Utilisateur:El Caro/0,999

In mathematics, 0.999… (also denoted $0.{\bar {9}}$ or $0.{\dot {9}}$) is a recurring decimal exactly equal to 1. In other words, the symbols '0.999…' and '1' represent the same real number. Mathematicians have formulated numerous proofs of this identity, which vary as to their level of rigor, preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality 0.999… = 1 has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often vocally reject the equality. The students' reasoning is often based on an expectation that infinitesimal quantities should exist, that arithmetic may be broken, or simply that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly less than 1.

That the number 1 has two decimal expansions is not a peculiarity of the decimal system. The same phenomenon occurs in integer bases other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all positional numeral systems contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of fractions and the structure of a simple fractal, the Cantor set. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

Digit manipulation

0.999… is a number written in decimal numeral system, and some of the simplest proofs that 0.999… = 1 rely on the convenient arithmetic properties of this system. Most of decimal arithmetic — addition, subtraction, multiplication, division, and comparison — uses manipulations at the digit level that are much the same as those for integers. And like integers, any two finite decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using fractions,

¹⁄₂ = ³⁄₆.

Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

Fraction proof

0.333…	= 1⁄3
3 × 0.333…	= 3 × 1⁄3
0.999…	= 1

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using long division, a simple division of integers like ¹⁄₃ becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × ¹⁄₃ equals 1, so 0.999… = 1. Another form of this proof multiplies ¹/₉ = 0.111… by 9.

This proof is flawed. If you cannot make the assumption that 0.9… = 1 then you also cannot make the assmuption that 1/3 = 0.3…. It is just as valid to argue that 0.3… is strictly less than 1/3 as it is to argue that 0.9… is strictly less than 1. This does not disprove the assumption and in fact the assumption (proven or otherwise) is an integral part of differential calculus. Imthedragn 16:53, 25 October 2006 (UTC)

Algebraic proof

c	= 0.999…
10c	= 9.999…
10c − c	= 9.999… − 0.999…
9c	= 9
c	= 1

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.

Subtraction proof

Another proof is to subtract 0.999… from 1, which yields 0.000… As none of the digits is greater than 0, the result is exactly 0. Since 1 − 0.999… = 0, 1 = 0.999….

Calculus and analysis

Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before the university level.

One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as b₀ and one can neglect negatives, so a decimal expansion has the form

b₀.b₁b₂b₃b₄b₅….

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a positional notation, so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

Infinite series and sequences

Perhaps the most common development of decimal expansions is to define them as sums of infinite series. In general:

$b_{0}.b_{1}b_{2}b_{3}b_{4}\ldots =b_{0}+b_{1}({\textstyle {\frac {1}{10}}})+b_{2}({\textstyle {\frac {1}{10}}})^{2}+b_{3}({\textstyle {\frac {1}{10}}})^{3}+b_{4}({\textstyle {\frac {1}{10}}})^{4}+\cdots .$

For 0.999… one can apply the powerful convergence theorem concerning infinite geometric series:

If $|r|<1$ then $ar+ar^{2}+ar^{3}+\cdots =\textstyle {\frac {ar}{1-r}}.$

Since 0.999… is such a sum with a common ratio $r=\textstyle {\frac {1}{10}}$, the theorem makes short work of the question:

$0.999\ldots =9({\textstyle {\frac {1}{10}}})+9({\textstyle {\frac {1}{10}}})^{2}+9({\textstyle {\frac {1}{10}}})^{3}+\cdots ={\frac {9({\textstyle {\frac {1}{10}}})}{1-{\textstyle {\frac {1}{10}}}}}=1.\,$

This proof (actually, that 10 equals "9·9999999, &c.") appears as early as 1770 in Leonard Euler's Elements of Algebra.

Limits: The unit interval, including the base-4 decimal sequence (.3, .33, .333, …) converging to 1.

The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebra 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…. A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.

A sequence (x₀, x₁, x₂, …) has a limit x if the distance |x − x_n| becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:

$0.999\ldots =\lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}=\lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)=1-\lim _{n\to \infty }{\frac {1}{10^{n}}}=1.$

The last step — that lim 1/10ⁿ = 0 — is often justified by the axiom that the real numbers have the Archimedean property. 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". Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see below.

Nested intervals and least upper bounds

Nested intervals: in base 3, 1 = 1.000… = 0.222…

The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b₀, b₁, b₂, b₃, …, and one writes

x = b₀.b₁b₂b₃…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.

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 b₀.b₁b₂b₃… is defined to be the unique number contained within all the intervals [b₀, b₀ + 1], [b₀.b₁, b₀.b₁ + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.

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 b₀.b₁b₂b₃… to be the least upper bound of the set of approximants {b₀, b₀.b₁, b₀.b₁b₂, …}. One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,

"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."

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.
• 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."
• 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 the sequence never reaches 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.
• Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount. (This is a more sophisticated response.)

These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated. Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.

Real numbers

Other approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the 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. More subtly, they include ordering, so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 that directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained. The following two examples come from less usual sources.

Dedekind cuts

In the Dedekind cut approach, each real number x is the infinite set of all rational numbers that are less than x. In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form 1 − (¹⁄₁₀)ⁿ. Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number a/b < 1, which implies a/b < 1 − (¹⁄₁₀)^b. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in Mathematics Magazine, which is targeted at undergraduate mathematicians. 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: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning." A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "Other number systems" below.

Cauchy sequences

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that are Cauchy using this distance. That is, in the sequence (x₀, x₁, x₂, …), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |x_m − x_n| ≤ δ for all m, n > N. (The distance between terms becomes arbitrarily small.)

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 b₀.b₁b₂b₃… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers

$\left(1-0,1-{9 \over 10},1-{99 \over 100},\dots \right)=\left(1,{1 \over 10},{1 \over 100},\dots \right)$

has the limit 0. Considering the nth term of the sequence, for n=0,1,2,…, it must therefore be shown that

$\lim _{n\rightarrow \infty }{\frac {1}{10^{n}}}=0.$

This limit is plain; one possible proof is that for ε = a/b > 0 one can take N = b in the definition of the limit of a sequence. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.

Other number systems

Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's reductio ad absurdum." If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

Infinitesimals

Some proofs that 0.999… = 1 rely on the Archimedean property of the standard real numbers: there are no nonzero infinitesimals. There are mathematically coherent ordered algebraic structures, including various alternatives to standard reals, which are non-Archimedean. For example, the dual numbers include a new infinitesimal element ε, analogous to the imaginary unit i in the complex numbers except that ε² = 0. The resulting structure is useful in automatic differentiation. The dual numbers can be given a lexicographic order, in which case the multiples of ε become non-Archimedean elements. Another way to construct alternatives to standard reals is to use topos theory and alternative logics rather than set theory and classical logic (which is a special case). For example, smooth infinitesimal analysis has infinitesimals with no reciprocals.

Non-standard analysis is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to calculus. A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:

0.333…;…000… does not exist, while

0.333…;…333… = 1/3 exactly.

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the surreal number 1/ω, where ω is the first infinite ordinal; the relevant game is LRRRR… or 0.000….

Breaking subtraction

Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999… + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to ¹⁄₃. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.

In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction d he allows both the cut (−∞, d) and the "principal cut" (−∞, d]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut D, but there is "a sort of negative infinitesimal," 0⁻, which has no decimal expansion. He concludes that 0.999… = 1 + 0⁻, while the equation "0.999… + x = 1" has no solution.

p-adic numbers

When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999…. For an infinite string of 9s including a last 9, one must look elsewhere.

The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.

The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to pⁿ, than it is to 1 . The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, 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: 1 + …999 = …000 = 0, and so …999 = −1. 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:

$\ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\ldots ={\frac {9}{1-10}}=-1.$

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = …999 then 10x = x − 9, hence x = −1 again.

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols" one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.

Generalizations

Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a doppelgänger 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.

Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111… equals 1, and in base 3 (the ternary numeral system) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.

Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including 2 and 10) for which there is only one base-q expansion of 1, other than the trivial 1.000…. This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, q = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the Thue-Morse sequence, which does not repeat.

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:

• In the balanced ternary system, 1/2 = 0.111… = 1.111….
• In the factoradic system, 1 = 1.000… = 0.1234….

Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any 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.

Applications

One application of 0.999… as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:

• 1/7 = 0.142857142857… and 142 + 857 = 999.
• 1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.

E. Midy proved a general result about such fractions, now called Midy's Theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.b₁b₂b₃… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.

Positions of 1/4, 2/3, and 1 in the Cantor set

Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a 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 nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point ²⁄₃ is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point ¹⁄₃ is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying 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. A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.

In popular culture

With the rise of the Internet, debates about 0.999… have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its FAQ. The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999… via 1/3 and limits, saying of misconceptions,

"The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense."

The Straight Dope cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company's president, Mike Morhaime, announced at a press conference on April 1, 2004 that it is 1:

"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."

Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

Related questions

• Zeno's paradoxes, particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.
• Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the Riemann sphere, it makes sense to define 1/0 to be infinity. In fact, some prominent mathematicians argued for such a definition long before either number system was developed.
• Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0. Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example the IEEE floating-point standard). In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

References

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  This introductory textbook on dynamics 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)

• (en) Bartle, R.G. and D.R. Sherbert, Introduction to real analysis, Wiley, 1982

  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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• Martin Berz « Automatic differentiation as nonarchimedean analysis » (1992) (lire en ligne)
  — « (ibid.) », dans Computer Arithmetic and Enclosure Methods, Elsevier, p. 439-450
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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 text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)

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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)

• (en) Leonard Euler, Elements of Algebra, Orme Longman, 1822 (1^re éd. 1770) (lire en ligne)
• <span class="ouvrage" id="DOI 10.2307/2687285">Paul Fjelstad, « The repeating integer paradox », The College Mathematics Journal, vol. 26,‎ janvier 1995, p. 11-15 (lire en ligne)
• (en) Anthony Gardiner, Understanding Infinity: The Mathematics of Infinite Processes, Dover, 2003 (1^re éd. 1982)
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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)

• A.J. Kempner, « Anormal Systems of Numeration », The American Mathematical Monthly, vol. 43,‎ décembre 1936, p. 610-617 (lire en ligne)
• Komornik, Vilmos; and Paola Loreti, « Unique Developments in Non-Integer Bases », The American Mathematical Monthly, vol. 105,‎ 1998, p. 636-639 (lire en ligne)
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• Joseph Lewittes, « Midy's Theorem for Periodic Decimals », New York Number Theory Workshop on Combinatorial and Additive Number Theory, arXiv, 2006
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• (en) Eli Maor, To infinity and beyond: a cultural history of the infinite, Birkhäuser, 1987

  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)

• (en) Joseph Mazur, Euclid in the Rainforest: Discovering Universal Truths in Logic and Math, Pearson: Pi Press, 2005
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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' 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)

• (en) George Pedrick, A First Course in Analysis, Springer, 1994
• Marko Petkovšek, « Ambiguous Numbers are Dense », American Mathematical Monthly, vol. 97,‎ mai 1990, p. 408-411 (lire en ligne)
• Pinto, Márcia and David Tall « Following students' development in a traditional university analysis course » (2001) (lire en ligne)
  — « (ibid.) », dans PME25, v4: 57-64
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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, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56-64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503-507)

• (en) Charles Chapman Pugh, Real mathematical analysis, Springer-Verlag, 2001

  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.

• Fred Richman, « Is 0.999… = 1? », Mathematics Magazine, vol. 72, n^o 5,‎ décembre 1999, p. 396-400 (lire en ligne) Free HTML preprint: Fred Richman, « Is 0.999… = 1? », 8 juin 1999 (consulté le 23 août 2006) Note: the journal article contains material and wording not found in the preprint.
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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)

• Maurice Shrader-Frechette, « Complementary Rational Numbers », Mathematics Magazine, vol. 51,‎ mars 1978, p. 90-98 (lire en ligne)
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• (en) Houshang Sohrab, Basic Real Analysis, Birkhäuser, 2003
• (en) Ian Stewart, The Foundations of Mathematics, Oxford UP, 1977
• (en) James Stewart, Calculus: Early transcendentals, Brooks/Cole, 1999

  This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.

• D.O. Tall and R.L.E. Schwarzenberger, « Conflicts in the Learning of Real Numbers and Limits », Mathematics Teaching, vol. 82,‎ 1978, p. 44-49 (lire en ligne)
• David Tall, « Conflicts and Catastrophes in the Learning of Mathematics », Mathematical Education for Teaching, vol. 2,‎ juillet 1976, p. 2-18 (lire en ligne)
• David Tall, « Cognitive Development In Advanced Mathematics Using Technology », Mathematics Education Research Journal, vol. 12,‎ 2000, p. 210-230 (lire en ligne)
• (de) Dr. Hans von Mangoldt, Einführung in die höhere Mathematik, Leipzig, Verlag von S. Hirzel, 1911, « Reihenzahlen »
• (en) David Foster Wallace, Everything and more: a compact history of infinity, Norton, 2003

External links

• .999999... = 1? from cut-the-knot
• Why does 0.9999… = 1 ?
• Ask A Scientist: Repeating Decimals
• Repeating Nines
• Point nine recurring equals one
• David Tall's research on mathematics cognition

Category:Mathematics paradoxes Category:Real analysis Category:Real numbers Category:Numeration Category:Proofs fr:Développement décimal de l'unité