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Файл:999 Perspective.svg
The repeating decimal continues infinitely.

In mathematics, 0.999... (also written as 0.Шаблон:Overline, 0.Шаблон:Overset or 0.(9)) is a notation for the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal is a numeral that represents the smallest number no less than every number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence.[1] This number is equal toШаблон:Nbs1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" Шаблон:NowrapШаблон:Tsprather, "0.999..." and "1" represent Шаблон:Em the same number.

There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.

More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

Elementary proof

Файл:Archimedean property for Achilles and Tortoise example.svg
The Archimedean property: any point x before the finish line lies between two of the points <math>P_n</math> (inclusive).

There is an elementary proof of the equation Шаблон:Math, which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, given below,[2] 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 Шаблон:Math.

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.

More precisely, the distance from 0.9 to 1 is Шаблон:Math, the distance from 0.99 to 1 is Шаблон:Math, and so on. The distance to 1 from the Шаблон:Mathth point (the one with Шаблон:Math 9s after the decimal point) is Шаблон:Math.

Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than Шаблон:Math for every integer Шаблон:Math. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than Шаблон:Math for all n. This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so Шаблон:Math.

Discussion on 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.: a 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. 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. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.

Rigorous proof

The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number Шаблон:Math, with Шаблон:Math nines after the decimal point, is denoted Шаблон:Math. Thus Шаблон:Math, Шаблон:Math, Шаблон:Math, and so on. As Шаблон:Math, with Шаблон:Math digits after the decimal point, the addition rule for decimal numbers implies Шаблон:Block indent and Шаблон:Block indent for every positive integer Шаблон:Math.

One has to show that 1 is the smallest number that is no less than all Шаблон:Math. For this, it suffices to prove that, if a number Шаблон:Math is not larger than 1 and no less than all Шаблон:Math, then Шаблон:Math. So let Шаблон:Math such that Шаблон:Block indent for every positive integer Шаблон:Math. Therefore, Шаблон:Block indent which, using basic arithmetic and the first equality established above, simplifies to Шаблон:Block indent This implies that the difference between Шаблон:Math and Шаблон:Math is less than the inverse of any positive integer. Thus this difference must be zero, and, thus Шаблон:Math; that is Шаблон:Block indent

This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers 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). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all Шаблон:Math. (This is implied by the fact that Шаблон:Math implies Шаблон:Math).

Algebraic argumentsШаблон:AnchorШаблон:Anchor

Many algebraic arguments have been provided, which suggest that <math>1=0.999\ldots</math> They are not mathematical proofs since they are typically based on the fact that the rules for adding and multiplying finite decimals extend to infinite decimals. This is true, but the proof is essentially the same as the proof of <math>1=0.999\ldots</math> So, all these arguments are essentially circular reasoning.

Nevertheless, the matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. Шаблон:Harvtxt discusses the argument that, in elementary school, one is taught that Шаблон:Math, so, ignoring all essential subtleties, "multiplying" this identity by Шаблон:Math gives Шаблон:Math. 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 Шаблон:Math." Most undergraduate mathematics majors encountered by Byers feel that while Шаблон:Math 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. Шаблон: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.

Byers also presents the following argument. Шаблон: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 for infinite decimals. Шаблон: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. Шаблон:Harvtxt, citing Шаблон:Harvtxt, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.

The same argument is also given by Шаблон:Harvtxt, who notes that skeptics may question whether Шаблон:Math is cancellableШаблон:Snd that is, whether it makes sense to subtract Шаблон:Math from both sides.

Analytic proofsШаблон:Anchor

Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more 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 Шаблон:Block indent

The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

Infinite series and sequences

Шаблон:Further

A common development of decimal expansions is to define them as sums of infinite series. In general: Шаблон:Block indent\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>}}

For 0.999... one can apply the convergence theorem concerning geometric series:[3] Шаблон:Block indent

Since 0.999... is such a sum with a = 9 and common ratio r = Шаблон:Frac, the theorem makes short work of the question: Шаблон:Block indent\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.</math>}} This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.[4]

Файл:Base4 333.svg
Limits: The unit interval, including the base-4 fraction 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 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...[5] 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.[6]

A sequence (x0, x1, x2, ...) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:[7] Шаблон:Block indent{}}{=} \ \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} \ \overset{\underset{\mathrm{def}}{}}{=} \ \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} \ = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1 \, - \, 0 = 1.</math>}} The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that Шаблон:Frac → 0 as n → ∞, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".[8] Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.

Nested intervals and least upper bounds

Шаблон:Further

Файл:999 Intervals C.svg
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) 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

Шаблон:Block indent

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.[9]

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.[10]

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, ...}.[11] 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,

Шаблон:Blockquote

Proofs from the construction of the real numbersШаблон:Anchor

Шаблон:Further

Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, 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 to be less than, greater than, or equal to another number.

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 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.[12]

Dedekind cuts

Шаблон:Further

In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers less than x.[13] In particular, the real number 1 is the set of all rational numbers that are less than 1.[14] 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[15] Шаблон:Block indent.</math>}}

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 Шаблон:Block indent with Шаблон:Math and Шаблон:Math. This implies Шаблон:Block indent and thus Шаблон:Block indent

and since

Шаблон:Block indent

by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.[16] 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,[17] which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.[18] 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 { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."[19] 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 Шаблон:Frac has no representation; see "Alternative number systems" below.

Cauchy sequences

Шаблон: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 |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)[20]

If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xn − yn) 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.[21] Thus in this formalism the task is to show that the sequence of rational numbers Шаблон:Block indent.</math>[22]}} 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....[23] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about Шаблон:Nowrap are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[24][25]

Hackenbush

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 0.0101012... = Шаблон:Frac. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number Шаблон:Frac, where ω is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.[26]

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to 3/4, but the first representation corresponds to the binary tree path LRLRLLL... while the second corresponds to the different path LRLLRRR....

Revisiting subtraction

Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman 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 Шаблон:Frac. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.[27]

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.[28]

p-adic numbers

Шаблон:Main

When asked about 0.999..., novices often believe there should be a "final 9", believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the final 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "final 9" in 0.999....[29] However, there is a system that contains an infinite string of 9s including a last 9.

Файл:4adic 333.svg
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.[30] 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: Шаблон:Block indent

(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 =  ...990, so 10x = x − 9, hence x = −1 again.[30]

As a final extension, since Шаблон:Nowrap begin0.999... = 1Шаблон:Nowrap end (in the reals) and Шаблон:Nowrap begin...999 = −1Шаблон:Nowrap end (in the 10-adics), then by "blind faith and unabashed juggling of symbols"[31] one may add the two equations and arrive at Шаблон:Nowrap begin...999.999... = 0.Шаблон:Nowrap end This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true in the doubly infinite decimal expansion of the 10-adic solenoid, with eventually repeating left ends to represent the real numbers[32] and eventually repeating right ends to represent the 10-adic numbers.

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.[33]
  • Division by zero occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has a "point at infinity". Here, it makes sense to define Шаблон:Frac to be infinity;[34] and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.[35]
  • 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.[36] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the sign and magnitude or ones' complement formats, or floating point numbers as specified by the IEEE floating-point standard).[37][38]

See also

Notes

Шаблон:Reflist

References

Шаблон:Refbegin

Шаблон:Refend

Further reading

Шаблон:Refbegin

Шаблон:Refend

External links

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  1. ↑ This definition is equivalent to the definition of decimal numbers as the limits of their summed components, which, in the case of 0.999..., is the limit of the sequence (0.9, 0.99, 0.999, ...). The equivalence is due to bounded increasing sequences having their limit always equal to their least upper bound.
  2. ↑ Шаблон:Harvtxt
  3. ↑ Rudin p. 61, Theorem 3.26; J. Stewart p. 706
  4. ↑ Euler p. 170
  5. ↑ Grattan-Guinness p. 69; Bonnycastle p. 177
  6. ↑ For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31
  7. ↑ The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) Thomas' Calculus: Early Transcendentals 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).
  8. ↑ Davies p. 175; Smith and Harrington p. 115
  9. ↑ Beals p. 22; I. Stewart p. 34
  10. ↑ Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46
  11. ↑ Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27
  12. ↑ The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30
  13. ↑ Enderton (p. 113) qualifies this description: "The idea behind Dedekind cuts is that a real number x can be named by giving an infinite set of rationals, namely all the rationals less than x. We will in effect define x to be the set of rationals smaller than x. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way..."
  14. ↑ Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".
  15. ↑ Richman p. 399
  16. ↑ Шаблон:Cite web
  17. ↑ Шаблон:Cite web
  18. ↑ Richman
  19. ↑ Richman pp. 398–399
  20. ↑ Griffiths & Hilton §24.2 "Sequences" p. 386
  21. ↑ Griffiths & Hilton pp. 388, 393
  22. ↑ Katz & Katz 2010
  23. ↑ Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.
  24. ↑ Katz & Katz (2010b)
  25. ↑ R. Ely (2010)
  26. ↑ Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and Шаблон:Frac and touch on Шаблон:Frac. The game for 0.111...2 follows directly from Berlekamp's Rule.
  27. ↑ Richman pp. 397–399
  28. ↑ Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.
  29. ↑ Gardiner p. 98; Gowers p. 60
  30. ↑ 30,0 30,1 Fjelstad p. 11
  31. ↑ DeSua p. 901
  32. ↑ DeSua pp. 902–903
  33. ↑ Wallace p. 51, Maor p. 17
  34. ↑ See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57
  35. ↑ Maor p. 54
  36. ↑ Munkres p. 34, Exercise 1(c)
  37. ↑ Шаблон:Cite book
  38. ↑ Шаблон:Cite web
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