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data-mw-deduplicate="TemplateStyles:r1135323948">body.skin-vector-2022 .mw-parser-output .skiptotalk,body.mw-mf .mw-parser-output .skiptotalk{display:none}.mw-parser-output .skiptotalk a{display:block;text-align:center;font-style:italic;line-height:1.9}.mw-parser-output .skiptotalk a::before,.mw-parser-output .skiptotalk a::after{content:"↓";font-size:larger;line-height:1.6;font-style:normal}.mw-parser-output .skiptotalk a::before{float:left}.mw-parser-output .skiptotalk a::after{float:right}</style><a href="#toc">Skip to table of contents</a></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><style data-mw-deduplicate="TemplateStyles:r1230750088">.mw-parser-output .talkheader{text-align:center}.mw-parser-output .talkheader-body{display:flex}.mw-parser-output .talkheader-help,.mw-parser-output .talkheader-policies,.mw-parser-output .talkheader-shortcuts{display:flex;align-items:center;padding:0.1em}.mw-parser-output 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to the <a href="/wiki/0.999..." title="0.999...">0.999...</a> article.<br/> This is <a href="/wiki/Wikipedia:What_Wikipedia_is_not#FORUM" title="Wikipedia:What Wikipedia is not">not a forum</a> for general discussion of the article's subject.</td> </tr> <tr> <td> <div class="talkheader-body"> <div class="talkheader-help"> <ul><li>Put new text under old text. <strong><a href="/wiki/Special:NewSection/Talk:0.999..." title="Special:NewSection/Talk:0.999...">Click here to start a new topic.</a></strong></li> <li>New to Wikipedia? 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padding: 1px 3px;"><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%220.999...%22">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%220.999...%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%220.999...%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%220.999...%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?safe=off&tbs=sur:fmc&tbm=isch&q=%220.999...%22+-site:wikipedia.org+-site:wikimedia.org">free images</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/custom?hl=en&cx=007734830908295939403%3Agalkqgoksq0&cof=FORID%3A13%3BAH%3Aleft%3BCX%3AWikipedia%2520Reference%2520Search&q=%220.999...%22">WP refs</a>) <b>·</b> <span title="Free English newspaper sources" style="border-bottom: 1px dotted;"><a class="external text" href="https://en.wikipedia.org/wiki/Wikipedia:Free_English_newspaper_sources">FENS</a></span> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%220.999...%22&acc=on&wc=on">JSTOR</a> <b>·</b> <span title="The Wikipedia Library" style="border-bottom: 1px dotted;"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/search/?q=%220.999...%22">TWL</a></span></span></td></tr><tr><td style="border-top: 1px solid #c0c090; padding: 1px 3px;"><b><a href="/wiki/Help:Archiving_a_talk_page" title="Help:Archiving a talk page">Archives</a>:</b> <a href="/wiki/Talk:0.999.../Archive_index" title="Talk:0.999.../Archive index">Index</a>, <a href="/wiki/Talk:0.999.../Archive_1" title="Talk:0.999.../Archive 1">1</a>, <a href="/wiki/Talk:0.999.../Archive_2" title="Talk:0.999.../Archive 2">2</a>, <a href="/wiki/Talk:0.999.../Archive_3" title="Talk:0.999.../Archive 3">3</a>, <a href="/wiki/Talk:0.999.../Archive_4" title="Talk:0.999.../Archive 4">4</a>, <a href="/wiki/Talk:0.999.../Archive_5" title="Talk:0.999.../Archive 5">5</a>, <a href="/wiki/Talk:0.999.../Archive_6" title="Talk:0.999.../Archive 6">6</a>, <a href="/wiki/Talk:0.999.../Archive_7" title="Talk:0.999.../Archive 7">7</a>, <a href="/wiki/Talk:0.999.../Archive_8" title="Talk:0.999.../Archive 8">8</a>, <a href="/wiki/Talk:0.999.../Archive_9" title="Talk:0.999.../Archive 9">9</a>, <a href="/wiki/Talk:0.999.../Archive_10" title="Talk:0.999.../Archive 10">10</a>, <a href="/wiki/Talk:0.999.../Archive_11" title="Talk:0.999.../Archive 11">11</a>, <a href="/wiki/Talk:0.999.../Archive_12" title="Talk:0.999.../Archive 12">12</a>, <a href="/wiki/Talk:0.999.../Archive_13" title="Talk:0.999.../Archive 13">13</a>, <a href="/wiki/Talk:0.999.../Archive_14" title="Talk:0.999.../Archive 14">14</a>, <a href="/wiki/Talk:0.999.../Archive_15" title="Talk:0.999.../Archive 15">15</a>, <a href="/wiki/Talk:0.999.../Archive_16" title="Talk:0.999.../Archive 16">16</a>, <a href="/wiki/Talk:0.999.../Archive_17" title="Talk:0.999.../Archive 17">17</a>, <a href="/wiki/Talk:0.999.../Archive_18" title="Talk:0.999.../Archive 18">18</a>, <a href="/wiki/Talk:0.999.../Archive_19" title="Talk:0.999.../Archive 19">19</a>, <a href="/wiki/Talk:0.999.../Archive_20" title="Talk:0.999.../Archive 20">20</a><span style="position: relative; 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padding: 0;"><div> <div class="mw-inputbox-centered" style=""><form name="searchbox" class="searchbox mw-inputbox-form-inline" action="/wiki/Special:Search"><div class="cdx-text-input"><input class="mw-searchInput searchboxInput cdx-text-input__input" name="search" placeholder="" size="50" dir="ltr"/></div><input type="hidden" value="Talk:0.999.../" name="prefix"/> <input type="submit" name="fulltext" value="Search archives" class="cdx-button"/><input type="hidden" value="Search" name="fulltext"/></form></div> </div></td> </tr> </tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><table class="box-Warning plainlinks tmbox tmbox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40"/></span></span></td><td class="mbox-text" style="text-align: left;">This is the <a href="/wiki/Wikipedia:Talk_pages" class="mw-redirect" title="Wikipedia:Talk pages">talk page</a> for discussing changes to the <a href="/wiki/0.999..." title="0.999...">0.999...</a> article itself. <i>Please place discussions on the underlying mathematical issues on the <a href="/wiki/Talk:0.999.../Arguments" title="Talk:0.999.../Arguments">arguments page</a></i>. For questions about the maths involved, try posting to the <a href="/wiki/Wikipedia:Reference_desk/Mathematics" title="Wikipedia:Reference desk/Mathematics">reference desk</a> instead.</td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><table class="plainlinks tmbox tmbox-notice" role="presentation"><tbody><tr><td class="mbox-empty-cell"></td><td class="mbox-text"><style data-mw-deduplicate="TemplateStyles:r1214851843">.mw-parser-output .hidden-begin{box-sizing:border-box;width:100%;padding:5px;border:none;font-size:95%}.mw-parser-output .hidden-title{font-weight:bold;line-height:1.6;text-align:left}.mw-parser-output .hidden-content{text-align:left}@media all and (max-width:500px){.mw-parser-output .hidden-begin{width:auto!important;clear:none!important;float:none!important}}</style><div class="hidden-begin mw-collapsible autocollapse" style="font-size: 105%"><div class="hidden-title skin-nightmode-reset-color" style="text-align: center"><span class="anchor" id="FAQ"></span><div style="float:left;"><span typeof="mw:File"><a href="//en.wikipedia.org/w/index.php?title=Talk:0.999.../FAQ&action=edit&preload=Template:FAQ/Preload" title="?"><img alt="?" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/OOjs_UI_icon_help-ltr-progressive.svg/24px-OOjs_UI_icon_help-ltr-progressive.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/OOjs_UI_icon_help-ltr-progressive.svg/36px-OOjs_UI_icon_help-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/OOjs_UI_icon_help-ltr-progressive.svg/48px-OOjs_UI_icon_help-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20"/></a></span> <small><a href="/wiki/Talk:0.999.../FAQ" title="Talk:0.999.../FAQ">view</a> <b>·</b> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Talk:0.999.../FAQ&action=edit&preload=Template:FAQ/Preload">edit</a></small></div> Frequently asked questions</div><div class="hidden-content mw-collapsible-content" style=""> <dl><dd><b>Q</b>: Are you positive that 0.999... equals 1 <i>exactly</i>, not approximately?</dd></dl> <dl><dd><b>A</b>: In the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>, yes. This is covered in the article. If you still have doubts, you can discuss it at <a href="/wiki/Talk:0.999.../Arguments" title="Talk:0.999.../Arguments">Talk:0.999.../Arguments</a>. However, please note that <a href="/wiki/Wikipedia:NOR" class="mw-redirect" title="Wikipedia:NOR">original research</a> should never be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only <a href="/wiki/Wikipedia:V" class="mw-redirect" title="Wikipedia:V">reputable secondary and tertiary sources</a> can do so.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: Can't "1 - 0.999..." be expressed as "0.000...1"?</dd></dl> <dl><dd><b>A</b>: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit <i>d</i> being <i>k</i> places past the decimal point is that the digit contributes <i>d</i> · 10<sup>-<i>k</i></sup> toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of <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 {\frac {0.000\dots 1}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>0.000</mn> <mo>…</mo> <mn>1</mn> </mrow> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {0.000\dots 1}{10}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab794a107eb4d83add089914554666b80af64e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.793ex; height:5.176ex;" alt="{\displaystyle {\frac {0.000\dots 1}{10}}}"/></span> would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1?</dd></dl> <dl><dd><b>A</b>: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit <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 =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.999</mn> <mo>…</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mn>0.</mn> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mn>99</mn> <mo>…</mo> <mn>9</mn> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc66bbf03eb4d56ee44779057be23f938b6b245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-right: -0.028ex; width:24.987ex; height:5.509ex;" alt="{\displaystyle 0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}}"/></span> is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1.</dd></dl> <dl><dd><b>A</b>: No. By this logic, 0.9 < 0.999...; 0.99 < 0.999... and so forth. Therefore 0.999... < 0.999..., which is absurd.</dd></dl> <dl><dd>Something that holds for various values need not hold for the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> of those values. For example, <i>f</i> (<i>x</i>)=<i>x</i><sup> 3</sup>/<i>x</i> is positive (>0) for all values in its implied domain (<i>x</i> ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught 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="{\displaystyle 0.x_{1}x_{2}x_{3}...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.x_{1}x_{2}x_{3}...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90be05fdf57b37cb05e33cd80e2b9785723068a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.676ex; height:2.509ex;" alt="{\displaystyle 0.x_{1}x_{2}x_{3}...}"/></span> must be less than <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 1.y_{1}y_{2}y_{3}...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.y_{1}y_{2}y_{3}...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0827ec1b5c1d56c969e44c7bd708c373dc614cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.104ex; height:2.509ex;" alt="{\displaystyle 1.y_{1}y_{2}y_{3}...}"/></span> for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: 0.999... is written differently from 1, so it can't be equal.</dd></dl> <dl><dd><b>A</b>: 1 can be written many ways: <a href="/wiki/Fraction_(mathematics)#Equivalent_fractions" class="mw-redirect" title="Fraction (mathematics)">1/1, 2/2</a>, <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">cos</a> 0, <a href="/wiki/Natural_logarithm" title="Natural logarithm">ln</a> <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)"><i>e</i></a>, <a href="/wiki/Imaginary_unit" title="Imaginary unit"><i>i</i><sup> 4</sup></a>, <a href="/wiki/Subtraction" title="Subtraction">2 - 1</a>, <a href="/wiki/Scientific_notation#Exponential_notation" title="Scientific notation">1e0</a>, <a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">1<sub>2</sub></a>, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a <a href="/wiki/Bijection" title="Bijection">bijection</a> from decimal representations to real numbers.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount?</dd></dl> <dl><dd><b>A</b>: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">ε</a>, which is not a fixed constant; <a href="/wiki/Differential_(calculus)" class="mw-redirect" title="Differential (calculus)">differentials</a>, which are not numbers at all; <a href="/wiki/Differential_form" title="Differential form">differential forms</a>, which are not real numbers and have <a href="/wiki/Anticommutativity" class="mw-redirect" title="Anticommutativity">anticommutativity</a>; 0<sup>+</sup>, which is not a number, but rather part of the expression <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 \lim _{x\rightarrow 0^{+}}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4edf62091a1e3cb0dcbafb83e544c3dc0483d3a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.412ex; height:4.343ex;" alt="{\displaystyle \lim _{x\rightarrow 0^{+}}f(x)}"/></span>, the <a href="/wiki/One-sided_limit" title="One-sided limit">right limit</a> of <i>x</i> (which can also be expressed without the "+" as <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 \lim _{x\downarrow 0}f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\downarrow 0}f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d8d664d1874c2e83e9812d15e75c0a0cca4a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.034ex; height:4.343ex;" alt="{\displaystyle \lim _{x\downarrow 0}f(x)}"/></span>); and values in number systems such as <a href="/wiki/Dual_number" title="Dual number">dual numbers</a> and <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreals</a>. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the <a href="/wiki/0.999..." title="0.999...">main article</a>, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: Are you sure 0.999... equals 1 in hyperreals?</dd></dl> <dl><dd><b>A</b>: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given <a href="/wiki/Hyperreal_number" title="Hyperreal number">here</a>, the problem is that almost same sequences give different hyperreal numbers, <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)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.9</mn> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.99</mn> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>99</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.9</mn> <mo stretchy="false">(</mo> <mn>99</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0.</mn> <mo stretchy="false">(</mo> <mn>999</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>1</mn> <mspace width="thickmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4455520a68cd832f08d9737ac1650733cb4856b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.384ex; height:2.843ex;" alt="{\displaystyle 0.(9)<0.9(9)<0.99(9)<0.(99)<0.9(99)<0.(999)<1\;}"/></span>, and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...).</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?</dd></dl> <dl><dd><b>A</b>: At the expense of abandoning many familiar features of mathematics, it is possible to construct <i>a system of notation</i> in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a <a href="/wiki/Wikipedia:POV" class="mw-redirect" title="Wikipedia:POV">WP:POV</a> desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right.</dd></dl> <dl><dd><b>A</b>: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in <a href="/wiki/Engineering" title="Engineering">engineering</a>, <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Computer_science" title="Computer science">computer science</a>, or a <a href="/wiki/Natural_science" title="Natural science">natural science</a>, it would probably help you in the future anyway.</dd></dl> <p><br/> </p> <dl><dd><b>Q</b>: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article?</dd></dl> <dl><dd><b>A</b>: The criteria for inclusion in Wikipedia is for information to be <a href="/wiki/Wikipedia:V" class="mw-redirect" title="Wikipedia:V">attributable</a> to a <a href="/wiki/Wikipedia:RS" class="mw-redirect" title="Wikipedia:RS">reliable published source</a>, not an <a href="/wiki/Wikipedia:NOR" class="mw-redirect" title="Wikipedia:NOR">editor's opinion</a>. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Wikipedia policy. Arguments posted on the <a href="/wiki/Talk:0.999.../Arguments" title="Talk:0.999.../Arguments">Talk:0.999.../Arguments</a> page are disqualified, as their inclusion would violate Wikipedia policy on <a href="/wiki/Wikipedia:OR" class="mw-redirect" title="Wikipedia:OR">original research</a>.</dd></dl> </div></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><style data-mw-deduplicate="TemplateStyles:r1135653291">@media(min-width:720px){.mw-parser-output .article-history{min-width:80%}}</style><table class="article-history tmbox tmbox-notice"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Featured_article_star_-_cross.svg" class="mw-file-description" title="Former featured article"><img alt="Former featured article" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Featured_article_star_-_cross.svg/48px-Featured_article_star_-_cross.svg.png" decoding="async" width="48" height="48" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Featured_article_star_-_cross.svg/72px-Featured_article_star_-_cross.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Featured_article_star_-_cross.svg/96px-Featured_article_star_-_cross.svg.png 2x" data-file-width="180" data-file-height="180"/></a></span></td><td class="mbox-text"><b>0.999...</b> is a <a href="/wiki/Wikipedia:Former_featured_articles" title="Wikipedia:Former featured articles">former featured article</a>. Please see the links under Article milestones below for its original nomination page (for older articles, check <a href="/wiki/Wikipedia:Featured_article_candidates/Archived_nominations/Index" title="Wikipedia:Featured article candidates/Archived nominations/Index">the nomination archive</a>) and why it was removed.</td></tr><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wikipedia-logo-v2.svg" class="mw-file-description" title="Main Page trophy"><img alt="Main Page trophy" src="//upload.wikimedia.org/wikipedia/en/thumb/8/80/Wikipedia-logo-v2.svg/30px-Wikipedia-logo-v2.svg.png" decoding="async" width="30" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/80/Wikipedia-logo-v2.svg/45px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/80/Wikipedia-logo-v2.svg/60px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94"/></a></span></td><td class="mbox-text">This article appeared on Wikipedia's Main Page as <a href="/wiki/Wikipedia:Today%27s_featured_article/October_25,_2006" title="Wikipedia:Today's featured article/October 25, 2006">Today's featured article</a> on October 25, 2006.</td></tr><tr><td colspan="2" style="width:100%"><table class="article-history-milestones mw-collapsible mw-collapsed" style="width:100%;font-size:90%"><tbody><tr><th colspan="3" style="font-size:110%">Article milestones</th></tr><tr style="text-align:left"><th>Date</th><th>Process</th><th>Result</th></tr><tr><td><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&oldid=51422447">May 5, 2006</a></span></td><td><b><a href="/wiki/Wikipedia:Articles_for_deletion/Proof_that_0.999..._equals_1" title="Wikipedia:Articles for deletion/Proof that 0.999... equals 1">Articles for deletion</a></b></td><td>Kept</td></tr><tr><td><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&oldid=80638011">October 10, 2006</a></span></td><td><b><a href="/wiki/Wikipedia:Featured_article_candidates/0.999..." title="Wikipedia:Featured article candidates/0.999...">Featured article candidate</a></b></td><td>Promoted</td></tr><tr><td><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&oldid=382076790">August 31, 2010</a></span></td><td><b><a href="/wiki/Wikipedia:Featured_article_review/0.999.../archive1" title="Wikipedia:Featured article review/0.999.../archive1">Featured article review</a></b></td><td>Kept</td></tr><tr><td><span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=0.999...&oldid=1243492081">September 24, 2024</a></span></td><td><b><a href="/wiki/Wikipedia:Featured_article_review/0.999.../archive2" title="Wikipedia:Featured article review/0.999.../archive2">Featured article review</a></b></td><td>Demoted</td></tr><tr><td colspan="3">Current status: <b>Former featured article</b></td></tr></tbody></table></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><style data-mw-deduplicate="TemplateStyles:r1243927654">.mw-parser-output .banner-shell{border-collapse:separate;border-spacing:4px}.mw-parser-output .banner-shell-header{text-align:center;font-weight:bold}.mw-parser-output .banner-shell-inner{padding:2px 4px;background:#fffaef;color:inherit;border:1px dotted gray}@media screen{html.skin-theme-clientpref-night .mw-parser-output .banner-shell-inner{background:#2e2505}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .banner-shell-inner{background:#2e2505}}.mw-parser-output .banner-shell .tmbox{margin:2px 0;width:100%}.mw-parser-output .banner-shell .tmbox.mbox-small{line-height:1.5em;font-size:100%}.mw-parser-output .banner-shell-inner 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.wpb-header{background-color:#302f2d}html.skin-theme-clientpref-os .mw-parser-output .wpbs .banner-shell-inner{background-color:#2e2505}}</style><table class="tmbox tmbox-notice mw-collapsible innercollapse wpb wpb-table"><tbody><tr class="wpb-header"><td class="wpb-header-icon"><span typeof="mw:File"><a href="/wiki/File:Sound-icon.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/33px-Sound-icon.svg.png" decoding="async" width="33" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/51px-Sound-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/67px-Sound-icon.svg.png 2x" data-file-width="128" data-file-height="96"/></a></span></td><td class="wpb-header-combined"><a href="/wiki/Wikipedia:WikiProject_Spoken_Wikipedia" title="Wikipedia:WikiProject Spoken Wikipedia">Spoken Wikipedia</a> </td></tr><tr><td class="mbox-text wpb-main" colspan="2"><table><tbody><tr><td class="mbox-image wpb-image"><span typeof="mw:File"><a href="/wiki/File:Sound-icon.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/100px-Sound-icon.svg.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/150px-Sound-icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Sound-icon.svg/200px-Sound-icon.svg.png 2x" data-file-width="128" data-file-height="96"/></a></span></td><td class="mbox-text">This article is within the scope of <b><a href="/wiki/Wikipedia:WikiProject_Spoken_Wikipedia" title="Wikipedia:WikiProject Spoken Wikipedia">WikiProject Spoken Wikipedia</a></b>, a collaborative effort to improve the coverage of articles that are <b><a href="/wiki/Speech" title="Speech">spoken</a></b> on Wikipedia. If you would like to participate, please visit the project page, where you can join <a href="/wiki/Wikipedia_talk:WikiProject_Spoken_Wikipedia" title="Wikipedia talk:WikiProject Spoken Wikipedia">the discussion</a> and see a list of open tasks.<span class="metadata wpb-metadata"><span class="wpb-project">Spoken Wikipedia</span><span class="wpb-project_link">Wikipedia:WikiProject Spoken Wikipedia</span><span class="wpb-banner_name">Template:WikiProject Spoken Wikipedia</span><span class="wpb-assessment_cat">Spoken Wikipedia articles</span></span></td><td class="mbox-empty-cell"></td></tr></tbody></table></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237879475"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1259570071"><table class="tmbox tmbox-notice mw-collapsible innercollapse wpb wpb-table"><tbody><tr class="wpb-header"><td class="wpb-header-icon"><span typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description"><img alt="WikiProject icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/25px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/38px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/50px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128"/></a></span></td><td class="wpb-header-combined"><a href="/wiki/Wikipedia:WikiProject_Mathematics" title="Wikipedia:WikiProject Mathematics">Mathematics</a> <span class="wpb-header-bubbles"><span class="mw-default-size" typeof="mw:File"><span title="This was a selected article on the Mathematics Portal."><img alt="" 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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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-XOR'easter-20240723023000-Dedhert.Jr-20240629061800"></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><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:0.999...&action=edit&section=3" title="Edit section: Two representations in every positional numeral system with one terminating?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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...#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 class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></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="Intuitive_explanation" data-mw-thread-id="h-Intuitive_explanation-20241004121600"><span data-mw-comment-start="" id="h-Intuitive_explanation-20241004121600"></span>Intuitive explanation<span data-mw-comment-end="h-Intuitive_explanation-20241004121600"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:0.999...&action=edit&section=4" title="Edit section: Intuitive explanation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><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-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600-Intuitive_explanation"></span>There seems to be an error in the intuitive explanation: </p> <pre>For any number x that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than x⁠⁠. </pre> <p>If we set x = 0.̅9 then the sequence will never reach a number <i>larger</i> than x. <a href="/wiki/Special:Contributions/2A01:799:39E:1300:F896:4392:8DAA:D475" title="Special:Contributions/2A01:799:39E:1300:F896:4392:8DAA:D475">2A01:799:39E:1300:F896:4392:8DAA:D475</a> (<a href="/w/index.php?title=User_talk:2A01:799:39E:1300:F896:4392:8DAA:D475&action=edit&redlink=1" class="new" title="User talk:2A01:799:39E:1300:F896:4392:8DAA:D475 (page does not exist)">talk</a>) <a href="https://en.wikipedia.org/wiki/Talk:0.999...#c-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600-Intuitive_explanation" class="ext-discussiontools-init-timestamplink">12:16, 4 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600-Intuitive_explanation"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600-Intuitive_explanation"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-MartinPoulter-20241004125000-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"></span>If x = 0.̅9 then x is not less than 1, so the conditional statement is true. What is the error? <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...#c-MartinPoulter-20241004125000-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600" class="ext-discussiontools-init-timestamplink">12:50, 4 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-MartinPoulter-20241004125000-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-MartinPoulter-20241004125000-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"></span> <dl><dd><span data-mw-comment-start="" id="c-Nø-20241004144700-MartinPoulter-20241004125000"></span>If you presuppose that 0.̅9 is less than one, the argument that should prove you wrong may apprear to be sort of circular. Would it be better to say "to the left of 1 on the number line" instead of "less than 1"? I know it's the same, but then the person believing 0.̅9 to be less than one would have to place it on the number line! <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...#c-Nø-20241004144700-MartinPoulter-20241004125000" class="ext-discussiontools-init-timestamplink">14:47, 4 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Nø-20241004144700-MartinPoulter-20241004125000"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Nø-20241004144700-MartinPoulter-20241004125000"></span></dd></dl></dd> <dd><span data-mw-comment-start="" id="c-Johnjbarton-20241004154300-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"></span>What does the notation 0.̅9 mean? <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...#c-Johnjbarton-20241004154300-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600" class="ext-discussiontools-init-timestamplink">15:43, 4 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20241004154300-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Johnjbarton-20241004154300-2A01:799:39E:1300:F896:4392:8DAA:D475-20241004121600"></span></dd></dl> <dl><dd><dl><dd><span data-mw-comment-start="" id="c-Mr_swordfish-20241005002400-Johnjbarton-20241004154300"></span>It means zero followed by the decimal point, followed by an infinite sequence of 9s. <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...#c-Mr_swordfish-20241005002400-Johnjbarton-20241004154300" class="ext-discussiontools-init-timestamplink">00:24, 5 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Mr_swordfish-20241005002400-Johnjbarton-20241004154300"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Mr_swordfish-20241005002400-Johnjbarton-20241004154300"></span> <dl><dd><span data-mw-comment-start="" id="c-Johnjbarton-20241005014700-Mr_swordfish-20241005002400"></span>Thanks! Seems a bit odd that this is curious combination of characters (which I don't know how to type) is not listed in the article on 0.999... <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...#c-Johnjbarton-20241005014700-Mr_swordfish-20241005002400" class="ext-discussiontools-init-timestamplink">01:47, 5 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Johnjbarton-20241005014700-Mr_swordfish-20241005002400"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Johnjbarton-20241005014700-Mr_swordfish-20241005002400"></span></dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2 ext-discussiontools-init-section"><h2 id="B_and_C" data-mw-thread-id="h-B_and_C-20241030110000"><span data-mw-comment-start="" id="h-B_and_C-20241030110000"></span>B and C<span data-mw-comment-end="h-B_and_C-20241030110000"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Talk:0.999...&action=edit&section=5" title="Edit section: B and C"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span><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-Dedhert.Jr-20241030110000-B_and_C"></span>@<a href="/wiki/User:Tito_Omburo" title="User:Tito Omburo">Tito Omburo</a>. There are other unsourced facts in the given sections. For example: </p> <ul><li>There is no source mentions about "Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as..." in Dedekind cuts.</li> <li>There is no source mentions about "Continuing this process yields an infinite sequence of <a href="/wiki/Nested_intervals" title="Nested intervals">nested intervals</a>, labeled by an infinite sequence of digits ⁠<span class="texhtml"><i>b</i><sub>1</sub>, <i>b</i><sub>2</sub>⁠⁠, <i>b</i><sub>3</sub>, ...</span>, and one writes..." in Nested intervals and least upper bounds. This is just one of them.</li></ul> <p><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...#c-Dedhert.Jr-20241030110000-B_and_C" class="ext-discussiontools-init-timestamplink">11:00, 30 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20241030110000-B_and_C"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dedhert.Jr-20241030110000-B_and_C"></span> </p> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20241030113500-Dedhert.Jr-20241030110000"></span>The section on Dedekind cuts is sourced to Richman throughout. The paragraph on nested intervals has three different sources attached to it. <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...#c-Tito_Omburo-20241030113500-Dedhert.Jr-20241030110000" class="ext-discussiontools-init-timestamplink">11:35, 30 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20241030113500-Dedhert.Jr-20241030110000"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tito_Omburo-20241030113500-Dedhert.Jr-20241030110000"></span> <dl><dd><span data-mw-comment-start="" id="c-Dedhert.Jr-20241030125200-Tito_Omburo-20241030113500"></span>Are you saying that citations in the latter paragraph supports the previous paragraphs? If that's the case, I prefer to attach the same citations into those previous ones. <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...#c-Dedhert.Jr-20241030125200-Tito_Omburo-20241030113500" class="ext-discussiontools-init-timestamplink">12:52, 30 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Dedhert.Jr-20241030125200-Tito_Omburo-20241030113500"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Dedhert.Jr-20241030125200-Tito_Omburo-20241030113500"></span> <dl><dd><span data-mw-comment-start="" id="c-Tito_Omburo-20241030130900-Dedhert.Jr-20241030125200"></span>Not sure what you mean. Both paragraphs have citations. <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...#c-Tito_Omburo-20241030130900-Dedhert.Jr-20241030125200" class="ext-discussiontools-init-timestamplink">13:09, 30 October 2024 (UTC)</a><span class="ext-discussiontools-init-replylink-buttons" data-mw-thread-id="c-Tito_Omburo-20241030130900-Dedhert.Jr-20241030125200"><span class="ext-discussiontools-init-replylink-bracket">[</span><a class="ext-discussiontools-init-replylink-reply" role="button" tabindex="0" href="">reply</a><span class="ext-discussiontools-init-replylink-bracket">]</span></span><span data-mw-comment-end="c-Tito_Omburo-20241030130900-Dedhert.Jr-20241030125200"></span></dd></dl></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...&oldid=1254337842">https://en.wikipedia.org/w/index.php?title=Talk:0.999...&oldid=1254337842</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Wikipedia_former_featured_articles" title="Category:Wikipedia former featured articles">Wikipedia former featured articles</a></li><li><a href="/wiki/Category:Featured_articles_that_have_appeared_on_the_main_page" title="Category:Featured articles that have appeared on the main page">Featured articles that have appeared on the main page</a></li><li><a href="/wiki/Category:Featured_articles_that_have_appeared_on_the_main_page_once" title="Category:Featured articles that have appeared on the main page once">Featured articles that have appeared on the main page once</a></li><li><a href="/wiki/Category:B-Class_mathematics_articles" title="Category:B-Class mathematics articles">B-Class mathematics articles</a></li><li><a href="/wiki/Category:Unknown-priority_mathematics_articles" title="Category:Unknown-priority mathematics articles">Unknown-priority mathematics articles</a></li><li><a href="/wiki/Category:Featured_articles_on_Mathematics_Portal" title="Category:Featured articles on Mathematics Portal">Featured articles on Mathematics Portal</a></li><li><a href="/wiki/Category:B-Class_Numbers_articles" title="Category:B-Class Numbers articles">B-Class Numbers articles</a></li><li><a href="/wiki/Category:Mid-importance_Numbers_articles" title="Category:Mid-importance Numbers articles">Mid-importance Numbers articles</a></li><li><a href="/wiki/Category:WikiProject_Numbers_articles" title="Category:WikiProject Numbers articles">WikiProject Numbers articles</a></li></ul></div></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 30 October 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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