{"history":[],"article":{"id":"9aed79e4-0e78-4e14-8171-693c1abc0cdf","title":"0.999...","slug":"0-999","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), ****0.9**99...** (also written as **0.9**, **0.. 9**, or **0.(9)**) denotes the smallest [number](https://bliptext.com/articles/number) greater than every [number](https://bliptext.com/articles/number) in the sequence (**0.9**, **0.9**9, **0.9**99, ...).\nIt can be proved that this number is [1](https://bliptext.com/articles/1); that is,\nDespite common misconceptions, 0.999... is not \"almost exactly 1\" or \"very, very nearly but not quite 1\"; rather, \"0.999...\"\nand \"1\" represent the same number.\nAn elementary proof is given below that involves only [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic) and the fact that there is no [positive real number](https://bliptext.com/articles/positive-real-number) less than all 1/10n, where\nn\nis a natural number, a property that results immediately from the [Archimedean property](https://bliptext.com/articles/archimedean-property) of the [real numbers](https://bliptext.com/articles/real-number).\nThere are many other ways of showing this equality, from [intuitive](https://bliptext.com/articles/intuitive) arguments to [mathematically rigorous](https://bliptext.com/articles/mathematical-rigor) [proofs](https://bliptext.com/articles/mathematical-proof).\nThe intuitive arguments are generally based on properties of [finite decimals](https://bliptext.com/articles/finite-decimal) that are extended without proof to in[finite decimals](https://bliptext.com/articles/finite-decimal).\nThe proofs are generally based on basic properties of real numbers and methods of [calculus](https://bliptext.com/articles/calculus), such as [series](https://bliptext.com/articles/series-mathematics) and [limits](https://bliptext.com/articles/limit-mathematics).\nA question studied in [mathematics education](https://bliptext.com/articles/mathematics-education) is why some people reject this equality.\nIn [other number systems](https://bliptext.com/articles/), 0.999... can have the same meaning, a different definition, or be undefined.\nEvery nonzero [terminating decimal](https://bliptext.com/articles/terminating-decimal) has two equal representations (for example, 8.32000... and 8.31999...).\nHaving values with multiple representations is a feature of all [positional numeral systems](https://bliptext.com/articles/positional-numeral-system) that represent the real numbers.\n# [Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely | upload.wikimedia.org/wikipedia/commons/a/ae/999///Perspective.svg]\n\n# Elementary proof\nArchimedean_property_for_Achilles_and_Tortoise_example.svg: any point\nx\nbefore the finish line lies between two of the points\nPn\n(inclusive).|class=skin-invert-image]]\nIt is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) [decimal numbers](https://bliptext.com/articles/decimal-number), without any reference to more advanced topics such as [series](https://bliptext.com/articles/series-mathematics) and [limits](https://bliptext.com/articles/limit-mathematics).\nThe proof given [below](https://bliptext.com/articles/) is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the [number line](https://bliptext.com/articles/number-line), there is no room left for placing a number between them and 1.\nThe 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.\n\n## Intuitive explanation\nIf one places 0.9, 0.99, 0.999, etc. on the [number line](https://bliptext.com/articles/number-line), one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.\nFor any number x that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than x. So, it does not make sense to identify 0.999... with any number smaller than 1.\nMeanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines.\nTherefore, 0.999... cannot be identified with any number larger than 1, either.\nBecause 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.\n\n## Rigorous proof\nDenote by 0.(9)n the number 0.999...9, with n nines after the decimal point.\nThus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on.\nOne 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](https://bliptext.com/articles/natural-number) n.\nLet x be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, 0.(9)n < x ≤ 1, for every n. By subtracting these inequalities from 1, one gets 0 ≤ 1 − x < \\textstyle \\frac{1}{10^n}.\nThe end of the proof requires that there is no positive number that is less than for all n. This is one version of the [Archimedean property](https://bliptext.com/articles/archimedean-property), which is true for real numbers.\nThis property implies that if 1 − x < \\textstyle \\frac{1}{10^n} for all n, then 1 − x can only be equal to 0.\nSo, x = 1 and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, 1 = 0.999....\nThis proof relies on the Archimedean property of rational and real numbers.\nReal numbers may be enlarged into [number systems](https://bliptext.com/articles/number-systems), such as [hyperreal numbers](https://bliptext.com/articles/hyperreal-number), with infinitely small numbers ([infinitesimals](https://bliptext.com/articles/infinitesimal)) and infinitely large numbers ([infinite numbers](https://bliptext.com/articles/infinite-number)).\nWhen 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.\n\n## Least upper bounds and completeness\nPart of what this argument shows is that there is a [least upper bound](https://bliptext.com/articles/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.\nOne of the [axioms](https://bliptext.com/articles/axiom) of the [real number system](https://bliptext.com/articles/real-number-system) is the [completeness axiom](https://bliptext.com/articles/completeness-axiom), which states that every bounded sequence has a least upper bound.\nThis 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.\nThe 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.\n\n# Algebraic arguments\nSimple algebraic illustrations of equality are a subject of pedagogical discussion and critique.\ndiscusses the argument that, in elementary school, one is taught that \\frac{1}{3} = 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](https://bliptext.com/articles/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.\ndiscusses how \"this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking\", but also suggests that the argument may lead skeptics to question this assumption.\nByers also presents the following argument.\nStudents 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.\n, 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](https://bliptext.com/articles/completeness-axiom).\n, citing, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.\nconcurs, 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.\nThe same argument is also given by, who notes that skeptics may question whether x is [cancellable](https://bliptext.com/articles/cancelling-out) – that is, whether it makes sense to subtract x from both sides.\nsimilarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.\n\n# Analytic proofs\n[Real analysis](https://bliptext.com/articles/real-analysis) is the study of the logical underpinnings of [calculus](https://bliptext.com/articles/calculus), including the behavior of sequences and series of real numbers.\nThe proofs in this section establish 0.999... = 1 using techniques familiar from real analysis.\n\n## Infinite series and sequences\nA common development of decimal expansions is to define them as sums of [infinite series](https://bliptext.com/articles/infinite-series).\nIn general:\nFor 0.999...\none can apply the [convergence](https://bliptext.com/articles/convergent-series) theorem concerning [geometric series](https://bliptext.com/articles/geometric-series), stating that if \\vert r \\vert < 1, then:\nSince 0.999...\nis such a sum with a = 9 and common ratio \\textstyle r = \\frac{1}{10}, the theorem makes short work of the question:\nThis proof appears as early as 1770 in [Leonhard Euler](https://bliptext.com/articles/leonhard-euler)'s [Elements of Algebra](https://bliptext.com/articles/elements-of-algebra).\nThe sum of a geometric series is itself a result even older than Euler.\nA typical 18th-century derivation used a term-by-term manipulation similar to the [algebraic proof](https://bliptext.com/articles/) 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... . 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.\nA corresponding proof of the theorem explicitly computes that sequence; it can be found in several proof-based introductions to calculus or analysis.\nA [sequence](https://bliptext.com/articles/sequence) (x_0, x_1, x_2, ...) has the value x as its [limit](https://bliptext.com/articles/limit-of-a-sequence) if the distance becomes arbitrarily small as n increases.\nThe statement that 0.999...\n= 1 can itself be interpreted and proven as a limit:\nThe first two equalities can be interpreted as symbol shorthand definitions.\nThe remaining equalities can be proven.\nThe last step, that 10n approaches 0 as n approaches infinity (\\infty), is often justified by the [Archimedean property](https://bliptext.com/articles/archimedean-property) of the real numbers.\nThis limit-based attitude towards 0.999... is often put in more evocative but less precise terms.\nFor 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\".\nSuch [heuristics](https://bliptext.com/articles/heuristic) are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.\n# [Limits: The unit interval, including the base-4 fraction sequence (.3, .33, .333, ...) converging to 1. | upload.wikimedia.org/wikipedia/commons/e/e0/base4///333.svg]\n\n## Nested intervals and least upper bounds\nThe series definition above defines the real number named by a decimal expansion.\nA complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.\nIf a real number x is known to lie in the [closed interval](https://bliptext.com/articles/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].\nThe 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].\nContinuing this process yields an infinite sequence of [nested intervals](https://bliptext.com/articles/nested-intervals), labeled by an infinite sequence of digits b_1, b_2, b_3, ..., and one writes\nIn this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1].\nand [1, 2], so one can choose either subinterval when finding its digits.\nTo ensure that this notation does not abuse the \"=\" sign, one needs a way to reconstruct a unique real number for each decimal.\nThis can be done with limits, but other constructions continue with the ordering theme.\nOne straightforward choice is the [nested intervals theorem](https://bliptext.com/articles/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](https://bliptext.com/articles/intersection-set-theory).\nSo b_1, b_2, b_3, ... is defined to be the unique number contained within all the intervals [b_0, b_0 + 1], [b_0.b_1, b_0.b_1 + 0.1], and so on.\n0.999... 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.\nSince 1 is an element of each of these intervals, 0.999... = 1.\nThe nested intervals theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [least upper bounds](https://bliptext.com/articles/least-upper-bound) or [suprema](https://bliptext.com/articles/suprema).\nTo 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.\nb_1, b_0.\nb_1 b_2, ... . One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again.\n[Tom Apostol](https://bliptext.com/articles/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.\"\n# [Nested intervals: in base 3, 1 = 1.000... = 0.222.... | upload.wikimedia.org/wikipedia/commons/f/f4/999///Intervals///C.svg]\n\n# Proofs from the construction of the real numbers\nSome approaches explicitly define real numbers to be certain [structures built upon the rational numbers](https://bliptext.com/articles/construction-of-the-real-numbers), using [axiomatic set theory](https://bliptext.com/articles/axiomatic-set-theory).\nThe [natural numbers](https://bliptext.com/articles/natural-number) begin with 0 and continue upwards so that every number has a successor.\nOne can extend the natural numbers with their negatives to give all the [integers](https://bliptext.com/articles/integer), and to further extend to ratios, giving the [rational numbers](https://bliptext.com/articles/rational-number).\nThese number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division.\nMore subtly, they include [ordering](https://bliptext.com/articles/order-theory), so that one number can be compared to another and found to be less than, greater than, or equal to another number.\nThe step from rationals to reals is a major extension.\nThere are at least two popular ways to achieve this step, both published in 1872: [Dedekind cuts](https://bliptext.com/articles/dedekind-cut) and [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence).\nProofs 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.\nEven when a construction is offered, it is usually applied toward proving the axioms of the real numbers, which then support the above proofs.\nHowever, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.\n\n## Dedekind cuts\nIn the [Dedekind cut](https://bliptext.com/articles/dedekind-cut) approach, each real number x is defined as the [infinite set](https://bliptext.com/articles/infinite-set) of all [rational numbers](https://bliptext.com/articles/rational-number) less than x. In particular, the real number 1 is the set of all [rational numbers](https://bliptext.com/articles/rational-number) that are less than 1.\nEvery positive decimal expansion easily determines a Dedekind cut: the set of rational numbers that are less than some stage of the expansion.\nSo the real number 0.999...\nis 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\nEvery element of 0.999... is less than 1, so it is an element of the real number 1.\nConversely, all elements of 1 are rational numbers that can be written as\nwith b > 0 and b > a. This implies\nand thus\nSince\nby 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.\nThe definition of real numbers as Dedekind cuts was first published by [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) in 1872.\nThe above approach to assigning a real number to each decimal expansion is due to an expository paper titled \"Is 0.999 ... = 1?\"\nby Fred Richman in [Mathematics Magazine](https://bliptext.com/articles/mathematics-magazine).\nRichman notes that taking Dedekind cuts in any [dense subset](https://bliptext.com/articles/dense-subset) of the rational numbers yields the same results; in particular, he uses [decimal fractions](https://bliptext.com/articles/decimal-fraction), for which the proof is more immediate.\nHe also notes that typically the definitions allow to be a cut but not (or vice versa).\nA further modification of the procedure leads to a different structure where the two are not equal.\nAlthough it is consistent, many of the common rules of decimal arithmetic no longer hold, for example, the fraction \\frac{1}{3} has no representation; see below.\n\n## Cauchy sequences\nAnother approach is to define a real number as the limit of a [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence) of rational numbers.\nThis construction of the real numbers uses the ordering of rationals less directly.\nFirst, the distance between x and y 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.\nThen the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance.\nThat is, in the sequence x_0, x_1, x_2, ..., a mapping from natural numbers to rationals, for any positive rational \\delta there is an N such that for all m, n > N; the distance between terms becomes smaller than any positive rational.\nIf (x_n) and (y_n) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (x_n - y_n) has the limit 0.\nTruncations of the decimal number b_0.\nb_1 b_2 b_3... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.\nThus in this formalism the task is to show that the sequence of rational numbers\nhas a limit 0.\nConsidering the nth term of the sequence, for n \\in \\mathbb{N}, it must therefore be shown that\nThis can be proved by the [definition of a limit](https://bliptext.com/articles/limit-of-a-sequence).\nSo again, 0.999... = 1.\nThe definition of real numbers as Cauchy sequences was first published separately by [Eduard Heine](https://bliptext.com/articles/eduard-heine) and [Georg Cantor](https://bliptext.com/articles/georg-cantor), also in 1872.\nThe 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.\n\n## Infinite decimal representation\nCommonly in [secondary schools](https://bliptext.com/articles/secondary-schools)' mathematics education, the real numbers are constructed by defining a number using an integer followed by a [radix point](https://bliptext.com/articles/radix-point) and an infinite sequence written out as a string to represent the [fractional part](https://bliptext.com/articles/fractional-part) of any given real number.\nIn 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.\nThis construction can be rigorously shown to satisfy all of the [real axioms](https://bliptext.com/articles/real-number) after defining an [equivalence relation](https://bliptext.com/articles/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.\nIn other words, the equality 0.999... = 1 holding true is a necessary condition for strings of digits to behave as real numbers should.\n\n## Dense order\nOne of the notions that can resolve the issue is the requirement that real numbers be densely ordered.\nDense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal.\nTherefore, if 0.99999... 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.\n\n# Generalizations\nThe result that 0.999... = 1 generalizes readily in two ways.\nFirst, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s.\nFor example, 0.24999... equals 0.25, exactly as in the special case considered.\nThese numbers are exactly the decimal fractions, and they are [dense](https://bliptext.com/articles/dense-set).\nSecond, a comparable theorem applies in each radix or [base](https://bliptext.com/articles/base-exponentiation).\nFor example, in base 2 (the [binary numeral system](https://bliptext.com/articles/binary-numeral-system)) 0.111... equals 1, and in base 3 (the [ternary numeral system](https://bliptext.com/articles/ternary-numeral-system)) 0.222... equals 1.\nIn general, any terminating base b expression has a counterpart with repeated trailing digits equal to b − 1.\nTextbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.\nAlternative representations of 1 also occur in non-integer bases.\nFor example, in the [golden ratio base](https://bliptext.com/articles/golden-ratio-base), the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s.\nGenerally, for [almost all](https://bliptext.com/articles/almost-all) q between 1 and 2, there are uncountably many base-q expansions of 1.\nIn contrast, 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](https://bliptext.com/articles/paul-erd-s), Miklos Horváth, and István Joó around 1990.\nIn 1998 Vilmos Komornik and [Paola Loreti](https://bliptext.com/articles/paola-loreti) determined the smallest such base, the [Komornik–Loreti constant](https://bliptext.com/articles/komornik-loreti-constant) q = 1.787231650... . In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [Thue–Morse sequence](https://bliptext.com/articles/thue-morse-sequence), which does not repeat.\nA more far-reaching generalization addresses [the most general positional numeral systems](https://bliptext.com/articles/non-standard-positional-numeral-systems).\nThey too have multiple representations, and in some sense, the difficulties are even worse.\nFor example:\nhas proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense.\nHe calls the proof \"an instructive exercise in elementary [point-set topology](https://bliptext.com/articles/point-set-topology)\"; it involves viewing sets of positional values as [Stone spaces](https://bliptext.com/articles/stone-space) and noticing that their real representations are given by [continuous functions](https://bliptext.com/articles/continuous-function-topology).\n\n# Applications\nOne application of 0.999... as a representation of 1 occurs in elementary [number theory](https://bliptext.com/articles/number-theory).\nIn 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [prime numbers](https://bliptext.com/articles/prime-number).\nExamples include:\nE. Midy proved a general result about such fractions, now called [Midy's theorem](https://bliptext.com/articles/midy-s-theorem), in 1836.\nThe 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.\nIf 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.\nInvestigations in this direction can motivate such concepts as [greatest common divisors](https://bliptext.com/articles/greatest-common-divisor), [modular arithmetic](https://bliptext.com/articles/modular-arithmetic), [Fermat primes](https://bliptext.com/articles/fermat-prime), [order](https://bliptext.com/articles/order-[group](https://bliptext.com/articles/group-mathematics)-theory) of [group](https://bliptext.com/articles/group-mathematics) elements, and [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity).\nReturning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in the characterization of one of the simplest [fractals](https://bliptext.com/articles/fractal), the middle-thirds [Cantor set](https://bliptext.com/articles/cantor-set): a point in the [unit interval](https://bliptext.com/articles/unit-interval) lies in the [Cantor set](https://bliptext.com/articles/cantor-set) if and only if it can be represented in ternary using only the digits 0 and 2.\nThe nth digit of the representation reflects the position of the point in the nth stage of the construction.\nFor example, the point \\frac{2}{3} 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.\nThe point \\frac{1}{3} 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.\nRepeating nines also turns up in yet another of Georg Cantor's works.\nThey must be taken into account to construct a valid proof, applying [his 1891 diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument) to decimal expansions, of the [uncountability](https://bliptext.com/articles/uncountability) of the unit interval.\nSuch 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.\nA 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.\n# [Positions of 1⁄4, 2⁄3, and 1 in the Cantor set | upload.wikimedia.org/wikipedia/commons/f/f4/Cantor///base///3.svg]\n\n# Skepticism in education\nStudents of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [limit](https://bliptext.com/articles/limit-of-a-sequence) concept and disagreements over the nature of [infinitesimals](https://bliptext.com/articles/infinitesimal).\nThere are many common contributing factors to the confusion:\nThese 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](https://bliptext.com/articles/counterexample) to better understand 0.999...; see below.\nMany of these explanations were found by [David Tall](https://bliptext.com/articles/david-tall), who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered with his college students.\nInterviewing 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.\nThe elementary argument of multiplying 0.333... = \\frac{1}{3} by 3 can convince reluctant students that 0.999... = 1.\nStill, 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.\nNor 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... = \\frac{1}{3} using a [supremum](https://bliptext.com/articles/supremum) definition but then insisted that 0.999... < 1 based on her earlier understanding of [long division](https://bliptext.com/articles/long-division).\nOthers still can prove that \\frac{1}{3} = 0.333..., but, upon being confronted by the [fractional proof](https://bliptext.com/articles/), insist that \"logic\" supersedes the mathematical calculations.\ntells 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.\nThe student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a \"wildly imagined infinite growing process\".\nAs part of the [APOS Theory](https://bliptext.com/articles/apos-theory) of mathematical learning, 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\".\nOther 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.\nThey also link this mental ability of encapsulation to viewing \\frac{1}{3} as a number in its own right and to dealing with the set of natural numbers as a whole.\n\n# Cultural phenomenon\nWith the rise of the [Internet](https://bliptext.com/articles/internet), debates about 0.999... have become commonplace on [newsgroups](https://bliptext.com/articles/newsgroup) and [message boards](https://bliptext.com/articles/message-board), including many that nominally have little to do with mathematics.\nIn 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](https://bliptext.com/articles/faq).\nThe FAQ briefly covers \\textstyle \\frac{1}{3}, multiplication by 10, and limits, and alludes to Cauchy sequences as well.\nA 2003 edition of the general-interest newspaper column [The Straight Dope](https://bliptext.com/articles/the-straight-dope) discusses 0.999...\nvia \\frac{1}{3} and limits, saying of misconceptions,\n\"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.\nNonsense.\"\nA [Slate](https://bliptext.com/articles/slate-magazine) article reports that the concept of 0.999... is \"hotly disputed on websites ranging from [World of Warcraft](https://bliptext.com/articles/world-of-warcraft) message boards to [Ayn Rand](https://bliptext.com/articles/ayn-rand) forums\".\n0.999... features also in [mathematical jokes](https://bliptext.com/articles/mathematical-joke), such as:\n\"Q: How many mathematicians does it take to screw in a lightbulb? A: 0.999999....\"\nThe fact that 0.999... is equal to 1 has been compared to [Zeno's paradox of the runner](https://bliptext.com/articles/zeno-s-paradoxes).\nThe runner paradox can be mathematically modeled and then, like 0.999..., resolved using a geometric series.\nHowever, it is not clear whether this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.\n\n# In alternative number systems\nAlthough the real numbers form an extremely useful [number system](https://bliptext.com/articles/number-system), the decision to interpret the notation \"0.999...\"\nas naming a real number is ultimately a convention, and [Timothy Gowers](https://bliptext.com/articles/timothy-gowers) argues in Mathematics: A Very Short Introduction that the resulting identity 0.999...\n= 1 is a convention as well:\n\"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.\"\n\n## Infinitesimals\nSome proofs that 0.999... = 1 rely on the [Archimedean property](https://bliptext.com/articles/archimedean-property) of the real numbers: that there are no nonzero [infinitesimals](https://bliptext.com/articles/infinitesimal).\nSpecifically, 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.\nHowever, there are mathematically coherent ordered [algebraic structures](https://bliptext.com/articles/algebraic-structure), including various alternatives to the real numbers, which are non-Archimedean.\n[Non-standard analysis](https://bliptext.com/articles/non-standard-analysis) provides a number system with a full array of infinitesimals (and their inverses).\n[A. H. Lightstone](https://bliptext.com/articles/a-h-lightstone) developed a decimal expansion for [hyperreal numbers](https://bliptext.com/articles/hyperreal-number) in (0, 1)∗.\nLightstone shows how to associate each number with a sequence of digits,\nindexed by the [hypernatural](https://bliptext.com/articles/hypernatural) numbers.\nWhile he does not directly discuss 0.999..., he shows the real number \\frac{1}{3} is represented by 0.333...;...333..., which is a consequence of the [transfer principle](https://bliptext.com/articles/transfer-principle).\nAs a consequence the number 0.999...;...999... = 1.\nWith this type of decimal representation, not every expansion represents a number.\nIn particular \"0.333...;...000...\"\nand \"0.999...;...000...\"\ndo not correspond to any number.\nThe standard definition of the number 0.999... is the [limit of the sequence](https://bliptext.com/articles/limit-of-a-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](https://bliptext.com/articles/ultrapower-construction), which is a number that falls short of 1 by an infinitesimal amount.\nMore generally, the hyperreal number u_H = 0.999...;...999000..., with last digit 9 at infinite [hypernatural](https://bliptext.com/articles/hypernatural) rank H, satisfies a strict inequality u_H < 1.\nAccordingly, an alternative interpretation for \"zero followed by infinitely many 9s\" could be\nAll such interpretations of \"0.999...\"\nare infinitely close to 1.\n[Ian Stewart](https://bliptext.com/articles/ian-stewart-mathematician) 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.... Along with, 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.\n\n## Hackenbush\n[Combinatorial game theory](https://bliptext.com/articles/combinatorial-game-theory) provides a generalized concept of number that encompasses the real numbers and much more besides.\nFor example, in 1974, [Elwyn Berlekamp](https://bliptext.com/articles/elwyn-berlekamp) described a correspondence between strings of red and blue segments in [Hackenbush](https://bliptext.com/articles/hackenbush) and binary expansions of real numbers, motivated by the idea of [data compression](https://bliptext.com/articles/data-compression).\nFor example, the value of the Hackenbush string LRRLRLRL... is 0.010101...2 = \\frac{1}{3}.\nHowever, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1.\nThe difference between the two is the [surreal number](https://bliptext.com/articles/surreal-number), where \\omega is the first [infinite ordinal](https://bliptext.com/articles/ordinal-number); the relevant game is LRRRR... or 0.000...2.\nThis 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.\nFor 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...\n.\n\n## Revisiting subtraction\nAnother manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist because subtraction is not always possible.\nMathematical structures with an addition operation but not a subtraction operation include [commutative](https://bliptext.com/articles/commutative) [semigroups](https://bliptext.com/articles/semigroup), [commutative](https://bliptext.com/articles/commutative) monoids, and [semirings](https://bliptext.com/articles/semiring).\nconsiders two such systems, designed so that 0.999... < 1.\nFirst, defines a nonnegative decimal number to be a literal decimal expansion.\nHe defines the [lexicographical order](https://bliptext.com/articles/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 canceled; another is that no decimal number corresponds to \\textstyle \\frac{1}{3}.\nAfter defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.\nIn the process of defining multiplication, Richman also defines another system he calls \"cut D\", which is the set of [Dedekind cuts](https://bliptext.com/articles/dedekind-cut) of decimal fractions.\nOrdinarily, this definition leads to the real numbers, but for a decimal fraction d 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... + x = 1\" has no solution.\n\n## p-adic numbers\nWhen 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\".\nWhether 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.\nAmong other reasons, this idea fails because there is no \"final 9\" in 0.999... . However, there is a system that contains an infinite string of 9s including a last 9.\nThe [p-adic numbers](https://bliptext.com/articles/p-adic-number) are an alternative number system of interest in [number theory](https://bliptext.com/articles/number-theory).\nLike the real numbers, the p-adic numbers can be built from the rational numbers via [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence); the construction uses a different metric in which 0 is closer to p, and much closer to p^n, than it is to 1.\nThe p-adic numbers form a [field](https://bliptext.com/articles/field-algebra) for prime p and a [ring](https://bliptext.com/articles/ring-mathematics) for other p, including 10.\nSo arithmetic can be performed in the p-adics, and there are no infinitesimals.\nIn the 10-adic numbers, the analogues of decimal expansions run to the left.\nThe 10-adic expansion ...999 does have a last 9, and it does not have a first 9.\nOne can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.\nAnother derivation uses a geometric series.\nThe 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:\nCompare with the series in the [section above](https://bliptext.com/articles/).\nA 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](https://bliptext.com/articles/) in the opposite direction: if x = ...999, then 10x = ...990, so 10x = x − 9, hence x = −1 again.\nAs 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\" one may add the two equations and arrive at ...999.999... = 0.\nThis equation does not make sense either as a 10-adic expansion or an ordinary [decimal expansion](https://bliptext.com/articles/positional-notation), but it turns out to be meaningful and true in the [doubly infinite](https://bliptext.com/articles/doubly-infinite) [decimal expansion](https://bliptext.com/articles/positional-notation) of the [10-adic solenoid](https://bliptext.com/articles/solenoid-mathematics), with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.\n# [The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1. | upload.wikimedia.org/wikipedia/commons/5/59/4adic///333.svg]\n\n# Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:41:02.465Z","updated_at":"2025-01-25T21:41:02.465Z","search_vector":"'0.999':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), ****0.9**99...** (also written as **0.9**, **0.. 9**, or **0.(9)**) denotes the smallest [number](https://bliptext.com/articles/number) greater than every [number](https://bliptext.com/articles/number) in the sequence (**0.9**, **0.9**9, **0.9**99, ...).\nIt can be proved that this number is [1](https://bliptext.com/articles/1); that is,\nDespite common misconceptions, 0.999... is not \"almost exactly 1\" or \"very, very nearly but not quite 1\"; rather, \"0.999...\"\nand \"1\" represent the same number.\nAn elementary proof is given below that involves only [elementary arithmetic](https://bliptext.com/articles/elementary-arithmetic) and the fact that there is no [positive real number](https://bliptext.com/articles/positive-real-number) less than all 1/10n, where\nn\nis a natural number, a property that results immediately from the [Archimedean property](https://bliptext.com/articles/archimedean-property) of the [real numbers](https://bliptext.com/articles/real-number).\nThere are many other ways of showing this equality, from [intuitive](https://bliptext.com/articles/intuitive) arguments to [mathematically rigorous](https://bliptext.com/articles/mathematical-rigor) [proofs](https://bliptext.com/articles/mathematical-proof).\nThe intuitive arguments are generally based on properties of [finite decimals](https://bliptext.com/articles/finite-decimal) that are extended without proof to in[finite decimals](https://bliptext.com/articles/finite-decimal).\nThe proofs are generally based on basic properties of real numbers and methods of [calculus](https://bliptext.com/articles/calculus), such as [series](https://bliptext.com/articles/series-mathematics) and [limits](https://bliptext.com/articles/limit-mathematics).\nA question studied in [mathematics education](https://bliptext.com/articles/mathematics-education) is why some people reject this equality.\nIn [other number systems](https://bliptext.com/articles/), 0.999... can have the same meaning, a different definition, or be undefined.\nEvery nonzero [terminating decimal](https://bliptext.com/articles/terminating-decimal) has two equal representations (for example, 8.32000... and 8.31999...).\nHaving values with multiple representations is a feature of all [positional numeral systems](https://bliptext.com/articles/positional-numeral-system) that represent the real numbers.\n# [Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely | upload.wikimedia.org/wikipedia/commons/a/ae/999///Perspective.svg]\n\n# Elementary proof\nArchimedean_property_for_Achilles_and_Tortoise_example.svg: any point\nx\nbefore the finish line lies between two of the points\nPn\n(inclusive).|class=skin-invert-image]]\nIt is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) [decimal numbers](https://bliptext.com/articles/decimal-number), without any reference to more advanced topics such as [series](https://bliptext.com/articles/series-mathematics) and [limits](https://bliptext.com/articles/limit-mathematics).\nThe proof given [below](https://bliptext.com/articles/) is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the [number line](https://bliptext.com/articles/number-line), there is no room left for placing a number between them and 1.\nThe 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.\n\n## Intuitive explanation\nIf one places 0.9, 0.99, 0.999, etc. on the [number line](https://bliptext.com/articles/number-line), one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.\nFor any number x that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than x. So, it does not make sense to identify 0.999... with any number smaller than 1.\nMeanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines.\nTherefore, 0.999... cannot be identified with any number larger than 1, either.\nBecause 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.\n\n## Rigorous proof\nDenote by 0.(9)n the number 0.999...9, with n nines after the decimal point.\nThus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on.\nOne 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](https://bliptext.com/articles/natural-number) n.\nLet x be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, 0.(9)n 0 and b > a. This implies\nand thus\nSince\nby 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.\nThe definition of real numbers as Dedekind cuts was first published by [Richard Dedekind](https://bliptext.com/articles/richard-dedekind) in 1872.\nThe above approach to assigning a real number to each decimal expansion is due to an expository paper titled \"Is 0.999 ... = 1?\"\nby Fred Richman in [Mathematics Magazine](https://bliptext.com/articles/mathematics-magazine).\nRichman notes that taking Dedekind cuts in any [dense subset](https://bliptext.com/articles/dense-subset) of the rational numbers yields the same results; in particular, he uses [decimal fractions](https://bliptext.com/articles/decimal-fraction), for which the proof is more immediate.\nHe also notes that typically the definitions allow to be a cut but not (or vice versa).\nA further modification of the procedure leads to a different structure where the two are not equal.\nAlthough it is consistent, many of the common rules of decimal arithmetic no longer hold, for example, the fraction \\frac{1}{3} has no representation; see below.\n\n## Cauchy sequences\nAnother approach is to define a real number as the limit of a [Cauchy sequence](https://bliptext.com/articles/cauchy-sequence) of rational numbers.\nThis construction of the real numbers uses the ordering of rationals less directly.\nFirst, the distance between x and y 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.\nThen the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance.\nThat is, in the sequence x_0, x_1, x_2, ..., a mapping from natural numbers to rationals, for any positive rational \\delta there is an N such that for all m, n > N; the distance between terms becomes smaller than any positive rational.\nIf (x_n) and (y_n) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (x_n - y_n) has the limit 0.\nTruncations of the decimal number b_0.\nb_1 b_2 b_3... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.\nThus in this formalism the task is to show that the sequence of rational numbers\nhas a limit 0.\nConsidering the nth term of the sequence, for n \\in \\mathbb{N}, it must therefore be shown that\nThis can be proved by the [definition of a limit](https://bliptext.com/articles/limit-of-a-sequence).\nSo again, 0.999... = 1.\nThe definition of real numbers as Cauchy sequences was first published separately by [Eduard Heine](https://bliptext.com/articles/eduard-heine) and [Georg Cantor](https://bliptext.com/articles/georg-cantor), also in 1872.\nThe 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.\n\n## Infinite decimal representation\nCommonly in [secondary schools](https://bliptext.com/articles/secondary-schools)' mathematics education, the real numbers are constructed by defining a number using an integer followed by a [radix point](https://bliptext.com/articles/radix-point) and an infinite sequence written out as a string to represent the [fractional part](https://bliptext.com/articles/fractional-part) of any given real number.\nIn 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.\nThis construction can be rigorously shown to satisfy all of the [real axioms](https://bliptext.com/articles/real-number) after defining an [equivalence relation](https://bliptext.com/articles/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.\nIn other words, the equality 0.999... = 1 holding true is a necessary condition for strings of digits to behave as real numbers should.\n\n## Dense order\nOne of the notions that can resolve the issue is the requirement that real numbers be densely ordered.\nDense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal.\nTherefore, if 0.99999... 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.\n\n# Generalizations\nThe result that 0.999... = 1 generalizes readily in two ways.\nFirst, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s.\nFor example, 0.24999... equals 0.25, exactly as in the special case considered.\nThese numbers are exactly the decimal fractions, and they are [dense](https://bliptext.com/articles/dense-set).\nSecond, a comparable theorem applies in each radix or [base](https://bliptext.com/articles/base-exponentiation).\nFor example, in base 2 (the [binary numeral system](https://bliptext.com/articles/binary-numeral-system)) 0.111... equals 1, and in base 3 (the [ternary numeral system](https://bliptext.com/articles/ternary-numeral-system)) 0.222... equals 1.\nIn general, any terminating base b expression has a counterpart with repeated trailing digits equal to b − 1.\nTextbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.\nAlternative representations of 1 also occur in non-integer bases.\nFor example, in the [golden ratio base](https://bliptext.com/articles/golden-ratio-base), the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s.\nGenerally, for [almost all](https://bliptext.com/articles/almost-all) q between 1 and 2, there are uncountably many base-q expansions of 1.\nIn contrast, 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](https://bliptext.com/articles/paul-erd-s), Miklos Horváth, and István Joó around 1990.\nIn 1998 Vilmos Komornik and [Paola Loreti](https://bliptext.com/articles/paola-loreti) determined the smallest such base, the [Komornik–Loreti constant](https://bliptext.com/articles/komornik-loreti-constant) q = 1.787231650... . In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [Thue–Morse sequence](https://bliptext.com/articles/thue-morse-sequence), which does not repeat.\nA more far-reaching generalization addresses [the most general positional numeral systems](https://bliptext.com/articles/non-standard-positional-numeral-systems).\nThey too have multiple representations, and in some sense, the difficulties are even worse.\nFor example:\nhas proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense.\nHe calls the proof \"an instructive exercise in elementary [point-set topology](https://bliptext.com/articles/point-set-topology)\"; it involves viewing sets of positional values as [Stone spaces](https://bliptext.com/articles/stone-space) and noticing that their real representations are given by [continuous functions](https://bliptext.com/articles/continuous-function-topology).\n\n# Applications\nOne application of 0.999... as a representation of 1 occurs in elementary [number theory](https://bliptext.com/articles/number-theory).\nIn 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [prime numbers](https://bliptext.com/articles/prime-number).\nExamples include:\nE. Midy proved a general result about such fractions, now called [Midy's theorem](https://bliptext.com/articles/midy-s-theorem), in 1836.\nThe 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.\nIf 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.\nInvestigations in this direction can motivate such concepts as [greatest common divisors](https://bliptext.com/articles/greatest-common-divisor), [modular arithmetic](https://bliptext.com/articles/modular-arithmetic), [Fermat primes](https://bliptext.com/articles/fermat-prime), [order](https://bliptext.com/articles/order-[group](https://bliptext.com/articles/group-mathematics)-theory) of [group](https://bliptext.com/articles/group-mathematics) elements, and [quadratic reciprocity](https://bliptext.com/articles/quadratic-reciprocity).\nReturning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in the characterization of one of the simplest [fractals](https://bliptext.com/articles/fractal), the middle-thirds [Cantor set](https://bliptext.com/articles/cantor-set): a point in the [unit interval](https://bliptext.com/articles/unit-interval) lies in the [Cantor set](https://bliptext.com/articles/cantor-set) if and only if it can be represented in ternary using only the digits 0 and 2.\nThe nth digit of the representation reflects the position of the point in the nth stage of the construction.\nFor example, the point \\frac{2}{3} 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.\nThe point \\frac{1}{3} 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.\nRepeating nines also turns up in yet another of Georg Cantor's works.\nThey must be taken into account to construct a valid proof, applying [his 1891 diagonal argument](https://bliptext.com/articles/cantor-s-diagonal-argument) to decimal expansions, of the [uncountability](https://bliptext.com/articles/uncountability) of the unit interval.\nSuch 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.\nA 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.\n# [Positions of 1⁄4, 2⁄3, and 1 in the Cantor set | upload.wikimedia.org/wikipedia/commons/f/f4/Cantor///base///3.svg]\n\n# Skepticism in education\nStudents of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [limit](https://bliptext.com/articles/limit-of-a-sequence) concept and disagreements over the nature of [infinitesimals](https://bliptext.com/articles/infinitesimal).\nThere are many common contributing factors to the confusion:\nThese 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](https://bliptext.com/articles/counterexample) to better understand 0.999...; see below.\nMany of these explanations were found by [David Tall](https://bliptext.com/articles/david-tall), who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered with his college students.\nInterviewing 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.\nThe elementary argument of multiplying 0.333... = \\frac{1}{3} by 3 can convince reluctant students that 0.999... = 1.\nStill, 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.\nNor 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... = \\frac{1}{3} using a [supremum](https://bliptext.com/articles/supremum) definition but then insisted that 0.999... < 1 based on her earlier understanding of [long division](https://bliptext.com/articles/long-division).\nOthers still can prove that \\frac{1}{3} = 0.333..., but, upon being confronted by the [fractional proof](https://bliptext.com/articles/), insist that \"logic\" supersedes the mathematical calculations.\ntells 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.\nThe student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a \"wildly imagined infinite growing process\".\nAs part of the [APOS Theory](https://bliptext.com/articles/apos-theory) of mathematical learning, 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\".\nOther 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.\nThey also link this mental ability of encapsulation to viewing \\frac{1}{3} as a number in its own right and to dealing with the set of natural numbers as a whole.\n\n# Cultural phenomenon\nWith the rise of the [Internet](https://bliptext.com/articles/internet), debates about 0.999... have become commonplace on [newsgroups](https://bliptext.com/articles/newsgroup) and [message boards](https://bliptext.com/articles/message-board), including many that nominally have little to do with mathematics.\nIn 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](https://bliptext.com/articles/faq).\nThe FAQ briefly covers \\textstyle \\frac{1}{3}, multiplication by 10, and limits, and alludes to Cauchy sequences as well.\nA 2003 edition of the general-interest newspaper column [The Straight Dope](https://bliptext.com/articles/the-straight-dope) discusses 0.999...\nvia \\frac{1}{3} and limits, saying of misconceptions,\n\"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.\nNonsense.\"\nA [Slate](https://bliptext.com/articles/slate-magazine) article reports that the concept of 0.999... is \"hotly disputed on websites ranging from [World of Warcraft](https://bliptext.com/articles/world-of-warcraft) message boards to [Ayn Rand](https://bliptext.com/articles/ayn-rand) forums\".\n0.999... features also in [mathematical jokes](https://bliptext.com/articles/mathematical-joke), such as:\n\"Q: How many mathematicians does it take to screw in a lightbulb? A: 0.999999....\"\nThe fact that 0.999... is equal to 1 has been compared to [Zeno's paradox of the runner](https://bliptext.com/articles/zeno-s-paradoxes).\nThe runner paradox can be mathematically modeled and then, like 0.999..., resolved using a geometric series.\nHowever, it is not clear whether this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.\n\n# In alternative number systems\nAlthough the real numbers form an extremely useful [number system](https://bliptext.com/articles/number-system), the decision to interpret the notation \"0.999...\"\nas naming a real number is ultimately a convention, and [Timothy Gowers](https://bliptext.com/articles/timothy-gowers) argues in Mathematics: A Very Short Introduction that the resulting identity 0.999...\n= 1 is a convention as well:\n\"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.\"\n\n## Infinitesimals\nSome proofs that 0.999... = 1 rely on the [Archimedean property](https://bliptext.com/articles/archimedean-property) of the real numbers: that there are no nonzero [infinitesimals](https://bliptext.com/articles/infinitesimal).\nSpecifically, 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.\nHowever, there are mathematically coherent ordered [algebraic structures](https://bliptext.com/articles/algebraic-structure), including various alternatives to the real numbers, which are non-Archimedean.\n[Non-standard analysis](https://bliptext.com/articles/non-standard-analysis) provides a number system with a full array of infinitesimals (and their inverses).\n[A. H. Lightstone](https://bliptext.com/articles/a-h-lightstone) developed a decimal expansion for [hyperreal numbers](https://bliptext.com/articles/hyperreal-number) in (0, 1)∗.\nLightstone shows how to associate each number with a sequence of digits,\nindexed by the [hypernatural](https://bliptext.com/articles/hypernatural) numbers.\nWhile he does not directly discuss 0.999..., he shows the real number \\frac{1}{3} is represented by 0.333...;...333..., which is a consequence of the [transfer principle](https://bliptext.com/articles/transfer-principle).\nAs a consequence the number 0.999...;...999... = 1.\nWith this type of decimal representation, not every expansion represents a number.\nIn particular \"0.333...;...000...\"\nand \"0.999...;...000...\"\ndo not correspond to any number.\nThe standard definition of the number 0.999... is the [limit of the sequence](https://bliptext.com/articles/limit-of-a-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](https://bliptext.com/articles/ultrapower-construction), which is a number that falls short of 1 by an infinitesimal amount.\nMore generally, the hyperreal number u_H = 0.999...;...999000..., with last digit 9 at infinite [hypernatural](https://bliptext.com/articles/hypernatural) rank H, satisfies a strict inequality u_H < 1.\nAccordingly, an alternative interpretation for \"zero followed by infinitely many 9s\" could be\nAll such interpretations of \"0.999...\"\nare infinitely close to 1.\n[Ian Stewart](https://bliptext.com/articles/ian-stewart-mathematician) 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.... Along with, 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.\n\n## Hackenbush\n[Combinatorial game theory](https://bliptext.com/articles/combinatorial-game-theory) provides a generalized concept of number that encompasses the real numbers and much more besides.\nFor example, in 1974, [Elwyn Berlekamp](https://bliptext.com/articles/elwyn-berlekamp) described a correspondence between strings of red and blue segments in [Hackenbush](https://bliptext.com/articles/hackenbush) and binary expansions of real numbers, motivated by the idea of [data compression](https://bliptext.com/articles/data-compression).\nFor example, the value of the Hackenbush string LRRLRLRL... is 0.010101...2 = \\frac{1}{3}.\nHowever, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1.\nThe difference between the two is the [surreal number](https://bliptext.com/articles/surreal-number), where \\omega is the first [infinite ordinal](https://bliptext.com/articles/ordinal-number); the relevant game is LRRRR... or 0.000...2.\nThis 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.\nFor 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...\n.\n\n## Revisiting subtraction\nAnother manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist because subtraction is not always possible.\nMathematical structures with an addition operation but not a subtraction operation include [commutative](https://bliptext.com/articles/commutative) [semigroups](https://bliptext.com/articles/semigroup), [commutative](https://bliptext.com/articles/commutative) monoids, and [semirings](https://bliptext.com/articles/semiring).\nconsiders two such systems, designed so that 0.999... < 1.\nFirst, defines a nonnegative decimal number to be a literal decimal expansion.\nHe defines the [lexicographical order](https://bliptext.com/articles/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 canceled; another is that no decimal number corresponds to \\textstyle \\frac{1}{3}.\nAfter defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.\nIn the process of defining multiplication, Richman also defines another system he calls \"cut D\", which is the set of [Dedekind cuts](https://bliptext.com/articles/dedekind-cut) of decimal fractions.\nOrdinarily, this definition leads to the real numbers, but for a decimal fraction d 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... + x = 1\" has no solution.\n\n## p-adic numbers\nWhen 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\".\nWhether 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.\nAmong other reasons, this idea fails because there is no \"final 9\" in 0.999... . However, there is a system that contains an infinite string of 9s including a last 9.\nThe [p-adic numbers](https://bliptext.com/articles/p-adic-number) are an alternative number system of interest in [number theory](https://bliptext.com/articles/number-theory).\nLike the real numbers, the p-adic numbers can be built from the rational numbers via [Cauchy sequences](https://bliptext.com/articles/cauchy-sequence); the construction uses a different metric in which 0 is closer to p, and much closer to p^n, than it is to 1.\nThe p-adic numbers form a [field](https://bliptext.com/articles/field-algebra) for prime p and a [ring](https://bliptext.com/articles/ring-mathematics) for other p, including 10.\nSo arithmetic can be performed in the p-adics, and there are no infinitesimals.\nIn the 10-adic numbers, the analogues of decimal expansions run to the left.\nThe 10-adic expansion ...999 does have a last 9, and it does not have a first 9.\nOne can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.\nAnother derivation uses a geometric series.\nThe 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:\nCompare with the series in the [section above](https://bliptext.com/articles/).\nA 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](https://bliptext.com/articles/) in the opposite direction: if x = ...999, then 10x = ...990, so 10x = x − 9, hence x = −1 again.\nAs 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\" one may add the two equations and arrive at ...999.999... = 0.\nThis equation does not make sense either as a 10-adic expansion or an ordinary [decimal expansion](https://bliptext.com/articles/positional-notation), but it turns out to be meaningful and true in the [doubly infinite](https://bliptext.com/articles/doubly-infinite) [decimal expansion](https://bliptext.com/articles/positional-notation) of the [10-adic solenoid](https://bliptext.com/articles/solenoid-mathematics), with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.\n# [The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1. | upload.wikimedia.org/wikipedia/commons/5/59/4adic///333.svg]\n\n# Sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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