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developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/24px-25_percent.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/36px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/48px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikibooks, open books for an open world</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:999_Perspective.svg" class="mw-file-description" title="0.999... repeating infinitely."><img alt="0.999... repeating infinitely." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/999_Perspective.svg/500px-999_Perspective.svg.png" decoding="async" width="500" height="59" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/999_Perspective.svg/750px-999_Perspective.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/999_Perspective.svg/1000px-999_Perspective.svg.png 2x" data-file-width="1300" data-file-height="153" /></a><figcaption>0.999... repeating infinitely.</figcaption></figure> <p>This <a href="/wiki/Wikibooks:What_is_Wikibooks%3F" class="mw-redirect" title="Wikibooks:What is Wikibooks?">Wikibook</a> is a reference guide to the theory behind the mathematical identity <b>0.999...</b> = 1. It includes select topics in <a href="/wiki/Real_Analysis" title="Real Analysis">real analysis</a>, <a href="/wiki/Calculus" title="Calculus">calculus</a>, and the arithmetic of infinite decimals, culminating in several proofs of the identity itself. The material cuts across the study of pure mathematics, mathematics for scientists and engineers, and high-school and middle-school mathematics. </p> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=1" title="Edit section: Introduction" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=1" title="Edit section's source code: Introduction"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A traditional textbook aimed at a single grade level will often present a proof that 0.999... = 1, but the textbook will rely upon assumptions whose proofs are deferred to more advanced textbooks. The reader who wants a full proof starting from basic algebra is therefore required to consult multiple references. However, different books take different approaches to real analysis, often presenting the material in different orders. Some textbooks assume 0.999... = 1 in their development of the real numbers. So a full proof that is pasted together introduces a real danger of circular reasoning. This book aims to alleviate worries about circular reasoning by staying self-contained and clearly indicating which results are used in which proofs. </p><p>The mathematics literature contains a wide variety of proofs that 0.999... = 1. This book provides enough background to understand all commonly encountered proofs. </p><p>This book sits somewhere between high school and university in difficulty. It will be most relevant to you if you've already completed an algebra curriculum, but not a course in real analysis. A working knowledge of the rational numbers from <a href="/wiki/Basic_Algebra" title="Basic Algebra">basic algebra</a> is assumed throughout. A background in <a href="/wiki/Set_Theory" title="Set Theory">set theory</a> will help for the more advanced modules. </p> <div class="mw-heading mw-heading3"><h3 id="How_to_read_this_book">How to read this book</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=2" title="Edit section: How to read this book" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=2" title="Edit section's source code: How to read this book"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Depending on what you're aiming to get out of this book, there are many ways to read it. </p> <ul><li>If you read from start to finish, you'll encounter topics in their logical order, and each proof will rely on concepts you've already seen. You'll also gain an understanding of all the most popular proofs of 0.999... = 1. However, you'll be hit with the most abstract concepts right away, and it won't be clear <i>why</i> each concept is introduced until later. This strategy is recommended for a more advanced reader.</li> <li>You can start by reading a proof of 0.999... = 1 from the last chapter of the book. When you encounter a step in the proof that you're unsure about, you can then go back to an earlier chapter and read the justification for that step. This way, you'll encounter topics in the same order as in a traditional mathematics curriculum, and you won't waste time on steps that are irrelevant to your chosen proof. This strategy is recommended for a less advanced reader.</li> <li>If you're already familiar with 0.999... = 1, but you've always wondered about a particular ingredient in a proof, you can jump directly to that ingredient in the below table of contents.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Contents">Contents</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=3" title="Edit section: Contents" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=3" title="Edit section's source code: Contents"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint" style="float:right; clear:right; margin:0 0 1em 1em; width:250px;"> <table> <tbody><tr> <td style="width:50px;"> </td> <td style="width:385px;"> <div class="mw-inputbox-centered" style=""><form name="searchbox" class="searchbox mw-inputbox-form" action="/wiki/Special:Search"><div class="cdx-text-input"><input class="mw-inputbox-input mw-searchInput searchboxInput cdx-text-input__input" name="search" placeholder="" size="13" dir="ltr" /></div><input type="hidden" value="0.999.../" name="prefix" /><br /><input type="submit" name="fulltext" value="Search this book" class="cdx-button" /><input type="hidden" value="Search" name="fulltext" /></form></div> </td></tr></tbody></table> </div> <div class="mw-heading mw-heading3"><h3 id="Real_analysis">Real analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=4" title="Edit section: Real analysis" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=4" title="Edit section's source code: Real analysis"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=0.999.../Construction_of_the_real_numbers_from_Dedekind_cuts&action=edit&redlink=1" class="new" title="0.999.../Construction of the real numbers from Dedekind cuts (does not exist)">Construction of the real numbers from Dedekind cuts</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Construction_of_the_real_numbers_from_Cauchy_sequences&action=edit&redlink=1" class="new" title="0.999.../Construction of the real numbers from Cauchy sequences (does not exist)">Construction of the real numbers from Cauchy sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Axioms_of_the_real_numbers&action=edit&redlink=1" class="new" title="0.999.../Axioms of the real numbers (does not exist)">Axioms of the real numbers</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Archimedean_property_of_the_real_numbers&action=edit&redlink=1" class="new" title="0.999.../Archimedean property of the real numbers (does not exist)">Archimedean property of the real numbers</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Nested_Intervals_Theorem&action=edit&redlink=1" class="new" title="0.999.../Nested Intervals Theorem (does not exist)">Nested Intervals Theorem</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Calculus">Calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=5" title="Edit section: Calculus" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=5" title="Edit section's source code: Calculus"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/0.999.../The_limit_of_a_sequence" title="0.999.../The limit of a sequence">The limit of a sequence</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Term-by-term_operations_on_sequences&action=edit&redlink=1" class="new" title="0.999.../Term-by-term operations on sequences (does not exist)">Term-by-term operations on sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../The_limit_of_a_geometric_sequence&action=edit&redlink=1" class="new" title="0.999.../The limit of a geometric sequence (does not exist)">The limit of a geometric sequence</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../The_sum_of_a_series&action=edit&redlink=1" class="new" title="0.999.../The sum of a series (does not exist)">The sum of a series</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Term-by-term_operations_on_series&action=edit&redlink=1" class="new" title="0.999.../Term-by-term operations on series (does not exist)">Term-by-term operations on series</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Shifting_a_series&action=edit&redlink=1" class="new" title="0.999.../Shifting a series (does not exist)">Shifting a series</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../The_geometric_series_formula" title="0.999.../The geometric series formula">The geometric series formula</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Definitions_of_infinite_decimals">Definitions of infinite decimals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=6" title="Edit section: Definitions of infinite decimals" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=6" title="Edit section's source code: Definitions of infinite decimals"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=0.999.../Definition_from_Dedekind_cuts&action=edit&redlink=1" class="new" title="0.999.../Definition from Dedekind cuts (does not exist)">Definition from Dedekind cuts</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Definition_from_Cauchy_sequences&action=edit&redlink=1" class="new" title="0.999.../Definition from Cauchy sequences (does not exist)">Definition from Cauchy sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Definition_from_series&action=edit&redlink=1" class="new" title="0.999.../Definition from series (does not exist)">Definition from series</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Definition_from_sequences&action=edit&redlink=1" class="new" title="0.999.../Definition from sequences (does not exist)">Definition from sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Definition_from_nested_intervals&action=edit&redlink=1" class="new" title="0.999.../Definition from nested intervals (does not exist)">Definition from nested intervals</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Properties_of_infinite_decimals">Properties of infinite decimals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=7" title="Edit section: Properties of infinite decimals" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=7" title="Edit section's source code: Properties of infinite decimals"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/0.999.../Decimal_addition_and_subtraction" title="0.999.../Decimal addition and subtraction">Decimal addition and subtraction</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="50% developed"><img alt="50% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/9px-50_percents.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/14px-50_percents.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/50_percents.svg/18px-50_percents.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Decimal_multiplication_by_a_small_number" title="0.999.../Decimal multiplication by a small number">Decimal multiplication by a small number</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Decimal_multiplication_by_10" title="0.999.../Decimal multiplication by 10">Decimal multiplication by 10</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/w/index.php?title=0.999.../Decimal_long_division&action=edit&redlink=1" class="new" title="0.999.../Decimal long division (does not exist)">Decimal long division</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="0% developed"><img alt="0% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/9px-00_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/14px-00_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/00_percent.svg/18px-00_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Proofs_that_0.999..._=_1"><span id="Proofs_that_0.999..._.3D_1"></span>Proofs that 0.999... = 1</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=8" title="Edit section: Proofs that 0.999... = 1" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=8" title="Edit section's source code: Proofs that 0.999... = 1"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/0.999.../Proof_from_0.333..." title="0.999.../Proof from 0.333...">Proof from 0.333...</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_from_0.111..." title="0.999.../Proof from 0.111...">Proof from 0.111...</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_multiplication_by_10" title="0.999.../Proof by multiplication by 10">Proof by multiplication by 10</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_the_geometric_series_formula" title="0.999.../Proof by the geometric series formula">Proof by the geometric series formula</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_limit_manipulation" title="0.999.../Proof by limit manipulation">Proof by limit manipulation</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_nested_intervals" title="0.999.../Proof by nested intervals">Proof by nested intervals</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_equality_of_Dedekind_cuts" title="0.999.../Proof by equality of Dedekind cuts">Proof by equality of Dedekind cuts</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li> <li><a href="/wiki/0.999.../Proof_by_equivalence_of_Cauchy_sequences" title="0.999.../Proof by equivalence of Cauchy sequences">Proof by equivalence of Cauchy sequences</a> <span class="noprint"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/9px-25_percent.svg.png" decoding="async" width="9" height="9" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/14px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/18px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Related_resources">Related resources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=0.999...&veaction=edit&section=9" title="Edit section: Related resources" class="mw-editsection-visualeditor"><span>edit</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=0.999...&action=edit&section=9" title="Edit section's source code: Related resources"><span>edit source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This book is narrowly focused on a handful of results that are directly useful for understanding 0.999.... It omits plenty of topics that are essential for a balanced understanding of calculus and real analysis. 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