# ০.৯৯৯...

## প্ৰমাণ

• {\begin{aligned}{\frac {1}{9}}&=0.111\dots \\9\times {\frac {1}{9}}&=9\times 0.111\dots \\1&=0.999\dots \end{aligned}}

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

{\begin{aligned}0.333\dots &={\frac {3}{9}}\\0.888\dots &={\frac {8}{9}}\\0.999\dots &={\frac {9}{9}}=1\end{aligned}}
• :{\begin{aligned}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{aligned}}

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। .

## তথ্য সংগ্ৰহ আৰু টোকা

• Alligood, K. T.; Sauer, T. D.; Yorke, J. A. (1996). "4.1 Cantor Sets". Chaos: An introduction to dynamical systems. Springer. ISBN 0-387-94677-2.
This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
• Apostol, Tom M. (1974). Mathematical analysis (2e সম্পাদনা). 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.; Sherbert, D. R. (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.; Conway, J. H.; Guy, R. K. (1982). Winning Ways for your Mathematical Plays. Academic Press. ISBN 0-12-091101-9.
• Berz, Martin (1992). "Automatic differentiation as nonarchimedean analysis". Computer Arithmetic and Enclosure Methods. Elsevier. pp. 439–450. সাঁচ:Citeseerx.
• Beswick, Kim (2004). "Why Does 0.999... = 1?: A Perennial Question and Number Sense". Australian Mathematics Teacher খণ্ড 60 (4): 7–9.
• 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) . Functions of one complex variable I (2e সম্পাদনা). 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। আহৰণ কৰা হৈছে: 4 July 2011.
• DeSua, Frank C. (November 1960). "A system isomorphic to the reals". The American Mathematical Monthly খণ্ড 67 (9): 900–903. doi:10.2307/2309468.
• 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 খণ্ড 60 (2): 253–266. doi:10.1007/s10649-005-0473-0.
• Edwards, Barbara; Ward, Michael (May 2004). "Surprises from mathematics education research: Student (mis)use of mathematical definitions". The American Mathematical Monthly খণ্ড 111 (5): 411–425. doi:10.2307/4145268. Archived from the original on 22 July 2011। আহৰণ কৰা হৈছে: 4 July 2011.
• 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) . John Hewlett and Francis Horner, English translators.. ed. Elements of Algebra (3rd English সম্পাদনা). Orme Longman. ISBN 0-387-96014-7। আহৰণ কৰা হৈছে: 4 July 2011.
• Fjelstad, Paul (January 1995). "The repeating integer paradox". The College Mathematics Journal খণ্ড 26 (1): 11–15. doi:10.2307/2687285.
• Gardiner, Anthony (2003) . 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.; Hilton, P. J. (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. প্ৰকাশক London: Van Nostrand Reinhold. সাঁচ:LCC. ISBN 0-442-02863-6.
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 খণ্ড 7 (1): 3–30. Archived from the original on 20 July 2011। আহৰণ কৰা হৈছে: 4 July 2011.
• Kempner, A. J. (December 1936). "Anormal Systems of Numeration". The American Mathematical Monthly খণ্ড 43 (10): 610–617. doi:10.2307/2300532.
• Komornik, Vilmos; Loreti, Paola (1998). "Unique Developments in Non-Integer Bases". The American Mathematical Monthly খণ্ড 105 (7): 636–639. doi:10.2307/2589246.
• Leavitt, W. G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly খণ্ড 74 (6): 669–673. doi:10.2307/2314251.
• Leavitt, W. G. (September 1984). "Repeating Decimals". The College Mathematics Journal খণ্ড 15 (4): 299–308. doi:10.2307/2686394.
• Lightstone, A. H. (March 1972). "Infinitesimals". The American Mathematical Monthly খণ্ড 79 (3): 242–251. doi:10.2307/2316619.
• 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) . Topology (2e সম্পাদনা). 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. পৃষ্ঠা. 160–181. ISBN 978-0-387-25717-4. Archived from the original on 18 July 2011। আহৰণ কৰা হৈছে: 4 July 2011.
• 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. Perspectives on Mathematical Practices. Logic, Epistemology, and the Unity of Science. 5. Springer. ISBN 978-1-4020-5033-6.
• Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly খণ্ড 97 (5): 408–411. doi:10.2307/2324393.
• Pinto, Márcia; Tall, David (2001). PME25: Following students' development in a traditional university analysis course. পৃষ্ঠা. v4: 57–64. Archived from the original on 30 May 2009। আহৰণ কৰা হৈছে: 2009-05-03.
• Protter, M. H.; Morrey, Jr., Charles B. (1991). A first course in real analysis (2e সম্পাদনা). Springer. ISBN 0-387-97437-7.
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 (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; Dundes, Allan (January 2005). "Foolproof: A Sampling of Mathematical Folk Humor". Notices of the AMS খণ্ড 52 (1): 24–34. Archived from the original on 25 February 2009। আহৰণ কৰা হৈছে: 2009-05-03.
• Richman, Fred (December 1999). "Is 0.999... = 1?". Mathematics Magazine volume=72 খণ্ড 72 (5): 396–400. doi:10.2307/2690798.  Free HTML preprint: Richman, Fred (June 1999). "Is 0.999... = 1?". Archived from the original on 2 September 2006। আহৰণ কৰা হৈছে: 2006-08-23.  Note: the journal article contains material and wording not found in the preprint.
• Robinson, Abraham (1996). Non-standard analysis (Revised সম্পাদনা). 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 (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
• Rudin, Walter (1976) . Principles of mathematical analysis (3e সম্পাদনা). 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 (March 1978). "Complementary Rational Numbers". Mathematics Magazine খণ্ড 51 (2): 90–98. doi:10.2307/2690144.
• Smith, Charles; Harrington, Charles (1895). Arithmetic for Schools. Macmillan. ISBN 0-665-54808-7। আহৰণ কৰা হৈছে: 4 July 2011.
• Sohrab, Houshang (2003). Basic Real Analysis. Birkhäuser. ISBN 0-8176-4211-0.
• Starbird, M.; Starbird, T. (March 1992). "Required Redundancy in the Representation of Reals". Proceedings of the American Mathematical Society (AMS) খণ্ড 114 (3): 769–774. doi:10.1090/S0002-9939-1992-1086343-5.
• 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 সম্পাদনা). 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.
• Tall, D. O.; Schwarzenberger, R. L. E. (1978). "Conflicts in the Learning of Real Numbers and Limits". Mathematics Teaching খণ্ড 82: 44–49. Archived from the original on 30 May 2009। আহৰণ কৰা হৈছে: 2009-05-03.
• Tall, David (1977). "Conflicts and Catastrophes in the Learning of Mathematics". Mathematical Education for Teaching খণ্ড 2 (4): 2–18. Archived from the original on 26 March 2009। আহৰণ কৰা হৈছে: 2009-05-03.
• Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology". Mathematics Education Research Journal খণ্ড 12 (3): 210–230. doi:10.1007/BF03217085. Archived from the original on 30 May 2009। আহৰণ কৰা হৈছে: 2009-05-03.
• von Mangoldt, Dr. Hans (1911). "Reihenzahlen" (German ভাষাত). Einführung in die höhere Mathematik (1st সম্পাদনা). প্ৰকাশক Leipzig: Verlag von S. Hirzel.
• Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 0-393-00338-8.
1. This argument is found in Peressini and Peressini p. 186. William Byers argues that a student who agrees that 0.999... = 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)