MediaWiki API result
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{
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"totitle": "0.999...",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Line 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Line 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><<del class=\"diffchange diffchange-inline\">strong</del>><del class=\"diffchange diffchange-inline\">MediaWiki has been installed</del>.<<del class=\"diffchange diffchange-inline\">/strong</del>></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><<ins class=\"diffchange diffchange-inline\">!-- NOTE: The content of this article is well-established. If you have an argument against one or more of the proofs listed here, please read the FAQ on [[Talk:0.999...]], or discuss it on [[Talk:0.999.../Arguments]]. However, please understand that the earlier, more naive proofs are not as rigorous as the later ones as they intend to appeal to intuition, and as such may require further justification to be complete.\u00a0 Thank you. --</ins>></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image:999 Perspective</ins>.<ins class=\"diffchange diffchange-inline\">png|300px|right]]</ins><<ins class=\"diffchange diffchange-inline\">!--[[Image:999 Perspective-color.png|300px|right]]--</ins>></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Consult </del>the [<del class=\"diffchange diffchange-inline\">https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage/Help:Contents User</del>'<del class=\"diffchange diffchange-inline\">s Guide</del>] <del class=\"diffchange diffchange-inline\">for information on using </del>the <del class=\"diffchange diffchange-inline\">wiki software</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In [[mathematics]], </ins>the [<ins class=\"diffchange diffchange-inline\">[repeating decimal]] '''0.999\u2026''' which may also be written as <math>0</ins>.<ins class=\"diffchange diffchange-inline\">\\bar{9} , 0</ins>.<ins class=\"diffchange diffchange-inline\">\\dot{9}<</ins>/<ins class=\"diffchange diffchange-inline\">math> or <math> 0.(9)\\,\\!<</ins>/<ins class=\"diffchange diffchange-inline\">math>\u00a0 denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|one]]. In other words</ins>: <ins class=\"diffchange diffchange-inline\">the notations ''0.999\u2026'' and ''1 '</ins>' <ins class=\"diffchange diffchange-inline\">actually represent the same real number. This [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. [[mathematical proof|Proofs]] have been formulated with varying degrees of [[mathematical rigour]</ins>]<ins class=\"diffchange diffchange-inline\">, taking into account preferred development of </ins>the <ins class=\"diffchange diffchange-inline\">real numbers, background assumptions, historical context, and target audience</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Getting started </del>==</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The fact that certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999\u2026. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://www.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Configuration_settings Configuration settings list</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://www.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:FAQ <del class=\"diffchange diffchange-inline\">MediaWiki FAQ</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999\u2026 eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are \"just shy\" of 1; these are generally unrelated to 0.999\u2026, but they are of considerable interest in [[mathematical analysis]].</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.org/<del class=\"diffchange diffchange-inline\">postorius</del>/<del class=\"diffchange diffchange-inline\">lists</del>/<del class=\"diffchange diffchange-inline\">mediawiki</del>-<del class=\"diffchange diffchange-inline\">announce</del>.<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/ <del class=\"diffchange diffchange-inline\">MediaWiki release mailing list</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://www.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Localise MediaWiki </del>for <del class=\"diffchange diffchange-inline\">your </del>language]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Introduction==</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://www.<del class=\"diffchange diffchange-inline\">mediawiki</del>.org/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Combating_spam Learn how to combat spam </del>on <del class=\"diffchange diffchange-inline\">your wiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.999\u2026 is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999\u2026 = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic\u2014[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]\u2014uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99\u20269, where the 9s eventually stop, is strictly less than 1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Misinterpreting the meaning of the use of the \"\u2026\" ([[ellipsis]]) in 0.999\u2026 accounts for some of the misunderstanding about its equality to 1.\u00a0 The use here is different from the usage in language or in 0.99\u20269, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted.\u00a0 When used to specify a [[recurring decimal]], \"\u2026\" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation \"0.999\u2026\" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, \u2026).</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, {{frac|1|3}} = {{frac|2|6}}. Infinite decimals, however, can express the same number in at most two different ways.\u00a0 If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">There are many proofs that 0.999\u2026 = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999\u2026 and 1, if it exists at all, must be very small.\u00a0 By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999\u2026 are identical.\u00a0 The same argument also explains why 0.333\u2026 = {{frac|1|3}}, 0.111\u2026 = {{frac|1|9}}, etc.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Proofs==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Algebraic===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Fractions and long division====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|3}} becomes a recurring decimal, 0.333\u2026, in which the digits repeat without end. This decimal yields a quick proof for 0.999\u2026 = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 \u00d7 0.333\u2026 equals 0.999\u2026. And 3 \u00d7 {{frac|1|3}} equals 1, so 0.999\u2026 = 1.<ref name=\"CME\">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Another form of this proof multiplies {{frac|1|9}} = 0.111\u2026 by 9.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:{| style=\"wikitable\"</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|<math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\begin{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.333\\dots\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &{} = \\frac{1}{3} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">3 \\times 0.333\\dots &{} = 3 \\times \\frac{1}{3} = \\frac{3 \\times 1}{3} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.999\\dots\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &{} = 1</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\end{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|width=\"25px\"|</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|width=\"25px\" style=\"border-left:1px solid silver;\"|</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|<math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\begin{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.111\\dots\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 & {} = \\frac{1}{9} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">9 \\times 0.111\\dots & {} = 9 \\times \\frac{1}{9} = \\frac{9 \\times 1}{9} \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.999\\dots\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 & {} = 1</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\end{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">|}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A more compact version of the same proof is given by the following equations:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">1 = \\frac{9}{9} = 9 \\times \\frac{1}{9} = 9 \\times 0.111\\dots = 0.999\\dots</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Since both equations are valid, 0.999\u2026 must equal 1 (by the [[transitive property]]). Similarly, {{frac|3|3}} = 1, and {{frac|3|3}} = 0.999\u2026. So, 0.999\u2026 must equal 1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Digit manipulation====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 \u00d7 0.999\u2026 equals 9.999\u2026, which is 9 greater than the original number.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">To see this, consider that subtracting 0.999\u2026 from 9.999\u2026 can proceed digit by digit; in each of the digits after the decimal separator the result is 9 \u2212 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999\u2026, be called ''x''. Then 10''x'' \u2212 ''x'' = 9. This is the same as 9''x'' = 9. Dividing both sides by 9 completes the proof: ''x'' = 1.<ref name=\"CME\"/> Written as a sequence of equations,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\begin{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">x\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &= 0.999\\ldots \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">10 x\u00a0 \u00a0 \u00a0 &= 9.999\\ldots \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">10 x - x\u00a0 \u00a0 &= 9.999\\ldots - 0.999\\ldots \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">9 x\u00a0 \u00a0 \u00a0 \u00a0 &= 9 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">x\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 &= 1 \\\\</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.999\\ldots &= 1</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">\\end{align}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999\u2026\u00a0 and 1.000\u2026 both represent the same number.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Analytic===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Since the question of 0.999\u2026 does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999\u2026, the integer part can be summarized as ''b''<sub>0</sub> and one can neglect negatives, so a decimal expansion has the form</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>b_0.b_1b_2b_3b_4b_5\\dots</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Infinite series and sequences====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{further|[[Decimal representation]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>b_0 . b_1 b_2 b_3 b_4 \\ldots = b_0 + b_1({\\tfrac{1}{10}}) + b_2({\\tfrac{1}{10}})^2 + b_3({\\tfrac{1}{10}})^3 + b_4({\\tfrac{1}{10}})^4 + \\cdots .</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">For 0.999\u2026 one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\\cdots = \\frac{ar}{1-r}.</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Since 0.999\u2026 is such a sum with a common ratio <math>r=\\textstyle\\frac{1}{10}</math>, the theorem makes short work of the question:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>0.999\\ldots = 9(\\tfrac{1}{10}) + 9({\\tfrac{1}{10}})^2 + 9({\\tfrac{1}{10}})^3 + \\cdots = \\frac{9({\\tfrac{1}{10}})}{1-{\\tfrac{1}{10}}} = 1.\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">This proof (actually, that 10 equals 9.999\u2026) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, \u2026) converging to 1.]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999\u2026.<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A [[sequence]] (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, \u2026) has a [[limit of a sequence|limit]] ''x'' if the distance |''x''&nbsp;\u2212&nbsp;''x''<sub>''n''</sub>| becomes arbitrarily small as ''n'' increases. The statement that 0.999\u2026&nbsp;=&nbsp;1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York.\u00a0 Section 8.1, example 2(a), example 6(b).</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>0.999\\ldots </ins>= <ins class=\"diffchange diffchange-inline\">\\lim_{n\\to\\infty}0.\\underbrace{ 99\\ldots9 }_{n} </ins>= <ins class=\"diffchange diffchange-inline\">\\lim_{n\\to\\infty}\\sum_{k </ins>= <ins class=\"diffchange diffchange-inline\">1}^n\\frac{9}{10^k}\u00a0 </ins>= <ins class=\"diffchange diffchange-inline\">\\lim_{n\\to\\infty}\\left(1-\\frac{1}{10^n}\\right) = 1-\\lim_{n\\to\\infty}\\frac{1}{10^n} = 1.\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The last step&nbsp;\u2014 that <math>\\lim_{n\\to\\infty} \\frac{1}{10^n} = 0</math>&nbsp;\u2014 is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999\u2026 is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, \".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1\"; the 1895 ''Arithmetic for Schools'' says, \"\u2026when a large number of 9s is taken, the difference between 1 and .99999\u2026 becomes inconceivably small\".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999\u2026 itself is less than 1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Nested intervals and least upper bounds====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{further|[[Nested intervals]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000\u2026 = 0.222\u2026]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit \"2\" and subdivides that interval into [2, 2.1], [2.1, 2.2], \u2026, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, ''b''<sub>3</sub>, \u2026, and one writes</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:''x'' = ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>\u2026</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In this formalism, the identities 1 = 0.999\u2026 and 1 = 1.000\u2026 reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the \"=\" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>\u2026 is defined to be the unique number contained within all the intervals [''b''<sub>0</sub>, ''b''<sub>0</sub> + 1], [''b''<sub>0</sub>.''b''<sub>1</sub>, ''b''<sub>0</sub>.''b''<sub>1</sub> + 0.1], and so on. 0.999\u2026 is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99\u20269, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999\u2026 = 1.<ref>Bartle and Sherbert pp.60\u201362; Pedrick p.29; Sohrab p.46</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>\u2026 to be the least upper bound of the set of approximants {''b''<sub>0</sub>, ''b''<sub>0</sub>.''b''<sub>1</sub>, ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>, \u2026}.<ref>Apostol pp.9, 11\u201312; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999\u2026 = 1 again. Tom Apostol concludes,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Based on the construction of the real numbers===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{further|[[Construction of the real numbers]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s&nbsp;\u2014 0, 1, 2, 3, and so on&nbsp;\u2014 begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999\u2026 = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by\u00a0 Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Dedekind cuts====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{further|[[Dedekind cut]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: \"The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way\u2026\"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17\u201320, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1</ins>*<ins class=\"diffchange diffchange-inline\">, 1<sup>\u2212</sup>, and 1<sub>''R''</sub>, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a \"nonprincipal Dedekind cut\".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999\u2026 is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\\begin{align}1-(\\tfrac{1}{10})^n\\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999\u2026 is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><math>\\begin{align}\\tfrac{a}{b}<1\\end{align}</math>, which implies <math>\\begin{align}\\tfrac{a}{b}<1-(\\tfrac{1}{10})^b\\end{align}</math>. Since 0.999\u2026 and 1 contain the same rational numbers, they are the same set: 0.999\u2026 = 1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The definition of real numbers as Dedekind cuts was first published by [</ins>[<ins class=\"diffchange diffchange-inline\">Richard Dedekind]] in 1872.<ref name=\"MacTutor2\">{{cite web |url=http</ins>://www<ins class=\"diffchange diffchange-inline\">-gap.dcs.st-and</ins>.<ins class=\"diffchange diffchange-inline\">ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The above approach to assigning a real number to each decimal expansion is due to an expository paper titled \"Is 0.999 \u2026 = 1?\" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa</ins>.org/<ins class=\"diffchange diffchange-inline\">pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: \"So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning.\"<ref>Richman pp.398\u2013399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see \"[[#Alternative number systems|Alternative number systems]]\" below.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Cauchy sequences====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{further|[[Cauchy sequence]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x''&nbsp;\u2212&nbsp;''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and \u2212''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, \u2026), a mapping from natural numbers to rationals, for any positive rational \u03b4 there is an ''N'' such that |''x''<sub>''m''</sub>&nbsp;\u2212&nbsp;''x''<sub>''n''</sub>|&nbsp;\u2264&nbsp;\u03b4 for all ''m'', ''n''&nbsp;>&nbsp;''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton \u00a724.2 \"Sequences\" p.386</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">If (''x''<sub>''n''</sub>) and (''y''<sub>''n''</sub>) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''<sub>''n''<</ins>/<ins class=\"diffchange diffchange-inline\">sub>&nbsp;\u2212&nbsp;''y''<sub>''n''</sub>) has the limit 0. Truncations of the decimal number ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>\u2026 generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\"><math>\\left(1 - 0, 1 - {9 \\over 10}, 1 - {99 \\over 100}, \\dots\\right)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">= \\left(1, {1 \\over 10}, {1 \\over 100}, \\dots \\right)<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,\u2026, it must therefore be shown that</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\"><math>\\lim_{n\\rightarrow\\infty}\\frac{1}{10^n} = 0.</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for \u03b5 = ''a''/''b'' > 0 one can take ''N''&nbsp;=&nbsp;''b'' in the definition of the [[limit of a sequence]]. So again 0.999\u2026&nbsp;=&nbsp;1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name=\"MacTutor2\" /> The above approach to decimal expansions, including the proof that 0.999\u2026 = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Generalizations==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The result that 0.999\u2026 = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999\u2026 equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkov\u0161ek p.408</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111\u2026 equals 1, and in base 3 (the [[ternary numeral system]]) 0.222\u2026 equals 1. Textbooks of real analysis are likely to skip the example of 0.999\u2026 and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000\u2026 and 0.101010\u2026, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000\u2026. This result was first obtained by [[Paul Erd\u0151s]], Miklos Horv\u00e1th, and Istv\u00e1n Jo\u00f3 around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650\u2026. In this base, 1 = 0.11010011001011010010110011010011\u2026; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkov\u0161ek p.409</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*In the [[balanced ternary]] system, <sup>1</sup>/<sub>2</sub> = 0.111\u2026 = 1.<u>111</u>\u2026.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*In the [[factoradic]] system, 1 = 1.000\u2026 = 0.1234\u2026.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Marko Petkov\u0161ek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof \"an instructive exercise in elementary [[point-set topology]]\"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkov\u0161ek pp.410\u2013411</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Applications==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">One application of 0.999\u2026 as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<sup>1</sup>/<sub>7</sub> = 0.142857142857\u2026 and 142 + 857 = 999.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*<sup>1</sup>/<sub>73</sub> = 0.0136986301369863\u2026 and 0136 + 9863 = 9999.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999\u2026, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>\u2026 is a positive integer, then it must be 0.999\u2026, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1\u20133; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96\u201398</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image:Cantor base 3.svg|right|thumb|Positions of <sup>1</sup>/<sub>4</sub>, <sup>2</sup>/<sub>3</sub>, and 1 in the [[Cantor set]]]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Returning to real analysis, the base-3 analogue 0.222\u2026 = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point \u00b2\u2044<sub>3</sub> is given the usual representation of 0.2 or 0.2000\u2026, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point <sup>1</sup>\u2044<sub>3</sub> is represented not as 0.1 but as 0.0222\u2026, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150\u2013152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999\u2026 . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Skepticism in education==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Students of mathematics often reject the equality of 0.999\u2026 and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Students are often \"mentally committed to the notion that a number can be represented in one and only one way by a decimal.\" Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): \"Perhaps the most reassuring of all numbers is 1.\u2026So it is particularly unsettling when someone tries to pass off 0.9~ as 1.\"</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Some students interpret \"0.999\u2026\" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 \"at infinity\".<ref>Tall and Schwarzenberger pp.6\u20137; Tall 2000 p.221</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read \"0.999\u2026\" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*Some students regard 0.999\u2026 as having a fixed value which is less than 1 by an [[infinitesimal]</ins>] <ins class=\"diffchange diffchange-inline\">but non-zero amount.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\\bar{9} \\approx 1</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999\u2026.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that \"students continued to conceive of 0.999\u2026 as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven\u2019t specified how many places there are' or 'it is the nearest possible decimal below 1'\".<ref>Tall 2000 p.221</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Of the elementary proofs, multiplying 0.333\u2026 = <sup>1</sup>\u2044<sub>3</sub> by 3 is apparently a successful strategy for convincing\u00a0 reluctant students that 0.999\u2026 = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10\u201314</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999\u2026. For example, one real analysis student was able to prove that 0.333\u2026 = <sup>1</sup>\u2044<sub>3</sub> using a [[supremum]] definition, but then insisted that 0.999\u2026 < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416\u2013417</ref> Others still are able to prove that <sup>1</sup>\u2044<sub>3</sub> = 0.333\u2026, but, upon being confronted by the [[#Fractions|fractional proof]], insist that \"logic\" supersedes the mathematical calculations.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who \"challenged almost everything I said in class but never questioned his calculator,\" and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99\u2026 = 10, calling it a \"wildly imagined infinite growing process.\"<ref>Mazur pp.137\u2013141</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">As part of Ed Dubinsky's \"[[APOS theory]]\" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999\u2026 as a finite, indeterminate string with an infinitely small distance from 1 have \"not yet constructed a complete process conception of the infinite decimal\". Other students who have a complete process conception of 0.999\u2026 may not yet be able to \"encapsulate\" that process into an \"object conception\", like the object conception they have of 1, and so they view the process 0.999\u2026 and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing <sup>1</sup>\u2044<sub>3</sub> as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261\u2013262</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==In popular culture==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">With the rise of the </ins>[<ins class=\"diffchange diffchange-inline\">[Internet]], debates about 0.999\u2026 have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999\u2026 is a \"popular sport\", and it is one of the questions answered in its [[FAQ]].<ref>As observed by Richman (p.396). {{cite web |url=http</ins>://www.<ins class=\"diffchange diffchange-inline\">faqs</ins>.org/<ins class=\"diffchange diffchange-inline\">faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999\u2026 = 1? |accessdate=2006-06-29}}</ref> The FAQ briefly covers <sup>1</sup>\u2044<sub>3</sub>, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999\u2026 via <sup>1</sup>\u2044<sub>3</sub> and limits, saying of misconceptions,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't\u00a0 .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><</ins>/<ins class=\"diffchange diffchange-inline\">blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified \"other message board \u2026 mostly about video games\". In the same vein, the question of 0.999\u2026 proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a \"press release\" on [[April Fools' Day]] 2004 that it is 1:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Two proofs are then offered, based on limits and multiplication by 10.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">0.999\u2026 features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref> </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Q: How many mathematicians does it take to screw in a lightbulb?</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A: 0.999999\u2026.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Alternative number systems==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation \"0.999\u2026\" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999\u2026 = 1 is a convention as well:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"></blockquote></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999\u2026 and 1 might not be identical.\u00a0 However, many number systems are extensions of&nbsp;\u2014 rather than independent alternatives to&nbsp;\u2014 the real number system, so 0.999\u2026 = 1 continues to hold.\u00a0 Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999\u2026 behaves (if, indeed, a number expressed as \"0.999\u2026\" is both meaningful and unambiguous), but also for the behavior of related phenomena.\u00a0 If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Infinitesimals===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{main|Infinitesimal}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Some proofs that 0.999\u2026 = 1 rely on the [[Archimedean property]] of the standard real numbers</ins>: <ins class=\"diffchange diffchange-inline\">there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999\u2026 depends on which structure we use.\u00a0 For example, the [[dual number]]s include a new infinitesimal element \u03b5, analogous to the imaginary unit ''i'' in the [[complex number]]s except that \u03b5\u00b2&nbsp;=&nbsp;0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of \u03b5 become non-Archimedean elements.<ref>Berz 439\u2013442<</ins>/<ins class=\"diffchange diffchange-inline\">ref>\u00a0 Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999\u2026=1.\u00a0 On a related note, while \u03b5 exists in dual numbers, so does \u03b5/2, so \u03b5 is not \"the smallest positive dual number,\" and, indeed, as in the reals, no such number exists.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)<sup>\u2217</sup>.<ref>Lightstone pp.245\u2013247</ref>\u00a0 Lightstone shows how to associate to each extended real number a sequence of digits</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">0.d<sub>1</sub>d<sub>2</sub>d<sub>3</sub>\u2026;\u2026d<sub>\u221e\u22121</sub>d<sub>\u221e</sub>d<sub>\u221e+1</sub>\u2026</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">indexed by the extended natural numbers. While he does not directly discuss 0.999\u2026, he shows the real number 1/3 is represented by 0.333\u2026;\u2026333\u2026 which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The hyperreal number ''u''<sub>''H''</sub>=0.999\u2026;\u2026999000\u2026 with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''<sub>''H''</sub> < 1.{{Fact|date=January 2009}}\u00a0 Indeed, the sequence u<sub>1</sub>=0.9,\u00a0 u<sub>2</sub>=0.99,\u00a0 u<sub>3</sub>=0.999,\u00a0 etc.\u00a0 satisfies u<sub>n</sub> = 1 - 1/n, hence by the transfer principle u<sub>H</sub> = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL\u2026 is 0.010101<sub>2</sub>\u2026&nbsp;=&nbsp;<sup>1</sup>/<sub>3</sub>. However, the value of LRLLL\u2026 (corresponding to 0.111\u2026<sub>2</sub>) is infinitesimally less than 1. The difference between the two is the [[surreal number]] <sup>1</sup>/<sub>\u03c9</sub>, where \u03c9 is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR\u2026 or 0.000\u2026<sub>2</sub>.<ref>Berlekamp, Conway, and Guy (pp.79\u201380, 307\u2013311) discuss 1 and <sup>1</sup>/<sub>3</sub> and touch on <sup>1</sup>/<sub>\u03c9</sub>. The game for 0.111\u2026<sub>2</sub> follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999\u2026 \u225f 1 </ins>FAQ <ins class=\"diffchange diffchange-inline\">|author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Breaking subtraction===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Another manner in which the proofs might be undermined is if 1&nbsp;\u2212&nbsp;0.999\u2026 simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999\u2026 < 1.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999\u2026&nbsp;<&nbsp;1 simply because 0&nbsp;<&nbsp;1 in the ones place, but for any nonterminating ''x'', one has 0.999\u2026&nbsp;+&nbsp;''x''&nbsp;=&nbsp;1&nbsp;+&nbsp;''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to <sup>1</sup>\u2044<sub>3</sub>. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397\u2013399</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In the process of defining multiplication, Richman also defines another system he calls \"cut ''D''\", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (\u2212\u221e,&nbsp;''d''&nbsp;) and the \"principal cut\" (\u2212\u221e,&nbsp;''d''&nbsp;]. The result is that the real numbers are \"living uneasily together with\" the decimal fractions. Again 0.999\u2026&nbsp;<&nbsp;1. There are no positive infinitesimals in cut ''D'', but there is \"a sort of negative infinitesimal,\" 0<sup>\u2212</sup>, which has no decimal expansion. He concludes that 0.999\u2026&nbsp;=&nbsp;1&nbsp;+&nbsp;0<sup>\u2212</sup>, while the equation \"0.999\u2026 + ''x'' = 1\"</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">has no solution.<ref>Richman pp.398\u2013400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===''p''-adic numbers===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{main|p-adic number}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">When asked about 0.999\u2026, novices often believe there should be a \"final 9,\" believing 1&nbsp;\u2212&nbsp;0.999\u2026 to be a positive number which they write as \"0.000\u20261\". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999\u2026 would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no \"last 9\" in 0.999\u2026.<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, \u2026) converging to \u22121. The 10-adic analogue is \u2026999 = \u22121.]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''p<sup>n</sup>'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion \u2026999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1&nbsp;+&nbsp;\u2026999&nbsp;=&nbsp;\u2026000&nbsp;=&nbsp;0, and so \u2026999&nbsp;=&nbsp;\u22121.<ref name=\"Fjelstad11\">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by \"\u2026999\" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:<math>\\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \\cdots = \\frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14\u201315</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999\u2026&nbsp;=&nbsp;1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x''&nbsp;=&nbsp;\u2026999 then 10''x''&nbsp;=&nbsp; \u2026990, so 10''x''&nbsp;=&nbsp;''x''&nbsp;\u2212&nbsp;9, hence ''x''&nbsp;=&nbsp;\u22121 again.<ref name=\"Fjelstad11\" /></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">As a final extension, since {{nowrap begin}}0.999\u2026 = 1{{nowrap end}} (in the reals) and {{nowrap begin}}\u2026999 = \u22121{{nowrap end}} (in the 10-adics), then by \"blind faith and unabashed juggling of symbols\"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}\u2026999.999\u2026 = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of \"double-decimals\" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902\u2013903</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Related questions==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><!--[[Intuitionism]] should be worked in somewhere and explained, not necessarily here.--></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999\u2026 and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999\u2026, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Division by zero]] occurs in some popular discussions of 0.999\u2026, and it also stirs up contention. While most authors choose to define 0.999\u2026, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a \"[[point at infinity]]\".\u00a0 Here, it makes sense to define <sup>1</sup>/<sub>0</sub> to be infinity;<ref>See, for example, J.B. Conway's treatment of M\u00f6bius transformations, pp.47\u201357</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics.\u00a0 Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where \"0\" denotes the additive identity and is neither positive nor negative, the usual interpretation of \"\u22120\" is that it should denote the additive inverse of 0, which forces \u22120&nbsp;=&nbsp;0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==See also==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Col-begin}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Col-1-of-3}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Decimal representation]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Infinity]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Limit (mathematics)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Col-2-of-3}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Informal mathematics|Naive mathematics]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Non-standard analysis]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Col-3-of-3}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Real analysis]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [[Series (mathematics)]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{col-end}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Notes==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{reflist|2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==References==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{refbegin|2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be \"honest, rigorous, up to date, and, at the same time, not too pedantic.\" (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.9\u201311)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This text aims to be \"an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis.\" Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |author=[</ins>[<ins class=\"diffchange diffchange-inline\">Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439\u2013450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, \"the rather tenuous relationship between mathematical reality and physical reality\". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999\u2026 is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This text assumes \"a stiff course in basic calculus\" as a prerequisite; its stated principles are to present complex analysis as \"An Introduction to Mathematics\" and to state the material clearly and precisely. (p.vii)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900\u2013903 |doi=10.2307/2309468}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253\u2013266 |doi=10.1007/s10649-005-0473-0}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411\u2013425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:An introductory undergraduate textbook in set theory that \"presupposes no specific background\". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11\u201315 |doi=10.2307/2687285}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*</ins>:<ins class=\"diffchange diffchange-inline\">This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], and the book is aimed at students \"who have reached roughly the level of completing one year of specialist mathematical study at a university\". The real numbers are constructed in Chapter 24, \"perhaps the most difficult chapter in the entire book\", although the authors ascribe much of the difficulty to their use of [[ideal theory]], which is not reproduced here. (pp.vii, xiv)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610\u2013617 |doi=10.2307</ins>/<ins class=\"diffchange diffchange-inline\">2300532 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636\u2013639 |doi=10.2307</ins>/<ins class=\"diffchange diffchange-inline\">2589246 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Leavitt |first=W</ins>.<ins class=\"diffchange diffchange-inline\">G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669\u2013673 |doi=10.2307/2314251 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299\u2013308 |doi=10.2307/2686394 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite web | url=http://arxiv</ins>.org/<ins class=\"diffchange diffchange-inline\">abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242\u2013251 |doi=10.2307/2316619 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:Mankiewicz seeks to represent \"the history of mathematics in an accessible style\" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkh\u00e4user |isbn=3-7643-3325-1}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:A topical rather than chronological review of infinity, this book is \"intended for the general reader\" but \"told from the point of view of a mathematician\". On the dilemma of rigor versus readable language, Maor comments, \"I hope I have succeeded in properly addressing this problem.\" (pp.x-xiii)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:Intended as an introduction \"at the senior or first-year graduate level\" with no formal prerequisites: \"I do not even assume the reader knows much set theory.\" (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, \"This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest.\" (p.30)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite conference |last=N\u00fa\u00f1ez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160\u2013181 |url=http:/</ins>/<ins class=\"diffchange diffchange-inline\">www.cogsci.ucsd.edu/~nunez</ins>/<ins class=\"diffchange diffchange-inline\">web/publications.html | id=ISBN 978-0-387-25717</ins>-<ins class=\"diffchange diffchange-inline\">4}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Petkov\u0161ek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408\u2013411 |doi=10</ins>.<ins class=\"diffchange diffchange-inline\">2307/2324393 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite conference |author=Pinto, M\u00e1rcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57\u201364 |year=2001 |url=http://www</ins>.<ins class=\"diffchange diffchange-inline\">warwick</ins>.<ins class=\"diffchange diffchange-inline\">ac.uk/staff/David.Tall/pdfs</ins>/<ins class=\"diffchange diffchange-inline\">dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This book aims to \"present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus.\" (p.vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56\u201364) Decimal expansions appear in Appendix 3, \"Expansions of real numbers in any base\". (pp.503\u2013507)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]</ins>]<ins class=\"diffchange diffchange-inline\">s as soon as possible, saying of the axiomatic treatment, \"This is something of a fraud, considering that the entire structure of analysis is built on the real number system.\" (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*<ins class=\"diffchange diffchange-inline\">{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[</ins>[<ins class=\"diffchange diffchange-inline\">Notices of the AMS]] |volume=52 |number=1 |pages=24\u201334 |url=http</ins>://www.<ins class=\"diffchange diffchange-inline\">ams</ins>.org/<ins class=\"diffchange diffchange-inline\">notices/200501</ins>/<ins class=\"diffchange diffchange-inline\">fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999\u2026 = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396\u2013400 }} Free HTML preprint: {{cite web |url=http</ins>:/<ins class=\"diffchange diffchange-inline\">/www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999\u2026 = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:A textbook for an advanced undergraduate course. \"Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe.\" (p.ix)</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90\u201398 }}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkh\u00e4user |isbn=0-8176-4211-0}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*:This book aims to \"assist students in discovering calculus\" and \"to foster conceptual understanding\". (p.v) It omits proofs of the foundations of calculus.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44\u201349 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education </ins>for <ins class=\"diffchange diffchange-inline\">Teaching |year=1976/7 |volume=2 |number=4 |pages=2\u201318 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210\u2013230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einf\u00fchrung in die h\u00f6here Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|</ins>language<ins class=\"diffchange diffchange-inline\">=German|chapter=Reihenzahlen}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{refend}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==External links==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Spoken Wikipedia|0.999....ogg|2006-10-19}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{commons|0.999...}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999\u2026 = 1?] from [[cut-the-knot]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* [<ins class=\"diffchange diffchange-inline\">http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999\u2026 = 1 ?]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http</ins>://www.<ins class=\"diffchange diffchange-inline\">newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://descmath.com/diag/nines.html Repeating Nines]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://qntm</ins>.org/<ins class=\"diffchange diffchange-inline\">pointnine Point nine recurring equals one]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http:/</ins>/<ins class=\"diffchange diffchange-inline\">www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http</ins>:/<ins class=\"diffchange diffchange-inline\">/www.dpmms.cam.ac.uk/~wtg10/decimals.html\u00a0 What is so wrong with thinking of real numbers as infinite decimals?]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http</ins>:<ins class=\"diffchange diffchange-inline\">//us.metamath.org/mpegif/0.999....html Theorem 0.999...] </ins>on <ins class=\"diffchange diffchange-inline\">[[Metamath]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{featured article}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\"><!-- [[en:0.999...]] --></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:One]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:Mathematics paradoxes]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:Real analysis]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:Real numbers]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:Numeration]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Category:Articles containing proofs]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Link FA|ja}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{Link FA|zh}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ar:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[be:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[be-x-old:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[bg:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[da:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[de:Eins#Periodischer Dezimalbruch]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[el:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[es:0,9 peri\u00f3dico]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[eo:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[fa:\u06f0\u066b\u06f9\u06f9\u06f9\u2026]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[fr:D\u00e9veloppement d\u00e9cimal de l'unit\u00e9]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ko:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[id:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[it:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[he:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ka:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[lv:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[hu:0,999\u2026]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ml:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ms:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ja:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[no:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[nov:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[uz:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[pl:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[pt:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ro:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ru:0,(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[sl:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[fi:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[sv:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[ta:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[th:0.999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[tr:0.(9)]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[vi:0,999...]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[zh:0.999\u2026]</ins>]</div></td></tr>\n"
}
}