-start="" id="c-Bera678-20231217135100-Nø-20231217132700"></span>OK <a href="/wiki/User:Bera678" title="User:Bera678">Bera678</a> (<a href="/wiki/User_talk:Bera678" title="User talk:Bera678">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Bera678-20231217135100-Nø-20231217132700" class="ext-discussiontools-init-timestamplink">13:51, 17 December 2023 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Bera678-20231217135100-Nø-20231217132700"></span><span data-mw-comment-end="c-Bera678-20231217135100-Nø-20231217132700"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="FAR_notice" data-mw-thread-id="h-FAR_notice-20240118022300"><span data-mw-comment-start="" id="h-FAR_notice-20240118022300"></span>FAR notice<span data-mw-comment-end="h-FAR_notice-20240118022300"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Voorts-20240118022300-FAR_notice"></span>This is an <a href="/wiki/Wikipedia:Featured_article_candidates/0.999..." title="Wikipedia:Featured article candidates/0.999...">FA</a> from 2006 that underwent <a href="/wiki/Wikipedia:Featured_article_review/0.999.../archive1" title="Wikipedia:Featured article review/0.999.../archive1">FAR</a> in 2010 and was kept. This article does not currently meet the <a href="/wiki/Wikipedia:FACR" class="mw-redirect" title="Wikipedia:FACR">featured article criteria</a>: </p> <ul><li>It uses a mixture of parenthetical referencing, which is <a href="/wiki/Wikipedia:HARV" class="mw-redirect" title="Wikipedia:HARV">deprecated</a>, and inline references, failing 2.c.</li> <li>The "Elementary proof" section is entirely unreferenced, and many other sections have unreferenced paragraphs, some of which appears to contain <a href="/wiki/Wikipedia:OR" class="mw-redirect" title="Wikipedia:OR">OR</a> (see, e.g., "Impossibility of unique representation"), failing 1.c.</li> <li>There are <a href="/wiki/Wikipedia:WEASEL" class="mw-redirect" title="Wikipedia:WEASEL">weasel words</a> and <a href="/wiki/MOS:OFCOURSE" class="mw-redirect" title="MOS:OFCOURSE">editorializing</a> throughout and the writing style is at times casual, failing 2.</li></ul> <p>Pinging @<a href="/wiki/User:JayBeeEll" title="User:JayBeeEll">JBL</a>; I saw your recent FA and hoped you might be able to take a look. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240118022300-FAR_notice" class="ext-discussiontools-init-timestamplink">02:23, 18 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240118022300-FAR_notice"></span><span data-mw-comment-end="c-Voorts-20240118022300-FAR_notice"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240118023100-Voorts-20240118022300"></span>@<a href="/wiki/User:Voorts" title="User:Voorts">Voorts</a> I'll deal the citation style. I'm changing to sfnp for all short citations. <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240118023100-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">02:31, 18 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240118023100-Voorts-20240118022300"></span><span data-mw-comment-end="c-Dedhert.Jr-20240118023100-Voorts-20240118022300"></span> <dl><dd><span data-mw-comment-start="" id="c-Voorts-20240118023600-Dedhert.Jr-20240118023100"></span>The rest of the cites need work too; many of them don't use any citation formats and some of them are ref tags with <span class="nowrap">{{</span><a href="/wiki/Template:Harv" class="mw-redirect" title="Template:Harv">harv</a><span class="nowrap">}}</span>s inside them. Since there are variations in citation style, I think they can all be changed to <span class="nowrap">{{</span><a href="/wiki/Template:Sfnp" title="Template:Sfnp">sfnp</a><span class="nowrap">}}</span> for conformity. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240118023600-Dedhert.Jr-20240118023100" class="ext-discussiontools-init-timestamplink">02:36, 18 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240118023600-Dedhert.Jr-20240118023100"></span><span data-mw-comment-end="c-Voorts-20240118023600-Dedhert.Jr-20240118023100"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Voorts-20240118024000-Voorts-20240118022300"></span>One additional thing: I don't see any kind of thorough source checking in either the FA or FAR discussions. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240118024000-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">02:40, 18 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240118024000-Voorts-20240118022300"></span><span data-mw-comment-end="c-Voorts-20240118024000-Voorts-20240118022300"></span> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240119054500-Voorts-20240118024000"></span>@<a href="/wiki/User:Voorts" title="User:Voorts">Voorts</a> An additional thing but optional likely. I do think that this article uses many types of math templates, math in TeX, and by simply just using HTML code. So I prefer to use Tex instead, right after completing the citations format problems. <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240119054500-Voorts-20240118024000" class="ext-discussiontools-init-timestamplink">05:45, 19 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240119054500-Voorts-20240118024000"></span><span data-mw-comment-end="c-Dedhert.Jr-20240119054500-Voorts-20240118024000"></span> <dl><dd><span data-mw-comment-start="" id="c-Voorts-20240119055600-Dedhert.Jr-20240119054500"></span>I'm not really well-versed in math templates on Wikipedia, so I can't really opine on what to use, but I agree that using plain html code is not the best. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240119055600-Dedhert.Jr-20240119054500" class="ext-discussiontools-init-timestamplink">05:56, 19 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240119055600-Dedhert.Jr-20240119054500"></span><span data-mw-comment-end="c-Voorts-20240119055600-Dedhert.Jr-20240119054500"></span></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-JayBeeEll-20240121205600-Voorts-20240118022300"></span>Thanks for the ping, <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a>. Unfortunately I've discovered about myself that I'm good at starting something more or less from scratch, and good at local spot-checking, but not very good at the kind of work needed here. I'll try to take a look, though. --<a href="/wiki/User:JayBeeEll" title="User:JayBeeEll">JBL</a> (<a href="/wiki/User_talk:JayBeeEll" title="User talk:JayBeeEll">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-JayBeeEll-20240121205600-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">20:56, 21 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-JayBeeEll-20240121205600-Voorts-20240118022300"></span><span data-mw-comment-end="c-JayBeeEll-20240121205600-Voorts-20240118022300"></span></dd> <dd><span data-mw-comment-start="" id="c-Hawkeye7-20240121212000-Voorts-20240118022300"></span>I am willing to help with this one. Ping me if want help with anything. I will conduct a source check. For the record: I do not see any problem with the casual writing style, given the readership of this article. <a href="/wiki/User:Hawkeye7" title="User:Hawkeye7"><span style="color:#800082">Hawkeye7</span></a> <a href="/wiki/User_talk:Hawkeye7" title="User talk:Hawkeye7"><span style="font-size:80%">(discuss)</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Hawkeye7-20240121212000-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">21:20, 21 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hawkeye7-20240121212000-Voorts-20240118022300"></span><span data-mw-comment-end="c-Hawkeye7-20240121212000-Voorts-20240118022300"></span> <dl><dd><span data-mw-comment-start="" id="c-Hawkeye7-20240121233600-Hawkeye7-20240121212000"></span>"we should convert this into the book being used as a reference (but that would require access to it to see how to use it)" Fortunately, I do. Which is why I said I would look at the sources. <a href="/wiki/User:Hawkeye7" title="User:Hawkeye7"><span style="color:#800082">Hawkeye7</span></a> <a href="/wiki/User_talk:Hawkeye7" title="User talk:Hawkeye7"><span style="font-size:80%">(discuss)</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Hawkeye7-20240121233600-Hawkeye7-20240121212000" class="ext-discussiontools-init-timestamplink">23:36, 21 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hawkeye7-20240121233600-Hawkeye7-20240121212000"></span><span data-mw-comment-end="c-Hawkeye7-20240121233600-Hawkeye7-20240121212000"></span></dd></dl></dd></dl> <dl><dd><span data-mw-comment-start="" id="c-Imaginatorium-20240122093800-Voorts-20240118022300"></span>Can you explain what "weasel words" means in this context? An example or two would help... <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> (<a href="/wiki/User_talk:Imaginatorium" title="User talk:Imaginatorium">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Imaginatorium-20240122093800-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">09:38, 22 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Imaginatorium-20240122093800-Voorts-20240118022300"></span><span data-mw-comment-end="c-Imaginatorium-20240122093800-Voorts-20240118022300"></span> <dl><dd><span data-mw-comment-start="" id="c-Voorts-20240122211900-Imaginatorium-20240122093800"></span>Sure, here's a couple: <ul><li>"While most authors choose to define"</li> <li>"Many algebraic arguments have been provided"</li></ul></dd> <dd><a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240122211900-Imaginatorium-20240122093800" class="ext-discussiontools-init-timestamplink">21:19, 22 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240122211900-Imaginatorium-20240122093800"></span><span data-mw-comment-end="c-Voorts-20240122211900-Imaginatorium-20240122093800"></span> <dl><dd><span data-mw-comment-start="" id="c-Voorts-20240122211900-Voorts-20240122211900"></span>See also <a href="/wiki/Wikipedia:WEASEL" class="mw-redirect" title="Wikipedia:WEASEL">WP:WEASEL</a>. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240122211900-Voorts-20240122211900" class="ext-discussiontools-init-timestamplink">21:19, 22 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240122211900-Voorts-20240122211900"></span><span data-mw-comment-end="c-Voorts-20240122211900-Voorts-20240122211900"></span> <dl><dd><span data-mw-comment-start="" id="c-JayBeeEll-20240122231000-Voorts-20240122211900"></span>I don't understand the assertion that either of these is weasel-y. These assertions might or might not be adequately sourced (to be clear: I haven't checked), but if they reflect the sources I don't see what's objectionable about them. --<a href="/wiki/User:JayBeeEll" title="User:JayBeeEll">JBL</a> (<a href="/wiki/User_talk:JayBeeEll" title="User talk:JayBeeEll">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-JayBeeEll-20240122231000-Voorts-20240122211900" class="ext-discussiontools-init-timestamplink">23:10, 22 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-JayBeeEll-20240122231000-Voorts-20240122211900"></span><span data-mw-comment-end="c-JayBeeEll-20240122231000-Voorts-20240122211900"></span> <dl><dd><span data-mw-comment-start="" id="c-Hawkeye7-20240123021000-JayBeeEll-20240122231000"></span>"While most authors choose to define" is <i>not</i> in the source, so I have removed it. I'm not seeing support for the assertion "Division by zero occurs in some popular discussions of 0.999..." either. Unless someone can find one, I suggest we remove the entire bullet point. Apart from that sentence though, it is correctly sourced. <a href="/wiki/User:Hawkeye7" title="User:Hawkeye7"><span style="color:#800082">Hawkeye7</span></a> <a href="/wiki/User_talk:Hawkeye7" title="User talk:Hawkeye7"><span style="font-size:80%">(discuss)</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Hawkeye7-20240123021000-JayBeeEll-20240122231000" class="ext-discussiontools-init-timestamplink">02:10, 23 January 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hawkeye7-20240123021000-JayBeeEll-20240122231000"></span><span data-mw-comment-end="c-Hawkeye7-20240123021000-JayBeeEll-20240122231000"></span> <dl><dd><span data-mw-comment-start="" id="c-Voorts-20240221224500-Hawkeye7-20240123021000"></span>@<a href="/wiki/User:Hawkeye7" title="User:Hawkeye7">Hawkeye7</a> @<a href="/wiki/User:JayBeeEll" title="User:JayBeeEll">JayBeeEll</a> @<a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a>: Where are we on this? Has enough been done to fix this, or should this proceed to FAR? <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240221224500-Hawkeye7-20240123021000" class="ext-discussiontools-init-timestamplink">22:45, 21 February 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240221224500-Hawkeye7-20240123021000"></span><span data-mw-comment-end="c-Voorts-20240221224500-Hawkeye7-20240123021000"></span> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240222133700-Voorts-20240221224500"></span>@<a href="/wiki/User:Voorts" title="User:Voorts">Voorts</a> I'm replying. Will trying to convert again as soon as possible, and copyedit; trying my best. <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240222133700-Voorts-20240221224500" class="ext-discussiontools-init-timestamplink">13:37, 22 February 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240222133700-Voorts-20240221224500"></span><span data-mw-comment-end="c-Dedhert.Jr-20240222133700-Voorts-20240221224500"></span> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240223133600-Dedhert.Jr-20240222133700"></span>I have converted the format footnotes into sfnp and harvtxt, and all math format in Tex. <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240223133600-Dedhert.Jr-20240222133700" class="ext-discussiontools-init-timestamplink">13:36, 23 February 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240223133600-Dedhert.Jr-20240222133700"></span><span data-mw-comment-end="c-Dedhert.Jr-20240223133600-Dedhert.Jr-20240222133700"></span> <dl><dd><span data-mw-comment-start="" id="c-Hawkeye7-20240223191600-Dedhert.Jr-20240223133600"></span>I have moved unused references to the Further reading section. <a href="/wiki/User:Hawkeye7" title="User:Hawkeye7"><span style="color:#800082">Hawkeye7</span></a> <a href="/wiki/User_talk:Hawkeye7" title="User talk:Hawkeye7"><span style="font-size:80%">(discuss)</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Hawkeye7-20240223191600-Dedhert.Jr-20240223133600" class="ext-discussiontools-init-timestamplink">19:16, 23 February 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hawkeye7-20240223191600-Dedhert.Jr-20240223133600"></span><span data-mw-comment-end="c-Hawkeye7-20240223191600-Dedhert.Jr-20240223133600"></span></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-XOR'easter-20240227014200-Hawkeye7-20240123021000"></span>The "Division by zero occurs in some popular discussions..." reads like <a href="/wiki/Wikipedia:SYNTH" class="mw-redirect" title="Wikipedia:SYNTH">WP:SYNTH</a> to me (that is, <a href="/wiki/Wikipedia:SYNTH" class="mw-redirect" title="Wikipedia:SYNTH">WP:SYNTH</a> dressed up with citations to the background topics being synthesized). There's maybe something to be said about how understanding limits can give a precise meaning to the intuitive idea of "division by zero" (or "division by infinity"), and limits are also important here, but without a source explicitly drawing that connection, we shouldn't include it. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240227014200-Hawkeye7-20240123021000" class="ext-discussiontools-init-timestamplink">01:42, 27 February 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240227014200-Hawkeye7-20240123021000"></span><span data-mw-comment-end="c-XOR'easter-20240227014200-Hawkeye7-20240123021000"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Voorts-20240326030600-Voorts-20240118022300"></span>Unless there are any objections, I plan on bringing this to FAR one week from now. <a href="/wiki/User:Voorts" title="User:Voorts">voorts</a> (<a href="/wiki/User_talk:Voorts" title="User talk:Voorts">talk</a>/<a href="/wiki/Special:Contributions/Voorts" title="Special:Contributions/Voorts">contributions</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Voorts-20240326030600-Voorts-20240118022300" class="ext-discussiontools-init-timestamplink">03:06, 26 March 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Voorts-20240326030600-Voorts-20240118022300"></span><span data-mw-comment-end="c-Voorts-20240326030600-Voorts-20240118022300"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Root_of_student_confusion?" data-mw-thread-id="h-Root_of_student_confusion?-20240414125500"><span id="Root_of_student_confusion.3F"></span><span data-mw-comment-start="" id="h-Root_of_student_confusion?-20240414125500"></span>Root of student confusion?<span data-mw-comment-end="h-Root_of_student_confusion?-20240414125500"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Chatul-20240414125500-Root_of_student_confusion?"></span>Is there any elementary education literature on confusion caused by teaching real numbers in terms of decimal expansions instead of axiomatically or geometrically? I believe that if such an <a href="/wiki/Wikipedia:RS" class="mw-redirect" title="Wikipedia:RS">RS</a> exits then the article should discuss the issue. -- <a href="/wiki/User:Chatul" title="User:Chatul">Shmuel (Seymour J.) Metz Username:Chatul</a> (<a href="/wiki/User_talk:Chatul" title="User talk:Chatul">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Chatul-20240414125500-Root_of_student_confusion?" class="ext-discussiontools-init-timestamplink">12:55, 14 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Chatul-20240414125500-Root_of_student_confusion?"></span><span data-mw-comment-end="c-Chatul-20240414125500-Root_of_student_confusion?"></span> </p> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Elementary_proof_supported_by_Stillwell?" data-mw-thread-id="h-Elementary_proof_supported_by_Stillwell?-20240411222000"><span id="Elementary_proof_supported_by_Stillwell.3F"></span><span data-mw-comment-start="" id="h-Elementary_proof_supported_by_Stillwell?-20240411222000"></span>Elementary proof supported by Stillwell?<span data-mw-comment-end="h-Elementary_proof_supported_by_Stillwell?-20240411222000"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Tito_Omburo-20240411222000-Elementary_proof_supported_by_Stillwell?"></span>It strikes me that the Stillwell reference for the section on the Elementary proof is not ideal. Can anyone find a better reference? <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240411222000-Elementary_proof_supported_by_Stillwell?" class="ext-discussiontools-init-timestamplink">22:20, 11 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240411222000-Elementary_proof_supported_by_Stillwell?"></span><span data-mw-comment-end="c-Tito_Omburo-20240411222000-Elementary_proof_supported_by_Stillwell?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240415172900-Tito_Omburo-20240411222000"></span>I looked once but didn't have any luck finding a source that spells it out with all the steps that <a href="/wiki/0.999...#Rigorous_proof" title="0.999...">this subsection</a> does. On the other hand, I'm not sure that subsection adds more clarity than it does notation. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240415172900-Tito_Omburo-20240411222000" class="ext-discussiontools-init-timestamplink">17:29, 15 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240415172900-Tito_Omburo-20240411222000"></span><span data-mw-comment-end="c-XOR'easter-20240415172900-Tito_Omburo-20240411222000"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240415181600-XOR'easter-20240415172900"></span>Are you satisfied that the section as a whole is well-supported? When I last checked, Stillwell was the only cited source, a situation you have now significantly improved. I'm less worried about whether all the steps are explicitly referenced, and we can cut the last section out if necessary. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240415181600-XOR'easter-20240415172900" class="ext-discussiontools-init-timestamplink">18:16, 15 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240415181600-XOR'easter-20240415172900"></span><span data-mw-comment-end="c-Tito_Omburo-20240415181600-XOR'easter-20240415172900"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240415214900-Tito_Omburo-20240415181600"></span>I'm significantly happier with that section than I was. The more I look at the "Rigorous proof" subsection, the more I think we could remove it without loss of clarity (perhaps even of rigor!). One thing that bothers me: the section heading "Elementary proof" is not very illuminating, and the proof is only "elementary" in a rather technical sense. The only argument for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.999\ldots =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.999</mn> <mo>…</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.999\ldots =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961087f58e21923379c69444a32c142e03308dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.668ex; height:2.176ex;" alt="{\displaystyle 0.999\ldots =1}"/></span> that I can recall being explicitly <i>called</i> an "elementary proof" is <a href="/wiki/0.999...#Algebraic_arguments" title="0.999...">this one</a>, in the Peressini and Peressini reference. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240415214900-Tito_Omburo-20240415181600" class="ext-discussiontools-init-timestamplink">21:49, 15 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240415214900-Tito_Omburo-20240415181600"></span><span data-mw-comment-end="c-XOR'easter-20240415214900-Tito_Omburo-20240415181600"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240416102100-XOR'easter-20240415214900"></span>It is stated in the linked section that Peressini and Peressini wrote that transforming this argument into a proof "would likely involve concepts of infinity and completeness". This is far from being elementary. On the other hand the proof given here is really elementary in the sense that it uses only elementary manipulation of (finite) decimal numbers and the Archimedean property, and it shows that the latter is unavoidable.</dd> <dd>Section <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Discussion on completeness</a> must be removed or moved elsewhere, since completeness is not involved in the proof considered in this section.</dd> <dd>This section "Elementary proof" was introduced by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&diff=prev&oldid=793249433">this edit</a>, in view of closing lenghty discussions on the talk page (see <a href="/wiki/Talk:0.999.../Archive_18" title="Talk:0.999.../Archive 18">Talk:0.999.../Archive 18</a> and more specially <a href="/wiki/Talk:0.999.../Archive_18#Elementary_proof" title="Talk:0.999.../Archive 18">Talk:0.999.../Archive 18#Elementary proof</a>. The subsections <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Intuitive explanation</a> and <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Rigorous proof</a> have been introduced by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&diff=prev&oldid=793894277">this edit</a> (the second heading has been improved since this edit).</dd> <dd>I am <b>strongly against</b> the removal of <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Rigorous proof</a>. Instead, we could reduce <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Intuitive explanation</a> to its first paragraph, since, all what follows "More precisely" is repeated in <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Rigorous proof</a>. The reason for keeping both subsections is that the common confusion about <span class="texhtml">0.999... = 1</span> results from a bad understanding of the difference between an intuitive explanation and a true proof. Since this article is aimed for young students, the distinction must be kept as clear as possible. Fortunately, with this proof, we have not to say them "wait to have learnt more mathematics for having a true proof", as it is the case with the other proofs given in this article. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240416102100-XOR'easter-20240415214900" class="ext-discussiontools-init-timestamplink">10:21, 16 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240416102100-XOR'easter-20240415214900"></span><span data-mw-comment-end="c-D.Lazard-20240416102100-XOR'easter-20240415214900"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240416173300-D.Lazard-20240416102100"></span>But if no one other than us <i>calls</i> the proof in this section "elementary", then doing so violates <a href="/wiki/Wikipedia:NOR" class="mw-redirect" title="Wikipedia:NOR">WP:NOR</a>. It's not our job to compare the existing arguments and proofs, evaluate the features that they each contain, and crown one of them as the most "elementary". And to a reader not familiar with how mathematicians use the word "elementary", applying it to a proof that invokes something called "the Archimedean property" is just confusing. (<a rel="nofollow" class="external text" href="https://xkcd.com/2501/">It's easy to forget that the average person probably only knows</a> that the rationals are dense in the reals.) Right now, our use of the term "Elementary proof" here is bad from the standpoint of policy (it's <a href="/wiki/Wikipedia:SYNTH" class="mw-redirect" title="Wikipedia:SYNTH">WP:SYNTH</a> until we find a source saying so), and it's not great from the standpoint of pedagogy either. <div class="paragraphbreak" style="margin-top:0.5em"></div> I moved the "Discussion on completeness" subsection to the end of the section, since it didn't really belong where it was. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240416173300-D.Lazard-20240416102100" class="ext-discussiontools-init-timestamplink">17:33, 16 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240416173300-D.Lazard-20240416102100"></span><span data-mw-comment-end="c-XOR'easter-20240416173300-D.Lazard-20240416102100"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-Tito_Omburo-20240416181900-Elementary_proof_supported_by_Stillwell?"></span>I think the term "elementary" is a bad one. Perhaps something indicating that the proof uses decimal representations? I think the rigorous proof should stay, and the new arrangement of content makes this clearer to me. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240416181900-Elementary_proof_supported_by_Stillwell?" class="ext-discussiontools-init-timestamplink">18:19, 16 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240416181900-Elementary_proof_supported_by_Stillwell?"></span><span data-mw-comment-end="c-Tito_Omburo-20240416181900-Elementary_proof_supported_by_Stillwell?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240417190400-Tito_Omburo-20240416181900"></span>I changed the section heading to "Proof by adding and comparing decimal numbers", which gets away from the term "Elementary" while still, I think, making it sound fairly easy. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240417190400-Tito_Omburo-20240416181900" class="ext-discussiontools-init-timestamplink">19:04, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240417190400-Tito_Omburo-20240416181900"></span><span data-mw-comment-end="c-XOR'easter-20240417190400-Tito_Omburo-20240416181900"></span></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Elementary_"proof"?" data-mw-thread-id="h-Elementary_"proof"?-20240529192000"><span id="Elementary_.22proof.22.3F"></span><span data-mw-comment-start="" id="h-Elementary_"proof"?-20240529192000"></span>Elementary "proof"?<span data-mw-comment-end="h-Elementary_"proof"?-20240529192000"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Nø-20240529192000-Elementary_"proof"?"></span>The article has </p> <dl><dd><i>It is possible to prove the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.999\ldots =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.999</mn> <mo>…</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.999\ldots =1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6961087f58e21923379c69444a32c142e03308dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.668ex; height:2.176ex;" alt="{\displaystyle 0.999\ldots =1}"/></span> using just the mathematical tools of comparison and addition of (finite) <a href="/wiki/Decimal_number" class="mw-redirect" title="Decimal number">decimal numbers</a>, without any reference to more advanced topics such as <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a>, <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limits</a>, or the formal <a href="/wiki/Construction_of_real_numbers" class="mw-redirect" title="Construction of real numbers">construction of real numbers</a>.</i></dd></dl> <p>I've changed this, but was reverted. I believe it makes no sense to talk about an <i>elementary proof</i> avoiding any formalism like limits or the construction of the real numbers; without these, the notation 0.999... has no meaning, and there is no such thing as a proof. Thoughts? Any good sources? <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240529192000-Elementary_"proof"?" class="ext-discussiontools-init-timestamplink">19:20, 29 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240529192000-Elementary_"proof"?"></span><span data-mw-comment-end="c-Nø-20240529192000-Elementary_"proof"?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240529210800-Nø-20240529192000"></span>Chapter 1 of Apostol defines decimal expansions with no reference to limits. (Just the completeness axiom.) <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240529210800-Nø-20240529192000" class="ext-discussiontools-init-timestamplink">21:08, 29 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240529210800-Nø-20240529192000"></span><span data-mw-comment-end="c-Tito_Omburo-20240529210800-Nø-20240529192000"></span></dd> <dd><small><span data-mw-comment-start="" id="c-D.Lazard-20240529211700-Nø-20240529192000"></span>(<a href="/wiki/Help:Edit_conflict" title="Help:Edit conflict">edit conflict</a>)</small> Read the proof: except some elementary manipulations of finite decimal numbers, the only tool that is used is that, if a real number <span class="texhtml mvar" style="font-style:italic;">x</span> is smaller than <span class="texhtml">1</span>, then there is a positive integer such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x<1-{\frac {1}{n}}<1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo><</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo><</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x<1-{\frac {1}{n}}<1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d0bee72b3abf96e5491967e67c101fdd9491f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.161ex; height:3.343ex;" alt="{\textstyle x<1-{\frac {1}{n}}<1.}"/></span> This does no involve any notion of limit or series. More, it does not involve the fact that a upper bounded set of real numbers admits a least upper bound. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240529211700-Nø-20240529192000" class="ext-discussiontools-init-timestamplink">21:17, 29 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240529211700-Nø-20240529192000"></span><span data-mw-comment-end="c-D.Lazard-20240529211700-Nø-20240529192000"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240529212000-D.Lazard-20240529211700"></span>I agree with this assessment. As for sources, a pretty clear version of this appears in Bartle and Sherbert. Basically, only <i>existence</i> of a real number with a given decimal expansion uses completeness. But here, of course, existence is not an issue. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240529212000-D.Lazard-20240529211700" class="ext-discussiontools-init-timestamplink">21:20, 29 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240529212000-D.Lazard-20240529211700"></span><span data-mw-comment-end="c-Tito_Omburo-20240529212000-D.Lazard-20240529211700"></span></dd> <dd><span data-mw-comment-start="" id="c-Nø-20240530071700-D.Lazard-20240529211700"></span>While the completeness theorem (involved in the so-called rigorous proof in the statement "This point would be at a positive distance from 1") intuitively makes sense (at least to anyone who has been used to real numbers, decimal notation, and the number line for a while), to call it an elementary topic (as opposed to an advanced one) seems quite a stretch to me. Am I missing something here? <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240530071700-D.Lazard-20240529211700" class="ext-discussiontools-init-timestamplink">07:17, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240530071700-D.Lazard-20240529211700"></span><span data-mw-comment-end="c-Nø-20240530071700-D.Lazard-20240529211700"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240530092300-Nø-20240530071700"></span>That is only the Archimedean property. Completeness in not needed. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240530092300-Nø-20240530071700" class="ext-discussiontools-init-timestamplink">09:23, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240530092300-Nø-20240530071700"></span><span data-mw-comment-end="c-Tito_Omburo-20240530092300-Nø-20240530071700"></span></dd> <dd><span data-mw-comment-start="" id="c-D.Lazard-20240530111700-Nø-20240530071700"></span>When I wrote "read the proof", I did not read it again. Indeed, numerous edits done since I introduced it several years ago made it confusing and much less elementary than needed. In particular, the proof was given twice and used the concept of number line and distance that may be useful in the explanation, but not in a rigourous proof. Also it was a <a href="/wiki/Proof_by_contradiction" title="Proof by contradiction">proof by contradiction</a> that I consider as not very elementary. I have fixed these issues, and restored the heading <a href="/w/index.php?title=0.999.../Archive_20&action=edit&redlink=1" class="new" title="0.999.../Archive 20 (page does not exist)">§ Rigorous proof</a>. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240530111700-Nø-20240530071700" class="ext-discussiontools-init-timestamplink">11:17, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240530111700-Nø-20240530071700"></span><span data-mw-comment-end="c-D.Lazard-20240530111700-Nø-20240530071700"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240530114300-D.Lazard-20240530111700"></span>This is definitely an improvement, however "elementary" is an adjective I suggest we avoid. Classically (according to Hardy), elementary means that it does not use complex variables. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240530114300-D.Lazard-20240530111700" class="ext-discussiontools-init-timestamplink">11:43, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240530114300-D.Lazard-20240530111700"></span><span data-mw-comment-end="c-Tito_Omburo-20240530114300-D.Lazard-20240530111700"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240530121700-Tito_Omburo-20240530114300"></span>"Elementary" refers also to <a href="/wiki/Elementary_school" class="mw-redirect" title="Elementary school">elementary school</a>, <a href="/wiki/Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a>, <a href="/wiki/Elementary_algebra" title="Elementary algebra">elementary algebra</a>. This is this meaning that is intended here. On the other hand, I never heard of the use of "elementary" as a synonym of "real context". <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240530121700-Tito_Omburo-20240530114300" class="ext-discussiontools-init-timestamplink">12:17, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240530121700-Tito_Omburo-20240530114300"></span><span data-mw-comment-end="c-D.Lazard-20240530121700-Tito_Omburo-20240530114300"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240530203100-D.Lazard-20240530121700"></span>It's rather common, in my experience; see, e.g., <a rel="nofollow" class="external autonumber" href="https://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf">[1]</a>. I'm not a fan of using "Elementary" in the section heading here for <a href="/wiki/Wikipedia:NOR" class="mw-redirect" title="Wikipedia:NOR">WP:NOR</a> reasons, as mentioned a few sections up. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240530203100-D.Lazard-20240530121700" class="ext-discussiontools-init-timestamplink">20:31, 30 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240530203100-D.Lazard-20240530121700"></span><span data-mw-comment-end="c-XOR'easter-20240530203100-D.Lazard-20240530121700"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Nø-20240531070600-Nø-20240529192000"></span>I suppose we agree that "advanced" essentially means the same as "not elementary" (however we delineate that).</dd> <dd>The point I - perhaps inadequately - tried to make with my original post above (and with the edit that was reverted, <a class="external text" href="https://en.wikipedia.org/w/index.php?diff=1225931460&oldid=1225926009&title=0.999...">diff</a>) is that there is no way to settle the question about the meaning of 0.999... that is entirely elementary. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240531070600-Nø-20240529192000" class="ext-discussiontools-init-timestamplink">07:06, 31 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240531070600-Nø-20240529192000"></span><span data-mw-comment-end="c-Nø-20240531070600-Nø-20240529192000"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20240531084200-Nø-20240531070600"></span>Here's an elementary "proof" why 0.999... is less than 1: <ul><li>0<1</li> <li>0.9<1</li> <li>0.99<1</li> <li>0.999<1</li> <li>...</li> <li>Hence, 0.999...<1</li></ul></dd> <dd>To prove me wrong, I believe you need something that is not elementary. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240531084200-Nø-20240531070600" class="ext-discussiontools-init-timestamplink">08:42, 31 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240531084200-Nø-20240531070600"></span><span data-mw-comment-end="c-Nø-20240531084200-Nø-20240531070600"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240531091800-Nø-20240531084200"></span>You need the archimedean property. You do not, in fact, need completeness or limits however. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240531091800-Nø-20240531084200" class="ext-discussiontools-init-timestamplink">09:18, 31 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240531091800-Nø-20240531084200"></span><span data-mw-comment-end="c-Tito_Omburo-20240531091800-Nø-20240531084200"></span></dd> <dd><span data-mw-comment-start="" id="c-D.Lazard-20240531094400-Nø-20240531084200"></span>{ec}If you read the proof, you will see that the only non-elementary step is the use of the Archimedean property that asserts that there is no positive real number that is less than all inverses of natural numbers, or, equivalently, that there is no real number that is greater than all integers. This is an axiom of the real numbers exactly as the <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a> is an axiom of geometry. Both cannot be proved, but both are easy to explain experimentally. If you consider this proof as non-elementary, you should consider also as non-elementary all proofs and constructions that use the parallel postulate and are taught in elementary geometry.</dd> <dd>By the way, there is something non-elementary here. This is the notation 0.999... and more generally the concept of infinite decimals. They are very non-elementary, since they use the concept of <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a> whose existence was refused by most mathematicians until the end of the 19th century. My opinion is that infinite decimals should never be taught in elementary classes. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240531094400-Nø-20240531084200" class="ext-discussiontools-init-timestamplink">09:44, 31 May 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240531094400-Nø-20240531084200"></span><span data-mw-comment-end="c-D.Lazard-20240531094400-Nø-20240531084200"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20240601091000-D.Lazard-20240531094400"></span>It seem we totally agree. There is no such thing as an elementary proof. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240601091000-D.Lazard-20240531094400" class="ext-discussiontools-init-timestamplink">09:10, 1 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240601091000-D.Lazard-20240531094400"></span><span data-mw-comment-end="c-Nø-20240601091000-D.Lazard-20240531094400"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240601105500-Nø-20240601091000"></span>No. This is an elementary proof of a result expressed with a non-elementary notation, namely that the least number greater than all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.(9)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>9</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.(9)_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb854b7cbce7be8bdaa9c2e5098fe0abf7727173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6ex; height:2.843ex;" alt="{\displaystyle 0.(9)_{n}}"/></span> is denoted with an infinite number of 9. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240601105500-Nø-20240601091000" class="ext-discussiontools-init-timestamplink">10:55, 1 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240601105500-Nø-20240601091000"></span><span data-mw-comment-end="c-D.Lazard-20240601105500-Nø-20240601091000"></span> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240601162600-D.Lazard-20240601105500"></span>The least number (if one exists), and it is also an elementary proof of existence. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240601162600-D.Lazard-20240601105500" class="ext-discussiontools-init-timestamplink">16:26, 1 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240601162600-D.Lazard-20240601105500"></span><span data-mw-comment-end="c-Tito_Omburo-20240601162600-D.Lazard-20240601105500"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20240602181700-Tito_Omburo-20240601162600"></span>Are you claiming one can give an elementary proof of someting that doesn't have an elementary definition? <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240602181700-Tito_Omburo-20240601162600" class="ext-discussiontools-init-timestamplink">18:17, 2 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240602181700-Tito_Omburo-20240601162600"></span><span data-mw-comment-end="c-Nø-20240602181700-Tito_Omburo-20240601162600"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240602193000-Nø-20240602181700"></span>The least number greater than all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.(9)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>9</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.(9)_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb854b7cbce7be8bdaa9c2e5098fe0abf7727173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6ex; height:2.843ex;" alt="{\displaystyle 0.(9)_{n}}"/></span> is an elementary concept, but the notation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.999...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.999...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.999...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/884a3ae7bfd73df603971cb0530cb343e67fb9d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.237ex; height:2.176ex;" alt="{\displaystyle 0.999...}"/></span> is not elementary, since it involves an <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a> of 9. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240602193000-Nø-20240602181700" class="ext-discussiontools-init-timestamplink">19:30, 2 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240602193000-Nø-20240602181700"></span><span data-mw-comment-end="c-D.Lazard-20240602193000-Nø-20240602181700"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20240605153100-D.Lazard-20240602193000"></span>I would not consider the <i>existence</i> of a least number greater than all numbers in an infinite sequence an elementary concept. I do not consider the <i>meaning</i> (definition) of 0.999... an elementary concept, and thus I think an argument avoiding advanced topics cannot be a proof. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240605153100-D.Lazard-20240602193000" class="ext-discussiontools-init-timestamplink">15:31, 5 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240605153100-D.Lazard-20240602193000"></span><span data-mw-comment-end="c-Nø-20240605153100-D.Lazard-20240602193000"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl> <div class="outdent-template" style="position:relative;left:1px;"><span style="display:block;width:16em;height:0.5em;border-bottom:1px solid #a2a9b1;border-right:1px solid #a2a9b1;"></span><span style="display:block;width:16em;height:0.5em;border-left:1px solid #a2a9b1;"></span></div><p><span data-mw-comment-start="" id="c-Tito_Omburo-20240605175000-Elementary_"proof"?"></span> And yet, the proof is elementary, which suggests you should revisit some preconceptions. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240605175000-Elementary_"proof"?" class="ext-discussiontools-init-timestamplink">17:50, 5 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240605175000-Elementary_"proof"?"></span><span data-mw-comment-end="c-Tito_Omburo-20240605175000-Elementary_"proof"?"></span> </p><dl><dd><span data-mw-comment-start="" id="c-Nø-20240606130400-Tito_Omburo-20240605175000"></span>The proof has this sentence:</dd> <dd><i>Let x be the smallest number greater than 0.9, 0.99, 0.999, etc.</i></dd> <dd>This presupposes the existence of such a number. As I said, I do not consider this elementary, but I acknowledge that we don't seem to have a clear and unambiguous consensus on what "elementary"/"advanced" really means. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240606130400-Tito_Omburo-20240605175000" class="ext-discussiontools-init-timestamplink">13:04, 6 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240606130400-Tito_Omburo-20240605175000"></span><span data-mw-comment-end="c-Nø-20240606130400-Tito_Omburo-20240605175000"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240606142300-Nø-20240606130400"></span>I changed recently the sentence for avoiding a proof by contradiction. The resulting proof, as stated, supposed the existence of a least upper bound, but it was easy to fix this. So, I edited the article for clarifying the proof, and making clear that it includes the proof that the numbers greater than all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.(9)_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>9</mn> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.(9)_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb854b7cbce7be8bdaa9c2e5098fe0abf7727173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6ex; height:2.843ex;" alt="{\displaystyle 0.(9)_{n}}"/></span> have a least element. By the way, this clarification simplifies the proof further. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240606142300-Nø-20240606130400" class="ext-discussiontools-init-timestamplink">14:23, 6 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240606142300-Nø-20240606130400"></span><span data-mw-comment-end="c-D.Lazard-20240606142300-Nø-20240606130400"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="efn?" data-mw-thread-id="h-efn?-20240417215400"><span id="efn.3F"></span><span data-mw-comment-start="" id="h-efn?-20240417215400"></span>efn?<span data-mw-comment-end="h-efn?-20240417215400"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-JayBeeEll-20240417215400-efn?"></span>Right now, we have footnotes that are references and footnotes that are explanatory notes or asides, the former using {{sfnp}} and <ref> tags, the latter using <ref> tags. I propose wrapping the second kind in {{efn}} instead, which has what I consider the advantage of distinguishing between the two types of notes (efn get labeled [a], [b], etc. instead of [1], [2]). One disadvantage is that there are clearly some judgement calls to be made. How do other people feel about this? (Obviously this is not urgent, am happy to have "I'm busy trying to preserve featured status and don't want to think about/deal with this" as an answer.) --<a href="/wiki/User:JayBeeEll" title="User:JayBeeEll">JBL</a> (<a href="/wiki/User_talk:JayBeeEll" title="User talk:JayBeeEll">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-JayBeeEll-20240417215400-efn?" class="ext-discussiontools-init-timestamplink">21:54, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-JayBeeEll-20240417215400-efn?"></span><span data-mw-comment-end="c-JayBeeEll-20240417215400-efn?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240417221500-JayBeeEll-20240417215400"></span>I'd be fine with that. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240417221500-JayBeeEll-20240417215400" class="ext-discussiontools-init-timestamplink">22:15, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240417221500-JayBeeEll-20240417215400"></span><span data-mw-comment-end="c-XOR'easter-20240417221500-JayBeeEll-20240417215400"></span></dd> <dd><span data-mw-comment-start="" id="c-Hawkeye7-20240417222900-JayBeeEll-20240417215400"></span>Fine with me. <a href="/wiki/User:Hawkeye7" title="User:Hawkeye7"><span style="color:#800082">Hawkeye7</span></a> <a href="/wiki/User_talk:Hawkeye7" title="User talk:Hawkeye7"><span style="font-size:80%">(discuss)</span></a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Hawkeye7-20240417222900-JayBeeEll-20240417215400" class="ext-discussiontools-init-timestamplink">22:29, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Hawkeye7-20240417222900-JayBeeEll-20240417215400"></span><span data-mw-comment-end="c-Hawkeye7-20240417222900-JayBeeEll-20240417215400"></span></dd> <dd><span data-mw-comment-start="" id="c-Trovatore-20240417223500-JayBeeEll-20240417215400"></span>Strongly support using efn. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240417223500-JayBeeEll-20240417215400" class="ext-discussiontools-init-timestamplink">22:35, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240417223500-JayBeeEll-20240417215400"></span><span data-mw-comment-end="c-Trovatore-20240417223500-JayBeeEll-20240417215400"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240417233100-Trovatore-20240417223500"></span>OK, I've made a stab at dividing them up. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240417233100-Trovatore-20240417223500" class="ext-discussiontools-init-timestamplink">23:31, 17 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240417233100-Trovatore-20240417223500"></span><span data-mw-comment-end="c-XOR'easter-20240417233100-Trovatore-20240417223500"></span> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240419040400-XOR'easter-20240417233100"></span>I think it's fine to add efn. Additionally, maybe both the notes and references sections should be merged into one section, containing three different lists (notes, footnotes, works cited)? <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240419040400-XOR'easter-20240417233100" class="ext-discussiontools-init-timestamplink">04:04, 19 April 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240419040400-XOR'easter-20240417233100"></span><span data-mw-comment-end="c-Dedhert.Jr-20240419040400-XOR'easter-20240417233100"></span></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627011100-JayBeeEll-20240417215400"></span>I assume that your goal is to eliminate footnotes that are in fact citations? For example, <ul><li>{{efn|{{harvtxt|Bunch|1982}}, p. 119; {{harvtxt|Tall|Schwarzenberger|1978}}, p. 6. The last suggestion is due to {{harvtxt|Burrell|1998}}, p. 28: "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."}}</li></ul></dd> <dd>These could be just ref tags with rp templates for page numbers and quotes, but I don't know if that is the style you want. <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627011100-JayBeeEll-20240417215400" class="ext-discussiontools-init-timestamplink">01:11, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627011100-JayBeeEll-20240417215400"></span><span data-mw-comment-end="c-Johnjbarton-20240627011100-JayBeeEll-20240417215400"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240627211600-Johnjbarton-20240627011100"></span>Gah, no. <span class="nowrap">{{</span><a href="/wiki/Template:Rp" class="mw-redirect" title="Template:Rp">rp</a><span class="nowrap">}}</span> tags are ugly enough when used in isolation. Stacking three in a row and then trying to fit in a quote as well would be a mess. We handled the concerns in this section <a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&diff=1219469250&oldid=1219457714">back in April</a>; nothing more in this regard needs to be done. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240627211600-Johnjbarton-20240627011100" class="ext-discussiontools-init-timestamplink">21:16, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240627211600-Johnjbarton-20240627011100"></span><span data-mw-comment-end="c-XOR'easter-20240627211600-Johnjbarton-20240627011100"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="lede_that_gets_to_the_point_without_jargon." data-mw-thread-id="h-lede_that_gets_to_the_point_without_jargon.-20240627145100"><span data-mw-comment-start="" id="h-lede_that_gets_to_the_point_without_jargon.-20240627145100"></span>lede that gets to the point without jargon.<span data-mw-comment-end="h-lede_that_gets_to_the_point_without_jargon.-20240627145100"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Johnjbarton-20240627145100-lede_that_gets_to_the_point_without_jargon."></span>I realize the mathematicians love precision and thus those special words that have meaning in math, but this article has an important point for a broader audience. I change the intro yesterday to concentrate the ideas that "It is the number one!" into the first paragraph and move the two (or is it three or maybe one) definitions to a separate section. The waffle-worded, footnoted definition will be completely opaque to naive readers. They will stop reading and never discover "This number is equal to 1.". Unfortunately my change was <a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&diff=prev&oldid=1231259750">reverted</a> by @<a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> with an edit summary, "Restored old lede. It is important that the lede refer to an actual number, not merely some notation.", which I do not understand. Note that my lede was </p> <ul><li>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>0.999...</b> (also written as <b>0.<span style="text-decoration:overline;">9</span></b>, <b>0.<span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;">.</span><span style="display:block; line-height:1em;">9</span></span></span></b> or <b>0.(9)</b>) is a notation for the number "1" represented as a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> consisting of an unending sequence of 9s after the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a>.</li></ul> <p>In my opinion we should change the content back towards the version I suggested. <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627145100-lede_that_gets_to_the_point_without_jargon." class="ext-discussiontools-init-timestamplink">14:51, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627145100-lede_that_gets_to_the_point_without_jargon."></span><span data-mw-comment-end="c-Johnjbarton-20240627145100-lede_that_gets_to_the_point_without_jargon."></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240627145800-Johnjbarton-20240627145100"></span>It's misleading to say that "0.999..." is notation referring to the number 1. The notation refers to a real number, namely the least real number greater than every truncation of the decimal. The fact is that <i>this real number</i> is equal to one. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240627145800-Johnjbarton-20240627145100" class="ext-discussiontools-init-timestamplink">14:58, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240627145800-Johnjbarton-20240627145100"></span><span data-mw-comment-end="c-Tito_Omburo-20240627145800-Johnjbarton-20240627145100"></span> <dl><dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627150900-Tito_Omburo-20240627145800"></span>Ok so how about <ul><li>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>0.999...</b> (also written as <b>0.<span style="text-decoration:overline;">9</span></b>, <b>0.<span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;">.</span><span style="display:block; line-height:1em;">9</span></span></span></b> or <b>0.(9)</b>) is a notation a <a href="/wiki/Real_number" title="Real number">real number</a> equal to "1". The real number is represented as a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> consisting of an unending sequence of 9s after the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a>.</li></ul></dd> <dd>and restoring the Definition section? <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627150900-Tito_Omburo-20240627145800" class="ext-discussiontools-init-timestamplink">15:09, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627150900-Tito_Omburo-20240627145800"></span><span data-mw-comment-end="c-Johnjbarton-20240627150900-Tito_Omburo-20240627145800"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240627152600-Johnjbarton-20240627150900"></span>This still boils the subject of the article down to a tautology, which it is not. 0.999... definitionally means something. It is not the same thing as the numeral 1. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240627152600-Johnjbarton-20240627150900" class="ext-discussiontools-init-timestamplink">15:26, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240627152600-Johnjbarton-20240627150900"></span><span data-mw-comment-end="c-Tito_Omburo-20240627152600-Johnjbarton-20240627150900"></span> <dl><dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627155200-Tito_Omburo-20240627152600"></span>Sorry I mis-edited. I know you disagreed with "notation" as it means definitional equivalence, but I accidentally left the word. Here is the alternative I should have written: <ul><li>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>0.999...</b> (also written as <b>0.<span style="text-decoration:overline;">9</span></b>, <b>0.<span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.8em; font-size:70%;">.</span><span style="display:block; line-height:1em;">9</span></span></span></b> or <b>0.(9)</b>) is a <a href="/wiki/Real_number" title="Real number">real number</a> equal to "1". The real number is represented as a <a href="/wiki/Repeating_decimal" title="Repeating decimal">repeating decimal</a> consisting of an unending sequence of 9s after the <a href="/wiki/Decimal_point" class="mw-redirect" title="Decimal point">decimal point</a>.</li></ul></dd> <dd><a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627155200-Tito_Omburo-20240627152600" class="ext-discussiontools-init-timestamplink">15:52, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627155200-Tito_Omburo-20240627152600"></span><span data-mw-comment-end="c-Johnjbarton-20240627155200-Tito_Omburo-20240627152600"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Mr_swordfish-20240627153100-Johnjbarton-20240627150900"></span>Agree that the simpler get-to-the-point jargon-free lede is better. The intended audience here is not mathematicians, it's lay people who likely are not familiar with the idea that the decimal representation of a real number is not unique in all cases (ie a "terminating" decimal that repeats zeros always has another representation that repeats nines).</dd> <dd>In particular, it's too early in the article to assume that the reader knows anything about infinite sequences and convergence. Statement like "The notation refers to a real number, namely the least real number greater than every truncation of the decimal." will be lost on the average reader.</dd> <dd>Similarly, it's not appropriate to assume that the reader knows the difference between a <i>numeral</i> and a <i>number</i>. We can explain all this later in the article.</dd> <dd>Agree with removing the technical details to a definition section. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20240627153100-Johnjbarton-20240627150900" class="ext-discussiontools-init-timestamplink">15:31, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20240627153100-Johnjbarton-20240627150900"></span><span data-mw-comment-end="c-Mr_swordfish-20240627153100-Johnjbarton-20240627150900"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240627153900-Mr_swordfish-20240627153100"></span>The problem is this is like telling lies to children. An unprepared reader has no idea what the notation 0.999... refers to. The current lede makes clear what that is. The proposed lede is actively misleading, in the name of being more accessible. The problem is that the subject of this article is <i>not</i> accessible to someone unwilling to grasp in some way with the concept of infinity. But this important aspect cannot be written out of the intro. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240627153900-Mr_swordfish-20240627153100" class="ext-discussiontools-init-timestamplink">15:39, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240627153900-Mr_swordfish-20240627153100"></span><span data-mw-comment-end="c-Tito_Omburo-20240627153900-Mr_swordfish-20240627153100"></span> <dl><dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627155900-Tito_Omburo-20240627153900"></span>@<a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> I gather your primary concern is the lede. My primary concern is the definition sentences. I think we should move that out of the intro.</dd> <dd>I agree that the concept of infinity is core to the article. How about a sentence in the first paragraph that explicitly calls out the concept of infinity? <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627155900-Tito_Omburo-20240627153900" class="ext-discussiontools-init-timestamplink">15:59, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627155900-Tito_Omburo-20240627153900"></span><span data-mw-comment-end="c-Johnjbarton-20240627155900-Tito_Omburo-20240627153900"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Trovatore-20240627161300-Johnjbarton-20240627145100"></span>One thing to consider is that opening sentences don't always have to have the form "foo is a bar"; when that's awkward, it's fine to pick a different structure. In this case, maybe something along the lines of <dl><dd><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">In mathematics, the notation 0.999..., with the digit 9 repeating endlessly, represents exactly the number 1.</span></dd></dl></dd> <dd>just as the first sentence, then we can continue on with elaborations. This way we can (as Johnjbarton put it) "get to the point" in the first sentence, and we haven't told any lies-to-children. By not insisting on including the phrase <span class="example deprecated-content example-deprecated" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-subtle, #696969);">0.999... is</span>, we <ul><li>don't have to say that it "is a notation"; we just put that part before the 0.999...</li> <li>don't have to say that it's a notation for a different (infinitely long) notation, which is true if you're super literal-minded, but is extremely confusing in the first sentence, and</li> <li>don't have to talk about least upper bounds before we give the punch line</li></ul></dd> <dd>I think this small tweak could open up a lot of possibilities for making the opening sentence (at least) more understandable to non-mathematicians, without saying anything false. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240627161300-Johnjbarton-20240627145100" class="ext-discussiontools-init-timestamplink">16:13, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240627161300-Johnjbarton-20240627145100"></span><span data-mw-comment-end="c-Trovatore-20240627161300-Johnjbarton-20240627145100"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240627161800-Trovatore-20240627161300"></span>I support this change. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240627161800-Trovatore-20240627161300" class="ext-discussiontools-init-timestamplink">16:18, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240627161800-Trovatore-20240627161300"></span><span data-mw-comment-end="c-Tito_Omburo-20240627161800-Trovatore-20240627161300"></span> <dl><dd><span data-mw-comment-start="" id="c-Mr_swordfish-20240627162200-Tito_Omburo-20240627161800"></span>Also support this change. <a href="/wiki/User:Mr_swordfish" title="User:Mr swordfish">Mr. Swordfish</a> (<a href="/wiki/User_talk:Mr_swordfish" title="User talk:Mr swordfish">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Mr_swordfish-20240627162200-Tito_Omburo-20240627161800" class="ext-discussiontools-init-timestamplink">16:22, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20240627162200-Tito_Omburo-20240627161800"></span><span data-mw-comment-end="c-Mr_swordfish-20240627162200-Tito_Omburo-20240627161800"></span> <dl><dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627163500-Mr_swordfish-20240627162200"></span>Great, I made that change. <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627163500-Mr_swordfish-20240627162200" class="ext-discussiontools-init-timestamplink">16:35, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627163500-Mr_swordfish-20240627162200"></span><span data-mw-comment-end="c-Johnjbarton-20240627163500-Mr_swordfish-20240627162200"></span></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Johnjbarton-20240627174100-Johnjbarton-20240627145100"></span>I reverted the change by @<a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> but I think it may have some things that are helpful. Unfortunately the comment by D.Lazard was added but not signed nor set as a Reply. <a href="/wiki/User:Johnjbarton" title="User:Johnjbarton">Johnjbarton</a> (<a href="/wiki/User_talk:Johnjbarton" title="User talk:Johnjbarton">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Johnjbarton-20240627174100-Johnjbarton-20240627145100" class="ext-discussiontools-init-timestamplink">17:41, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20240627174100-Johnjbarton-20240627145100"></span><span data-mw-comment-end="c-Johnjbarton-20240627174100-Johnjbarton-20240627145100"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240627174400-Johnjbarton-20240627174100"></span>Yeah, I didn't like the new version as a whole. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240627174400-Johnjbarton-20240627174100" class="ext-discussiontools-init-timestamplink">17:44, 27 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240627174400-Johnjbarton-20240627174100"></span><span data-mw-comment-end="c-Tito_Omburo-20240627174400-Johnjbarton-20240627174100"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-D.Lazard-20240627192300-Johnjbarton-20240627145100"></span>Independently from Trovatore's post, I have rewritten the lead for removing jargon (in particular "denotes" is less jargonny/pedantic than "is a notation for") and unneeded technicalities from the beginning. This required a complete restructuration. By the way, I have removed some editorial considerations that do not belong here. By doing this, I deleted the last Johnjbarton's edit, but I think that my version is better for the intended audience. <small class="autosigned">— Preceding <a href="/wiki/Wikipedia:Signatures" title="Wikipedia:Signatures">unsigned</a> comment added by <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard#top" title="User talk:D.Lazard">talk</a> • <a href="/wiki/Special:Contributions/D.Lazard" title="Special:Contributions/D.Lazard">contribs</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240627192300-Johnjbarton-20240627145100" class="ext-discussiontools-init-timestamplink">19:23, 27 June 2024 (UTC)</a></small><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240627192300-Johnjbarton-20240627145100"></span><span data-mw-comment-end="c-D.Lazard-20240627192300-Johnjbarton-20240627145100"></span> <dl><dd><small><span data-mw-comment-start="" id="c-D.Lazard-20240628103700-D.Lazard-20240627192300"></span>Sorry to have forgotten to sign. I did not use the "reply" button, because this is an answer to the opening post, and, as such, should not be indented.</small></dd> <dd>I rewrote the first paragraph of the lead that used many terms (technical or jargon) that are useless for people that know infinite decimals and are confusing for others. Also, I added that the equality can be proved, and therefore that is is not a convention that may be rejected by people who do not like it. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240628103700-D.Lazard-20240627192300" class="ext-discussiontools-init-timestamplink">10:37, 28 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240628103700-D.Lazard-20240627192300"></span><span data-mw-comment-end="c-D.Lazard-20240628103700-D.Lazard-20240627192300"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240628104100-D.Lazard-20240628103700"></span>Looks fine to me. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240628104100-D.Lazard-20240628103700" class="ext-discussiontools-init-timestamplink">10:41, 28 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240628104100-D.Lazard-20240628103700"></span><span data-mw-comment-end="c-Tito_Omburo-20240628104100-D.Lazard-20240628103700"></span></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-MartinPoulter-20240628113400-Johnjbarton-20240627145100"></span>Glad to see the work done to make this article more readable to a general audience. I hope "The utilitarian preference for the terminating decimal representation..." (last para of lede) can also be simplified, as I see what it means but as written it's well above most of the population's reading level. I'm confused by the use of the {{spaces}} template before the 1 in the first paragraph: it looks like a formatting error. What's the point of that big space? <a href="/wiki/User:MartinPoulter" title="User:MartinPoulter">MartinPoulter</a> (<a href="/wiki/User_talk:MartinPoulter" title="User talk:MartinPoulter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-MartinPoulter-20240628113400-Johnjbarton-20240627145100" class="ext-discussiontools-init-timestamplink">11:34, 28 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-MartinPoulter-20240628113400-Johnjbarton-20240627145100"></span><span data-mw-comment-end="c-MartinPoulter-20240628113400-Johnjbarton-20240627145100"></span> <dl><dd><span data-mw-comment-start="" id="c-D.Lazard-20240628142100-MartinPoulter-20240628113400"></span>I have removed this sentence per <a href="/wiki/Wikipedia:NPOV" class="mw-redirect" title="Wikipedia:NPOV">WP:NPOV</a>: authors in mathematical education have different explanation on the difficulties of the students with the equality; this is not to Wikipedia to select one amongst several. I did several other edits that are explained in the edit summaries. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240628142100-MartinPoulter-20240628113400" class="ext-discussiontools-init-timestamplink">14:21, 28 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240628142100-MartinPoulter-20240628113400"></span><span data-mw-comment-end="c-D.Lazard-20240628142100-MartinPoulter-20240628113400"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Two_representations_in_every_positional_numeral_system_with_one_terminating?" data-mw-thread-id="h-Two_representations_in_every_positional_numeral_system_with_one_terminating?-20240630015600"><span id="Two_representations_in_every_positional_numeral_system_with_one_terminating.3F"></span><span data-mw-comment-start="" id="h-Two_representations_in_every_positional_numeral_system_with_one_terminating?-20240630015600"></span>Two representations in every positional numeral system with one terminating?<span data-mw-comment-end="h-Two_representations_in_every_positional_numeral_system_with_one_terminating?-20240630015600"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-Quondum-20240630015600-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span>The article contains the statement </p> <dl><dd><span class="example" style="font-family: Georgia, 'DejaVu Serif', serif; color: var(--color-content-added, #006400);">... every nonzero terminating decimal has two equal representations ... all positional numeral systems have this property.</span></dd></dl> <p>Every positional numeral system has two representations for certain numbers, but is this necessarily true of terminating representations? A counterexample would seem to be <a href="/wiki/Balanced_ternary" title="Balanced ternary">balanced ternary</a>: the numbers that have two representations seem to be nonterminating, e.g. 1 = 1.000...<sub>bal3</sub> has no other representation, but 1/2 = 0.111...<sub>bal3</sub> = 1.TTT...<sub>bal3</sub> (where T = −1) has two. Or maybe I need some coffee? —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630015600-Two_representations_in_every_positional_numeral_system_with_one_terminating?" class="ext-discussiontools-init-timestamplink">01:56, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630015600-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span><span data-mw-comment-end="c-Quondum-20240630015600-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Imaginatorium-20240630040700-Quondum-20240630015600"></span>Well, the trouble is that "balanced ternary" is not a "usual" positional numeral system. Perhaps it might be better to write, "and this is true of all bases, not just decimal". In the end it depends whether you think Wikipedia is here for the benefit of lawyers, or just to help people understand things. <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">Imaginatorium</a> (<a href="/wiki/User_talk:Imaginatorium" title="User talk:Imaginatorium">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Imaginatorium-20240630040700-Quondum-20240630015600" class="ext-discussiontools-init-timestamplink">04:07, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Imaginatorium-20240630040700-Quondum-20240630015600"></span><span data-mw-comment-end="c-Imaginatorium-20240630040700-Quondum-20240630015600"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240630053600-Imaginatorium-20240630040700"></span>A better viewpoint is that such systems have no "terminating" representations at all, but only ones that eventually repeat the digit 0. That's beyond the scope of this article. Still, I think we should de-emphasize the notion of "terminating" representations. We don't really <i>need</i> to talk about them. We can say, for example, that 3.4999... is the same as 3.5000.... --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240630053600-Imaginatorium-20240630040700" class="ext-discussiontools-init-timestamplink">05:36, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240630053600-Imaginatorium-20240630040700"></span><span data-mw-comment-end="c-Trovatore-20240630053600-Imaginatorium-20240630040700"></span> <dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630151100-Trovatore-20240630053600"></span>While I agree that "terminating representations" are a little peripheral, including that to answer the immediate question "How easy is it to find such values?" seems reasonable, although extrapolating from the example would seem obvious to us, and the phrase is adequately defined as linked. I don't feel strongly about keeping "terminating representations" or any other specific description of the class, though. I have clarified the statement in a way that fits the section <i><a href="/wiki/0.999...#Generalizations" title="0.999...">0.999... § Generalizations</a></i>.</dd> <dd><del>That aside, it is interesting that having multiple representations depends on the definition of a positional numeral system as having position weights and values associated with symbols that do not depend on the value of other digits; I say this, because <a href="/wiki/Gray_code" title="Gray code">Gray codes</a> are remarkably close to being a positional system and (extended to a fractional part) they evidently have a unique representation for each real number.</del> —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630151100-Trovatore-20240630053600" class="ext-discussiontools-init-timestamplink">15:11, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630151100-Trovatore-20240630053600"></span><span data-mw-comment-end="c-Quondum-20240630151100-Trovatore-20240630053600"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240630183000-Quondum-20240630151100"></span>Not sure quite what you mean about the Gray codes. The key point here is that representations of this sort are <a href="/wiki/Zero-dimensional" class="mw-redirect" title="Zero-dimensional">zero-dimensional</a> in the <a href="/wiki/Product_topology" title="Product topology">product topology</a>, which seems to be the natural one to put on them, whereas the reals are one-dimensional. So a continuous surjection from the representations to the real numbers cannot be an injection <s>with a continuous inverse</s>, because otherwise it would be a homeomorphism, contradicting the previous observation about dimension. Therefore there must be reals with non-unique representations.</dd> <dd>Maybe the representation by Gray codes you're talking about isn't continuous (or its inverse is not); would need to see what you mean. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240630183000-Quondum-20240630151100" class="ext-discussiontools-init-timestamplink">18:30, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240630183000-Quondum-20240630151100"></span><span data-mw-comment-end="c-Trovatore-20240630183000-Quondum-20240630151100"></span> <dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630191200-Trovatore-20240630183000"></span>I intended that a digit incrementing reverses the interpretation of all subsequent digits – in my quest to get rid of the infinite number of digits 'rolling over' (e.g. 0.999... to 1.000...). Looking more closely, I see that my supposition was incorrect: you still get two representations for a (presumably the same) set of real values, but now a pair of equal representations differs in one digit instead of infinitely many. This leads me to wonder whether a representation composed of an infinite sequence of discrete symbols could avoid doubles, once we have the the restriction that every real number must have a representation. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630191200-Trovatore-20240630183000" class="ext-discussiontools-init-timestamplink">19:12, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630191200-Trovatore-20240630183000"></span><span data-mw-comment-end="c-Quondum-20240630191200-Trovatore-20240630183000"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240630195200-Quondum-20240630191200"></span>If you put no restrictions at all on what you mean by a "representation", the answer is clearly yes, you can avoid duplicates. For example the set of all countably infinite decimal strings and the set of all reals have the same cardinality, so there's a bijection between them. You can even make that bijection pretty explicit, by playing games with (the proof of) the <a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a>.</dd> <dd>However, if you care at all about continuity, you're going to need to deal with the dimension issue I mentioned in my previous comment. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240630195200-Quondum-20240630191200" class="ext-discussiontools-init-timestamplink">19:52, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240630195200-Quondum-20240630191200"></span><span data-mw-comment-end="c-Trovatore-20240630195200-Quondum-20240630191200"></span> <dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630214000-Trovatore-20240630195200"></span>I was having difficulty framing the issue properly, and I thought about the cardinality argument after my post. I this context, all I would care about in this context is finding a simple constructive definition of a class of surjective maps from sequences of symbols to the reals for which multiple representations occur, ideally independent of the axiom of choice and it would be nice if it gave a sense of the lengths one might have to go to to avoid these multiples. The class of standard positional systems suffices, but clearly the class for which this is true is larger, and this gives room for possibly finding a class that is simpler to define. Improving on just copying the second sentence in <i><a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a></i> seems to me to be challenge. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630214000-Trovatore-20240630195200" class="ext-discussiontools-init-timestamplink">21:40, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630214000-Trovatore-20240630195200"></span><span data-mw-comment-end="c-Quondum-20240630214000-Trovatore-20240630195200"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240630221300-Quondum-20240630214000"></span>In this context, a nice way to think of continuity is that, if you want finitely much information about the answer, you need only finitely much information about the input. On the "real" side, finitely much information means an open interval. On the "representation" side, it means finitely many digits. If that's true in both directions, then there must be duplicates.</dd> <dd>I'm a <i>little</i> skeptical that this can (or should) be worked into <i>this</i> article, but it would be satisfying if it could be. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240630221300-Quondum-20240630214000" class="ext-discussiontools-init-timestamplink">22:13, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240630221300-Quondum-20240630214000"></span><span data-mw-comment-end="c-Trovatore-20240630221300-Quondum-20240630214000"></span> <dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630233500-Trovatore-20240630221300"></span>I'm not of the opinion that anything more complicated than a sentence defining 'positional numeral system' is called for here to address the concerns that have been expressed, and it seems that my hope that a broader class could be defined more easily was too much. I have to confess that I do not know what you mean by 'finitely many digits': we are dealing with a countably infinite sequence of digits, each of which is an element of a finite (or possibly infinite) set of symbols. I'm afraid educating me on this is, for the moment, a lost cause, so I suggest that we just focus on language to include in the article. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630233500-Trovatore-20240630221300" class="ext-discussiontools-init-timestamplink">23:35, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630233500-Trovatore-20240630221300"></span><span data-mw-comment-end="c-Quondum-20240630233500-Trovatore-20240630221300"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240701030600-Quondum-20240630233500"></span>I tend to agree that this is getting off-scope for the talk page. I'll drop a note on your talk page. I don't think it's a lost cause; I probably just haven't found the right way of explaining myself.</dd> <dd><small>But as long as we're here, I do want to correct the record for the benefit of any lurkers. Turns out my maunderings about the continuity of the inverse mapping were unnecessary. As long as <ul><li>The alphabet is finite (or at least there are only finitely many choices for a digit at any given position),</li> <li>The mapping is continuous, and</li> <li>The mapping is injective</li></ul></small></dd><small> </small><dd><small>you get continuity of the inverse mapping, and therefore a contradiction, for free. That's because the representation space is compact (by <a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a>), so any closed set is compact. Then the continuous image of a closed set is compact and therefore closed, which in the injective case implies that the inverse map is continuous. </small></dd> <dd>So if we can source it, we <i>could</i> say that any continuous interpretation of the representations would have to have duplicates. Is that appropriate for <i>this</i> article? I doubt it. This article ought to be pitched considerably lower. Anyone who understands the above argument isn't looking to understand 0.999.... --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240701030600-Quondum-20240630233500" class="ext-discussiontools-init-timestamplink">03:06, 1 July 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240701030600-Quondum-20240630233500"></span><span data-mw-comment-end="c-Trovatore-20240701030600-Quondum-20240630233500"></span> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240701105900-Trovatore-20240701030600"></span>Very nice. This also in some sense explains how p-adic integers can have unique expansions: the p-adic integers are compact. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240701105900-Trovatore-20240701030600" class="ext-discussiontools-init-timestamplink">10:59, 1 July 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240701105900-Trovatore-20240701030600"></span><span data-mw-comment-end="c-Tito_Omburo-20240701105900-Trovatore-20240701030600"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240701200500-Tito_Omburo-20240701105900"></span>Or more importantly, the p-adic integers are zero-dimensional, or it might be easier to think of it in terms of the p-adic integers are totally disconnected (not quite equivalent but it gets at the same point for our purposes).</dd> <dd>The representations are totally disconnected whereas the reals are connected, so intuitively, to map the representations to the reals, you have to "connect something", which is where the duplicates come from. The p-adics are already totally disconnected, so the problem doesn't come up. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240701200500-Tito_Omburo-20240701105900" class="ext-discussiontools-init-timestamplink">20:05, 1 July 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240701200500-Tito_Omburo-20240701105900"></span><span data-mw-comment-end="c-Trovatore-20240701200500-Tito_Omburo-20240701105900"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240701231600-Trovatore-20240701200500"></span>Also, the product space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,...,9\}^{\mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mn>9</mn> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,...,9\}^{\mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c11cb4a1694d8c4290f660293d24d1a3e541420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:3.176ex;" alt="{\displaystyle \{0,...,9\}^{\mathbb {N} }}"/></span> is totally disconnected, and so is not the continuous image of a real interval. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240701231600-Trovatore-20240701200500" class="ext-discussiontools-init-timestamplink">23:16, 1 July 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240701231600-Trovatore-20240701200500"></span><span data-mw-comment-end="c-Tito_Omburo-20240701231600-Trovatore-20240701200500"></span></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Nø-20240630153200-Imaginatorium-20240630040700"></span>As <a href="/wiki/User:Imaginatorium" title="User:Imaginatorium">user:Imaginatorium</a> point out, the statement is valid in usual <a href="/wiki/Positional_numeral_systems" class="mw-redirect" title="Positional numeral systems">positional numeral systems</a>, but not in all <a href="/wiki/Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">non-standard positional numeral systems</a>. Luckily, <a href="/wiki/Positional_numeral_systems" class="mw-redirect" title="Positional numeral systems">positional numeral systems</a> redirects to a section of <a href="/wiki/List_of_numeral_systems" title="List of numeral systems">List of numeral systems</a> on "Standard positional numeral systems", so I suggest we simply use that wikilink. <a href="/wiki/User:N%C3%B8" title="User:Nø">Nø</a> (<a href="/wiki/User_talk:N%C3%B8" title="User talk:Nø">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Nø-20240630153200-Imaginatorium-20240630040700" class="ext-discussiontools-init-timestamplink">15:32, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20240630153200-Imaginatorium-20240630040700"></span><span data-mw-comment-end="c-Nø-20240630153200-Imaginatorium-20240630040700"></span> <dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630155400-Nø-20240630153200"></span>We should never have a redirect and its plural linking to different places, so that is not a solution. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630155400-Nø-20240630153200" class="ext-discussiontools-init-timestamplink">15:54, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630155400-Nø-20240630153200"></span><span data-mw-comment-end="c-Quondum-20240630155400-Nø-20240630153200"></span></dd> <dd><span data-mw-comment-start="" id="c-Quondum-20240630164300-Nø-20240630153200"></span>I have changed that redirect to be consistent with the singular form, after verifying that there are no mainspace uses. In any event, the definition at <i><a href="/wiki/Positional_notation" title="Positional notation">Positional notation</a></i> (essentially a weighted sum) is precisely correct for the statement as it now stands (i.e. including all nonstandard positional systems that meet this definition, with the proviso that they can represent all real numbers), and as supported by the text of the article. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630164300-Nø-20240630153200" class="ext-discussiontools-init-timestamplink">16:43, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630164300-Nø-20240630153200"></span><span data-mw-comment-end="c-Quondum-20240630164300-Nø-20240630153200"></span></dd></dl></dd></dl></dd></dl> <p><span data-mw-comment-start="" id="c-D.Lazard-20240630164400-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span>As far as I understand, this section discusses supposed properties of <style data-mw-deduplicate="TemplateStyles:r1248585704">@media screen{html.skin-theme-clientpref-night .mw-parser-output .inline-quote-talk{color:#00B785}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .inline-quote-talk{color:#00B785}}.mw-parser-output .inline-quote-talk{font-family:Georgia,"DejaVu Serif",serif;color:#008560;quotes:none}.mw-parser-output .inline-quote-talk-italic{font-family:inherit;font-style:italic}.mw-parser-output .inline-quote-talk-marks{quotes:"\"""\""}</style><q class="inline-quote-talk inline-quote-talk-marks">all positional numeral systems</q>. But this supposes a precise definition of a positional numeral system, and of a positional numeral system that accepts infinite strings. Without such a definition, everything is original research. </p><p>As an example, the standard <span class="texhtml mvar" style="font-style:italic;">p</span>-adic representation of <a href="/wiki/P-adic_number" title="P-adic number"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic numbers</a> is an example of a positional numeral system such that there is always a unique representation. </p><p>By the way it is astonishing that nobody mention what is, in my opinion, the main reason for which there is so much confusion with the subject of the article: it is that "infinite decimals" make a systematic use of <a href="/wiki/Actual_infinity" title="Actual infinity">actual infinity</a>, a concept that is so counterintuitive that, before the 20th century, it was refused by most mathematicians. It seems that some teachers hope that kids could understand easily concept that were refused by mathematicians and philosophers a century ago. <a href="/wiki/User:D.Lazard" title="User:D.Lazard">D.Lazard</a> (<a href="/wiki/User_talk:D.Lazard" title="User talk:D.Lazard">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-D.Lazard-20240630164400-Two_representations_in_every_positional_numeral_system_with_one_terminating?" class="ext-discussiontools-init-timestamplink">16:44, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-D.Lazard-20240630164400-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span><span data-mw-comment-end="c-D.Lazard-20240630164400-Two_representations_in_every_positional_numeral_system_with_one_terminating?"></span> </p> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240630185300-D.Lazard-20240630164400"></span>I think you're a bit off on that point, Prof. Lazard. My impression is that the rejection of the actual infinite was more in theory than in practice, and its systematic use considerably predated the 20th century, since real analysis was developing in the mid-19th century and used the actual infinite implicitly. It took Cantor to make it <i>explicit</i>, but the ideas of Bolzano and Cauchy and Weierstrass and Dedekind were already laying the groundwork.</dd> <dd>That said, sure, it's a key psychological point.</dd> <dd><a rel="nofollow" class="external text" href="https://www.smbc-comics.com/comic/how-math-works">ObSMBC</a>. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240630185300-D.Lazard-20240630164400" class="ext-discussiontools-init-timestamplink">18:53, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240630185300-D.Lazard-20240630164400"></span><span data-mw-comment-end="c-Trovatore-20240630185300-D.Lazard-20240630164400"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Quondum-20240630165700-D.Lazard-20240630164400"></span>This is interesting, but does it apply? As I understand it (and admittedly this is outside my area of knowledge), <i>p</i>-adic numbers do not embed the reals. The ability to represent all reals is core to the statement that there are necessarily multiple representations. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630165700-D.Lazard-20240630164400" class="ext-discussiontools-init-timestamplink">16:57, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630165700-D.Lazard-20240630164400"></span><span data-mw-comment-end="c-Quondum-20240630165700-D.Lazard-20240630164400"></span></dd> <dd><span data-mw-comment-start="" id="c-Tito_Omburo-20240630170100-D.Lazard-20240630164400"></span>Interesting point about p-adic numbers. I think the lead should mention infinity somewhere. I think the issues are resolved if the article is clear on what a "positional number system" is. I am unclear exactly what is meant. <a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a> (<a href="/wiki/User_talk:Tito_Omburo" title="User talk:Tito Omburo">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Tito_Omburo-20240630170100-D.Lazard-20240630164400" class="ext-discussiontools-init-timestamplink">17:01, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20240630170100-D.Lazard-20240630164400"></span><span data-mw-comment-end="c-Tito_Omburo-20240630170100-D.Lazard-20240630164400"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Quondum-20240630180900-Tito_Omburo-20240630170100"></span>So we have two people saying that a clear definition of a 'positional number system' is needed in the article, and I tend to agree in the context of this claim. I imagine that this can be omitted from the lead, but it might make sense in <i><a href="/wiki/0.999...#Generalizations" title="0.999...">0.999... § Generalizations</a></i>. —<a href="/wiki/User_talk:Quondum" title="User talk:Quondum">Quondum</a> <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Quondum-20240630180900-Tito_Omburo-20240630170100" class="ext-discussiontools-init-timestamplink">18:09, 30 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Quondum-20240630180900-Tito_Omburo-20240630170100"></span><span data-mw-comment-end="c-Quondum-20240630180900-Tito_Omburo-20240630170100"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="Sourcing_question" data-mw-thread-id="h-Sourcing_question-20240629051000"><span data-mw-comment-start="" id="h-Sourcing_question-20240629051000"></span>Sourcing question<span data-mw-comment-end="h-Sourcing_question-20240629051000"></span></h2><div class="ext-discussiontools-init-section-bar"><div class="ext-discussiontools-init-section-metadata"></div><div class="ext-discussiontools-init-section-actions"></div></div></div> <p><span data-mw-comment-start="" id="c-XOR'easter-20240629051000-Sourcing_question"></span>Does the argument in <a href="/wiki/0.999...#Impossibility_of_unique_representation" title="0.999...">0.999...#Impossibility of unique representation</a> come from somewhere? Other than that, the sourcing seems OK. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240629051000-Sourcing_question" class="ext-discussiontools-init-timestamplink">05:10, 29 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240629051000-Sourcing_question"></span><span data-mw-comment-end="c-XOR'easter-20240629051000-Sourcing_question"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20240629061800-XOR'easter-20240629051000"></span>The first and the second, as well as the bullet list, remain unsourced. <a href="/wiki/User:Dedhert.Jr" title="User:Dedhert.Jr">Dedhert.Jr</a> (<a href="/wiki/User_talk:Dedhert.Jr" title="User talk:Dedhert.Jr">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Dedhert.Jr-20240629061800-XOR'easter-20240629051000" class="ext-discussiontools-init-timestamplink">06:18, 29 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20240629061800-XOR'easter-20240629051000"></span><span data-mw-comment-end="c-Dedhert.Jr-20240629061800-XOR'easter-20240629051000"></span> <dl><dd><span data-mw-comment-start="" id="c-XOR'easter-20240629174700-Dedhert.Jr-20240629061800"></span>I have provisionally trimmed the passage I couldn't find support for. It wasn't technically wrong, as far as I could tell, but we aren't a repository for everything that <i>could</i> be said about a math topic. <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240629174700-Dedhert.Jr-20240629061800" class="ext-discussiontools-init-timestamplink">17:47, 29 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240629174700-Dedhert.Jr-20240629061800"></span><span data-mw-comment-end="c-XOR'easter-20240629174700-Dedhert.Jr-20240629061800"></span> <dl><dd><span data-mw-comment-start="" id="c-Trovatore-20240629215000-XOR'easter-20240629174700"></span>Probably a correct removal, but sort of a pity, since it's the only bit of actual <i>mathematical</i> interest.</dd> <dd>No matter. This isn't really a math article, or shouldn't be. Mathematicians are unlikely to care about 0.999... per se. We should keep that in mind when thinking about how to present the material. I'm totally against <a href="/wiki/Lies_to_children" class="mw-redirect" title="Lies to children">lies to children</a>, but I also don't see the point in making this an article about real analysis. If you understand real analysis you don't need this article. --<a href="/wiki/User:Trovatore" title="User:Trovatore">Trovatore</a> (<a href="/wiki/User_talk:Trovatore" title="User talk:Trovatore">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-Trovatore-20240629215000-XOR'easter-20240629174700" class="ext-discussiontools-init-timestamplink">21:50, 29 June 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Trovatore-20240629215000-XOR'easter-20240629174700"></span><span data-mw-comment-end="c-Trovatore-20240629215000-XOR'easter-20240629174700"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-XOR'easter-20240723023000-Dedhert.Jr-20240629061800"></span>Are any of the concerns in the <a href="/wiki/Wikipedia:Featured_article_review/0.999.../archive2" title="Wikipedia:Featured article review/0.999.../archive2">FA review</a> still outstanding, then? <a href="/wiki/User:XOR%27easter" title="User:XOR'easter">XOR'easter</a> (<a href="/wiki/User_talk:XOR%27easter" class="mw-redirect" title="User talk:XOR'easter">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999.../Archive_20#c-XOR'easter-20240723023000-Dedhert.Jr-20240629061800" class="ext-discussiontools-init-timestamplink">02:30, 23 July 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-XOR'easter-20240723023000-Dedhert.Jr-20240629061800"></span><span data-mw-comment-end="c-XOR'easter-20240723023000-Dedhert.Jr-20240629061800"></span></dd></dl></dd></dl>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Talk:0.999.../Archive_20&oldid=1260211219">https://en.wikipedia.org/w/index.php?title=Talk:0.999.../Archive_20&oldid=1260211219</a>"</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 November 2024, at 14:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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