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| | My name is Lamar (28 years old) and my hobbies are Fencing and Computer programming.<br><br>Feel free to surf to my blog; [http://sbsproduction.dk/fifa-coin-generator-0 Fifa 15 coin generator] |
| {{Use dmy dates|date=December 2012}}
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| [[File:999 Perspective.png|300px|thumbnail|The repeating decimal continues with an infinite number of nines.]]
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| In [[mathematics]], the [[repeating decimal]] '''0.999...''' (sometimes written with more or fewer 9s before the final [[ellipsis]], or as '''0.<span style="text-decoration: overline;">9</span>''', '''0.(9)''', or {{nowrap|<math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math>}}) denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols "0.999..." and "1" represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigor]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.
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| Every nonzero, terminating decimal has an equal twin representation with infinitely many trailing 9s (for example, 8.32 and 8.31999...). The terminating decimal representation is almost always preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other [[radix|bases]] or in any similar representation of the real numbers.
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| The equality of 0.999... and 1 is closely related to the absence of nonzero [[infinitesimal]]s in the real number system, the most commonly used system in [[mathematical analysis]]. Some [[#In alternative number systems|alternative number systems]], such as the [[hyperreal number|hyperreals]], do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.
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| The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently [[counterintuitive]] that they question or reject it, commonly enough that the difficulty of convincing them of the validity of this identity has been the subject of numerous studies in mathematics education.
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| ==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
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| Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.
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| ===Fractions and long division{{anchor|Fractions}}===
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| 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|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for {{nowrap|1=0.999... = 1}}. Multiplication of 9 times 1 produces 9 in each digit, so {{nowrap|9 × 0.111...}} equals 0.999... and {{nowrap|9 × {{frac|1|9}}}} equals 1, so {{nowrap|1=0.999... = 1}}:
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| :<math>
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| \begin{align}
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| \frac{1}{9} & = 0.111\dots \\
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| 9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
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| 1 & = 0.999\dots
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| \end{align}
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| </math>
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| Another form of this proof multiplies {{nowrap|1= {{frac|1|3}} = 0.333...}} by 3.
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| ===Digit manipulation===
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| When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:
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| :<math>
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| \begin{align}
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| x &= 0.999\ldots \\
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| 10 x &= 9.999\ldots \\
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| 10 x - x &= 9.999\ldots - 0.999\ldots \\
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| 9 x &= 9 \\
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| x &= 1
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| \end{align}
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| </math>
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| ===Discussion===
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| Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999... = 1 but 0.333... < 0.34. In introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186. William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation (Byers pp. 39–41). Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".(p. 396)</ref>
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| Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.
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| ==Analytic proofs{{anchor|Analytic}}==
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| Since the question of 0.999... 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..., the integer part can be summarized as ''b''<sub>0</sub> and one can neglect negatives, so a decimal expansion has the form
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| :<math>b_0.b_1b_2b_3b_4b_5 \dots.</math>
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| It should be noted 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 digit 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.
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| ===Infinite series and sequences===
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| {{further2|[[Decimal representation]]}}
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| Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
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| :<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>
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| For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
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| :If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>
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| Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
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| :<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
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| This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p. 170</ref>
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| [[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
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| 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 [[#Digit manipulation|algebraic 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...<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>
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| A [[sequence]] (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''<sub>''n''</sub>| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 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. Section 8.1, example 2(a), example 6(b).</ref>
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| :<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>
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| The last step, that {{frac|1|10<sup>''n''</sup>}} → 0 as ''n'' → ∞, is often justified by the [[Archimedean property]] of the real numbers. This limit-based attitude towards 0.999... 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, "...when a large number of 9s is taken, the difference between 1 and .99999... 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... itself is less than 1.
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| ===Nested intervals and least upper bounds===
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| {{further2|[[Nested intervals]]}}
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| [[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
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| 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.
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| 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], ..., [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>, ..., and one writes
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| :<math>x = b_0.b_1b_2b_3 \dots.</math>
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| In this formalism, the identities 1 = 0.999... and 1 = 1.000... 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>
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| 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>... 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... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>
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| 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>... 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>, ...}.<ref>Apostol pp. 9, 11–12; 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... = 1 again. Tom Apostol concludes,
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| <blockquote>
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| 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>
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| </blockquote>
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| ==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
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| {{further2|[[Construction of the real numbers]]}}
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| 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 – 0, 1, 2, 3, and so on – 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.
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| 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... = 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 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>
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| ===Dedekind cuts===
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| {{further2|[[Dedekind cut]]}}
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| In the [[Dedekind cut]] approach, each real number ''x'' is defined as the '''[[infinite set]] of all rational numbers 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..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1<sup>−</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... 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
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| :<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}.</math><ref>Richman p. 399</ref>
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| Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
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| :<math>\begin{align}\tfrac{a}{b}<1,\end{align}</math>
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| which implies
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| :<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
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| Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.
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| The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |first1 = J. J. | last1 = O'Connor | first2 = E. F. | last2 = Robertson |work=MacTutor History of Mathematics |accessdate=2006-08-30 |archiveurl=http://web.archive.org/web/20070929095431/http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |archivedate=2007-09-29 |deadurl=no |date=October 2005}}</ref>
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| The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 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>Richman</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. He also notes that typically the definitions allow
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| { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] 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. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.
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| ===Cauchy sequences===
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| {{further2|[[Cauchy sequence]]}}
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| Another approach is to define a real number as the '''limit of a Cauchy sequence of rational numbers'''. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''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>, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''<sub>''m''</sub> − ''x''<sub>''n''</sub>| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>
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| 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''</sub> − ''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>... 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
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| :<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
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| = \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>
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| has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
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| :<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>
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| This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.
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| 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... = 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>
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| === Infinite decimal representation ===
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| {{further2|[[Construction of the real numbers#Stevin.27s construction|Stevin's construction]]}}
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| Commonly in [[secondary schools]]' mathematics education, the real numbers are constructed by defining a number using an integer followed by a [[radix point]] and an infinite sequence written out as a string to represent the [[fractional part]] of any given real number. In this construction, the set of any combination of integers and digits after the decimal point (or radix point in non-base 10 systems) are the set of real numbers. This construction can too be rigorously shown to satisfy all of the [[Real number#Axiomatic approach|real axioms]] after defining an [[equivalence relation]] over the set that '''defines''' 1 =<sub>eq</sub> 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version.<ref>{{cite arXiv |title=A new approach to the real numbers |author=Liangpan Li |deadurl=no |eprint=1101.1800 |class=math.CA |date=March 2011}}</ref> With this construction of the reals, all proofs of the statement 1 = .999... can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.
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| ==Generalizations==
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| The result that 0.999... = 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... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>
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| Second, a comparable theorem applies in each radix or [[base (exponentiation)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>
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| Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., 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.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>
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| 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; Petkovšek p. 409</ref>
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| *In the [[balanced ternary]] system, <sup>1</sup>/<sub>2</sub> = 0.111... = 1.<u>111</u>....
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| *In the reverse [[factorial number system]] (using bases 2!,3!,4!,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....
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| ===Impossibility of unique representation===
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| That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:
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| * If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
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| * The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''<sub>1</sub>, ''p''<sub>2</sub> of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''<sub>1</sub>, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''<sub>2</sub>. Then ''L'' has a largest element, starting with ''p''<sub>1</sub> and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''<sub>2</sub> by the smallest symbol in all positions.
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| The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''<sub>1</sub> = "0", ''p''<sub>2</sub> = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.
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| Marko Petkovšek has proven that for any positional 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>Petkovšek pp. 410–411</ref>
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| ==Applications==
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| One application of 0.999... 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:
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| *<sup>1</sup>/<sub>7</sub> = 0.142857142857... and 142 + 857 = 999.
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| *<sup>1</sup>/<sub>73</sub> = 0.0136986301369863... and 0136 + 9863 = 9999.
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| 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..., but at least one modern proof by W. G. Leavitt does. If it can be proved that a decimal of the form 0.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>... is a positive integer, then it must be 0.999..., 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. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>
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| [[File: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]]]]
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| Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
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| *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.
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| 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 <sup>2</sup>⁄<sub>3</sub> is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point <sup>1</sup>⁄<sub>3</sub> is represented not as 0.1 but as 0.0222..., 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. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>
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| 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... 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>
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| ==Skepticism in education==
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| Students of mathematics often reject the equality of 0.999... 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:
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| *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 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
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| *Some students interpret "0.999..." (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. 6–7; Tall 2000 p. 221</ref>
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| *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..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>
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| 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...
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| Many of these explanations were found by [[David 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... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>
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| Of the elementary proofs, multiplying 0.333... = <sup>1</sup>⁄<sub>3</sub> by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 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. 10–14</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.... For example, one real analysis student was able to prove that 0.333... = <sup>1</sup>⁄<sub>3</sub> using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that <sup>1</sup>⁄<sub>3</sub> = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.
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| [[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... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>
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| As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, he and his collaborators (2005) propose that students who conceive of 0.999... 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... 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... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing <sup>1</sup>⁄<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.'' pp. 261–262</ref>
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| ==In popular culture==
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| With the rise of the [[Internet]], debates about 0.999... 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>sci.math</tt>, arguing over 0.999... is described as 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://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ | first = Hans | last = de Vreught |year=1994 |title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29 |archiveurl=http://web.archive.org/web/20070929122649/http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |archivedate=2007-09-29 |deadurl=no}}</ref> The FAQ briefly covers <sup>1</sup>⁄<sub>3</sub>, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.
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| A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via <sup>1</sup>⁄<sub>3</sub> and limits, saying of misconceptions,
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| <blockquote>
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| 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.
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| Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |authorlink=Cecil Adams | first = Cecil | last = Adams |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06 |archiveurl= http://web.archive.org/web/20060815010844/http://www.straightdope.com/columns/030711.html |archivedate= 15 August 2006 |deadurl=no}}</ref>
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| </blockquote>
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| ''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... 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:
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| <blockquote>
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| 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 press release |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16| archiveurl= http://web.archive.org/web/20091104222830/http://us.blizzard.com/en-us/company/press/pressreleases.html?040401| archivedate= 4 November 2009 }}</ref>
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| </blockquote>
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| Two proofs are then offered, based on limits and multiplication by 10.
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| 0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
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| <blockquote>
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| Q: How many mathematicians does it take to [[Lightbulb joke|screw in a lightbulb]]?
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| </blockquote>
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| <blockquote>
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| A: 0.999999....
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| </blockquote>
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| ==In alternative number systems{{anchor|Alternative number systems}}==
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| Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." 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... = 1 is a convention as well:
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| <blockquote>
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| 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>
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| </blockquote>
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| 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... and 1 might not be identical. However, many number systems are extensions of —rather than independent alternatives to— the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. 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.
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| ===Infinitesimals===
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| {{main|Infinitesimal}}
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| Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.
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| However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to the real numbers, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε<sup>2</sup> = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note however that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.
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| [[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 [[hyperreal number]]s in (0, 1)<sup>∗</sup>.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,
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| :<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>
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| indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.
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| The standard definition of the number 0.999... is the [[limit of a sequence|limit of the sequence]] 0.9, 0.99, 0.999, ... A different definition considers the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the [[ultrapower construction]], which corresponds to a number that is infinitesimally smaller than 1. More generally, the hyperreal number {{nowrap|1 = ''u''<sub>''H''</sub>=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''<sub>''H''</sub> < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative interpretation of "0.999...":
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| :<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}.</math><ref>Katz & Katz 2010</ref>
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| All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>
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| Jose Benardete in his book ''Infinity: An essay in metaphysics'' argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:
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| :The intelligibility of the continuum has been found—many times over—to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a ''number'' ... though not as a ''real'' number.<ref>{{cite book|first=José Amado |last=Benardete |title=Infinity: An essay in metaphysics |publisher=Clarendon Press |year=1964 |page=279 |url=http://books.google.com/?id=wMgtAAAAMAAJ| accessdate= 27 November 2011 }}</ref>
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| ===Hackenbush===
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| [[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... is 0.010101<sub>2</sub>... = <sup>1</sup>/<sub>3</sub>. However, the value of LRLLL... (corresponding to 0.111...<sub>2</sub>) is infinitesimally less than 1. The difference between the two is the [[surreal number]] <sup>1</sup>/<sub>ω</sub>, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...<sub>2</sub>.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and <sup>1</sup>/<sub>3</sub> and touch on <sup>1</sup>/<sub>ω</sub>. The game for 0.111...<sub>2</sub> follows directly from Berlekamp's Rule.</ref>
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| This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...<sub>2</sub> = 0.11000...<sub>2</sub>, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....
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| ===Revisiting subtraction===
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| Another manner in which the proofs might be undermined is if 1 − 0.999... 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... < 1.
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| 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... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''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>⁄<sub>3</sub>. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>
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| In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of [[Dedekind cut]]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞, ''d'' ) and the "principal cut" (−∞, ''d'' ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999... < 1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0<sup>−</sup>, which has no decimal expansion. He concludes that 0.999... = 1 + 0<sup>−</sup>, while the equation "0.999... + ''x'' = 1"
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| has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>
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| ===''p''-adic numbers===
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| {{main|p-adic number}}
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| When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... 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....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.
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| [[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]
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| 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.
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| In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 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 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
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| :<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>
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| (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... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' = ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />
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| As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{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}}...999.999... = 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. 902–903</ref>
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| ==Related questions==
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| <!--[[Intuitionism]] should be worked in somewhere and explained, not necessarily here.-->
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| * [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., 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>
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| * [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., 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]]". 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 Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
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| * [[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 "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the [[sign and magnitude]] or [[ones' complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book | last1 = Kroemer | first1 = Herbert | last2 = Kittel | first2 = 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| archiveurl= http://web.archive.org/web/20060824085452/http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp| archivedate= 24 August 2006}}</ref>
| |
| | |
| ==See also==
| |
| * [[Limit (mathematics)]]
| |
| * [[Series (mathematics)]]
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| * [[Informal mathematics|Naive mathematics]]
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| * [[Finitism]]
| |
| | |
| ==Notes==
| |
| {{reflist|colwidth=30em}}
| |
| | |
| ==References==
| |
| {{refbegin|colwidth=30em}}
| |
| * {{cite book |author=Alligood, K. T.; Sauer, T. D.; Yorke, J. A. |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
| |
| *: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
| |
| * {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
| |
| *: 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. 9–11)
| |
| * {{cite book |author=Bartle, R. G.; Sherbert, D. R. |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
| |
| *: 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)
| |
| * {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
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| * {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|Conway, J. H.]]; [[Richard K. Guy|Guy, R. K.]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
| |
| * {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |conference=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |id = {{citeseerx|10.1.1.31.3019}} |year=1992}}
| |
| *{{Cite journal|title=Why Does 0.999... = 1?: A Perennial Question and Number Sense|last1=Beswick|first1=Kim|journal=Australian Mathematics Teacher|volume=60|pages=7–9|year=2004|issue=4|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
| |
| *: 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... 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)
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| * {{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}}
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| * {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
| |
| * {{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}}
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| *: 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)
| |
| * {{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| accessdate= 4 July 2011 <!--Added by DASHBot-->}}
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| * {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |jstor=2309468 |journal=The American Mathematical Monthly |volume=67 |pages=900–903 |doi=10.2307/2309468 |issue=9 |date=November 1960}}
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| * {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |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=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2}}
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| * {{cite journal |author=Edwards, Barbara; Ward, Michael |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |jstor=4145268| accessdate= 4 July 2011 <!--Added by DASHBot-->| archiveurl= http://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf| archivedate= 22 July 2011 <!--Added by DASHBot--> |issue=5 |date=May 2004}}
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| * {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
| |
| *: 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)
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| * {{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/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7| accessdate= 4 July 2011 <!--Added by DASHBot-->}}
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| * {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |jstor=2687285 |journal=The College Mathematics Journal |volume=26 |pages=11–15 |doi=10.2307/2687285 |issue=1 |date=January 1995}}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
| |
| *: 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)
| |
| * {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/| accessdate= 4 July 2011 <!--Added by DASHBot-->| archiveurl= http://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/| archivedate= 20 July 2011 <!--Added by DASHBot--> |deadurl=no |bibcode=2010arXiv1007.3018U |arxiv=1007.3018 }}
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| * {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |jstor=2300532 |journal=The American Mathematical Monthly |volume=43 |pages=610–617 |doi=10.2307/2300532 |issue=10 |date=December 1936}}
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| * {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |jstor=2589246 |journal=The American Mathematical Monthly |volume=105 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
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| * {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=The American Mathematical Monthly |volume=74 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
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| * {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |jstor=2686394 |journal=The College Mathematics Journal |volume=15 |pages=299–308 |doi=10.2307/2686394 |issue=4 |date=September 1984}}
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| * {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=The American Mathematical Monthly |volume=79 |pages=242–251 |doi=10.2307/2316619 |issue=3 |date=March 1972}}
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| * {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
| |
| *: 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)
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| * {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
| |
| *: 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)
| |
| * {{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}}
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| * {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
| |
| *: 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)
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| * {{cite conference |last=Núñez |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=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | isbn=978-0-387-25717-4| accessdate= 4 July 2011 <!--Added by DASHBot-->| archiveurl= http://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html| archivedate= 18 July 2011 <!--Added by DASHBot-->}}
| |
| * {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
| |
| * {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
| |
| * {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |jstor=2324393 |journal=[[American Mathematical Monthly]] |volume=97 |pages=408–411 |doi=10.2307/2324393 |issue=5 |date=May 1990}}
| |
| * {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf| archivedate= 30 May 2009 <!--Added by DASHBot-->}}
| |
| * {{cite book |author=Protter, M. H.; [[Charles B. Morrey, Jr.|Morrey, C. B.]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
| |
| *: 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. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)
| |
| * {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
| |
| *: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]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.
| |
| * {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf| archivedate= 25 February 2009 <!--Added by DASHBot--> |date=January 2005}}
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| * {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] volume=72 |issue=5 |pages=396–400 |date=December 1999}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred |last=Richman |title=Is 0.999... = 1? |accessdate=2006-08-23| archiveurl= http://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm| archivedate= 2 September 2006 <!--Added by DASHBot--> |deadurl=no|date=June 1999}} Note: the journal article contains material and wording not found in the preprint.
| |
| * {{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}}
| |
| * {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
| |
| * {{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}}
| |
| *: 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)
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| * {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |pages=90–98 |date=March 1978}}
| |
| * {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7| accessdate= 4 July 2011 <!--Added by DASHBot-->}}
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| * {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
| |
| * {{cite journal | last1=Starbird | first1=M. | last2=Starbird | first2=T. | title= Required Redundancy in the Representation of Reals| volume=114 | pages=769–774|journal=Proceedings of the American Mathematical Society|jstor=2159403|publisher=AMS | issue=3 | doi=10.1090/S0002-9939-1992-1086343-5 |date=March 1992}}
| |
| * {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
| |
| * {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
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| * {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
| |
| *: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
| |
| * {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf| archivedate= 30 May 2009 <!--Added by DASHBot-->}}
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| * {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1977 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf| archivedate= 26 March 2009 <!--Added by DASHBot-->}}
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| * {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf| archivedate= 30 May 2009 <!--Added by DASHBot--> |doi=10.1007/BF03217085}}
| |
| * {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
| |
| * {{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}}
| |
| {{refend}}
| |
| | |
| ==Further reading==
| |
| {{refbegin|colwidth=30em}}
| |
| *{{Cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
| |
| *{{Cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |date=March 1997}}
| |
| *{{Cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |first1=J. B.|last1=Calvert|first2=E. R.|last2=Tuttle|first3=Michael S.|last3=Martin|first4=Peter|last4=Warren |journal=The History Teacher |volume=14 |issue=2 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |date=February 1981}}
| |
| *{{Cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |pages=13–15, 36 |jstor=40248503 |date=November 2005}}
| |
| *{{Cite journal |title=Rational Approximations to π |first1=K. Y.|last1=Choong|first2=D. E.|last2=Daykin|first3=C. R.|last3=Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |date=April 1971}}
| |
| *{{Cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student's understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22}}
| |
| *{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf| accessdate= 4 July 2011 <!--Added by DASHBot-->}}
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| *{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
| |
| *: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
| |
| *{{Cite journal |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |chapter=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |title=Research issues in undergraduate mathematics learning}}
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| * {{cite arXiv | eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 | class=math.NT }}
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| *{{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |issue=3}}
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| *{{Cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |date=June 1985}}
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| *{{Cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |date=December 1988}}
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| *{{Cite journal |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999...=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf| accessdate= 4 July 2011 <!--Added by DASHBot-->}}
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| {{refend}}
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| | |
| ==External links==
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| {{Spoken Wikipedia|0.999....ogg|2006-10-19}}
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| {{commons|0.999...}}
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| * [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
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| * [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1 ?]
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| * [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals]
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| * [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
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| * [http://descmath.com/diag/nines.html Repeating Nines]
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| * [http://qntm.org/pointnine Point nine recurring equals one]
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| * [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
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| * [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
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| * [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
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| * [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]
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| * [https://www.youtube.com/watch?v=TINfzxSnnIE 9.999... reasons why 0.999... = 1] by [[Vi Hart]]
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| {{featured article}}
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| [[Category:One]]
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| [[Category:Mathematics paradoxes]]
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| [[Category:Real numbers]]
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| [[Category:Articles containing proofs]]
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| {{Link FA|hu}}
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| {{Link FA|ja}}
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| {{Link FA|zh}}
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| {{Link FA|ca}}
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| [[de:Eins#Periodischer Dezimalbruch]]
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| [[nl:Repeterende breuk#Repeterende negens]]
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