0.999...

This is an old revision of this page, as edited by 213.151.217.129 (talk) at 10:49, 24 October 2007 (→Applications). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Revision as of 10:49, 24 October 2007 by 213.151.217.129 (talk) (→Applications)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

In mathematics, the recurring decimal 0.999… , which is also written as $0.{\bar {9}},0.{\dot {9}}$ or $\ 0.(9)$ , denotes a real number equal to 1. In other words, "0.999…" represents the same number as the symbol "1". The equality has long been accepted by professional mathematicians and taught in textbooks. Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.

In the last few decades, researchers of mathematics education have studied the reception of this equality among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique decimal expansion, that nonzero infinitesimal quantities should exist, or that the expansion of 0.999… eventually terminates.

The non-uniqueness of such expansions is not limited to the decimal system. The same phenomenon occurs in integer bases other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. For reasons of simplicity, the terminating decimal is almost always the preferred representation, further contributing to the misconception that it is the only representation. In fact, once infinite expansions are allowed, all positional numeral systems contain an infinity of ambiguous numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations. These various identities have been applied to better understand patterns in the decimal expansions of fractions and the structure of a simple fractal, the Cantor set. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

Number systems in which 0.999… is strictly less than 1 can be constructed, but only outside the standard real number system which is used in elementary mathematics.

Introduction

0.999… is a number written in the decimal numeral system, and some of the simplest proofs that 0.999… = 1 rely on the convenient arithmetic properties of this system. Most of decimal arithmetic — addition, subtraction, multiplication, division, and comparison — uses manipulations at the digit level that are much the same as those for integers. As with integers, any two finite decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

The meaning of "…" (ellipsis) in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some finite portion is left unstated or otherwise omitted. When used to specify a recurring decimal, "…" means that some infinite portion is left unstated. In particular, 0.999… indicates the limit of the sequence (0.9,0.99,0.999,0.9999,…) (or, equivalently, the sum of all terms of the form 9 × 0.1 for integers k=1 to infinity). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1.

There are many proofs that 0.999… = 1. Before demonstrating this using algebraic methods, consider that two real numbers are identical if and only if their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a closed interval defined by the above sequence and the triangle inequality). Thus the difference is 0 and the numbers are identical. This also explains why 0.333… = ⁄₃, etc.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using fractions, ⁄₃ = ⁄₆. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

Skepticism in education

Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:

Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.
Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount. (i.e. 1 - 0.999… = 10)

Note: It is important to know that 10 has no mathematical meaning, a common error made by students. You may however determine the limit as k→-∞ of 10, which is 0.

Some students believe that the value of a convergent series is an approximation, not the actual value.

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".

Of the elementary proofs, multiplying 0.333… = ⁄₃ by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated. Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = ⁄₃ using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division. Others still are able to prove that ⁄₃ = 0.333…, but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing ⁄₃ as a number in its own right and to dealing with the set of natural numbers as a whole.

In popular culture

With the rise of the Internet, debates about 0.999… have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its FAQ. The FAQ briefly covers ⅓, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999… via ⅓ and limits, saying of misconceptions,

The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

Nonsense.

The Straight Dope cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company issued a "press release" on April Fool's Day 2004 that it is 1:

We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.

Two proofs are then offered, based on limits and multiplication by 10.

Different answers from alternative number systems

Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in Mathematics: A Very Short Introduction that the resulting identity 0.999… = 1 is a convention as well:

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.

One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

Infinitesimals

Main article: Infinitesimal

Some proofs that 0.999… = 1 rely on the Archimedean property of the standard real numbers: there are no nonzero infinitesimals. There are mathematically coherent ordered algebraic structures, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the dual numbers include a new infinitesimal element ε, analogous to the imaginary unit i in the complex numbers except that ε² = 0. The resulting structure is useful in automatic differentiation. The dual numbers can be given a lexicographic order, in which case the multiples of ε become non-Archimedean elements. Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use topos theory and alternative logics rather than set theory and classical logic (which is a special case). For example, smooth infinitesimal analysis has infinitesimals with no reciprocals.

Non-standard analysis is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to calculus. A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of /₃ by an infinitesimal:

0.333…;…000… does not exist, while

0.333…;…333… = /₃ exactly.

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = /₃. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the surreal number /_ω, where ω is the first infinite ordinal; the relevant game is LRRRR… or 0.000….

Breaking subtraction

Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999… + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to ⁄₃. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.

In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction d he allows both the cut (−∞, d ) and the "principal cut" (−∞, d ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut D, but there is "a sort of negative infinitesimal," 0, which has no decimal expansion. He concludes that 0.999… = 1 + 0, while the equation "0.999… + x = 1" has no solution.

p-adic numbers

Main article: p-adic number

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999…. For an infinite string of 9s including a last 9, one must look elsewhere.

The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to p, than it is to 1 . The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1. Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

\ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\cdots ={\frac {9}{1-10}}=-1.

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = …999 then 10x = …990, so 10x = x − 9, hence x = −1 again.

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols" one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.

Notes

Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (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."
Tall and Schwarzenberger pp.6–7; Tall 2000 p.221
Tall and Schwarzenberger p.6; Tall 2000 p.221
Tall 2000 p.221
Tall 1976 pp.10–14
Pinto and Tall p.5, Edwards and Ward pp.416–417
Mazur pp.137–141
Dubinsky et al. 261–262
As observed by Richman (p.396). Hans de Vreught (1994). "sci.math FAQ: Why is 0.9999… = 1?". Retrieved 2006-06-29.
Cecil Adams (2003-07-11). "An infinite question: Why doesn't .999~ = 1?". The Straight Dope. The Chicago Reader. Retrieved 2006-09-06.
"Blizzard Entertainment® Announces .999~ (Repeating) = 1". Press Release. Blizzard Entertainment. 2004-04-01. Retrieved 2006-09-03.
Gowers p.60
Berz 439–442
John L. Bell (2003). "An Invitation to Smooth Infinitesimal Analysis" (PDF). Retrieved 2006-06-29. {{cite journal}}: Cite journal requires |journal= (help)
For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis.
Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.
Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and /₃ and touch on /_ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by A. N. Walker (1999). "Hackenstrings and the 0.999… ≟ 1 FAQ". Retrieved 2006-06-29.
Richman pp.397–399
Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.
Gardiner p.98; Gowers p.60
^ Fjelstad p.11
Fjelstad pp.14–15
DeSua p.901
DeSua pp.902–903
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.
Conway, John B. (1978) . 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. {{cite journal}}: |format= requires |url= (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. {{cite journal}}: 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) . John Hewlett and Francis Horner, English translators. (ed.). Elements of Algebra (3rd English edition ed.). Orme Longman. {{cite book}}: |edition= has extra text (help); |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}}: |format= requires |url= (help); Unknown parameter |month= ignored (help)
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. (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. London: Van Nostrand Reinhold. ISBN 0-442-02863-6. LCC QA37.2 G75. {{cite book}}: Unknown parameter |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)
Kempner, A.J. (1936). "Anormal Systems of Numeration". The American Mathematical Monthly. 43 (10): 610–617. {{cite journal}}: |format= requires |url= (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. {{cite journal}}: |format= requires |url= (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. {{cite journal}}: |format= requires |url= (help)
Leavitt, W.G. (1984). "Repeating Decimals". The College Mathematics Journal. 15 (4): 299–308. {{cite journal}}: |format= requires |url= (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. {{cite journal}}: |format= requires |url= (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) . 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)
Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 0-387-94108-8.
Petkovšek, Marko (1990). "Ambiguous Numbers are Dense". American Mathematical Monthly. 97 (5): 408–411. {{cite journal}}: |format= requires |url= (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. {{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.
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.
Richman, Fred (1999). "Is 0.999… = 1?". Mathematics Magazine. 72 (5): 396–400. {{cite journal}}: |format= requires |url= (help); 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 edition ed.). Princeton University Press. ISBN 0-691-04490-2. {{cite book}}: |edition= has extra text (help)
Rosenlicht, Maxwell (1985). Introduction to Analysis. Dover. ISBN 0-486-65038-3.
Rudin, Walter (1976) . 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}}: |format= requires |url= (help); 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, 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.
Tall, David (1976/7). "Conflicts and Catastrophes in the Learning of Mathematics" (PDF). Mathematical Education for Teaching. 2 (4): 2–18. {{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.
von Mangoldt, Dr. Hans (1911). "Reihenzahlen". Einführung in die höhere Mathematik (in German) (1st ed. ed.). Leipzig: Verlag von S. Hirzel. {{cite book}}: |edition= has extra text (help)
Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 0-393-00338-8.

External links

Listen to this article
(2 parts, 50 minutes)

Part 2

These audio files were created from a revision of this article dated Error: no date provided, and do not reflect subsequent edits.(Audio help · More spoken articles)

Template:Link FA

Categories:

Misplaced Pages