References analysis of the Wikipedia article "0.999..." in the English version

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The limit follows, for example, from Rudin (1976), p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir & Giordano (2001), section 8.1, example 2(a), example 6(b). 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)

Finney, Ross L.; Weir, Maurice D.; Giordano, Frank R. (2001). Thomas' Calculus: Early Transcendentals (10th ed.). New York: Addison-Wesley.

The historical synthesis is claimed by Griffiths & Hilton (1970), p. xiv and again by Pugh (2002), p. 10; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh (2002), p. 17 or Rudin (1976), p. 17. For viewpoints on logic, see Pugh (2002), p. 10, Rudin (1976), p.ix, or Munkres (2000), p. 30. 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. 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)

Pugh, Charles Chapman (2002). 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.

Pugh, Charles Chapman (2002). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.

Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

Pugh, Charles Chapman (2002). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.

Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

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'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)

Enderton (1977), 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 ..." 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)

Rudin (1976), pp. 17–20, Richman (1999), p. 399, or Enderton (1977), 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 "non-principal Dedekind cut". Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

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. Enderton, Herbert B. (1977). Elements of Set Theory. Elsevier. ISBN 978-0-12-238440-0.

Maor (1987), p. 60 and Mankiewicz (2000), p. 151 review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (2000), p. 50 mentions the latter method. 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)

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)

Munkres, James R. (2000) [1975]. Topology (2e ed.). Prentice-Hall. ISBN 978-0-13-181629-9.

Bunch (1982), p. 119; Tall & Schwarzenberger (1978), p. 6. The last suggestion is due to Burrell (1998), 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." 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)

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. Burrell, Brian (1998). Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster. ISBN 978-0-87779-621-3.

Stewart (2009), p. 175; the full discussion of 0.999... is spread through pp. 172–175. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6.

Richman (1999), pp. 398–400. Rudin (1976), p. 23 assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1. 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. Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

Stewart (2009), p. 175. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6.

Rosenlicht (1985), p. 27. 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.

Byers (2007), p. 39. Byers, William (2007). How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton University Press. ISBN 978-0-691-12738-5.

Richman (1999). 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.

Peressini & Peressini (2007), p. 186. 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.

Rudin (1976), p. 61, Theorem 3.26; Stewart (1999), p. 706. Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

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.

Euler (1822), p. 170. Euler, Leonhard (1822) [1770]. Elements of Algebra. John Hewlett and Francis Horner, English translators (3rd English ed.). Orme Longman. p. 170. Retrieved 4 July 2011.

Grattan-Guinness (1970), p. 69; Bonnycastle (1806), p. 177. Grattan-Guinness, Ivor (1970). The Development of the Foundations of Mathematical Analysis from Euler to Riemann. MIT Press. ISBN 978-0-262-07034-8. Bonnycastle, John (1806). An introduction to algebra; with notes and observations: designed for the use of schools and places of public education (First American ed.). Philadelphia. hdl:2027/mdp.39015063620382.

Stewart (1999), p. 706; Rudin (1976), p. 61; Protter & Morrey (1991), p. 213; Pugh (2002), p. 180; Conway (1978), p. 31. 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.

Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

Protter, Murray H.; Morrey, Charles B. Jr. (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 (2002). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.

Conway, John B. (1978) [1973]. Functions of One Complex Variable I (2e ed.). Springer-Verlag. ISBN 978-0-387-90328-6.

Davies (1846), p. 175; Smith & Harrington (1895), p. 115. Davies, Charles (1846). The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications. A.S. Barnes. p. 175. Retrieved 4 July 2011. Smith, Charles; Harrington, Charles (1895). Arithmetic for Schools. Macmillan. p. 115. Retrieved 4 July 2011.

Beals (2004), p. 22; Stewart (2009), p. 34. Beals, Richard (2004). Analysis: An Introduction. Cambridge University Press. ISBN 978-0-521-60047-7. Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6.

Bartle & Sherbert (1982), pp. 60–62; Pedrick (1994), p. 29; Sohrab (2003), p. 46. 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)

Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 978-0-387-94108-0. Sohrab, Houshang (2003). Basic Real Analysis. Birkhäuser. ISBN 978-0-8176-4211-2.

Apostol (1974), pp. 9, 11–12; Beals (2004), p. 22; Rosenlicht (1985), p. 27. 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)

Beals, Richard (2004). Analysis: An Introduction. Cambridge University Press. ISBN 978-0-521-60047-7. 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.

Apostol (1974), p. 12. Apostol, Tom M. (1974). Mathematical Analysis (2e ed.). Addison-Wesley. ISBN 978-0-201-00288-1.

Rudin (1976), pp. 3, 8. Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

Richman (1999), p. 399. 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.

Richman (1999), p. 398–399. "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." 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.

Griffiths & Hilton (1970), p. 386, §24.2 "Sequences". 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. LCC QA37.2 G75.

Griffiths & Hilton (1970), pp. 388, 393. 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. LCC QA37.2 G75.

Griffiths & Hilton (1970), p. 395. 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. LCC QA37.2 G75.

Griffiths & Hilton (1970), pp. viii, 395. 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. LCC QA37.2 G75.

Petkovšek (1990), p. 408. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Protter & Morrey (1991), p. 503; Bartle & Sherbert (1982), p. 61. Protter, Murray H.; Morrey, Charles B. Jr. (1991). A First Course in Real Analysis (2e ed.). Springer. ISBN 978-0-387-97437-8.

Bartle, R. G.; Sherbert, D. R. (1982). Introduction to Real Analysis. Wiley. ISBN 978-0-471-05944-8.

Kempner (1936), p. 611; Petkovšek (1990), p. 409. Kempner, Aubrey J. (December 1936). "Anormal Systems of Numeration". The American Mathematical Monthly. 43 (10): 610–617. doi:10.2307/2300532. JSTOR 2300532. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Petkovšek (1990), pp. 410–411. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Goodwyn (1802); Dickson (1919), pp. 161. Goodwyn, H. (1802). "Curious properties of prime Numbers, taken as the Divisors of unity. By a Correspondent". Journal of Natural Philosophy, Chemistry, and the Arts. New Series. 1: 314–316. Dickson, Leonard Eugene (1919). History of the Theory of Numbers. Vol. 1. Carnegie Institution of Washington.

Leavitt (1984), p. 301. Leavitt, William G. (September 1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. doi:10.2307/2686394. JSTOR 2686394.

Pugh (2002), p. 97; Alligood, Sauer & Yorke (1996), pp. 150–152; Protter & Morrey (1991), p. 507; Pedrick (1994), p. 29. Pugh, Charles Chapman (2002). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.

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)

Protter, Murray H.; Morrey, Charles B. Jr. (1991). A First Course in Real Analysis (2e ed.). Springer. ISBN 978-0-387-97437-8.

Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 978-0-387-94108-0.

Rudin (1976), p. 50; Pugh (2002), p. 98. Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

Pugh, Charles Chapman (2002). Real Mathematical Analysis. Springer-Verlag. ISBN 978-0-387-95297-0.

Dubinsky et al. (2005), pp. 261–262. 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. S2CID 45937062.

Richman (1999), p. 396. 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.

Richman (1999); Adams (2003); Ellenberg (2014). 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. Ellenberg, Jordan (6 June 2014). "Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?". Slate. Archived from the original on 8 August 2023.

Wallace (2003), p. 51; Maor (1987), p. 17. Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 978-0-393-00338-3. Maor, Eli (1987). To Infinity and Beyond: A Cultural History of the Infinite. Birkhäuser. ISBN 978-3-7643-3325-6.

Lightstone (1972), pp. 245–247. Lightstone, Albert H. (March 1972). "Infinitesimals". The American Mathematical Monthly. 79 (3): 242–251. doi:10.2307/2316619. JSTOR 2316619.

Katz & Katz (2010b); Ely (2010). 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. S2CID 115168622. 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.

Richman (1999), pp. 397–399. 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.

DeSua (1960), p. 901. 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.

DeSua (1960), p. 902–903. 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.

doi.org

Diamond (1955). Diamond, Louis E. (1955). "Irrational Numbers". Mathematics Magazine. 29 (2). Mathematical Association of America: 89–99. doi:10.2307/3029588. JSTOR 3029588.

Baldwin & Norton (2012); Katz & Katz (2010a). Baldwin, Michael; Norton, Anderson (2012). "Does 0.999... Really Equal 1?". The Mathematics Educator. 21 (2): 58–67. Katz, Karin Usadi; Katz, Mikhail G. (2010a). "When is .999... less than 1?". The Montana Mathematics Enthusiast. 7 (1): 3–30. arXiv:1007.3018. Bibcode:2010arXiv1007.3018U. doi:10.54870/1551-3440.1381. S2CID 11544878. Archived from the original on 20 July 2011. Retrieved 4 July 2011.

Tall (2000), p. 221. 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. S2CID 143438975. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.

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).". Artigue, Michèle (2002). Holton, Derek; Artigue, Michèle; Kirchgräber, Urs; Hillel, Joel; Niss, Mogens; Schoenfeld, Alan (eds.). The Teaching and Learning of Mathematics at University Level. New ICMI Study Series. Vol. 7. Springer, Dordrecht. doi:10.1007/0-306-47231-7. ISBN 978-0-306-47231-2.

Petkovšek (1990), p. 408. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Komornik & Loreti (1998), p. 636. 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.

Petkovšek (1990), pp. 410–411. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Leavitt (1984), p. 301. Leavitt, William G. (September 1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. doi:10.2307/2686394. JSTOR 2686394.

Ginsberg (2004), pp. 26–30; Lewittes (2006), pp. 1–3; Leavitt (1967), pp. 669, 673; Shrader-Frechette (1978), pp. 96–98. Ginsberg, Brian (2004). "Midy's (nearly) secret theorem – an extension after 165 years". The College Mathematics Journal. 35 (1): 26–30. doi:10.1080/07468342.2004.11922047. Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". arXiv:math.NT/0605182. Leavitt, William G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly. 74 (6): 669–673. doi:10.2307/2314251. JSTOR 2314251. Shrader-Frechette, Maurice (March 1978). "Complementary Rational Numbers". Mathematics Magazine. 51 (2): 90–98. doi:10.2307/2690144. JSTOR 2690144.

Tall & Schwarzenberger (1978), pp. 6–7; Tall (2000), p. 221. 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 (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. S2CID 143438975. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.

Tall & Schwarzenberger (1978), p. 6; Tall (2000), p. 221. 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 (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. S2CID 143438975. Archived from the original (PDF) on 30 May 2009. Retrieved 3 May 2009.

Pinto & Tall (2001), p. 5; Edwards & Ward (2004), pp. 416–417. Pinto, Márcia; Tall, David O. (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. 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.

Lightstone (1972), pp. 245–247. Lightstone, Albert H. (March 1972). "Infinitesimals". The American Mathematical Monthly. 79 (3): 242–251. doi:10.2307/2316619. JSTOR 2316619.

Katz & Katz (2010a). Katz, Karin Usadi; Katz, Mikhail G. (2010a). "When is .999... less than 1?". The Montana Mathematics Enthusiast. 7 (1): 3–30. arXiv:1007.3018. Bibcode:2010arXiv1007.3018U. doi:10.54870/1551-3440.1381. S2CID 11544878. Archived from the original on 20 July 2011. Retrieved 4 July 2011.

Mascari & Miola (1988), p. 83–84. 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. doi:10.1007/BFb0039172. ISBN 978-3-540-39133-3.

Fjelstad (1995), p. 11. Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285.

Fjelstad (1995), pp. 14–15. Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285.

DeSua (1960), p. 901. 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.

DeSua (1960), p. 902–903. 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.

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For a full treatment of non-standard numbers, see Robinson (1996). Robinson, Abraham (1996). Non-standard Analysis (Revised ed.). Princeton University Press. ISBN 978-0-691-04490-3. JSTOR j.ctt1cx3vb6.

Diamond (1955). Diamond, Louis E. (1955). "Irrational Numbers". Mathematics Magazine. 29 (2). Mathematical Association of America: 89–99. doi:10.2307/3029588. JSTOR 3029588.

Petkovšek (1990), p. 408. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Petkovšek (1990), pp. 410–411. Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393.

Leavitt (1984), p. 301. Leavitt, William G. (September 1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. doi:10.2307/2686394. JSTOR 2686394.

Lightstone (1972), pp. 245–247. Lightstone, Albert H. (March 1972). "Infinitesimals". The American Mathematical Monthly. 79 (3): 242–251. doi:10.2307/2316619. JSTOR 2316619.

Fjelstad (1995), p. 11. Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285.

Fjelstad (1995), pp. 14–15. Fjelstad, Paul (January 1995). "The Repeating Integer Paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285.

DeSua (1960), p. 901. 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.

DeSua (1960), p. 902–903. 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.

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Ellenberg (2014). Ellenberg, Jordan (6 June 2014). "Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?". Slate. Archived from the original on 8 August 2023.

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0.999... (English Wikipedia)

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