0.999...

${\displaystyle \{x\colon x<1\}}$

${\displaystyle \{x\colon x\leq 1\}}$

${\textstyle {\frac {1}{3}}}$

Cauchy sequences

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 ${\displaystyle x}$ and ${\displaystyle y}$ is defined as the absolute value ${\displaystyle \left|x-y\right|}$, where the absolute value ${\displaystyle \left|z\right|}$ is defined as the maximum of ${\displaystyle z}$ and ${\displaystyle -z}$, 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 ${\displaystyle x_{0},x_{1},x_{2},\ldots }$, a mapping from natural numbers to rationals, for any positive rational ${\displaystyle \delta }$ there is an ${\displaystyle N}$ such that ${\displaystyle \left|x_{m}-x_{n}\right|\leq \delta }$ for all ${\displaystyle m,n>N}$; the distance between terms becomes smaller than any positive rational.^[30]

If ${\displaystyle (x_{n})}$ and ${\displaystyle (y_{n})}$ are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence ${\displaystyle (x_{n}-y_{n})}$ has the limit 0. Truncations of the decimal number ${\displaystyle b_{0}.b_{1}b_{2}b_{3}\ldots }$ generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.^[31] Thus in this formalism the task is to show that the sequence of rational numbers

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

${\displaystyle n}$-

${\displaystyle n\in \mathbb {N} }$

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

${\displaystyle 0.999\ldots =1}$

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.^[28] The above approach to decimal expansions, including the proof that ${\displaystyle 0.999\ldots =1}$, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation.^[33]

Infinite decimal representation

Commonly in

One of the notions that can resolve the issue is the requirement that real numbers be densely ordered. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal. Therefore, if ${\displaystyle 0.99999...}$ were to be different from ${\displaystyle 1}$, there would have to be another real number in between them but there is none: a single digit cannot be changed in either of the two to obtain such a number.^[35]

Generalizations

The result that ${\displaystyle 0.999\ldots =1}$ generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.^[36][9]

Second, a comparable theorem applies in each radix or

) 0.222... equals 1. In general, any terminating base ${\displaystyle b}$ expression has a counterpart with repeated trailing digits equal to ${\displaystyle b-1}$. 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 ${\displaystyle 1.000\ldots }$ and ${\displaystyle 0.101010\ldots }$, and there are infinitely many more representations that include adjacent 1s. Generally, for almost all ${\displaystyle q}$ between 1 and 2, there are uncountably many base-${\displaystyle q}$ expansions of 1. In contrast, there are still uncountably many ${\displaystyle q}$, including all natural numbers greater than 1, for which there is only one base-${\displaystyle q}$ expansion of 1, other than the trivial ${\displaystyle 1.000\ldots }$. This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik–Loreti constant ${\displaystyle q=1.787231650\ldots }$. In this base, ${\displaystyle 1=0.11010011001011010010110011010011\ldots }$; the digits are given by the Thue–Morse sequence, which does not repeat.^[38]

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:^[39]

• In the balanced ternary system, ${\textstyle {\frac {1}{2}}=0.111\ldots =1.{\underline {111}}\ldots }$.
• In the reverse factorial number system (using bases ${\displaystyle 2!,3!,4!,\ldots }$ for positions after the decimal point), ${\displaystyle 1=1.000\ldots =0.1234\ldots }$.

Impossibility of unique representation

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered

• If an interval of the real numbers is partitioned into two non-empty parts ${\displaystyle L}$, ${\displaystyle R}$, such that every element of ${\displaystyle L}$ is (strictly) less than every element of ${\displaystyle R}$, then either ${\displaystyle L}$ contains a largest element or ${\displaystyle R}$ contains a smallest element, but not both.
• The collection of infinite strings of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ${\displaystyle L}$, ${\displaystyle R}$, such that every element of ${\displaystyle L}$ is less than every element of ${\displaystyle R}$, while ${\displaystyle L}$ contains the largest element and ${\displaystyle R}$ contains the smallest element. Indeed, it suffices to take two finite
  prefixes
  (initial substrings) ${\displaystyle p_{1},p_{2}}$ of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ${\displaystyle L}$ the set of all strings in the collection whose corresponding prefix is at most ${\displaystyle p_{1}}$, and for ${\displaystyle R}$ the remainder, the strings in the collection whose corresponding prefix is at least ${\displaystyle p_{2}}$. Then ${\displaystyle L}$ has the largest element, starting with ${\displaystyle p_{1}}$ and choosing the largest available symbol in all following positions. In contrast, ${\displaystyle R}$ has the smallest element obtained by following ${\displaystyle p_{2}}$ by the smallest symbol in all positions.

The first point follows from the basic properties of real numbers: ${\displaystyle L}$ has a

${\displaystyle R}$

${\displaystyle R}$

${\displaystyle L}$

${\displaystyle L}$

${\displaystyle R}$

${\displaystyle p_{1}=0}$

${\displaystyle p_{2}=1}$

One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers.^[41] Examples include:

• ${\textstyle {\frac {1}{7}}=0.{\overline {142857}}}$ and ${\displaystyle 142+857=999}$.
• ${\textstyle {\frac {1}{73}}=0.{\overline {01369863}}}$ and ${\displaystyle 0136+9863=9999}$.

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 it can be proved that if a decimal of the form ${\displaystyle 0.b_{1}b_{2}b_{3}\ldots }$ is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.

Returning to real analysis, the base-3 analogue ${\displaystyle 0.222\ldots =1}$ plays a key role in the characterization of one of the simplest fractals, the middle-thirds Cantor set: a point in the unit interval lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ${\displaystyle n}$-th digit of the representation reflects the position of the point in the ${\displaystyle n}$-th stage of the construction. For example, the point ${\textstyle {\frac {2}{3}}}$ is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and the left of every deletion thereafter. The point ${\textstyle {\frac {1}{3}}}$ is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and the right of every deletion thereafter.^[44]

Repeating nines also turns up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying

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.^[h]
• 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".^[46]
• Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.^[47]

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999...

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered with his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven't specified how many places there are' or 'it is the nearest possible decimal below 1'".^[48]

The elementary argument of multiplying ${\textstyle 0.333\ldots ={\frac {1}{3}}}$ by 3 can convince reluctant students that ${\displaystyle 0.999\ldots =1}$. Still, when confronted with the conflict between their belief in the first equation and their disbelief in the second, some students either begin to disbelieve the first equation or simply become frustrated.^[49] 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 ${\displaystyle 0.999\ldots }$. For example, one real analysis student was able to prove that ${\textstyle 0.333\ldots ={\frac {1}{3}}}$ using a

${\displaystyle 0.999\ldots <1}$

${\textstyle {\frac {1}{3}}=0.333\ldots }$

Mazur (2005) 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 ${\displaystyle 9.99\ldots =10}$, calling it a "wildly imagined infinite growing process."^[51]

As part of Ed Dubinsky's APOS theory of mathematical learning, he 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 ${\textstyle {\frac {1}{3}}}$ as a number in its own right and to dealing with the set of natural numbers as a whole.^[52]

Cultural phenomenon

With the rise of the

The FAQ briefly covers ${\textstyle {\frac {1}{3}}}$, 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 ${\textstyle {\frac {1}{3}}}$ 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.^[55]

A Slate article reports that the concept of 0.999... is "hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums".^[56] 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 issued a "press release" on April Fools' Day 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.^[57]

Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical jokes, such as:^[58]

Q: How many mathematicians does it take to screw in a lightbulb?

A: 0.999999....

In alternative number systems

Although the real numbers form an extremely useful

argues in Mathematics: A Very Short Introduction that the resulting identity ${\displaystyle 0.999\ldots =1}$ 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.[59]

Infinitesimals

Some proofs that ${\displaystyle 0.999\ldots =1}$ rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference ${\displaystyle 1-0.999\ldots }$ must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is zero, and therefore the two values are the same.

However, there are mathematically coherent ordered

in (0, 1)^∗. Lightstone shows how to associate each number with a sequence of digits,

$${\displaystyle 0.d_{1}d_{2}d_{3}\dots ;\dots d_{\infty -1}d_{\infty }d_{\infty +1}\dots ,}$$

indexed by the

hypernatural

numbers. While he does not directly discuss 0.999..., he shows the real number ${\textstyle {\frac {1}{3}}}$ is represented by ${\displaystyle 0.333\ldots ;\ldots 333\ldots }$, which is a consequence of the

${\displaystyle 0.999\ldots ;\ldots 999\ldots =1}$

${\displaystyle 0.333\ldots ;\ldots 000\ldots }$

${\displaystyle 0.999\ldots ;\ldots 000\ldots }$

The standard definition of the number 0.999... is the limit of the sequence ${\displaystyle 0.9,0.99,0.999,\ldots }$. A different definition involves an ultralimit, i.e., the equivalence class ${\displaystyle [(0.9,0.99,0.999,\ldots )]}$ of this sequence in the

More generally, the hyperreal number ${\displaystyle u_{H}=0.999\ldots ;\ldots 999000\ldots }$, with last digit 9 at infinite

hypernatural

rank ${\displaystyle H}$, satisfies a strict inequality ${\displaystyle u_{H}<1}$. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be

$${\displaystyle {\underset {H}{0.\underbrace {999\ldots } }}\;=1\;-\;{\frac {1}{10^{H}}}.}$$

${\displaystyle 0.999\ldots <1}$

Hackenbush

Combinatorial game theory provides a generalized concept of number that encompasses the real numbers and much more besides.^[64] For example, in 1974, Elwyn Berlekamp described a correspondence between strings of red and blue segments in Hackenbush and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string ${\displaystyle LRRLRLRL\ldots }$ is ${\textstyle 0.010101_{2}\ldots ={\frac {1}{3}}}$. However, the value of ${\displaystyle LRLLL\ldots }$ (corresponding to ${\displaystyle 0.111\ldots _{2}}$ is infinitesimally less than 1. The difference between the two is the surreal number ${\textstyle {\frac {1}{\omega }}}$, where ${\displaystyle \omega }$ is the first infinite ordinal; the relevant game is ${\displaystyle LRRRR\ldots }$ or ${\displaystyle 0.000\ldots _{2}}$.^[k]

This is true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, ${\displaystyle 0.10111\ldots _{2}=0.11000\ldots _{2}}$, which are both equal to ${\textstyle {\frac {3}{4}}}$, but the first representation corresponds to the binary tree path ${\displaystyle LRLRLLL\ldots }$, while the second corresponds to the different path ${\displaystyle LRLLRRR\ldots }$.

Revisiting subtraction

Another manner in which the proofs might be undermined is if ${\displaystyle 1-0.999\ldots }$ simply does not exist because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include

. Richman considers two such systems, designed so that ${\displaystyle 0.999\ldots <1}$.

First,

lexicographical order

and an addition operation, noting that ${\displaystyle 0.999\ldots <1}$ simply because ${\displaystyle 0<1}$ in the ones place, but for any nonterminating ${\displaystyle x}$, one has ${\displaystyle 0.999\ldots +x=1+x}$. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to ${\textstyle {\frac {1}{3}}}$. 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 ${\displaystyle D}$", which is the set of Dedekind cuts of decimal fractions. Ordinarily, this definition leads to the real numbers, but for a decimal fraction ${\displaystyle d}$ he allows both the cut ${\displaystyle (-\infty ,d)}$ and the "principal cut" ${\displaystyle (-\infty ,d]}$. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again ${\displaystyle 0.999\ldots <1}$. There are no positive infinitesimals in cut ${\displaystyle D}$, but there is "a sort of negative infinitesimal," ${\displaystyle 0^{-}}$, which has no decimal expansion. He concludes that ${\displaystyle 0.999\ldots =1+0^{-}}$, while the equation "${\displaystyle 0.999\ldots +x=1}$" has no solution.^[l]

p-adic numbers

When asked about 0.999..., novices often believe there should be a "final 9", believing ${\displaystyle 1-0.999\ldots }$ 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....^[66] However, there is a system that contains an infinite string of 9s including a last 9.

The ${\displaystyle p}$-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the ${\displaystyle p}$-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to ${\displaystyle p}$, and much closer to ${\displaystyle p^{n}}$, than it is to 1.^[67] The ${\displaystyle p}$-adic numbers form a

${\displaystyle p}$

${\displaystyle p}$

${\displaystyle p}$-

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: ${\displaystyle 1+\ldots 999=\ldots 000=0}$, and so ${\displaystyle \ldots 999=-1}$.^[68] 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:^[69]

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

Compare with the series in the section above. A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if ${\displaystyle x=\ldots 999}$, then ${\displaystyle 10x=\ldots 990}$, so ${\displaystyle 10x=x-9}$, hence ${\displaystyle x=-1}$ again.^[68]

As a final extension, since ${\displaystyle 0.999\ldots =1}$ (in the reals) and ${\displaystyle \ldots 999=-1}$ (in the 10-adics), then by "blind faith and unabashed juggling of symbols"^[70] one may add the two equations and arrive at ${\displaystyle \ldots 999.999\ldots =0}$. 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

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.^[72]

${\displaystyle -0=0}$

• Finitism
• Informal mathematics
• Limit (mathematics)
• Series (mathematics)

Notes

1. ^ Cheng (2023), p. 141.
2. ^ Diamond (1955).
3. ^ Baldwin & Norton (2012).
4. ^ Meier & Smith (2017), §8.2.
5. ^ Stewart (2009), p. 175.
6. ^ Propp (2023).
7. ^ Stillwell (1994), p. 42.
8. ^ Earl & Nicholson (2021), "bound".
9. ^ ^{a b} Rosenlicht (1985), p. 27.
10. ^ Byers (2007), p. 39.
11. ^ ^{a b c} Richman (1999).
12. ^ Peressini & Peressini (2007), p. 186.
13. ^ Baldwin & Norton (2012); Katz & Katz (2010a).
14. ^ Cheng (2023), p. 136.
15. ^ Rudin (1976), p. 61, Theorem 3.26; Stewart (1999), p. 706.
16. ^ Euler (1822), p. 170.
17. ^ Grattan-Guinness (1970), p. 69; Bonnycastle (1806), p. 177.
18. ^ Stewart (1999), p. 706; Rudin (1976), p. 61; Protter & Morrey (1991), p. 213; Pugh (2002), p. 180; Conway (1978), p. 31.
19. ^ Davies (1846), p. 175; Smith & Harrington (1895), p. 115.
20. ^ Beals (2004), p. 22; Stewart (2009), p. 34.
21. ^ Bartle & Sherbert (1982), pp. 60–62; Pedrick (1994), p. 29; Sohrab (2003), p. 46.
22. ^ Apostol (1974), pp. 9, 11–12; Beals (2004), p. 22; Rosenlicht (1985), p. 27.
23. ^ Apostol (1974), p. 12.
24. ^ Cheng (2023), pp. 153–156.
25. ^ Conway (2001), pp. 25–27.
26. ^ Rudin (1976), pp. 3, 8.
27. ^ Richman (1999), p. 399.
28. ^ ^{a b} O'Connor & Robertson (2005).
29. ^ Richman (1999), p. 398–399. "Why do that? Precisely to rule out the existence of distinct numbers ${\textstyle 0.{\overline {9}}}$ and ${\displaystyle 1}$. [...] So we see that in the traditional definition of the real numbers, the equation ${\textstyle 0.{\overline {9}}=1}$ is built in at the beginning."
30. ^ Griffiths & Hilton (1970), p. 386, §24.2 "Sequences".
31. ^ Griffiths & Hilton (1970), pp. 388, 393.
32. ^ Griffiths & Hilton (1970), p. 395.
33. ^ Griffiths & Hilton (1970), pp. viii, 395.
34. ^ Li (2011).
35. ^ Artigue (2002), p. 212, "... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1).".
36. ^ Petkovšek (1990), p. 408.
37. ^ Protter & Morrey (1991), p. 503; Bartle & Sherbert (1982), p. 61.
38. ^ Komornik & Loreti (1998), p. 636.
39. ^ Kempner (1936), p. 611; Petkovšek (1990), p. 409.
40. ^ Petkovšek (1990), pp. 410–411.
41. ^ Goodwyn (1802); Dickson (1919), pp. 161.
42. ^ Leavitt (1984), p. 301.
43. ^ Ginsberg (2004), pp. 26–30; Lewittes (2006), pp. 1–3; Leavitt (1967), pp. 669, 673; Shrader-Frechette (1978), pp. 96–98.
44. ^ Pugh (2002), p. 97; Alligood, Sauer & Yorke (1996), pp. 150–152; Protter & Morrey (1991), p. 507; Pedrick (1994), p. 29.
45. ^ Rudin (1976), p. 50; Pugh (2002), p. 98.
46. ^ Tall & Schwarzenberger (1978), pp. 6–7; Tall (2000), p. 221.
47. ^ Tall & Schwarzenberger (1978), p. 6; Tall (2000), p. 221.
48. ^ Tall (2000), p. 221.
49. ^ Tall (1976), pp. 10–14.
50. ^ Pinto & Tall (2001), p. 5; Edwards & Ward (2004), pp. 416–417.
51. ^ Mazur (2005), pp. 137–141.
52. ^ Dubinsky et al. (2005), pp. 261–262.
53. ^ Richman (1999), p. 396.
54. ^ de Vreught (1994).
55. ^ Adams (2003).
56. ^ Ellenberg, Jordan (6 June 2014). "Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?". Slate.
57. ^ "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.
58. ^ Renteln & Dundes (2005), p. 27.
59. ^ Gowers (2002), p. 60.
60. ^ Lightstone (1972), pp. 245–247.
61. ^ Tao (2012), pp. 156–180.
62. ^ Katz & Katz (2010a).
63. ^ Katz & Katz (2010b); Ely (2010).
64. ^ Conway (2001), pp. 3–5, 12–13, 24–27.
65. ^ Richman (1999), pp. 397–399.
66. ^ Gardiner (2003), p. 98; Gowers (2002), p. 60.
67. ^ Mascari & Miola (1988), p. 83–84.
68. ^ ^{a b} Fjelstad (1995), p. 11.
69. ^ Fjelstad (1995), pp. 14–15.
70. ^ DeSua (1960), p. 901.
71. ^ DeSua (1960), p. 902–903.
72. ^ Wallace (2003), p. 51; Maor (1987), p. 17.
73. ^ Munkres (2000), p. 34, Exercise 1(c).
74. ^ Kroemer & Kittel (1980), p. 462. "Floating point types". MSDN C# Language Specification. Archived from the original on 24 August 2006. Retrieved 29 August 2006.

References

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• Dickson, Leonard Eugene (1919). History of the Theory of Numbers. Vol. 1. Carnegie Institution of Washington.
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• Earl, Richard; Nicholson, James (2021). The Concise Oxford Dictionary of Mathematics (6th ed.).
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• 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.
  JSTOR 4145268. Archived from the original
  (PDF) on 22 July 2011. Retrieved 4 July 2011.
• Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals". Journal for Research in Mathematics Education. 41 (2): 117–146.
  doi:10.5951/jresematheduc.41.2.0117
  .

  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.

• 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)

• ISBN 978-0-387-96014-2
  . Retrieved 4 July 2011.
• Finney, Ross L.; Weir, Muarice D.; Giordano, Frank tR. (2001). Thomas' Calculus: Early Transcendentals (10th ed.). New York: Addison-Wesley.
• Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15.
  JSTOR 2687285
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• ISBN 978-0-486-42538-2
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• Ginsberg, Brian (2004). "Midy's (nearly) secret theorem – an extension after 165 years". College Mathematics Journal. 35 (1).
• Goodwyn, H. (1802). "Curious properties of prime Numbers, taken as the Divisors of unity. By a Correspondent". Jour. Nat. Phil. Chem. Arts new series. 1: 314–316.
• ISBN 978-0-19-285361-5
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• ISBN 978-0-262-07034-8
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• Griffiths, H. B.;
  LCC QA37.2 G75
  .

  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.;
  S2CID 11544878. 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".
  S2CID 115168622
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• Kempner, A. J. (December 1936). "Anormal Systems of Numeration". The American Mathematical Monthly. 43 (10): 610–617.
  JSTOR 2300532
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• Komornik, Vilmos; Loreti, Paola (1998). "Unique Developments in Non-Integer Bases". The American Mathematical Monthly. 105 (7): 636–639.
  JSTOR 2589246
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• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2e ed.). W. H. Freeman. p. 462.
  ISBN 978-0-7167-1088-2
  .
• Li, Liangpan (March 2011). "A new approach to the real numbers".
  arXiv:1101.1800 [math.CA
  ].
• Leavitt, W. G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly. 74 (6): 669–673.
  JSTOR 2314251
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• Leavitt, W. G. (September 1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308.
  JSTOR 2686394
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• Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals".
  arXiv:math.NT/0605182
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• JSTOR 2316619
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• Mankiewicz, Richard (2000). The Story of Mathematics. Cassell.
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  Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)

• Mascari, Gianfranco; Miola, Alfonso (1988). "On the integration of numeric and algebraic computations". In Beth, Thomas; Clausen, Michael (eds.). Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra.
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• ISBN 978-3-7643-3325-6
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  A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)

• ISBN 978-0-13-147994-4
  .
• Meier, John; Smith, Derek (2017). Exploring Mathematics: An Engaging Introduction to Proof. Cambridge University Press.
  ISBN 978-1-107-12898-9
  .
• 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's treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)

• 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.
• 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.
• 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. Vol. 5. Springer.
  ISBN 978-1-4020-5033-6
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• JSTOR 2324393
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• 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.
• Propp, James (17 January 2023). "Denominators and Doppelgängers". Mathematical Enchantments. Retrieved 16 April 2024.
• 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)

• 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;
  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?".
  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.
• JSTOR j.ctt1cx3vb6
  .
• 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.
• 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.
  JSTOR 2690144
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• Smith, Charles; Harrington, Charles (1895). Arithmetic for Schools. Macmillan. p. 115.
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• Sohrab, Houshang (2003). Basic Real Analysis. Birkhäuser.
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• ISBN 978-1-84668-292-6
  .
• 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.

• ISBN 9783540942900
• 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.
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  S2CID 143438975. Archived from the original
  (PDF) on 30 May 2009. Retrieved 3 May 2009.
• Tao, Terence (2012). Higher order Fourier analysis (PDF). American Mathematical Society.
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Further reading

• Beswick, Kim (2004). "Why Does 0.999... = 1?: A Perennial Question and Number Sense". Australian Mathematics Teacher. 60 (4): 7–9.
• Burkov, S. E. (1987). "One-dimensional model of the quasicrystalline alloy". Journal of Statistical Physics. 47 (3/4): 409–438.
  S2CID 120281766
  .
• Burn, Bob (March 1997). "81.15 A Case of Conflict". The Mathematical Gazette. 81 (490): 109–112.
  S2CID 187823601
  .
• 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.
  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.
  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. Vol. 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.
• 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.
• Gardiner, Tony (June 1985). "Infinite processes in elementary mathematics: How much should we tell the children?". The Mathematical Gazette. 69 (448): 77–87.
  S2CID 125222118
  .
• Monaghan, John (December 1988). "Real Mathematics: One Aspect of the Future of A-Level". The Mathematical Gazette. 72 (462): 276–281.
  S2CID 125825964
  .
• ISBN 978-0-387-25717-4. Archived from the original
  on 18 July 2011. 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.
  S2CID 120453706
  .
• Sandefur, James T. (February 1996). "Using Self-Similarity to Find Length, Area, and Dimension". The American Mathematical Monthly. 103 (2): 107–120.
  JSTOR 2975103
  .
• S2CID 144880659
  .
• JSTOR 2159403
  .
• 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.
  JSTOR 749807
  .
• Tall, David O. (2009). "Dynamic mathematics and the blending of knowledge structures in the calculus". ZDM Mathematics Education. 41 (4): 481–492.
  S2CID 14289039
  .
• Tall, David O. (May 1981). "Intuitions of infinity". Mathematics in School. 10 (3): 30–33.
  JSTOR 30214290
  .

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