০.৯৯৯...ৰ উৎস চাওক

==প্ৰমাণ==
* <math>
\begin{align}
 \frac{1}{9} & = 0.111\dots \\
 9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
 1 & = 0.999\dots
\end{align}
</math>

এই প্ৰক্ৰিয়াই ৯ বিভাজক হিচাপে থকা অন্য সংখ্যাৰ সৈতে এই সংখ্যাৰ মিল আনে--

:<math>
\begin{align}
0.333\dots & = \frac{3}{9} \\
0.888\dots & = \frac{8}{9} \\
0.999\dots & = \frac{9}{9} = 1
\end{align}
</math>

* :<math>
\begin{align}
 x &= 0.999\ldots \\
10x &= 9.999\ldots & \text{multiply by }10\\
 10x &= 9+0.999\ldots \\
 10x &= 9 + x & \text{definition of }x\\
 9x &= 9 & \text{subtract }x\\
 x &= 1 & \text{divide by }9
\end{align}
</math>

এই প্ৰক্ৰিয়াই এই সত্য ব্যৱহাৰ কৰিছে যে কোনো সংখ্যাক ১০ৰে পুৰণ কৰিলে সংখ্যাটোৰ দশমিক পইণ্টৰ স্থান এক স্থান সোঁফালে যায়। 

===আলোচনা===
যদিও প্ৰাৰম্ভিক স্তৰত এই প্ৰক্ৰিয়াই ০.৯৯৯.... ১ৰ সমান বুলি প্ৰমাণিত কৰে, এনে প্ৰক্ৰিয়াই দেচিমেল আৰু সিহঁতে বুজোৱা সংখ্যাৰ মাজৰ সম্পৰ্কৰ বিষয়ে একোৱেই নকয়, যাক নাজানিলে দুটি সংখ্যানো কেতিয়া সমান হয় তাকেই জানিব পৰা নাযায়। .<ref>This argument is found in Peressini and Peressini p. 186. William Byers argues that a student who agrees that 0.999...&nbsp;=&nbsp;1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation (Byers pp. 39–41). Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".(p. 396)</ref>

==তথ্য সংগ্ৰহ আৰু টোকা==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, K. T.; Sauer, T. D.; Yorke, J. A. |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=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.&nbsp;9–11)
* {{cite book |author=Bartle, R. G.; Sherbert, D. R. |year=1982 |title=Introduction to real analysis |publisher=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)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9 |author1 = Berlekamp|first1 = E. R.|last2 = Conway|first2=J. H. |last3=Guy |first3=R. K. |authorlink1=Elwyn Berlekamp |authorlink2=John Horton Conway |authorlink3=Richard k. Guy }}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |conference=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |id = {{citeseerx|10.1.1.31.3019}} |year=1992}}
* {{cite journal|title=Why Does 0.999... = 1?: A Perennial Question and Number Sense|last1=Beswick|first1=Kim|journal=Australian Mathematics Teacher|volume=60|pages=7–9|year=2004|issue=4}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=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)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |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)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175 |accessdate=4 July 2011 }}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |jstor=2309468 |journal=The American Mathematical Monthly |volume=67 |pages=900–903 |doi=10.2307/2309468 |issue=9 |date=November 1960}}
* {{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 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2}}
* {{cite journal |author=Edwards, Barbara; Ward, 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 |accessdate=4 July 2011 |archiveurl=https://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |archivedate=22 July 2011 |issue=5 |date=May 2004 |deadurl=no }}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=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)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7 |accessdate=4 July 2011 }}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |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 |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{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 [[mathematics education|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)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/ |accessdate=4 July 2011 |archiveurl=https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/ |archivedate=20 July 2011 |deadurl=yes |bibcode=2010arXiv1007.3018U |arxiv=1007.3018 }}
* {{cite journal |last=Kempner |first=A. J. |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 |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=W. 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 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |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=A. H. |title=Infinitesimals |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 |title=The story of mathematics|publisher=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.&nbsp;8)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=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)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=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.&nbsp;30)
* {{cite book |last=Núñez |first=Rafael |chapter=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |title=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html |isbn=978-0-387-25717-4 |accessdate=4 July 2011 |archiveurl=https://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html |archivedate=18 July 2011 }}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |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 |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 |accessdate=2009-05-03 |archiveurl=https://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |archivedate=2009-05-30 }}
* {{cite book |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7 |author1 = Protter|first1 = M. H.|last2 = Morrey, Jr.|first2=Charles B.|authorlink2=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=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s 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.&nbsp;10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf |doi= |accessdate=2009-05-03 |archiveurl=https://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf |archivedate=2009-02-25 |date=January 2005 |deadurl=no }}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] volume=72 |volume=72 |issue=5 |pages=396–400 |date=December 1999}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred |last=Richman |title=Is 0.999... = 1? |accessdate=2006-08-23 |archiveurl=https://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm |archivedate=2006-09-02 |deadurl=no |date=June 1999 }} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=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 (pp.&nbsp;27–31) as infinite decimals with 0.999...&nbsp;=&nbsp;1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=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)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |pages=90–98 |date=March 1978}}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7 |accessdate=4 July 2011 }}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite journal | last1=Starbird | first1=M. | last2=Starbird | first2=T. | title= Required Redundancy in the Representation of Reals| volume=114 | pages=769–774|journal=Proceedings of the American Mathematical Society|jstor=2159403|publisher=AMS | issue=3 | doi=10.1090/S0002-9939-1992-1086343-5 |date=March 1992}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=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.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E. |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 |accessdate=2009-05-03 |archiveurl=https://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |archivedate=2009-05-30 |deadurl=no }}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1977 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03 |archiveurl=https://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |archivedate=2009-03-26 |deadurl=no }}
* {{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 |accessdate=2009-05-03 |archiveurl=https://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |archivedate=2009-05-30 |doi=10.1007/BF03217085 |bibcode=2000MEdRJ..12..196T |deadurl=no }}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

[[শ্ৰেণী:সংখ্যা]]