০.৯৯৯...

**০.৯৯৯...** বা **০.(৯)** বা **০.৯...** বা **** বা **0.<span style="text-decoration: overline;">৯</span>** এটি বাস্তৱ সংখ্যা। এই সংখ্যাৰ মূল্য ১ৰ সমান বুলি দেখুৱাব পৰা যায়।

## প্ৰমাণ
* 
\begin{align}
 \frac{1}{9} & = 0.111\dots \\
 9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
 1 & = 0.999\dots
\end{align}


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

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


* :
\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}


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

* 
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* 
*: 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)
* 
*: 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)
* 
* 
* 
* 
* 
*: 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)
* 
* 
* 
*: 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)
* 
* 
* 
* 
* 
*: 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)
* 
* 
* 
* 
* 
* 
*: 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)
* 
* 
* 
* 
* 
* 
* 
*: 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)
* 
*: 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)
* 
* 
*: 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)
* 
* 
* 
* 
* 
* 
*: 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)
* 
*: 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.&nbsp;10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* 
*  Free HTML preprint:  Note: the journal article contains material and wording not found in the preprint.
* 
*  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.
* 
*: 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)
* 
* 
* 
* 
* 
* 
* 
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* 
* 
* 
* 
* 

শ্ৰেণী:সংখ্যা