0.999...

Template:Short description Template:Use dmy dates

Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely

In mathematics, 0.999... (also written as 0.Template:Overline, 0.Template:Overset, or 0.(9)) is a repeating decimal that is an alternative way of writing the number 1. Following the standard rules for representing numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence Template:Nowrap. It can be proved that this number isTemplate:Spaces1; that is,

0.999 \dots = 1 .

Despite common misconceptions, 0.999... is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent Template:Em the same number.

An elementary proof is given below that involves only elementary arithmetic and the fact that there is no positive real number less than all $\frac{1}{1 0^{n}}$ where Template:Math is a natural number, a property that results immediately from the Archimedean property of the real numbers.

There are many other 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. The proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

In other number systems, 0.999... can have the same meaning, a different definition, or be undefined. Every nonzero terminating decimal has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all positional numeral systems that represent the real numbers.

Elementary proof

It is possible to prove the equation Template:Nowrap using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits. 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 Template:Nowrap.

Intuitive explanation

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 $x$ that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than Template:Tmath. 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.Template:Sfnp Template:Sfnp

Rigorous proof

Denote by 0.(9)_{Template:Math} the number 0.999...9, with $n$ nines after the decimal point. Thus Template:Nowrap, Template:Nowrap, Template:Nowrap, and so on. One has Template:Nowrap, Template:Nowrap, and so on; that is, Template:Nowrap for every natural number Template:Tmath.

Let $x$ be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, Template:Nowrap, for every Template:Tmath. By subtracting these inequalities from 1, one gets Template:Nowrap.

The end of the proof requires that there is no positive number that is less than $\frac{1}{1 0^{n}}$ for all Template:Tmath. This is one version of the Archimedean property, which is true for real numbers.Template:Sfnp Template:Sfnp This property implies that if Template:Nowrap for all Template:Tmath, then Template:Nowrap can only be equal to 0. So, Template:Nowrap and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, Template:Nowrap.

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).Template:Sfnp Template:Sfnp 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)_{Template:Math}.Template:Efn

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.Template:Sfnp Template:Sfnp 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.Template:Sfnp 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.Template:Sfnp

Algebraic arguments

Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. Template:Harvtxt discusses the argument that, in elementary school, one is taught that Template:Nowrap, so, ignoring all essential subtleties, "multiplying" this identity by 3 gives Template:Nowrap. 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...Template:Px2." 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.Template:Sfnp Template:Harvtxt discusses 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.Template:Sfnp

Byers also presents the following argument. Template:Block indent

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. Template:Harvtxt, 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.Template:Sfnp Template:Harvtxt, citing Template:Harvtxt, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.Template:Sfnmp Template:Harvtxt 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.Template:Sfnp The same argument is also given by Template:Harvtxt, who notes that skeptics may question whether $x$ is cancellable Template:Snd that is, whether it makes sense to subtract $x$ from both sides.Template:Sfnp Template:Harvtxt similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.Template:Sfnp

Analytic proofs

Real analysis is the study of the logical underpinnings of calculus, including the behavior of sequences and series of real numbers.Template:Sfnp The proofs in this section establish Template:Nowrap using techniques familiar from real analysis.

Infinite series and sequences

Template:Further

A common development of decimal expansions is to define them as sums of infinite series. In general: $b_{0} . b_{1} b_{2} b_{3} b_{4} \dots = b_{0} + b_{1} (\frac{1}{10}) + b_{2} {(\frac{1}{10})}^{2} + b_{3} {(\frac{1}{10})}^{3} + b_{4} {(\frac{1}{10})}^{4} + \dots .$

For 0.999... one can apply the convergence theorem concerning geometric series, stating that if Template:Nowrap, then:Template:Sfnmp $a r + a r^{2} + a r^{3} + \dots = \frac{a r}{1 - r} .$

Since 0.999... is such a sum with $a = 9$ and common ratio Template:Tmath, the theorem makes short work of the question: $0.999 \dots = 9 (\frac{1}{10}) + 9 {(\frac{1}{10})}^{2} + 9 {(\frac{1}{10})}^{3} + \dots = \frac{9 (\frac{1}{10})}{1 - \frac{1}{10}} = 1 .$ This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.Template:Sfnp

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...Template:Px2.Template:Sfnmp 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.Template:Sfnmp

A sequence Template:Nowrap has the value $x$ as its limit if the distance $| x - x_{n} |$ becomes arbitrarily small as $n$ increases. The statement that Template:Nowrap can itself be interpreted and proven as a limit:Template:Efn $0.999 \dots \overset{d e f}{=} \lim_{n \to \infty} 0 . \underset{n}{\underset{⏟}{99 \dots 9}} \overset{d e f}{=} \lim_{n \to \infty} \sum_{k = 1}^{n} \frac{9}{1 0^{k}} = \lim_{n \to \infty} (1 - \frac{1}{1 0^{n}}) = 1 - \lim_{n \to \infty} \frac{1}{1 0^{n}} = 1 - 0 = 1 .$ The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10^{Template:Math} approaches 0 as $n$ approaches infinity (Template:Tmath), 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".Template:Sfnmp Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.Template:Sfnp

Nested intervals and least upper bounds

Template:Further

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 $x$ is known to lie in the closed interval Template:Nowrap (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: Template:Nowrap, Template:Nowrap, Template:Nowrap, and so on up to Template:Nowrap. The number $x$ must belong to one of these; if it belongs to Template:Nowrap, then one records the digit "2" and subdivides that interval into Template:Nowrap, Template:Nowrap, ..., Template:Nowrap, Template:Nowrap. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits Template:Tmath, Template:Tmath, Template:Tmath, ..., and one writes $x = b_{0} . b_{1} b_{2} b_{3} \dots .$

In this formalism, the identities Template:Nowrap and Template:Nowrap reflect, respectively, the fact that 1 lies in both Template:Nowrap. and Template:Nowrap, 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.Template:Sfnmp

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 Template:Tmath, Template:Tmath, Template:Tmath, ... is defined to be the unique number contained within all the intervals Template:Nowrap, Template:Nowrap, and so on. 0.999... is then the unique real number that lies in all of the intervals Template:Nowrap, Template:Nowrap, Template:Nowrap, and Template:Nowrap for every finite string of 9s. Since 1 is an element of each of these intervals, Template:Nowrap.Template:Sfnmp

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 Template:Tmath... to be the least upper bound of the set of approximants Template:Tmath, Template:Tmath, Template:Tmath, ...Template:Px2.Template:Sfnmp One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying Template:Nowrap 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."Template:Sfnp

Proofs from the construction of the real numbers

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Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers Template:Nowrap 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.Template:Sfnp Template:Sfnp 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.Template:Sfnp

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 Template:Nowrap 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.Template:Efn

Dedekind cuts

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In the Dedekind cut approach, each real number $x$ is defined as the infinite set of all rational numbers less than Template:Tmath.Template:Efn In particular, the real number 1 is the set of all rational numbers that are less than 1.Template:Efn 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 $r$ such that Template:Nowrap, or Template:Nowrap, or Template:Nowrap, or $r$ is less than some other number of the formTemplate:Sfnp $1 - \frac{1}{1 0^{n}} = 0 . (9)_{n} = 0 . \underset{n nines}{\underset{⏟}{99 \dots 9}} .$

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 $\frac{a}{b} < 1,$ with $b > 0$ and Template:Tmath. This implies $1 - \frac{a}{b} = \frac{b - a}{b} \geq \frac{1}{b} > \frac{1}{1 0^{b}},$ and thus $\frac{a}{b} < 1 - \frac{1}{1 0^{b}} .$

Since $1 - \frac{1}{1 0^{b}} = 0 . (9)_{b} < 0.999 \dots,$ 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, Template:Nowrap.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.Template:Sfnp The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is Template:Nowrap?" by Fred Richman in Mathematics Magazine.Template:Sfnp 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 Template:Nowrap to be a cut but not Template:Nowrap (or vice versa).^[1] 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 $\frac{1}{3}$ has no representation; see Template:Slink below.

Cauchy sequences

Template:Further

Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between $x$ and $y$ is defined as the absolute value Template:Tmath, where the absolute value $| z |$ is defined as the maximum of $z$ and Template:Tmath, 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 Template:Tmath, Template:Tmath, Template:Tmath, ..., a mapping from natural numbers to rationals, for any positive rational $δ$ there is an $N$ such that $| x_{m} - x_{n} | \leq δ$ for all Template:Tmath; the distance between terms becomes smaller than any positive rational.Template:Sfnp

If $(x_{n})$ 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. Truncations of the decimal number Template:Tmath... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.Template:Sfnp Thus in this formalism the task is to show that the sequence of rational numbers $(1 - 0, 1 - \frac{9}{10}, 1 - \frac{99}{100}, \dots) = (1, \frac{1}{10}, \frac{1}{100}, \dots)$ has a limit 0. Considering the Template:Tmathth term of the sequence, for Template:Tmath, it must therefore be shown that $\lim_{n \to \infty} \frac{1}{1 0^{n}} = 0 .$ This can be proved by the definition of a limit. So again, Template:Nowrap.Template:Sfnp

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.Template:Sfnp The above approach to decimal expansions, including the proof that Template:Nowrap, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation.Template:Sfnp

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 Template:Nowrap 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 Template:Nowrap holding true is a necessary condition for strings of digits to behave as real numbers should.Template:Sfnp Template:Sfnp

Dense order

Template:Further 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 0.99999... 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.Template:Sfnp

Generalizations

The result that Template:Nowrap 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.Template:Sfnp Template:Sfnp

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 $b$ expression has a counterpart with repeated trailing digits equal to Template:Nowrap. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.Template:Sfnmp

Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for almost all $q$ between 1 and 2, there are uncountably many Template:Nowrap expansions of 1. In contrast, there are still uncountably many Template:Tmath, including all natural numbers greater than 1, for which there is only one Template:Nowrap expansion of 1, other than the trivial 1.000...Template:Px2. 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 Template:Nowrap Template:Px2. In this base, Template:Nowrap; the digits are given by the Thue–Morse sequence, which does not repeat.Template:Sfnp

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:Template:Sfnmp

In the balanced ternary system, Template:Nowrap.
In the reverse factorial number system (using bases 2!, 3!, 4!, ... for positions after the decimal point), Template:Nowrap Template:Px2.

Template:Harvtxt 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.Template:Sfnp

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.Template:Sfnmp Examples include:

$\frac{1}{7}$ = 0.Template:Overline and Template:Nowrap.
$\frac{1}{73}$ = 0.Template:Overline and Template:Nowrap.

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 Template:Tmath... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.Template:Sfnp Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.Template:Sfnmp

Returning to real analysis, the base-3 analogue Template:Nowrap 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 Template:Tmathth digit of the representation reflects the position of the point in the Template:Tmathth stage of the construction. For 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. The 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.Template:Sfnmp

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.Template:Efn 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.Template:Sfnmp

Skepticism in education

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.Template:Efn
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".Template:Sfnmp
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.Template:Sfnmp

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 Template:Slink 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 1Template:'".Template:Sfnp

The elementary argument of multiplying Template:Nowrap 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.Template:Sfnp 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...Template:Px2. For example, one real analysis student was able to prove that Template:Nowrap using a supremum definition but then insisted that Template:Nowrap based on her earlier understanding of long division.Template:Sfnmp Others still can prove that Template:Nowrap, but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.

Template:Harvtxt 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 Template:Nowrap, calling it a "wildly imagined infinite growing process".Template:Sfnp

As part of the APOS Theory of mathematical learning, Template:Harvtxt 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 $\frac{1}{3}$ as a number in its own right and to dealing with the set of natural numbers as a whole.Template:Sfnp

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 Template:Mono in the 1990s, arguing over 0.999... became a "popular sport", and was one of the questions answered in its FAQ.Template:Sfnp Template:Sfnp The FAQ briefly covers Template:Tmath, 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 $\frac{1}{3}$ and limits, saying of misconceptions, Template:Blockquote

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".Template:Sfnp 0.999... features also in mathematical jokes, such as:Template:Sfnp Template:Blockquote

The fact that 0.999... is equal to 1 has been compared to Zeno's paradox of the runner.Template:Sfnmp 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.Template:Sfnmp

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 Template:Nowrap is a convention as well: Template:Blockquote

Infinitesimals

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Some proofs that Template:Nowrap rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference Template:Nowrap 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).Template:Efn A. H. Lightstone developed a decimal expansion for hyperreal numbers in Template:Nowrap. Lightstone shows how to associate each number with a sequence of digits, $0 . d_{1} d_{2} d_{3} \dots; \dots d_{\infty - 1} d_{\infty} d_{\infty + 1} \dots,$ indexed by the hypernatural numbers. While 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. As a consequence the number Template:Nowrap. 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.Template:Sfnp

The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ...Template:Px2. A different definition involves an ultralimit, i.e., the equivalence class Template:Nowrap of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount.Template:Sfnp More generally, the hyperreal number Template:Nowrap, with last digit 9 at infinite hypernatural rank Template:Tmath, satisfies a strict inequality Template:Tmath. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could beTemplate:Sfnp $\underset{H}{0 . \underset{⏟}{999 \dots}} = 1 - \frac{1}{1 0^{H}} .$ 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....Template:Efn Along with Template:Harvtxt, Template:Harvtxt also questions the assumption that students' ideas about Template:Nowrap are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.Template:Sfnmp

Hackenbush

Combinatorial game theory provides a generalized concept of number that encompasses the real numbers and much more besides.Template:Sfnp 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 Template:Nowrap However, the value of LRLLL... (corresponding to 0.111...₂) is infinitesimally less than 1. The difference between the two is the surreal number $\frac{1}{ω}$ , where $ω$ is the first infinite ordinal; the relevant game is LRRRR... or 0.000...₂.Template:Efn

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, Template:Nowrap, which are both equal to Template:Tmath, but the first representation corresponds to the binary tree path LRLRLLL..., while the second corresponds to the different path LRLLRRR...Template:Px2.

Revisiting subtraction

Another manner in which the proofs might be undermined is if Template:Nowrap 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. Template:Harvtxt considers two such systems, designed so that Template:Nowrap.Template:Sfnp

First, Template:Harvtxt defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that Template:Nowrap simply because Template:Nowrap in the ones place, but for any nonterminating Template:Tmath, one has Template:Nowrap. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to Template:Tmath. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.Template:Sfnp

In the process of defining multiplication, Richman also defines another system he calls "cut Template:Tmath", which is the set of Dedekind cuts of decimal fractions. Ordinarily, this definition leads to the real numbers, but for a decimal fraction $d$ he allows both the cut Template:Nowrap and the "principal cut" Template:Nowrap. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again Template:Nowrap. There are no positive infinitesimals in cut Template:Tmath, but there is "a sort of negative infinitesimal", 0⁻, which has no decimal expansion. He concludes that Template:Nowrap, while the equation "Template:Nowrap" has no solution.Template:Efn

p-adic numbers

Template:Main

When asked about 0.999..., novices often believe there should be a "final 9", believing Template:Nowrap 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...Template:Px2.Template:Sfnmp However, there is a system that contains an infinite string of 9s including a last 9.

The [[p-adic number|Template:Nowrapadic numbers]] are an alternative number system of interest in number theory. Like the real numbers, the Template:Nowrapadic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to Template:Tmath, and much closer to Template:Tmath, than it is to 1.Template:Sfnp The Template:Nowrapadic numbers form a field for prime $p$ and a ring for other Template:Tmath, including 10. So arithmetic can be performed in the Template:Nowrapadics, and there are no infinitesimals.

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: Template:Nowrap, and so Template:Nowrap.Template:Sfnp 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:Template:Sfnp $\dots 999 = 9 + 9 (10) + 9 (10)^{2} + 9 (10)^{3} + \dots = \frac{9}{1 - 10} = - 1 .$

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 Template:Nowrap but was inspired to take the multiply-by-10 proof above in the opposite direction: if Template:Nowrap, then Template:Nowrap, so Template:Nowrap, hence Template:Nowrap again.Template:Sfnp

As a final extension, since Template:Nowrap (in the reals) and Template:Nowrap (in the 10-adics), then by "blind faith and unabashed juggling of symbols"Template:Sfnp one may add the two equations and arrive at Template:Nowrap. 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.Template:Sfnp

Notes

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References

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Sources

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This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
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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)
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This book is intended as introduction to real analysis aimed at upper- undergraduate and graduate-level. (pp. xi-xii)
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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)
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This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
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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 Template:Nowrap falling short of 1 by an infinitesimal Template:Nowrap
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An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
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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)
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Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)
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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)
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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)
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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)
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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.
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Template:Cite journal Free HTML preprint: Template:Cite web Note: the journal article contains material and wording not found in the preprint.
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Template:Cite book 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.
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A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
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This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
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External links

Template:Spoken Wikipedia Template:Commons category

.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

Template:Real numbers Template:Portal bar

↑ Template:Harvtxt, p. 398–399. "Why do that? Precisely to rule out the existence of distinct numbers 0.Template:Overline and 1. [...] So we see that in the traditional definition of the real numbers, the equation Template:Nowrap is built in at the beginning."

[1] Template:Harvtxt, p. 398–399. "Why do that? Precisely to rule out the existence of distinct numbers 0.Template:Overline and 1. [...] So we see that in the traditional definition of the real numbers, the equation Template:Nowrap is built in at the beginning."

[1]

0.999...

Contents