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In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the notations ''0.999…'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. |
In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the notations ''0.999…'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. |
Revision as of 21:44, 18 April 2010
- 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.[1]
- Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has a "point at infinity". Here, it makes sense to define 1/0 to be infinity;[2] and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.[3]
- Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.[4] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the sign and magnitude or one's complement formats, or floating point numbers as specified by the IEEE floating-point standard).[5][6]
See also
|
Notes
- ^ Wallace p.51, Maor p.17
- ^ See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57
- ^ Maor p.54
- ^ Munkres p.34, Exercise 1(c)
- ^ Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2e ed.). W. H. Freeman. p. 462. ISBN 0-7167-1088-9.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - ^ "Floating point types". MSDN C# Language Specification. Retrieved 2006-08-29.
References
- Alligood, Sauer, and Yorke (1996). "4.1 Cantor Sets". Chaos: An introduction to dynamical systems. Springer. ISBN 0-387-94677-2.
{{cite book}}
: CS1 maint: multiple names: authors list (link)- This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
- Apostol, Tom M. (1974). Mathematical analysis (2e ed.). Addison-Wesley. ISBN 0-201-00288-4.
- 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)
- Bartle, R.G. and D.R. Sherbert (1982). Introduction to real analysis. Wiley. ISBN 0-471-05944-7.
- This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
- Beals, Richard (2004). Analysis. Cambridge UP. ISBN 0-521-60047-2.
- Berlekamp, E.R.; J.H. Conway; and R.K. Guy (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 0-12-091101-9.
{{cite book}}
: CS1 maint: multiple names: authors list (link) - Berz, Martin (1992). "Automatic differentiation as nonarchimedean analysis". Computer Arithmetic and Enclosure Methods. Elsevier. pp. 439–450.
{{cite conference}}
: Unknown parameter|booktitle=
ignored (|book-title=
suggested) (help) - Bunch, Bryan H. (1982). Mathematical fallacies and paradoxes. Van Nostrand Reinhold. ISBN 0-442-24905-5.
- This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
- Burrell, Brian (1998). Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster. ISBN 0-87779-621-1.
- Byers, William (2007). How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton UP. ISBN 0-691-12738-7.
- Conway, John B. (1978) [1973]. Functions of one complex variable I (2e ed.). Springer-Verlag. ISBN 0-387-90328-3.
- This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
- Davies, Charles (1846). The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications. A.S. Barnes.
- DeSua, Frank C. (1960). "A system isomorphic to the reals". The American Mathematical Monthly. 67 (9): 900–903. doi:10.2307/2309468.
{{cite journal}}
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and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown (2005). "Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2". Educational Studies in Mathematics. 60: 253–266. doi:10.1007/s10649-005-0473-0.
{{cite journal}}
: CS1 maint: multiple names: authors list (link) - Edwards, Barbara and Michael Ward (2004). "Surprises from mathematics education research: Student (mis)use of mathematical definitions" (PDF). The American Mathematical Monthly. 111 (5): 411–425. doi:10.2307/4145268.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Enderton, Herbert B. (1977). Elements of set theory. Elsevier. ISBN 0-12-238440-7.
- An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
- Euler, Leonhard (1822) [1770]. John Hewlett and Francis Horner, English translators. (ed.). Elements of Algebra (3rd English ed.). Orme Longman. ISBN 0387960147.
{{cite book}}
:|editor=
has generic name (help) - Fjelstad, Paul (1995). "The repeating integer paradox". The College Mathematics Journal. 26 (1): 11–15. doi:10.2307/2687285.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Gardiner, Anthony (2003) [1982]. Understanding Infinity: The Mathematics of Infinite Processes. Dover. ISBN 0-486-42538-X.
- Gowers, Timothy (2002). Mathematics: A Very Short Introduction. Oxford UP. ISBN 0-19-285361-9.
- Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 0-262-07034-0.
- Griffiths, H.B. (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. London: Van Nostrand Reinhold. ISBN [[Special:BookSources/0-442-02863-6. LCC QA37.2 G75|0-442-02863-6. [[LCC (identifier)|LCC]] [https://catalog.loc.gov/vwebv/search?searchCode=CALL%2B&searchArg=QA37.2+G75&searchType=1&recCount=25 QA37.2 G75]]].
{{cite book}}
: Check|isbn=
value: invalid character (help); External link in
(help); Unknown parameter|isbn=
|coauthors=
ignored (|author=
suggested) (help)- This book grew out of a course for Birmingham-area grammar school mathematics teachers. The course was intended to convey a university-level perspective on school mathematics, and the book is aimed at students "who have reached roughly the level of completing one year of specialist mathematical study at a university". The real numbers are constructed in Chapter 24, "perhaps the most difficult chapter in the entire book", although the authors ascribe much of the difficulty to their use of ideal theory, which is not reproduced here. (pp.vii, xiv)
- Katz, K.; Katz, M. (2010a). "When is .999… less than 1?". The Montana Mathematics Enthusiast. 7 (1): 3–30.
- Kempner, A.J. (1936). "Anormal Systems of Numeration". The American Mathematical Monthly. 43 (10): 610–617. doi:10.2307/2300532.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Komornik, Vilmos; and Paola Loreti (1998). "Unique Developments in Non-Integer Bases". The American Mathematical Monthly. 105 (7): 636–639. doi:10.2307/2589246.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help)CS1 maint: multiple names: authors list (link) - Leavitt, W.G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly. 74 (6): 669–673. doi:10.2307/2314251.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help) - Leavitt, W.G. (1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. doi:10.2307/2686394.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". New York Number Theory Workshop on Combinatorial and Additive Number Theory. arXiv.
- Lightstone, A.H. (1972). "Infinitesimals". The American Mathematical Monthly. 79 (3): 242–251. doi:10.2307/2316619.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Mankiewicz, Richard (2000). The story of mathematics. Cassell. ISBN 0-304-35473-2.
- Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.8)
- Maor, Eli (1987). To infinity and beyond: a cultural history of the infinite. Birkhäuser. ISBN 3-7643-3325-1.
- A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
- Mazur, Joseph (2005). Euclid in the Rainforest: Discovering Universal Truths in Logic and Math. Pearson: Pi Press. ISBN 0-13-147994-6.
- Munkres, James R. (2000) [1975]. Topology (2e ed.). Prentice-Hall. ISBN 0-13-181629-2.
- Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
- Núñez, Rafael (2006). "Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics". 18 Unconventional Essays on the Nature of Mathematics. Springer. pp. 160–181. ISBN 978-0-387-25717-4.
{{cite conference}}
: Unknown parameter|booktitle=
ignored (|book-title=
suggested) (help) - Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 0-387-94108-8.
- Peressini, Anthony; Peressini, Dominic (2007). "Philosophy of Mathematics and Mathematics Education". In Bart van Kerkhove, Jean Paul van Bendegem (ed.). Perspectives on Mathematical Practices. Logic, Epistemology, and the Unity of Science. Vol. 5. Springer. ISBN 978-1-4020-5033-6.
- Petkovšek, Marko (1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. doi:10.2307/2324393.
{{cite journal}}
: More than one of|number=
and|issue=
specified (help); Unknown parameter|month=
ignored (help) - Pinto, Márcia and David Tall (2001). "Following students' development in a traditional university analysis course" (PDF). PME25. pp. v4: 57–64. Retrieved 2009-05-03.
{{cite conference}}
: Unknown parameter|booktitle=
ignored (|book-title=
suggested) (help) - Protter, M.H. and C.B. Morrey (1991). A first course in real analysis (2e ed.). Springer. ISBN 0-387-97437-7.
{{cite book}}
: CS1 maint: multiple names: authors list (link)- 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, nodecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.503–507)
- Pugh, Charles Chapman (2001). Real mathematical analysis. Springer-Verlag. ISBN 0-387-95297-7.
- 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 and Allan Dundes (2005). "Foolproof: A Sampling of Mathematical Folk Humor" (PDF). Notices of the AMS. 52 (1): 24–34. Retrieved 2009-05-03.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Richman, Fred (1999). "Is 0.999… = 1?". Mathematics Magazine. 72 (5): 396–400.
{{cite journal}}
: Unknown parameter|month=
ignored (help) Free HTML preprint: Richman, Fred (1999-06-08). "Is 0.999… = 1?". Retrieved 2006-08-23. Note: the journal article contains material and wording not found in the preprint. - Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2.
- Rosenlicht, Maxwell (1985). Introduction to Analysis. Dover. ISBN 0-486-65038-3. This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (p 27-31) as infinite decimals with 0.999…=1 as part of the definition.
- Rudin, Walter (1976) [1953]. Principles of mathematical analysis (3e ed.). McGraw-Hill. ISBN 0-07-054235-X.
- 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 (1978). "Complementary Rational Numbers". Mathematics Magazine. 51 (2): 90–98.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Smith, Charles and Charles Harrington (1895). Arithmetic for Schools. Macmillan.
- Sohrab, Houshang (2003). Basic Real Analysis. Birkhäuser. ISBN 0-8176-4211-0.
- Stewart, Ian (1977). The Foundations of Mathematics. Oxford UP. ISBN 0-19-853165-6.
- Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6.
- Stewart, James (1999). Calculus: Early transcendentals (4e ed.). Brooks/Cole. ISBN 0-534-36298-2.
- This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
- D.O. Tall and R.L.E. Schwarzenberger (1978). "Conflicts in the Learning of Real Numbers and Limits" (PDF). Mathematics Teaching. 82: 44–49. Retrieved 2009-05-03.
- Tall, David (1976/7). "Conflicts and Catastrophes in the Learning of Mathematics" (PDF). Mathematical Education for Teaching. 2 (4): 2–18. Retrieved 2009-05-03.
{{cite journal}}
: Check date values in:|year=
(help)CS1 maint: year (link) - Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology" (PDF). Mathematics Education Research Journal. 12 (3): 210–230. Retrieved 2009-05-03.
- von Mangoldt, Dr. Hans (1911). "Reihenzahlen". Einführung in die höhere Mathematik (in German) (1st ed.). Leipzig: Verlag von S. Hirzel.
- Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 0-393-00338-8.
Further reading
- Burkov, S. E. (1987). "One-dimensional model of the quasicrystalline alloy". Journal of Statistical Physics. 47 (3/4): 409. doi:10.1007/BF01007518.
- Burn, Bob (1997). "81.15 A Case of Conflict". The Mathematical Gazette. 81 (490): 109–112. doi:10.2307/3618786.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren (1981). "The Age of Newton: An Intensive Interdisciplinary Course". The History Teacher. 14 (2): 167–190. doi:10.2307/493261.
{{cite journal}}
: Unknown parameter|month=
ignored (help)CS1 maint: multiple names: authors list (link) - Choi, Younggi; Do, Jonghoon (2005). "Equality Involved in 0.999… and (-8)⅓". For the Learning of Mathematics. 25 (3): 13–15, 36.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - K. Y. Choong, D. E. Daykin, C. R. Rathbone (1971). "Rational Approximations to π". Mathematics of Computation. 25 (114): 387–392. doi:10.2307/2004936.
{{cite journal}}
: Unknown parameter|month=
ignored (help)CS1 maint: multiple names: authors list (link) - Ely, Robert (2010). "Nonstandard student conceptions about infinitesimals". Journal for Research in Mathematics Education. 41 (2): 117–146.
- This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for 0.999… falling short of 1 by an infinitesimal 0.000…1.
- Katz, Karin Usadi; Katz, Mikhail G. (2010b). "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era". Educational Studies in Mathematics. doi:10.1007/s10649-010-9239-4. See also arXiv:1003.1501.
- Gardiner, Tony (1985). "Infinite processes in elementary mathematics: How much should we tell the children?". The Mathematical Gazette. 69 (448): 77–87. doi:10.2307/3616921.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Monaghan, John (1988). "Real Mathematics: One Aspect of the Future of A-Level". The Mathematical Gazette. 72 (462): 276–281. doi:10.2307/3619940.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Przenioslo, Malgorzata (2004). "Images of the limit of function formed in the course of mathematical studies at the university". Educational Studies in Mathematics. 55 (1–3): 103–132. doi:10.1023/B:EDUC.0000017667.70982.05.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Sandefur, James T. (1996). "Using Self-Similarity to Find Length, Area, and Dimension". The American Mathematical Monthly. 103 (2): 107–120. doi:10.2307/2975103.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Sierpińska, Anna (1987). "Humanities students and epistemological obstacles related to limits". Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Szydlik, Jennifer Earles (2000). "Mathematical Beliefs and Conceptual Understanding of the Limit of a Function". Journal for Research in Mathematics Education. 31 (3): 258–276. doi:10.2307/749807.
{{cite journal}}
: Unknown parameter|month=
ignored (help) - Tall, David O. (2009). "Dynamic mathematics and the blending of knowledge structures in the calculus". ZDM Mathematics Education. 41 (4): 481–492. doi:10.1007/s11858-009-0192-6.
- Tall, David O. (1981). "Intuitions of infinity". Mathematics in School: 30–33.
{{cite journal}}
: Unknown parameter|month=
ignored (help)
External links
- .999999… = 1? from cut-the-knot
- Why does 0.9999… = 1 ?
- Ask A Scientist: Repeating Decimals
- Proof of the equality based on arithmetic
- Repeating Nines
- Point nine recurring equals one
- David Tall's research on mathematics cognition
- What is so wrong with thinking of real numbers as infinite decimals?
- Theorem 0.999… on Metamath
- Hackenstrings, and the 0.999… ?= 1 FAQ
- A Friendly Chat About Whether 0.999… = 1