# 0.999...

In mathematics, **0.999...** (also written as **0.9**, among other ways) denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...).^{[1]} This number is equal to 1. In other words, "0.999..." and "1" represent 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 base representations. 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[edit source | edit]

There is an elementary proof of the equation 0.999... = 1, 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, an exercise given by Stillwell (1994, p. 42), is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1.

### Intuitive explanation[edit source | edit]

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 0.1 = 1/10, the distance from 0.99 to 1 is 0.01 = 1/10^{2}, and so on. The distance to 1 from the *n*th point (the one with *n* 9s after the decimal point) is 1/10^{n}.

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 1/10^{n} for every integer *n*. In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than 1/10^{n} 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 1 = 0.999....

### Discussion on completeness[edit source | edit]

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.

### Formal proof[edit source | edit]

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 0.999...9, with *n* nines after the decimal point, is denoted 0.(9)_{n}. Thus 0.(9)_{1} = 0.9, 0.(9)_{2} = 0.99, 0.(9)_{3} = 0.999, and so on. As 1/10^{n} = 0.0...01, with *n* digits after the decimal point, the addition rule for decimal numbers implies

and

for every positive integer *n*.

One has to show that 1 is the smallest number that is no less than all 0.(9)_{n}. For this, it suffices to prove that, if a number *x* is not larger than 1 and no less than all 0.(9)_{n}, then *x* = 1. So let *x* such that

for every positive integer *n*.
Therefore,

This implies that the difference between 1 and *x* is less than the inverse of any positive integer. Thus this difference must be zero, and, thus *x* = 1; that is

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 0.(9)_{n}. (This is implied by the fact that 0.(9)_{n} ≤ *x* < 1 implies 0.(9)_{n–1} ≤ 2*x* – 1 < *x* < 1).

## Algebraic arguments[edit source | edit]

The matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. Byers (2007, p. 39) discusses the argument that, in elementary school, one is taught that ^{1}⁄_{3}=0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1=0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. Richman (1999) 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. Let

Students who did not accept the first argument sometimes accept the second argument, but, in Byers' opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.

The same argument is also given by Richman (1999), who notes that skeptics may question whether *x* is cancellable – that is, whether it makes sense to subtract *x* from both sides.

## Analytic proofs[edit source | edit]

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 *b*_{0} and one can neglect negatives, so a decimal expansion has the form

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[edit source | edit]

*For further information, see Decimal representation*

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

\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:^{[2]}

Since 0.999... is such a sum with *a* = 9 and common ratio *r* = ^{1}⁄_{10}, the theorem makes short work of the question:

\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*.^{[3]}

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...^{[4]} 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.^{[5]}

A sequence (*x*_{0}, *x*_{1}, *x*_{2}, ...) 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:^{[6]}

{}}{=} \ \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 ^{1}⁄_{10n} → 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".^{[7]} Such heuristics are often interpreted by students as implying that 0.999... itself is less than 1.

### Nested intervals and least upper bounds[edit source | edit]

*For further information, see Nested intervals*

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 *b*_{0}, *b*_{1}, *b*_{2}, *b*_{3}, ..., and one writes

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.^{[8]}

One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So *b*_{0}.*b*_{1}*b*_{2}*b*_{3}... is defined to be the unique number contained within all the intervals [*b*_{0}, *b*_{0} + 1], [*b*_{0}.*b*_{1}, *b*_{0}.*b*_{1} + 0.1], and so on. 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.^{[9]}

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or *suprema*. To directly exploit these objects, one may define *b*_{0}.*b*_{1}*b*_{2}*b*_{3}... to be the least upper bound of the set of approximants {*b*_{0}, *b*_{0}.*b*_{1}, *b*_{0}.*b*_{1}*b*_{2}, ...}.^{[10]} 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.

^{[11]}

## Proofs from the construction of the real numbers[edit source | edit]

*For further information, see Construction of the real numbers*

Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 0, 1, 2, 3, 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[edit source | edit]

*For further information, see Dedekind cut*

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]}

.</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

with *b* > 0 and *b* > *a*.
This implies

and thus

and since

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

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.^{[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 ^{1}⁄_{3} has no representation; see "Alternative number systems" below.

### Cauchy sequences[edit source | edit]

*For further information, see Cauchy sequence*

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 (*x*_{0}, *x*_{1}, *x*_{2}, ...), 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 smaller than any positive rational.)^{[20]}

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 *b*_{0}.*b*_{1}*b*_{2}*b*_{3}... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.^{[21]} Thus in this formalism the task is to show that the sequence of rational numbers

.</math>^{[51]}}}

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....^{[52]} Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999... < 1 are erroneous intuitions about the real numbers, interpreting them rather as *nonstandard* intuitions that could be valuable in the learning of calculus.^{[53]}^{[54]}
Jose Benardete in his book *Infinity: An essay in metaphysics* argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:

The intelligibility of the continuum has been found–many times over–to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a

number... though not as arealnumber.^{[55]}

### Hackenbush[edit source | edit]

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_{2}... = ^{1}⁄_{3}. However, the value of LRLLL... (corresponding to 0.111..._{2}) 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..._{2}.^{[56]}

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[edit source | edit]

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.^{[57]}

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.^{[58]}

*p*-adic numbers[edit source | edit]

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....^{[59]} However, there is a system that contains an infinite string of 9s including a last 9.

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 *p ^{n}*, 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.^{[60]} 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:

(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 10*x* = ...990, so 10*x* = *x* − 9, hence *x* = −1 again.^{[60]}

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"^{[62]} one may add the two equations and arrive at ...999.999... = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true in the doubly infinite decimal expansion of the 10-adic solenoid, with eventually repeating left ends to represent the real numbers^{[63]} and eventually repeating right ends to represent the 10-adic numbers.

### Ultrafinitism[edit source | edit]

The philosophy of ultrafinitism rejects as meaningless concepts dealing with infinite sets, such as idea that the notation <math>0.999\ldots</math> might stand for a decimal number with an *infinite sequence of nines*, as well as the summation of infinitely many numbers <math>9/10+9/100+\cdots</math> corresponding to the positional values of the decimal digits in that infinite string. In this approach to mathematics, only some particular (fixed) number of finite decimal digits is meaningful. Instead of "equality", one has "approximate equality", which is equality up to the number of decimal digits that one is permitted to compute.^{[64]} Although Katz and Katz argue that ultrafinitism may capture the student intuition that 0.999... ought to be less than 1, the ideas of ultrafinitism do not enjoy widespread acceptance in the mathematical community, and the philosophy lacks a generally agreed-upon formal mathematical foundation.^{[65]}

## Related questions[edit source | edit]

- 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.
^{[66]} - 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
^{1}⁄_{0}to be infinity;^{[67]}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.^{[68]} - 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.
^{[69]}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).^{[70]}^{[71]}

## See also[edit source | edit]

## Notes[edit source | edit]

- ↑ 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.
- ↑ Rudin p. 61, Theorem 3.26; J. Stewart p. 706
- ↑ Euler p. 170
- ↑ Grattan-Guinness p. 69; Bonnycastle p. 177
- ↑ For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31
- ↑ 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). - ↑ Davies p. 175; Smith and Harrington p. 115
- ↑ Beals p. 22; I. Stewart p. 34
- ↑ Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46
- ↑ Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27
- ↑ Apostol p. 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
- ↑ 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..." - ↑ Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1
^{−}, and 1_{R}, 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". - ↑ Richman p. 399
- ↑
^{16.0}^{16.1}O'Connor, J. J.; Robertson, E. F. (October 2005). "History topic: The real numbers: Stevin to Hilbert".*MacTutor History of Mathematics*. Archived from the original on 29 September 2007. Retrieved 30 August 2006. - ↑ Fred Richman (December 1999). "Is 0.999... = 1?".
*Mathematics Magazine*. Mathematical Association of America. pp. 396–400. - ↑ Richman
- ↑ Richman pp. 398–399
- ↑ Griffiths & Hilton §24.2 "Sequences" p. 386
- ↑ Griffiths & Hilton pp. 388, 393
- ↑ Griffiths & Hilton p. 395
- ↑ Griffiths & Hilton pp.viii, 395
- ↑ Liangpan Li (March 2011). "A new approach to the real numbers". arXiv:1101.1800 [math.CA].
- ↑ Petkovšek p. 408
- ↑ Protter and Morrey p. 503; Bartle and Sherbert p. 61
- ↑ Komornik and Loreti p. 636
- ↑ Kempner p. 611; Petkovšek p. 409
- ↑ Petkovšek pp. 410–411
- ↑ Leavitt 1984 p. 301
- ↑ Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98
- ↑ Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.
- ↑ Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.
- ↑ Rudin p. 50, Pugh p. 98
- ↑ Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."
- ↑ Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221
- ↑ Tall and Schwarzenberger p. 6; Tall 2000 p. 221
- ↑ Tall 2000 p. 221
- ↑ Tall 1976 pp. 10–14
- ↑ Pinto and Tall p. 5, Edwards and Ward pp. 416–417
- ↑ Mazur pp. 137–141
- ↑ Dubinsky
*et al.*pp. 261–262 - ↑ As observed by Richman (p. 396). de Vreught, Hans (1994). "sci.math FAQ: Why is 0.9999... = 1?". Archived from the original on 29 September 2007. Retrieved 29 June 2006.
- ↑ Adams, Cecil (11 July 2003). "An infinite question: Why doesn't .999~ = 1?".
*The Straight Dope*. Chicago Reader. Archived from the original on 15 August 2006. Retrieved 6 September 2006. - ↑ Ellenberg, Jordan (6 June 2014). "Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?".
*Slate*. - ↑ "Blizzard Entertainment Announces .999~ (Repeating) = 1" (Press release). Blizzard Entertainment. 1 April 2004. Archived from the original on 4 November 2009. Retrieved 16 November 2009.
- ↑ Renteln and Dundes, p. 27
- ↑ Gowers p. 60
- ↑ For a full treatment of non-standard numbers see for example Robinson's
*Non-standard Analysis*. - ↑ Lightstone pp. 245–247
- ↑ Katz & Katz 2010
- ↑ Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.
- ↑ Katz & Katz (2010b)
- ↑ R. Ely (2010)
- ↑ Benardete, José Amado (1964).
*Infinity: An essay in metaphysics*. Clarendon Press. p. 279. Retrieved 27 November 2011. - ↑ Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and
^{1}⁄_{3}and touch on ^{1}⁄_{ω}. The game for 0.111..._{2}follows directly from Berlekamp's Rule. - ↑ Richman pp. 397–399
- ↑ Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.
- ↑ Gardiner p. 98; Gowers p. 60
- ↑
^{60.0}^{60.1}Fjelstad p. 11 - ↑ Fjelstad pp. 14–15
- ↑ DeSua p. 901
- ↑ DeSua pp. 902–903
- ↑ Sazonov, Vladimir (1995), "On feasible numbers",
*Logic and computational complexity*, Springer, pp. 30–50 - ↑ Katz & Katz (2010a)
- ↑ Wallace p. 51, Maor p. 17
- ↑ See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57
- ↑ Maor p. 54
- ↑ Munkres p. 34, Exercise 1(c)
- ↑ Kroemer, Herbert; Kittel, Charles (1980).
*Thermal Physics*(2e ed.). W. H. Freeman. p. 462. ISBN 978-0-7167-1088-2. - ↑ "Floating point types".
*MSDN C# Language Specification*. Archived from the original on 24 August 2006. Retrieved 29 August 2006.

## References[edit source | edit]

- Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). "4.1 Cantor Sets".
*Chaos: An introduction to dynamical systems*. Springer. ISBN 978-0-387-94677-1.- This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)

- Apostol, Tom M. (1974).
*Mathematical analysis*(2e ed.). Addison-Wesley. ISBN 978-0-201-00288-1.- A transition from calculus to advanced analysis,
*Mathematical analysis*is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)

- A transition from calculus to advanced analysis,
- Baldwin, Michael; Norton, Anderson (2012). "Does 0.999... Really Equal 1?".
*The Mathematics Educator*.**21**(2): 58–67. - Bartle, R. G.; Sherbert, D. R. (1982).
*Introduction to real analysis*. Wiley. ISBN 978-0-471-05944-8.- This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)

- Beals, Richard (2004).
*Analysis*. Cambridge UP. ISBN 978-0-521-60047-7. - Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (1982).
*Winning Ways for your Mathematical Plays*. Academic Press. ISBN 978-0-12-091101-1. - Berz, Martin (1992).
*Automatic differentiation as nonarchimedean analysis*. Computer Arithmetic and Enclosure Methods. Elsevier. pp. 439–450. CiteSeerX 10.1.1.31.3019. - Beswick, Kim (2004). "Why Does 0.999... = 1?: A Perennial Question and Number Sense".
*Australian Mathematics Teacher*.**60**(4): 7–9. - Bunch, Bryan H. (1982).
*Mathematical fallacies and paradoxes*. Van Nostrand Reinhold. ISBN 978-0-442-24905-2.- This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)

- Burrell, Brian (1998).
*Merriam-Webster's Guide to Everyday Math: A Home and Business Reference*. Merriam-Webster. ISBN 978-0-87779-621-3. - Byers, William (2007).
*How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics*. Princeton UP. ISBN 978-0-691-12738-5. - Conway, John B. (1978) [1973].
*Functions of one complex variable I*(2e ed.). Springer-Verlag. ISBN 978-0-387-90328-6.- 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)

- Davies, Charles (1846).
*The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications*. A.S. Barnes. p. 175. Retrieved 4 July 2011. - DeSua, Frank C. (November 1960). "A system isomorphic to the reals".
*The American Mathematical Monthly*.**67**(9): 900–903. doi:10.2307/2309468. JSTOR 2309468. - Dubinsky, Ed; Weller, Kirk; McDonald, Michael; Brown, Anne (2005). "Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2".
*Educational Studies in Mathematics*.**60**(2): 253–266. doi:10.1007/s10649-005-0473-0. - Edwards, Barbara; Ward, Michael (May 2004). "Surprises from mathematics education research: Student (mis)use of mathematical definitions" (PDF).
*The American Mathematical Monthly*.**111**(5): 411–425. CiteSeerX 10.1.1.453.7466. doi:10.2307/4145268. JSTOR 4145268. Archived from the original (PDF) on 22 July 2011. Retrieved 4 July 2011. - Enderton, Herbert B. (1977).
*Elements of set theory*. Elsevier. ISBN 978-0-12-238440-0.- An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)

- Euler, Leonhard (1822) [1770].
*Elements of Algebra*. John Hewlett and Francis Horner, English translators (3rd English ed.). Orme Longman. p. 170. ISBN 978-0-387-96014-2. Retrieved 4 July 2011. - Fjelstad, Paul (January 1995). "The repeating integer paradox".
*The College Mathematics Journal*.**26**(1): 11–15. doi:10.2307/2687285. JSTOR 2687285. - Gardiner, Anthony (2003) [1982].
*Understanding Infinity: The Mathematics of Infinite Processes*. Dover. ISBN 978-0-486-42538-2. - Gowers, Timothy (2002).
*Mathematics: A Very Short Introduction*. Oxford UP. ISBN 978-0-19-285361-5. - Grattan-Guinness, Ivor (1970).
*The development of the foundations of mathematical analysis from Euler to Riemann*. MIT Press. ISBN 978-0-262-07034-8. - Griffiths, H. B.; Hilton, P. J. (1970).
*A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation*. London: Van Nostrand Reinhold. ISBN 978-0-442-02863-3. Template:LCC.- This book grew out of a course for Birmingham-area grammar school mathematics teachers. The course was intended to convey a university-level perspective on school mathematics, and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of ideal theory, which is not reproduced here. (pp. vii, xiv)

- Katz, K.; Katz, M. (2010a). "When is .999... less than 1?".
*The Montana Mathematics Enthusiast*.**7**(1): 3–30. arXiv:1007.3018. Bibcode:2010arXiv1007.3018U. Archived from the original on 20 July 2011. Retrieved 4 July 2011. - Katz, Karin Usadi; Katz, Mikhail G. (2010b). "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era".
*Educational Studies in Mathematics*.**74**(3): 259. arXiv:1003.1501. Bibcode:2010arXiv1003.1501K. doi:10.1007/s10649-010-9239-4. - Kempner, A. J. (December 1936). "Anormal Systems of Numeration".
*The American Mathematical Monthly*.**43**(10): 610–617. doi:10.2307/2300532. JSTOR 2300532. - Komornik, Vilmos; Loreti, Paola (1998). "Unique Developments in Non-Integer Bases".
*The American Mathematical Monthly*.**105**(7): 636–639. doi:10.2307/2589246. JSTOR 2589246. - Leavitt, W. G. (1967). "A Theorem on Repeating Decimals".
*The American Mathematical Monthly*.**74**(6): 669–673. doi:10.2307/2314251. JSTOR 2314251. - Leavitt, W. G. (September 1984). "Repeating Decimals".
*The College Mathematics Journal*.**15**(4): 299–308. doi:10.2307/2686394. JSTOR 2686394. - Lightstone, A. H. (March 1972). "Infinitesimals".
*The American Mathematical Monthly*.**79**(3): 242–251. doi:10.2307/2316619. JSTOR 2316619. - Mankiewicz, Richard (2000).
*The story of mathematics*. Cassell. ISBN 978-0-304-35473-3.- Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)

- Maor, Eli (1987).
*To infinity and beyond: a cultural history of the infinite*. Birkhäuser. ISBN 978-3-7643-3325-6.- A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)

- Mazur, Joseph (2005).
*Euclid in the Rainforest: Discovering Universal Truths in Logic and Math*. Pearson: Pi Press. ISBN 978-0-13-147994-4. - Munkres, James R. (2000) [1975].
*Topology*(2e ed.). Prentice-Hall. ISBN 978-0-13-181629-9.- Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' 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)

- Núñez, Rafael (2006). "Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics".
*18 Unconventional Essays on the Nature of Mathematics*. Springer. pp. 160–181. ISBN 978-0-387-25717-4. Archived from the original on 18 July 2011. Retrieved 4 July 2011. - Pedrick, George (1994).
*A First Course in Analysis*. Springer. ISBN 978-0-387-94108-0. - Peressini, Anthony; Peressini, Dominic (2007). "Philosophy of Mathematics and Mathematics Education". In van Kerkhove, Bart; van Bendegem, Jean Paul (eds.).
*Perspectives on Mathematical Practices*. Logic, Epistemology, and the Unity of Science.**5**. Springer. ISBN 978-1-4020-5033-6. - Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense".
*American Mathematical Monthly*.**97**(5): 408–411. doi:10.2307/2324393. JSTOR 2324393. - Pinto, Márcia; Tall, David (2001).
*PME25: Following students' development in a traditional university analysis course*(PDF). pp. v4: 57–64. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009. - Protter, M. H.; Morrey, Jr., Charles B. (1991).
*A first course in real analysis*(2e ed.). Springer. ISBN 978-0-387-97437-8.- This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nondecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)

- Pugh, Charles Chapman (2001).
*Real mathematical analysis*. Springer-Verlag. ISBN 978-0-387-95297-0.- While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.

- Renteln, Paul; Dundes, Alan (January 2005). "Foolproof: A Sampling of Mathematical Folk Humor" (PDF).
*Notices of the AMS*.**52**(1): 24–34. Archived from the original (PDF) on 25 February 2009. Retrieved 3 May 2009. - Richman, Fred (December 1999). "Is 0.999... = 1?".
*Mathematics Magazine*.**72**(5): 396–400. doi:10.2307/2690798. JSTOR 2690798. Free HTML preprint: Richman, Fred (June 1999). "Is 0.999... = 1?". Archived from the original on 2 September 2006. Retrieved 23 August 2006. Note: the journal article contains material and wording not found in the preprint. - Robinson, Abraham (1996).
*Non-standard analysis*(Revised ed.). Princeton University Press. ISBN 978-0-691-04490-3. - Rosenlicht, Maxwell (1985).
*Introduction to Analysis*. Dover. ISBN 978-0-486-65038-8. This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition. - Rudin, Walter (1976) [1953].
*Principles of mathematical analysis*(3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.- A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)

- Shrader-Frechette, Maurice (March 1978). "Complementary Rational Numbers".
*Mathematics Magazine*.**51**(2): 90–98. doi:10.2307/2690144. JSTOR 2690144. - Smith, Charles; Harrington, Charles (1895).
*Arithmetic for Schools*. Macmillan. p. 115. ISBN 978-0-665-54808-6. Retrieved 4 July 2011. - Sohrab, Houshang (2003).
*Basic Real Analysis*. Birkhäuser. ISBN 978-0-8176-4211-2. - Starbird, M.; Starbird, T. (March 1992). "Required Redundancy in the Representation of Reals".
*Proceedings of the American Mathematical Society*.**114**(3): 769–774. doi:10.1090/S0002-9939-1992-1086343-5. JSTOR 2159403. - Stewart, Ian (1977).
*The Foundations of Mathematics*. Oxford UP. ISBN 978-0-19-853165-4. - Stewart, Ian (2009).
*Professor Stewart's Hoard of Mathematical Treasures*. Profile Books. ISBN 978-1-84668-292-6. - Stewart, James (1999).
*Calculus: Early transcendentals*(4e ed.). Brooks/Cole. ISBN 978-0-534-36298-0.- This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.

- Stillwell, John (1994),
*Elements of algebra: geometry, numbers, equations*, Springer - Tall, David; Schwarzenberger, R. L. E. (1978). "Conflicts in the Learning of Real Numbers and Limits" (PDF).
*Mathematics Teaching*.**82**: 44–49. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009. - Tall, David (1977). "Conflicts and Catastrophes in the Learning of Mathematics" (PDF).
*Mathematical Education for Teaching*.**2**(4): 2–18. Archived from the original (PDF) on 26 March 2009. Retrieved 3 May 2009. - Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology" (PDF).
*Mathematics Education Research Journal*.**12**(3): 210–230. Bibcode:2000MEdRJ..12..196T. doi:10.1007/BF03217085. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009. - von Mangoldt, Dr. Hans (1911). "Reihenzahlen".
*Einführung in die höhere Mathematik*(in German) (1st ed.). Leipzig: Verlag von S. Hirzel. - Wallace, David Foster (2003).
*Everything and more: a compact history of infinity*. Norton. ISBN 978-0-393-00338-3.

## Further reading[edit source | edit]

- Burkov, S. E. (1987). "One-dimensional model of the quasicrystalline alloy".
*Journal of Statistical Physics*.**47**(3/4): 409–438. Bibcode:1987JSP....47..409B. doi:10.1007/BF01007518. - Burn, Bob (March 1997). "81.15 A Case of Conflict".
*The Mathematical Gazette*.**81**(490): 109–112. doi:10.2307/3618786. JSTOR 3618786. - Calvert, J. B.; Tuttle, E. R.; Martin, Michael S.; Warren, Peter (February 1981). "The Age of Newton: An Intensive Interdisciplinary Course".
*The History Teacher*.**14**(2): 167–190. doi:10.2307/493261. JSTOR 493261. - Choi, Younggi; Do, Jonghoon (November 2005). "Equality Involved in 0.999... and (-8)
^{1}⁄_{3}".*For the Learning of Mathematics*.**25**(3): 13–15, 36. JSTOR 40248503. - Choong, K. Y.; Daykin, D. E.; Rathbone, C. R. (April 1971). "Rational Approximations to π".
*Mathematics of Computation*.**25**(114): 387–392. doi:10.2307/2004936. JSTOR 2004936. - Edwards, B. (1997). "An undergraduate student's understanding and use of mathematical definitions in real analysis". In Dossey, J.; Swafford, J.O.; Parmentier, M.; Dossey, A.E. (eds.).
*Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*.**1**. Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education. pp. 17–22. - Eisenmann, Petr (2008). "Why is it not true that 0.999... < 1?" (PDF).
*The Teaching of Mathematics*.**11**(1): 35–40. Retrieved 4 July 2011. - Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals".
*Journal for Research in Mathematics Education*.**41**(2): 117–146.- This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for 0.999... falling short of 1 by an infinitesimal 0.000...1.

- Ferrini-Mundy, J.; Graham, K. (1994). Kaput, J.; Dubinsky, E. (eds.). "Research in calculus learning: Understanding of limits, derivatives and integrals".
*MAA Notes: Research Issues in Undergraduate Mathematics Learning*.**33**: 31–45. - Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". arXiv:math.NT/0605182.
- Gardiner, Tony (June 1985). "Infinite processes in elementary mathematics: How much should we tell the children?".
*The Mathematical Gazette*.**69**(448): 77–87. doi:10.2307/3616921. JSTOR 3616921. - Monaghan, John (December 1988). "Real Mathematics: One Aspect of the Future of A-Level".
*The Mathematical Gazette*.**72**(462): 276–281. doi:10.2307/3619940. JSTOR 3619940. - Navarro, Maria Angeles; Carreras, Pedro Pérez (2010). "A Socratic methodological proposal for the study of the equality 0.999...=1" (PDF).
*The Teaching of Mathematics*.**13**(1): 17–34. Retrieved 4 July 2011. - Przenioslo, Malgorzata (March 2004). "Images of the limit of function formed in the course of mathematical studies at the university".
*Educational Studies in Mathematics*.**55**(1–3): 103–132. doi:10.1023/B:EDUC.0000017667.70982.05. - Sandefur, James T. (February 1996). "Using Self-Similarity to Find Length, Area, and Dimension".
*The American Mathematical Monthly*.**103**(2): 107–120. doi:10.2307/2975103. JSTOR 2975103. - Sierpińska, Anna (November 1987). "Humanities students and epistemological obstacles related to limits".
*Educational Studies in Mathematics*.**18**(4): 371–396. doi:10.1007/BF00240986. JSTOR 3482354. - Szydlik, Jennifer Earles (May 2000). "Mathematical Beliefs and Conceptual Understanding of the Limit of a Function".
*Journal for Research in Mathematics Education*.**31**(3): 258–276. doi:10.2307/749807. JSTOR 749807. - Tall, David O. (2009). "Dynamic mathematics and the blending of knowledge structures in the calculus".
*ZDM Mathematics Education*.**41**(4): 481–492. doi:10.1007/s11858-009-0192-6. - Tall, David O. (May 1981). "Intuitions of infinity".
*Mathematics in School*.**10**(3): 30–33. JSTOR 30214290.

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