{{Short description|Alternative decimal expansion of 1}}

{{Use dmy dates|date=May 2024}}

 

 

There are many other ways of showing this equality, from [[intuitive]] arguments to [[mathematical rigor|mathematically rigorous]] [[mathematical proof|proofs]]. The intuitive arguments are generally based on properties of [[finite decimal]]s that are extended without proof to infinite decimals. The proofs are generally based on basic properties of real numbers and methods of [[calculus]], such as [[series (mathematics)|series]] and [[limit (mathematics)|limit]]s. A question studied in [[mathematics education]] is why some people reject this equality.

 

== Elementary proof ==

 

It is possible to prove the equation {{nowrap|1=0.999... = 1}} using just the mathematical tools of comparison and addition of (finite) [[decimal number]]s, without any reference to more advanced topics such as [[series (mathematics)|series]] and [[limit (mathematics)|limits]]. The proof given [[#Rigorous proof|below]] is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the [[number line]], there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so {{nowrap|1=0.999... = 1}}.

=== Rigorous proof ===

 

 

 

 

This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into [[number systems]], such as [[hyperreal number]]s, with infinitely small numbers ([[infinitesimal]]s) and infinitely large numbers ([[infinite number]]s).{{sfnp|Stewart|2009|p=175}}{{sfnp|Propp|2023}} When using such systems, the notation 0.999... is generally not used, as there is no smallest number among the numbers larger than all 0.(9)_{{{math|''n''}}}.{{efn|For example, one can show this as follows: if {{math|''x''}} is any number such that {{nowrap|0.(9)_{{{math|''n''}}} ≤ {{math|''x'' < 1}}}}, then {{nowrap|0.(9)_{{{math|''n''}}−1} ≤ 10{{math|''x''}} − 9 < {{math|''x''}} < 1}}. Thus if {{math|''x''}} has this property for all {{math|''n''}}, the smaller number {{nowrap|10{{math|''x''}} − 9}} does, as well.}}

=== Least upper bounds and completeness ===

 

 

== Algebraic arguments <span class="anchor" id="Proofs"></span><span class="anchor" id="Algebraic"></span> ==

Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. {{harvtxt|Byers|2007}} discusses the argument that, in elementary school, one is taught that {{nowrap|1=<math display="inline"> \frac{1}{3} </math> = 0.333...}}, so, ignoring all essential subtleties, "multiplying" this identity by 3 gives {{nowrap|1=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...{{px2}}." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to&nbsp;1.{{sfnp|Byers|2007|p=39}} {{harvtxt|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.{{sfnp|Richman|1999}}

 

| 1a1 = Baldwin | 1a2 = Norton | 1y = 2012

| 2a1 = Katz | 2a2 = Katz | 2y = 2010a

 

== Analytic proofs <span class="anchor" id="Analytic"></span> ==

<math display="block"> b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\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>

 

| 1a1 = Rudin | 1y = 1976 | 1p = 61 | 1loc = Theorem 3.26

| 2a1 = Stewart | 2y = 1999 | 2p = 706

<math display="block"> ar + ar^2 + ar^3 + \cdots = \frac{ar}{1-r}.</math>

 

<math display="block"> 0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\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]]''.{{sfnp|Euler|1822|p=170}}

 

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 arguments|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...{{px2}}.{{sfnmp

| 1a1 = Grattan-Guinness | 1y = 1970 | 1p = 69

| 2a1 = Bonnycastle |2y=1806 | 2p = 177

| 1a1 = Stewart | 1y = 1999 | 1p = 706

| 2a1 = Rudin | 2y = 1976 | 2p = 61

}}

 

<math display="block"> 0.999\ldots \ \overset{\underset{\mathrm{def}}{}}{=} \ \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>

| 1a1 = Davies | 1y = 1846 | 1p = 175

| 2a1 = Smith | 2a2 = Harrington | 2y = 1895 | 2p = 115

{{further|Nested intervals}}

 

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

 

<math display="block"> x = b_0.b_1b_2b_3 \ldots \,. </math>

 

| 2a1 = Beals | 2y= 2004 | 2p = 22

| 3a1 = Rosenlicht | 3y = 1985 | 3p = 27

 

== Proofs from the construction of the real numbers <span class="anchor" id="Based on the construction of the real numbers"></span> ==

{{further|Cauchy sequence}}

 

 

If <math> (x_n) </math> and <math> (y_n) </math> are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence <math> (x_n - y_n) </math> has the limit 0. Truncations of the decimal number {{tmath|1= 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.{{sfnp|Griffiths|Hilton|1970|pp=388, 393}} Thus in this formalism the task is to show that the sequence of rational numbers

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

</math>

=== Infinite decimal representation ===

 

 

=== Dense order ===

{{further|Dense order}}

 

== Generalizations ==

 

| 1a1 = Protter | 1a2 = Morrey | 1y = 1991 | 1p = 503

| 2a1 = Bartle | 2a2 = Sherbert | 2y = 1982 | 2p = 61

}}

 

 

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense, the difficulties are even worse. For example:{{sfnmp

| 2a1 = Petkovšek | 2y = 1990 | 2p = 409

}}

* In the reverse [[factorial number system]] (using bases 2!, 3!, 4!,&nbsp;... for positions ''after'' the decimal point), {{nowrap|1=1 = 1.000... = 0.1234...}}{{px2}}.

 

== Applications ==

One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s.{{sfnmp|1a1=Goodwyn|1y=1802|2a1=Dickson|2y=1919|2pp=161}} Examples include:

| 1a1 = Ginsberg | 1y = 2004 | 1pp = 26–30

| 2a1 = Lewittes | 2y = 2006 | 2pp = 1–3

}}

 

Returning to real analysis, the base-3 analogue {{nowrap|1=0.222... = 1}} plays a key role in the characterization of one of the simplest [[fractal]]s, 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.

 

| 1a1 = Pinto | 1a2 = Tall | 1y = 2001 | 1p = 5

| 2a1 = Edwards | 2a2 = Ward | 2y = 2004 | 2pp = 416–417

 

{{harvtxt|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 {{nowrap|1=9.99... = 10}}, calling it a "wildly imagined infinite growing process".{{sfnp|Mazur|2005|pp=137–141}}

 

 

== Cultural phenomenon ==

 

A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via <math display="inline"> \frac{1}{3} </math> and limits, saying of misconceptions,

}}

 

0.999... features also in [[mathematical joke]]s, such as:{{sfnp|Renteln|Dundes|2005|p=27}}

{{blockquote|

A: 0.999999....

}}

 

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number <math display="inline"> \frac{1}{3} </math> is represented by 0.333...;...333..., which is a consequence of the [[transfer principle]]. As a consequence the number {{nowrap|1=0.999...;...999... = 1}}. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.{{sfnp|Lightstone|1972|pp=245–247}}

 

<math display="block"> \underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}. </math>

All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|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....{{efn|{{harvtxt|Stewart|2009}}, p.&nbsp;175; the full discussion of 0.999... is spread through pp. 172–175.}} Along with {{harvtxt|Katz|Katz|2010b}}, {{harvtxt|Ely|2010}} also questions the assumption that students' ideas about {{nowrap|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.{{sfnmp

[[Combinatorial game theory]] provides a generalized concept of number that encompasses the real numbers and much more besides.{{sfnp|Conway|2001|pp=3–5,12–13,24–27}} For example, in 1974, [[Elwyn Berlekamp]] described a correspondence between strings of red and blue segments in [[Hackenbush]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is {{nowrap|1=0.010101...₂ = <math display="inline"> \frac{1}{3} </math>.}} However, the value of LRLLL... (corresponding to 0.111...₂ is infinitesimally less than 1. The difference between the two is the [[surreal number]] <math display="inline"> \frac{1}{\omega} </math>, where <math> \omega </math> is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...₂.{{efn|{{harvtxt|Berlekamp|Conway|Guy|1982}}, pp. 79–80, 307–311 discuss 1 and {{sfrac|1|3}} and touch on {{sfrac|1|{{math|''ω''}}}}. The game for 0.111...₂ follows directly from Berlekamp's Rule.}}

 

 

=== Revisiting subtraction ===

First, {{harvtxt|Richman|1999}} defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that {{nowrap|0.999... < 1}} simply because {{nowrap|0 < 1}} in the ones place, but for any nonterminating {{tmath|1= x }}, one has {{nowrap|1=0.999... + <math> x </math> = 1 + {{tmath|1= x }}}}. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to {{tmath|1= \textstyle \frac{1}{3} }}. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.{{sfnp|Richman|1999|pp=397–399}}

 

 

=== ''p''-adic numbers ===

}} However, there is a system that contains an infinite string of 9s including a last 9.

 

 

The [[p-adic number|{{nowrap|1=<math> p </math>-}}adic numbers]] are an alternative number system of interest in [[number theory]]. Like the real numbers, the {{nowrap|1=<math> p </math>-}}adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to {{tmath|1= p }}, and much closer to {{tmath|1= p^n }}, than it is to 1.{{sfnp|Mascari|Miola|1988|p=[https://books.google.com/books?id=YVKzPSseyu4C&pg=PA83 83&ndash;84]}} The {{nowrap|1=<math> p </math>-}}adic numbers form a [[field (algebra)|field]] for prime <math> p </math> and a [[ring (mathematics)|ring]] for other {{tmath|1= p }}, including 10. So arithmetic can be performed in the {{nowrap|1=<math> p </math>-}}adics, and there are no infinitesimals.

<math display="block">\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math>

 

 

As a final extension, since {{nowrap|1=0.999... = 1}} (in the reals) and {{nowrap|1=...999 = −1}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"{{sfnp|DeSua|1960|p=901}} one may add the two equations and arrive at {{nowrap|1=...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]] [[positional notation|decimal expansion]] of the [[Solenoid (mathematics)#p-adic solenoids|10-adic solenoid]], with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.{{sfnp|DeSua|1960|p=902&ndash;903}}

 

== See also ==

* [[Informal mathematics]]

 

{{notelist|colwidth=30em}}

 

{{reflist|colwidth=30em}}

 

{{refbegin|colwidth=30em}}

* {{cite book |last1=Alligood |first1=K. T. |last2=Sauer |first2=T. D. |last3=Yorke |first3=J. A. |author-link3=James A. Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=978-0-387-94677-1}}

*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)

* {{cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |year=1974 |url=https://archive.org/details/mathematicalanal02edtomm |title=Mathematical Analysis |edition=2e |publisher=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.&nbsp;9–11)

* {{cite journal |last1=Baldwin |first1=Michael |last2=Norton |first2=Anderson |title=Does 0.999... Really Equal 1? |url=https://eric.ed.gov/?id=EJ961516 |year=2012 |journal=[[The Mathematics Educator]] |volume=21 |issue=2 |pages=58–67}}

* {{cite book |last1=Bartle |first1=R. G. |author-link1=Robert G. Bartle |last2=Sherbert |first2=D. R. |year=1982 |url=https://archive.org/details/introductiontore0000bart |title=Introduction to Real Analysis |publisher=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)

* {{cite book |last=Beals |first=Richard |author-link=Richard Beals (mathematician) |url=https://books.google.com/books?id=cXAqJUYqXx0C |title=Analysis: An Introduction |year=2004 |publisher=Cambridge University Press |isbn=978-0-521-60047-7}}

* {{cite book |year=1982 |title=Winning Ways for your Mathematical Plays |publisher=Academic Press |isbn=978-0-12-091101-1 |last1=Berlekamp |first1=E. R. |last2=Conway |first2=J. H. |last3=Guy |first3=R. K. |author-link1=Elwyn Berlekamp |author-link2=John Horton Conway |title-link=Winning Ways for your Mathematical Plays}}

* {{cite book |last=Bunch |first=Bryan H. |url=https://archive.org/details/mathematicalfall0000bunc |title=Mathematical Fallacies and Paradoxes |year=1982 |publisher=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)

* {{cite book |last=Conway |first=John H. |author-link=John Horton Conway |title=On Numbers and Games |title-link=On Numbers and Games |publisher=A K Peters |year=2001 |edition=2nd |isbn=1-56881-127-6}}

* {{cite book |last=Davies |first=Charles |author-link=Charles Davies (professor) |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=https://archive.org/details/universityarith00davigoog |page=[https://archive.org/details/universityarith00davigoog/page/n181 175] |access-date=4 July 2011}}

* {{cite journal |last=DeSua |first=Frank C. |title=A System Isomorphic to the Reals |url=https://archive.org/details/sim_american-mathematical-monthly_1960-11_67_9/page/900 |jstor=2309468 |journal=[[The American Mathematical Monthly]] |volume=67 |pages=900–903 |doi=10.2307/2309468 |issue=9 |date=November 1960}}

* {{cite journal |last=Diamond |first=Louis E. |title=Irrational Numbers |journal=[[Mathematics Magazine]] |publisher=Mathematical Association of America |year=1955 |volume=29 |number=2 |pages=89–99 |doi=10.2307/3029588 |jstor=3029588}}

* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |url=https://archive.org/details/sim_educational-studies-in-mathematics_2005_60_2/page/253 |journal=[[Educational Studies in Mathematics]] |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2 |s2cid=45937062}}

* {{cite book |last1=Earl |first1=Richard |last2=Nicholson |first2=James |title=The Concise Oxford Dictionary of Mathematics |edition=6th |year=2021 |publisher=Oxford University Press |isbn=978-0-192-58405-2}}

* {{cite journal |last1=Edwards |first1=Barbara |last2=Ward |first2=Michael |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=[[The American Mathematical Monthly]] |volume=111 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |jstor=4145268 |access-date=4 July 2011 |archive-url=https://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |archive-date=22 July 2011 |issue=5 |date=May 2004 |citeseerx=10.1.1.453.7466}}

*: 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 {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}

* {{cite book |last=Enderton |first=Herbert B. |author-link=Herbert Enderton |year=1977 |url=https://archive.org/details/elementsofsetthe0000ende |title=Elements of Set Theory |publisher=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)

* {{cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |orig-year=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |others=John Hewlett and Francis Horner, English translators |publisher=Orme Longman |url=https://archive.org/details/elementsalgebra00lagrgoog |page=[https://archive.org/details/elementsalgebra00lagrgoog/page/n205 170] |isbn=978-0-387-96014-2 |access-date=4 July 2011}}

* {{cite journal |last=Fjelstad |first=Paul |title=The Repeating Integer Paradox |url=https://www.tandfonline.com/doi/abs/10.1080/07468342.1995.11973659 |jstor=2687285 |journal=[[The College Mathematics Journal]] |volume=26 |pages=11–15 |doi=10.2307/2687285 |issue=1 |date=January 1995}}

* {{cite book |last=Gardiner |first=Anthony |author-link=Tony Gardiner |url=https://books.google.com/books?id=NiDCYJ8vrGQC |title=Understanding Infinity: The Mathematics of Infinite Processes |orig-year=1982 |year=2003 |publisher=Dover |isbn=978-0-486-42538-2}}

* {{cite book |last=Gowers |first=Timothy |author-link=William Timothy Gowers |url=https://books.google.com/books?id=DBxSM7TIq48C |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford University Press |isbn=978-0-19-285361-5}}

* {{cite book |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |year=1970 |title=The Development of the Foundations of Mathematical Analysis from Euler to Riemann |publisher=MIT Press |isbn=978-0-262-07034-8 |url=https://archive.org/details/developmentoffo00ivor}}

* {{cite journal |last=Kempner |first=Aubrey J. |author-link=Aubrey Kempner |title=Anormal Systems of Numeration |jstor=2300532 |journal=[[The American Mathematical Monthly]] |volume=43 |pages=610–617 |doi=10.2307/2300532 |issue=10 |date=December 1936}}

* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |author-link2=Paola Loreti |jstor=2589246 |journal=[[The American Mathematical Monthly]] |volume=105 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7}}

* {{cite journal |last=Leavitt |first=William G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=[[The American Mathematical Monthly]] |volume=74 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 |url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1047&context=mathfacpub}}

* {{cite journal |last=Leavitt |first=William G. |title=Repeating Decimals |url=https://archive.org/details/sim_college-mathematics-journal_1984-09_15_4/page/299 |jstor=2686394 |journal=[[The College Mathematics Journal]] |volume=15 |pages=299–308 |doi=10.2307/2686394 |issue=4 |date=September 1984}}

* {{cite journal |last=Lightstone |first=Albert H. |author-link=A. H. Lightstone |title=Infinitesimals |url=https://archive.org/details/sim_american-mathematical-monthly_1972-03_79_3/page/242 |jstor=2316619 |journal=[[The American Mathematical Monthly]] |volume=79 |pages=242–251 |doi=10.2307/2316619 |issue=3 |date=March 1972}}

* {{cite book |last=Mankiewicz |first=Richard |year=2000 |url=https://archive.org/details/storyofmathemati0000mank_k4e8 |title=The Story of Mathematics |publisher=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.&nbsp;8)

* {{cite book |last=Maor |first=Eli |author-link=Eli Maor |title=To Infinity and Beyond: A Cultural History of the Infinite |url=https://archive.org/details/toinfinitybeyond0000maor |url-access=registration |year=1987 |publisher=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)

* {{cite book |last=Mazur |first=Joseph |author-link=Joseph Mazur |url=https://books.google.com/books?id=7KSxBwAAQBAJ |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=978-0-13-147994-4}}

* {{cite book |last=Munkres |first=James R. |author-link=James Munkres |title=Topology |year=2000 |orig-year=1975 |edition=2e |publisher=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'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.&nbsp;30)

* {{cite journal |last=Petkovšek |first=Marko |author-link=Marko Petkovšek |title=Ambiguous Numbers are Dense |url=https://archive.org/details/sim_american-mathematical-monthly_1990-05_97_5/page/408 |jstor=2324393 |journal=[[American Mathematical Monthly]] |volume=97 |pages=408–411 |doi=10.2307/2324393 |issue=5 |date=May 1990}}

* {{cite book |last1=Pinto |first1=Márcia |last2=Tall |first2=David O. |author-link2=David Tall |title=PME25: Following students' development in a traditional university analysis course |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |archive-date=30 May 2009}}

* {{cite book |year=1991 |edition=2e |url=https://archive.org/details/firstcourseinrea0000prot |title=A First Course in Real Analysis |publisher=Springer |isbn=978-0-387-97437-8 |last1=Protter |first1=Murray H. |author-link1=Murray H. Protter |last2=Morrey |first2=Charles B. Jr. |author-link2=Charles B. Morrey}}

*: 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.&nbsp;56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.&nbsp;503–507)

* {{cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=978-0-534-36298-0 |url=https://archive.org/details/calculusearlytra00stew}}

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

* {{cite journal |last1=Tall |first1=David |author-link1=David Tall |last2=Schwarzenberger |first2=R. L. E. |author-link2=Rolph Ludwig Edward Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |archive-date=30 May 2009}}

* {{cite journal |last=Tall |first=David O. |author-link=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |archive-date=26 March 2009}}

* {{cite journal |last=Tall |first=David |title=Cognitive Development in Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |archive-date=30 May 2009 |doi=10.1007/BF03217085 |bibcode=2000MEdRJ..12..196T |s2cid=143438975}}

* {{cite web |last=Tao |first=Terence |author-link=Terence Tao |title=Math 131AH: Week 1 |url=https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf |website=Honors Analysis |publisher=UCLA Mathematics |year=2003 |access-date=23 May 2024}}

* {{cite book |last=Wallace |first=David Foster |author-link=David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=978-0-393-00338-3 |url=https://archive.org/details/everythingmore00davi}}

{{refend}}

 

=== Further reading ===

{{refbegin|colwidth=30em}}

* {{cite journal |journal=[[Journal of Statistical Physics]] |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409–438 |bibcode=1987JSP....47..409B |s2cid=120281766 |ref=none}}

* {{cite journal |title=81.15 A Case of Conflict |url=https://archive.org/details/sim_mathematical-gazette_1997-03_81_490/page/109 |first=Bob |last=Burn |journal=[[The Mathematical Gazette]] |volume=81 |issue=490 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |date=March 1997 |s2cid=187823601 |ref=none}}

{{Real numbers}}

{{portal bar|Mathematics}}

 

[[Category:1 (number)]]

[[Category:Mathematical paradoxes]]

[[Category:Real numbers]]

[[Category:Articles containing proofs]]