0.999...

Alternative decimal expansion of 1 May 2024 upright=1.35|class=skin-invert-image|thumb|alt=Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely 0.999... is a repeating decimal that represents the number 1.Stillwell 1994Apostol 1974 The three dots represent an infinite list of "9" digits.This is also written as 0.(9), 0.9, or 0.. 9. Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is1; that is, Despite common misconceptions, 0.999... is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent exactly the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only elementary arithmetic and the Archimedean property: for each real number, there is a natural number that is greater (for example, by rounding up). Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits. Why some people reject this equality is a question studied in mathematics education. The equality 0.999... = 1 is a special case of the fact that every non-zero terminating decimal has two equal representations (for example, 8.32000... and 8.31999...). Everything that precedes refers to the decimal number system, that is to base 10. For other bases, everything remains true with 9 replaced with the largest digit. For example, in the binary number system, one has 0.111... = 1. In some other number systems, 0.999... can have the same meaning, a different definition, or be undefined. Elementary proof right|thumb|The Archimedean property: any point x before the finish line lies between two of the points Pn (inclusive).|class=skin-invert-image It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics. The proof given below is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line, there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1. Intuitive explanation Zeno's paradoxes If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. For any number that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than x . So, it does not make sense to identify 0.999... with any number smaller than 1. Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines. Therefore, 0.999... cannot be identified with any number larger than 1, either. Because 0.999... cannot be bigger than 1 or smaller than 1, it must equal 1 if it is to be any real number at all.Cheng 2023 141Diamond 1955 Rigorous proof Denote by 0.(9)n the number 0.999...9, with nines after the decimal point. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. One has 1 − 0.(9)1 = 0.1 = \textstyle \frac{1}{10} , 1 − 0.(9)2 = 0.01 = \textstyle \frac{1}{10^2} , and so on; that is, 1 − 0.(9)n = for every natural number n . Let be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, 0.(9)n < ≤ 1, for every n . By subtracting these inequalities from 1, one gets 0 ≤ 1 − < \textstyle \frac{1}{10^n} . The end of the proof requires that there be no positive number that is less than for all n . This follows from the Archimedean property, which can be expressed as, "for every real number, there is a natural number that is greater". By computing the reciprocal, this implies that for every positive real number, there are natural numbers whose reciprocals are smaller. Therefore, for any positive real number, there must be some n such that is smaller.Baldwin Norton 2012Meier Smith 2017 §8.2 This property implies that if 1 − < \textstyle \frac{1}{10^n} for all n, then 1 − can only be equal to 0. So, = 1 and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, 1 = 0.999..., as claimed.Stewart Tall 2015 38–39 This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers).Stewart 2009 175Propp 2023 When using such systems, the notation 0.999... is generally not used, as there is no smallest number among the numbers larger than all 0.(9)n.For example, one can show this as follows: if x is any number such that 0.(9)n ≤ x < 1, then 0.(9)n−1 ≤ 10x − 9 < x < 1. Thus if x has this property for all n, the smaller number 10x − 9 does, as well. Least upper bounds and completeness Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound.Stillwell 1994 42Earl Nicholson 2021 "bound" This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations.Rosenlicht 1985 27 The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers has a least upper bound and that this least upper bound is equal to one.Bauldry 2009 47 Algebraic arguments Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. Byers 2007 discusses the argument that, in elementary school, one is taught that = 0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1 = 0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1.Byers 2007 39 Richman 1999 discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation [i.e.,that = 0.333...] without thinking", but also suggests that the argument may lead skeptics to question this assumption.Richman 1999 Byers also presents the following argument. Students who did not accept the first argument sometimes accept the second argument, but, in Byers's opinion, still have not resolved the ambiguity, and therefore do not understand the representation of infinite decimals. Peressini Peressini 2007, presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness.Peressini Peressini 2007 186 Baldwin Norton 2012, citing Katz Katz 2010a, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature. Baldwin Norton 2012 Katz Katz 2010a Cheng 2023 concurs, arguing that knowing one can multiply 0.999... by 10 by shifting the decimal point presumes an answer to the deeper question of how one gives a meaning to the expression 0.999... at all.Cheng 2023 136 The same argument is also given by Richman 1999, who notes that skeptics may question whether is cancellable that is, whether it makes sense to subtract from both sides.Richman 1999 Eisenmann 2008 similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.Eisenmann 2008 38 Analytic proofs Real analysis is the study of the logical underpinnings of calculus, including the behavior of sequences and series of real numbers.Tao 2003 The proofs in this section establish 0.999... = 1 using techniques familiar from real analysis. Infinite series and sequences Decimal representation A common development of decimal expansions is to define them as infinite series. In general For 0.999... one can apply the convergence theorem concerning geometric series, stating that if \vert r \vert < 1, then Rudin 1976 61 Theorem 3.26 Stewart 1999 706 Since 0.999... is such a sum with and common ratio \textstyle r = \frac{1}{10} , the theorem makes short work of the question: This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.Euler 1828 170 thumb|upright=1.1|Limits: The unit interval, including the base-4 fraction sequence (.3, .33, .333, ...) converging to 1.|class=skin-invert-image 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 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.... Grattan-Guinness 1970 69 Bonnycastle 1806 177 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 several proof-based introductions to calculus or analysis. Stewart 1999 706 Rudin 1976 61 Protter Morrey 1991 213 Pugh 2002 180 Conway 1978 31 A sequence (, , , ...) has the value as its limit if the distance becomes arbitrarily small as increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:The limit follows, for example, from Rudin 1976, p. 57, Theorem 3.20e. For a more direct approach, see also Finney Weir Giordano 2001, section 8.1, example 2(a), example 6(b). The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10−n approaches 0 as approaches infinity ( \infty ), 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". Davies 1846 175 Smith Harrington 1895 115 Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.Tall 2000 221 Nested intervals and least upper bounds Nested intervals right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222....|class=skin-invert-image The series definition above defines 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. If a real number is known to lie in the closed interval [0, 10] (that is, 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 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_1 , b_2 , b_3 , ..., and one writes 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. Beals 2004 22 Stewart 2009 34 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. So b_1 , b_2 , b_3 , ... is defined to be the unique number contained within all the intervals , + 1, , + 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. Bartle Sherbert 1982 60–62 Pedrick 1994 29 Sohrab 2003 46 The nested intervals theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b_0.b_1 b_2 b_3 ... to be the least upper bound of the set of approximants b_0 , b_0. b_1 , b_0. b_1 b_2 , .... Apostol 1974 9, 11–12 Beals 2004 22 Rosenlicht 1985 27 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, "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."Apostol 1974 12 Proofs from the construction of the real numbers Construction of the real numbers Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers 0, 1, 2, 3, ... 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 integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division.Cheng 2023 153–156Conway 2001 25–27 More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.Rudin 1976 3,8 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 cuts and Cauchy sequences. Proofs that 0.999... = 1 that directly uses 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 toward 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.The historical synthesis is claimed by Griffiths Hilton 1970, p. xiv and again by Pugh 2002, p. 10; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh 2002, p. 17 or Rudin 1976, p. 17. For viewpoints on logic, see Pugh 2002, p. 10, Rudin 1976, p.ix, or Munkres 2000, p. 30. Dedekind cuts In the Dedekind cut approach, each real number is defined as the infinite set of all rational numbers less than x.Enderton 1977, 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 ..." In particular, the real number 1 is the set of all rational numbers that are less than 1.Rudin 1976, pp. 17–20, Richman 1999, p. 399, or Enderton 1977, p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1∗, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "non-principal Dedekind cut". Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers that are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers such that < 0, or < 0.9, or < 0.99, or is less than some other number of the formRichman 1999 399 Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as with and b > a . This implies and thus Since by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1. The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.O'Connor Robertson 2005 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.Richman 1999 Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate. He also notes that typically the definitions allow < 1 to be a cut but not ≤ 1 (or vice versa).Richman 1999, p. 398–399. "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." 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 has no representation; see Alternative number systems below. Cauchy sequences 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 and is defined as the absolute value \left\vert x - y \right\vert , where the absolute value 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_0 , x_1 , x_2 , ..., a mapping from natural numbers to rationals, for any positive rational there is an such that for all m, n > N ; the distance between terms becomes smaller than any positive rational.Griffiths Hilton 1970 386 §24.2 "Sequences" If and are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence has the limit 0. Truncations of the decimal number b_0. b_1 b_2 b_3 ... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.Griffiths Hilton 1970 388, 393 Thus in this formalism the task is to show that the sequence of rational numbers has a limit 0. Considering the n th term of the sequence, for n \in \mathbb{N} , it must therefore be shown that This can be proved by the definition of a limit. So again, 0.999... = 1.Griffiths Hilton 1970 395 The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.O'Connor Robertson 2005 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.Griffiths Hilton 1970 viii, 395 Infinite decimal representation 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 an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 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. In other words, the equality 0.999... = 1 holding true is a necessary condition for strings of digits to behave as real numbers should.Gowers 2001Li 2011 Dense order One of the notions that can resolve the issue is the requirement that real numbers be densely ordered. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal. Therefore, if 0.999... were to be different from 1, there would have to be another real number in between them but there is none: a single digit cannot be changed in either of the two to obtain such a number.Artigue 2002 212 "... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1)." Generalizations 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.Petkovšek 1990 408Rosenlicht 1985 27 Second, a comparable theorem applies in each radix (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. In general, any terminating base expression has a counterpart with repeated trailing digits equal to . Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start. Protter Morrey 1991 503 Bartle Sherbert 1982 61 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 between 1 and 2, there are uncountably many base- expansions of 1. In contrast, there are still uncountably many q , including all natural numbers greater than 1, for which there is only one base- 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 = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the Thue–Morse sequence, which does not repeat.Komornik Loreti 1998 636 A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense, the difficulties are even worse. For example: Kempner 1936 611 Petkovšek 1990 409 In the balanced ternary system, = 0.111... = 1.111.... In the reverse factorial number system (using bases 2!, 3!, 4!, ... for positions after the decimal point), 1 = 1.000... = 0.1234.... Petkovšek 1990 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 spaces and noticing that their real representations are given by continuous functions.Petkovšek 1990 410–411 Applications One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers.Goodwyn 1802 Dickson 1919 161 Examples include: = 0.142857 and 142 + 857 = 999. = 0.01369863 and 0136 + 9863 = 9999. E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear whether his proof directly involved 0.999..., but at least one modern proof by William G. Leavitt does. If it can be proved that if a decimal of the form 0.b_1 b_2 b_3 ... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.Leavitt 1984 301 Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity. Ginsberg 2004 26–30 Lewittes 2006 1–3 Leavitt 1967 669, 673 Shrader-Frechette 1978 96–98 right|class=skin-invert-image|thumb|Positions of 1 4, 2 3, and 1 in the Cantor set Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in the characterization of one of the simplest fractals, the middle-thirds Cantor set: 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. 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 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and the left of every deletion thereafter. The point is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and the right of every deletion thereafter. Pugh 2002 97 Alligood Sauer Yorke 1996 150–152 Protter Morrey 1991 507 Pedrick 1994 29 Repeating nines also turns up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying 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.Maor 1987, p. 60 and Mankiewicz 2000, p. 151 review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres 2000, p. 50 mentions the latter method. A variant that may be closer to Cantor's original argument uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well. Rudin 1976 50 Pugh 2002 98 Skepticism among students 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 concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion: 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.Bunch 1982, p. 119; Tall Schwarzenberger 1978, p. 6. The last suggestion is due to Burrell 1998, p. 28: "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1." 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". Tall Schwarzenberger 1978 6–7 Tall 2000 221 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. Tall Schwarzenberger 1978 6 Tall 2000 221 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 counterexamples to better understand 0.999...; see #In alternative number systems below. 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 with 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.Tall 2000 221 The elementary argument of multiplying 0.333... = by 3 can convince reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief in the first equation and their disbelief in the second, some students either begin to disbelieve the first equation or simply become frustrated.Tall 1976 10–14 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... = using a supremum definition but then insisted that 0.999... < 1 based on her earlier understanding of long division. Pinto Tall 2001 5 Edwards Ward 2004 416–417 Others still can prove that = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations. Mazur 2005 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".Mazur 2005 137–141 As part of the APOS Theory of mathematical learning, Dubinsky Weller McDonald Brown 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. They also link this mental ability of encapsulation to viewing as a number in its own right and to dealing with the set of natural numbers as a whole.Dubinsky Weller McDonald Brown 2005 261–262 Cultural phenomenon With the rise of the Internet, debates about 0.999... have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math in the 1990s, arguing over 0.999... became a "popular sport", and was one of the questions answered in its FAQ.Richman 1999 396de Vreught 1994 The FAQ briefly covers \textstyle \frac{1}{3} , multiplication by 10, and limits, and alludes to Cauchy sequences as well. A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999... via and limits, saying of misconceptions, 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. Nonsense.Adams 2003 A Slate article reports that the concept of 0.999... is "hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums".Ellenberg 2014 0.999... features also in mathematical jokes, such as:Renteln Dundes 2005 27 Q: How many mathematicians does it take to screw in a lightbulb? A: 0.999999.... The fact that 0.999... is equal to 1 has been compared to Zeno's paradox of the runner.Richman 1999 Adams 2003 Ellenberg 2014 The runner paradox can be mathematically modeled and then, like 0.999..., resolved using a geometric series. However, it is not clear whether this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring. Wallace 2003 51 Maor 1987 17 In alternative number systems 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: 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.Gowers 2002 60 Infinitesimals Infinitesimal Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. 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 zero, and therefore the two values are the same. However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).For a full treatment of non-standard numbers, see Robinson 1996. A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗. Lightstone shows how to associate each number with a sequence of digits, indexed by the hypernatural numbers. While he does not directly discuss 0.999..., he shows the real number 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.Lightstone 1972 245–247 The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, .... A different definition involves an ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount.Tao 2012 156–180 More generally, the hyperreal number = 0.999...;...999000..., with last digit 9 at infinite hypernatural rank H , satisfies a strict inequality u_H < 1 . Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could beKatz Katz 2010a All such interpretations of "0.999..." are infinitely close to 1. 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....Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175. Along with Katz Katz 2010b, Ely 2010 also questions the assumption that students' ideas about 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. Katz Katz 2010b Ely 2010 Hackenbush Combinatorial game theory provides a generalized concept of number that encompasses the real numbers and much more besides.Conway 2001 3–5,12–13,24–27 For example, in 1974, Elwyn Berlekamp described a correspondence between strings of red and blue segments in Hackenbush 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...2 = . However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number , where is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.Berlekamp Conway Guy 1982, pp. 79–80, 307–311 discuss 1 and 1 3 and touch on 1 ω. The game for 0.111...2 follows directly from Berlekamp's Rule. This is 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...2 = 0.11000...2, which are both equal to \textstyle \frac{3}{4} , but the first representation corresponds to the binary tree path LRLRLLL..., while the second corresponds to the different path LRLLRRR.... Revisiting subtraction 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 semigroups, commutative monoids, and semirings. Richman 1999 considers two such systems, designed so that 0.999... < 1.Richman 1999 First, Richman 1999 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... + = 1 + x . So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to \textstyle \frac{1}{3} . After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.Richman 1999 397–399 In the process of defining multiplication, Richman also defines another system he calls "cut D ", which is the set of Dedekind cuts of decimal fractions. Ordinarily, this definition leads to the real numbers, but for a decimal fraction he allows both the cut ( -\infty , d ) and the "principal cut" ( -\infty , 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−, which has no decimal expansion. He concludes that 0.999... = 1 + 0−, while the equation "0.999... + = 1" has no solution.Richman 1999, pp. 398–400. Rudin 1976, p. 23 assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1. p-adic numbers p-adic number 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 final 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 "final 9" in 0.999.... Gardiner 2003 98 Gowers 2002 60 However, there is a system that contains an infinite string of 9s including a last 9, but where the decimal point is to the right of the nines rather than to the left. right|thumb|class=skin-invert-image|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1. The -adic numbers are an alternative number system of interest in number theory. Like the real numbers, the -adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p , and much closer to p^n , than it is to 1.Mascari Miola 1988 83–84 The -adic numbers form a field for prime and a ring for other p , including 10. So arithmetic can be performed in the -adics. 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.Fjelstad 1995 11 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:Fjelstad 1995 14–15 Compare with the series in the section 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 above in the opposite direction: if = ...999, then 10 = ...990, so 10 = − 9 , hence = −1 again.Fjelstad 1995 11 In the 10-adics, 0.999... is not a meaningful expansion, because the partial sums do not converge. As a final extension, since 0.999... = 1 (in the reals) and ...999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols"DeSua 1960 901 one may add the two equations and arrive at ...999.999... = 0. 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 in the doubly infinite decimal expansion of the 10-adic solenoid, with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.DeSua 1960 902–903 See also Finitism Informal mathematics Dichotomy paradox, a paradox formed based on not intuitively understanding infinite sequences Notes 30em References 30em Sources 30em http://www.straightdope.com/columns/030711.html An infinite question: Why doesn't .999~ = 1? 11 July 2003 Cecil Adams Cecil Adams The Straight Dope Chicago Reader 2006-09-06 https://web.archive.org/web/20060815010844/http://www.straightdope.com/columns/030711.html 15 August 2006 live Alligood K. T. Sauer T. D. Yorke J. A. James A. Yorke 1996 Chaos: An introduction to dynamical systems 4.1 Cantor Sets Springer 978-0-387-94677-1 This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix) Apostol Tom M. Tom M. Apostol 1974 https://archive.org/details/mathematicalanal02edtomm Mathematical Analysis 2e Addison-Wesley 978-0-201-00288-1 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) Artigue Michèle Derek Holton Michèle Artigue Urs Kirchgräber Joel Hillel Mogens Niss Alan Schoenfeld The Teaching and Learning of Mathematics at University Level New ICMI Study Series 2002 7 Springer, Dordrecht 10.1007/0-306-47231-7 978-0-306-47231-2 Baldwin Michael Norton Anderson Does 0.999... Really Equal 1? https://eric.ed.gov/?id=EJ961516 2012 The Mathematics Educator 21 2 58–67 Bauldry William C. 2009 https://books.google.com/books?id=ab-2vpx0FyYC&pg=PA47 Introduction to Real Analysis: An Educational Approach John Wiley & Sons 978-0-470-37136-7 This book is intended as introduction to real analysis aimed at upper- undergraduate and graduate-level. (pp. xi-xii) Bartle R. G. Robert G. Bartle Sherbert D. R. 1982 https://archive.org/details/introductiontore0000bart Introduction to Real Analysis Wiley 978-0-471-05944-8 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) Beals Richard Richard Beals (mathematician) https://books.google.com/books?id=cXAqJUYqXx0C Analysis: An Introduction 2004 Cambridge University Press 978-0-521-60047-7 1982 Winning Ways for your Mathematical Plays Academic Press 978-0-12-091101-1 Berlekamp E. R. Conway J. H. Guy R. K. Elwyn Berlekamp John Horton Conway Winning Ways for your Mathematical Plays Bonnycastle John 1806 An introduction to algebra; with notes and observations: designed for the use of schools and places of public education http://hdl.handle.net/2027/mdp.39015063620382 First American Philadelphia 2027/mdp.39015063620382 Bunch Bryan H. https://archive.org/details/mathematicalfall0000bunc Mathematical Fallacies and Paradoxes 1982 Van Nostrand Reinhold 978-0-442-24905-2 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) Burrell Brian Merriam-Webster's Guide to Everyday Math: A Home and Business Reference 1998 Merriam-Webster 978-0-87779-621-3 https://archive.org/details/merriamwebstersg00burr Byers William How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics 2007 Princeton University Press 978-0-691-12738-5 https://archive.org/details/howmathematician00byer Cheng Eugenia Eugenia Cheng Is Math Real? How Simple Questions Lead Us To Mathematics' Deepest Truths 2023 Basic Books 978-1-541-6-01826 Conway John B. John B. Conway https://archive.org/details/isbn_9781461263142 Functions of One Complex Variable I 2e Springer-Verlag 1973 1978 978-0-387-90328-6 Conway John H. John Horton Conway On Numbers and Games On Numbers and Games A K Peters 2001 2nd 1-56881-127-6 Davies Charles Charles Davies (professor) 1846 The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications A.S. Barnes https://archive.org/details/universityarith00davigoog 175 4 July 2011 http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ Hans de Vreught 1994 sci.math FAQ: Why is 0.9999... = 1? 29 June 2006 https://web.archive.org/web/20070929122649/http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ 29 September 2007 live DeSua Frank C. A System Isomorphic to the Reals https://archive.org/details/sim_american-mathematical-monthly_1960-11_67_9/page/900 2309468 The American Mathematical Monthly 67 900–903 10.2307/2309468 9 November 1960 Diamond Louis E. Irrational Numbers Mathematics Magazine Mathematical Association of America 1955 29 2 89–99 10.2307/3029588 3029588 Dickson Leonard Eugene History of the Theory of Numbers 1 Carnegie Institution of Washington 1919 Dubinsky Ed Weller Kirk McDonald Michael Brown Anne Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 https://archive.org/details/sim_educational-studies-in-mathematics_2005_60_2/page/253 Educational Studies in Mathematics 2005 60 253–266 10.1007/s10649-005-0473-0 2 45937062 Earl Richard Nicholson James The Concise Oxford Dictionary of Mathematics 6th 2021 Oxford University Press 978-0-192-58405-2 Edwards Barbara Ward Michael Surprises from mathematics education research: Student (mis)use of mathematical definitions The American Mathematical Monthly 111 411–425 http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf 10.2307/4145268 4145268 4 July 2011 https://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf 22 July 2011 5 May 2004 http://www.slate.com/blogs/how_not_to_be_wrong/2014/06/06/does_0_999_1_and_are_divergent_series_the_invention_of_the_devil.html Slate Ellenberg Jordan Does 0.999... = 1? And Are Divergent Series the Invention of the Devil? 6 June 2014 Jordan Ellenberg 8 August 2023 https://web.archive.org/web/20230808022739/https://slate.com/human-interest/2014/06/does-0-999-1-and-are-divergent-series-the-invention-of-the-devil.html Ely Robert 2010 Nonstandard student conceptions about infinitesimals https://archive.org/details/sim_journal-for-research-in-mathematics-education_2010-03_41_2/page/117 Journal for Research in Mathematics Education 41 2 117–146 10.5951/jresematheduc.41.2.0117 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 0.999... falling short of 1 by an infinitesimal 0.000...1. Enderton Herbert B. Herbert Enderton 1977 https://archive.org/details/elementsofsetthe0000ende Elements of Set Theory Elsevier 978-0-12-238440-0 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) Euler Leonhard Leonhard Euler 1770 1828 4th English Elements of Algebra John Hewlett Longman, Rees, Orme, and Co. https://archive.org/details/elementsofalgebr00eule/page/164/ Of Geometrical Progressions Chapter 3.11, 164–170 Finney Ross L. Weir Maurice D. Giordano Frank R. 2001 Thomas' Calculus: Early Transcendentals 10th Addison-Wesley New York Fjelstad Paul The Repeating Integer Paradox https://www.tandfonline.com/doi/abs/10.1080/07468342.1995.11973659 2687285 The College Mathematics Journal 26 11–15 10.2307/2687285 1 January 1995 subscription Gardiner Anthony Tony Gardiner https://books.google.com/books?id=NiDCYJ8vrGQC Understanding Infinity: The Mathematics of Infinite Processes 1982 2003 Dover 978-0-486-42538-2 Midy's (nearly) secret theorem – an extension after 165 years The College Mathematics Journal 35 1 2004 Brian Ginsberg 26–30 10.1080/07468342.2004.11922047 https://www.tandfonline.com/doi/abs/10.1080/07468342.2004.11922047 subscription H. Goodwyn 1802 Journal of Natural Philosophy, Chemistry, and the Arts New Series 1 314–316 Curious properties of prime Numbers, taken as the Divisors of unity. By a Correspondent https://archive.org/details/journalofnatural01lond/page/314/mode/2up Gowers Timothy William Timothy Gowers https://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals? Department of Pure Mathematics and Mathematical Statistics Cambridge University 2024-10-03 2001 Gowers Timothy William Timothy Gowers https://books.google.com/books?id=DBxSM7TIq48C Mathematics: A Very Short Introduction 2002 Oxford University Press 978-0-19-285361-5 Grattan-Guinness Ivor Ivor Grattan-Guinness 1970 The Development of the Foundations of Mathematical Analysis from Euler to Riemann MIT Press 978-0-262-07034-8 https://archive.org/details/developmentoffo00ivor Griffiths H. B. Hilton P. J. Peter Hilton A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation https://archive.org/details/comprehensivetex0000grif registration 1970 Van Nostrand Reinhold London 978-0-442-02863-3 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 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) Katz Karin Usadi Katz Mikhail G. Mikhail Katz 2010a When is .999... less than 1? The Montana Mathematics Enthusiast 7 1 3–30 10.54870/1551-3440.1381 http://www.math.umt.edu/TMME/vol7no1/ 4 July 2011 https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/ 20 July 2011 dead 2010arXiv1007.3018U 1007.3018 11544878 Karin Usadi Katz Mikhail G. Katz Mikhail Katz 2010b 259 Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era 74 Educational Studies in Mathematics 10.1007/s10649-010-9239-4 1003.1501 2010arXiv1003.1501K 3 115168622 Kempner Aubrey J. Aubrey Kempner Anormal Systems of Numeration 2300532 The American Mathematical Monthly 43 610–617 10.2307/2300532 10 December 1936 Komornik Vilmos Loreti Paola Unique Developments in Non-Integer Bases Paola Loreti 2589246 The American Mathematical Monthly 105 1998 636–639 10.2307/2589246 7 Li Liangpan A new approach to the real numbers 1101.1800 math.CA March 2011 Leavitt William G. A Theorem on Repeating Decimals 2314251 The American Mathematical Monthly 74 1967 669–673 10.2307/2314251 6 http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1047&context=mathfacpub subscription Leavitt William G. Repeating Decimals https://archive.org/details/sim_college-mathematics-journal_1984-09_15_4/page/299 2686394 The College Mathematics Journal 15 299–308 10.2307/2686394 4 September 1984 math.NT/0605182 Midy's Theorem for Periodic Decimals Lewittes Joseph 2006 Lightstone Albert H. A. H. Lightstone Infinitesimals https://archive.org/details/sim_american-mathematical-monthly_1972-03_79_3/page/242 2316619 The American Mathematical Monthly 79 242–251 10.2307/2316619 3 March 1972 Mankiewicz Richard 2000 https://archive.org/details/storyofmathemati0000mank_k4e8 The Story of Mathematics Cassell 978-0-304-35473-3 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) Mascari Gianfranco Miola Alfonso 1988 Beth Thomas Clausen Michael Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra On the integration of numeric and algebraic computations 10.1007/BFb0039172 978-3-540-39133-3 Maor Eli Eli Maor To Infinity and Beyond: A Cultural History of the Infinite https://archive.org/details/toinfinitybeyond0000maor registration 1987 Birkhäuser 978-3-7643-3325-6 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) Mazur Joseph Joseph Mazur https://books.google.com/books?id=7KSxBwAAQBAJ Euclid in the Rainforest: Discovering Universal Truths in Logic and Math 2005 New York Pi Press 978-0-13-147994-4 Meier John Smith Derek Exploring Mathematics: An Engaging Introduction to Proof Cambridge University Press 2017 978-1-107-12898-9 Munkres James R. James Munkres Topology 2000 1975 2e Prentice-Hall 978-0-13-181629-9 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's 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) Maria Angeles Navarro Pedro Pérez Carreras 2010 A Socratic methodological proposal for the study of the equality 0.999...=1 The Teaching of Mathematics 13 1 17–34 http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf 4 July 2011 none HistTopics The real numbers: Stevin to Hilbert October 2005CITEREFO'ConnorRobertson2005 Pedrick George A First Course in Analysis https://archive.org/details/firstcourseinana0000pedr registration 1994 Springer 978-0-387-94108-0 Anthony Peressini Dominic Peressini Bart van Kerkhove Jean Paul van Bendegem Jean Paul Van Bendegem Philosophy of Mathematics and Mathematics Education https://archive.org/details/perspectivesonma0000unse_f3x1 Perspectives on Mathematical Practices Springer 978-1-4020-5033-6 2007 Logic, Epistemology, and the Unity of Science 5 Petkovšek Marko Marko Petkovšek Ambiguous Numbers are Dense https://archive.org/details/sim_american-mathematical-monthly_1990-05_97_5/page/408 2324393 American Mathematical Monthly 97 408–411 10.2307/2324393 5 May 1990 Pinto Márcia Tall David O. David Tall PME25: Following students' development in a traditional university analysis course v4: 57–64 2001 http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf 3 May 2009 https://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf 30 May 2009 Propp James Jim Propp The One About .999... Mathematical Enchantments https://mathenchant.wordpress.com/2015/09/17/the-one-about-999/ 17 September 2015 24 May 2024 Propp James Jim Propp Denominators and Doppelgängers Mathematical Enchantments https://mathenchant.wordpress.com/2023/01/17/denominators-and-doppelgangers/ 17 January 2023 16 April 2024 1991 2e https://archive.org/details/firstcourseinrea0000prot A First Course in Real Analysis Springer 978-0-387-97437-8 Protter Murray H. Murray H. Protter Morrey Charles B. Jr. Charles B. Morrey 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, nondecreasing 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) Pugh Charles Chapman Charles C. Pugh Real Mathematical Analysis 2002 Springer-Verlag 978-0-387-95297-0 https://archive.org/details/realmathematical00char While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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. Renteln Paul Dundes Alan Alan Dundes Foolproof: A Sampling of Mathematical Folk Humor Notices of the AMS 52 1 24–34 https://www.ams.org/notices/200501/fea-dundes.pdf 3 May 2009 https://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf 25 February 2009 January 2005 10.2307/2690798 Fred Richman Is 0.999... = 1? https://archive.org/details/sim_mathematics-magazine_1999-12_72_5/page/396 2690798 Mathematics Magazine 72 5 396–400 December 1999 Free HTML preprint: http://www.math.fau.edu/Richman/HTML/999.htm Fred Richman Is 0.999... = 1? 23 August 2006 https://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm 2 September 2006 dead June 1999 none Note: the journal article contains material and wording not found in the preprint. Robinson Abraham j.ctt1cx3vb6 Abraham Robinson Non-standard Analysis 1996 Revised Princeton University Press 978-0-691-04490-3 Rosenlicht Maxwell Maxwell Rosenlicht 1985 https://archive.org/details/introductiontoan0000rose Introduction to Analysis Dover 978-0-486-65038-8 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. Rudin Walter Walter Rudin Principles of Mathematical Analysis https://archive.org/details/principlesofmath00rudi registration 3e 1976 1953 McGraw-Hill 978-0-07-054235-8 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) 10.2307/2690144 Shrader-Frechette Maurice Complementary Rational Numbers 2690144 Mathematics Magazine 51 2 90–98 March 1978 Smith Charles Harrington Charles 1895 Arithmetic for Schools Macmillan https://archive.org/details/arithmeticforsc02smitgoog 115 4 July 2011 Sohrab Houshang https://books.google.com/books?id=gBPI_oYZoMMC Basic Real Analysis 2003 Birkhäuser 978-0-8176-4211-2 Stewart Ian Ian Stewart (mathematician) https://archive.org/details/professorstewart0000stew_i8r6 Professor Stewart's Hoard of Mathematical Treasures 2009 Profile Books 978-1-84668-292-6 Stewart Ian Ian Stewart (mathematician) Tall David David Tall 2015 The Foundations of Mathematics 38–39 https://books.google.com/books?id=xSN-BwAAQBAJ&pg=PA38 Oxford University Press 978-0-19-870644-1 2nd Stewart James James Stewart (mathematician) Calculus: Early transcendentals 4e 1999 Brooks/Cole 978-0-534-36298-0 https://archive.org/details/calculusearlytra00stew This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus. John Stillwell John Stillwell Elements of Algebra: Geometry, Numbers, Equations https://books.google.com/books?id=jWgPAQAAMAAJ 1994 Springer 9783540942900 Tall David David Tall Schwarzenberger R. L. E. Rolph Ludwig Edward Schwarzenberger Conflicts in the Learning of Real Numbers and Limits Mathematics Teaching 1978 82 44–49 http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf 3 May 2009 https://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf 30 May 2009 Tall David O. David Tall Conflicts and Catastrophes in the Learning of Mathematics Mathematical Education for Teaching 1976 2 4 2–18 http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf 3 May 2009 https://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf 26 March 2009 Tall David Cognitive Development in Advanced Mathematics Using Technology Mathematics Education Research Journal 2000 12 3 210–230 http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf 3 May 2009 https://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf 30 May 2009 10.1007/BF03217085 2000MEdRJ..12..196T 143438975 Tao Terence Terence Tao Higher order Fourier analysis American Mathematical Society 2012 https://terrytao.files.wordpress.com/2011/03/higher-book.pdf Tao Terence Terence Tao Math 131AH: Week 1 https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf Honors Analysis UCLA Mathematics 2003 23 May 2024 Wallace David Foster David Foster Wallace Everything and more: a compact history of infinity 2003 Norton 978-0-393-00338-3 https://archive.org/details/everythingmore00davi Eisenmann Petr 2008 Why is not true that 0.999 . . . < 1? http://teaching.math.rs/vol/tm1114.pdf The Teaching of Mathematics 11 1 38 Further reading 30em Why Does 0.999... = 1?: A Perennial Question and Number Sense https://eric.ed.gov/?id=EJ717818 Beswick Kim Australian Mathematics Teacher 60 7–9 2004 4 none Journal of Statistical Physics 47 3/4 1987 One-dimensional model of the quasicrystalline alloy S. E. Burkov 10.1007/BF01007518 409–438 1987JSP....47..409B 120281766 none 81.15 A Case of Conflict https://archive.org/details/sim_mathematical-gazette_1997-03_81_490/page/109 Bob Burn The Mathematical Gazette 81 490 109–112 3618786 10.2307/3618786 March 1997 187823601 none The Age of Newton: An Intensive Interdisciplinary Course https://archive.org/details/sim_history-teacher_1981-02_14_2/page/167 J. B. Calvert E. R. Tuttle Michael S. Martin Peter Warren The History Teacher 14 2 167–190 493261 10.2307/493261 February 1981 none Younggi Choi Jonghoon Do Equality Involved in 0.999... and (-8)1/3 For the Learning of Mathematics 25 3 13–15, 36 40248503 November 2005 none Rational Approximations to π https://archive.org/details/sim_mathematics-of-computation_1971-04_25_114/page/387 K. Y. Choong D. E. Daykin C. R. Rathbone Mathematics of Computation 25 114 387–392 2004936 10.2307/2004936 April 1971 none Edwards B. 1997 An undergraduate student's understanding and use of mathematical definitions in real analysis Dossey J. Swafford J.O. Parmentier M. Dossey A.E. Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education 1 ERIC Clearinghouse for Science, Mathematics and Environmental Education Columbus, OH 17–22 none Eisenmann Petr 2008 Why is it not true that 0.999... < 1? The Teaching of Mathematics 11 1 35–40 http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf 4 July 2011 none Ferrini-Mundy J. Graham K. 1994 Research in calculus learning: Understanding of limits, derivatives and integrals MAA Notes: Research Issues in Undergraduate Mathematics Learning 33 31–45 J. Kaput E. Dubinsky none Infinite processes in elementary mathematics: How much should we tell the children? https://archive.org/details/sim_mathematical-gazette_1985-06_69_448/page/77 Tony Gardiner The Mathematical Gazette 69 448 77–87 3616921 10.2307/3616921 June 1985 125222118 none Real Mathematics: One Aspect of the Future of A-Level https://archive.org/details/sim_mathematical-gazette_1988-12_72_462/page/276 John Monaghan The Mathematical Gazette 72 462 276–281 3619940 10.2307/3619940 December 1988 125825964 none Núñez Rafael Rafael E. Núñez Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics 2006 18 Unconventional Essays on the Nature of Mathematics Springer 160–181 http://www.cogsci.ucsd.edu/~nunez/web/publications.html 978-0-387-25717-4 4 July 2011 https://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html 18 July 2011 none Malgorzata Przenioslo Images of the limit of function formed in the course of mathematical studies at the university Educational Studies in Mathematics 55 1–3 10.1023/B:EDUC.0000017667.70982.05 103–132 March 2004 120453706 none Using Self-Similarity to Find Length, Area, and Dimension https://archive.org/details/sim_american-mathematical-monthly_1996-02_103_2/page/107 James T. Sandefur The American Mathematical Monthly 103 2 107–120 2975103 10.2307/2975103 February 1996 none Sierpińska Anna Anna Sierpińska Humanities students and epistemological obstacles related to limits Educational Studies in Mathematics 18 4 371–396 10.1007/BF00240986 3482354 November 1987 144880659 none Starbird Michael Michael Starbird Starbird Thomas Required Redundancy in the Representation of Reals 114 769–774 Proceedings of the American Mathematical Society 2159403 3 10.1090/S0002-9939-1992-1086343-5 March 1992 free none Mathematical Beliefs and Conceptual Understanding of the Limit of a Function Jennifer Earles Szydlik Journal for Research in Mathematics Education 31 3 258–276 749807 10.2307/749807 May 2000 none David O. Tall David Tall Dynamic mathematics and the blending of knowledge structures in the calculus ZDM – Mathematics Education 2009 41 4 481–492 10.1007/s11858-009-0192-6 14289039 none David O. Tall David Tall Intuitions of infinity https://archive.org/details/sim_mathematics-in-school_1981-05_10_3/page/30 Mathematics in School May 1981 10 3 30–33 30214290 none External links En-0.999...-article.oga 2006-10-19 .999999... = 1? from Cut-the-Knot Why does 0.9999... = 1 ? Proof of the equality based on arithmetic from Math Central David Tall's research on mathematics cognition What is so wrong with thinking of real numbers as infinite decimals? Theorem 0.999... on Metamath Mathematics Category:1 (number) Category:Mathematical paradoxes Category:Real numbers Category:Real analysis Category:Articles containing proofs