0.999...

From Wikipedia, the free encyclopedia

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, the notations "0.999…" and "1" represent the same real number. 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.

The non-uniqueness of real expansions such as 0.999… 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 infinite number of alternative representations of 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.

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 real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard real number system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in mathematical analysis.

Contents

[edit] 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.

Misinterpreting the meaning of the use of the "…" (ellipsis) in 0.999… accounts for some of the misunderstanding about its equality to 1. 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, which can only be interpreted as a number by using the mathematical concept of limits. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is the real number which is the limit of the convergent sequence (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using fractions, 1⁄3 = 2⁄6. 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).

There are many proofs that 0.999… = 1, of varying degrees of mathematical rigour. A short sketch of one rigorous proof can be simply stated as follows. Consider that two real numbers are identical if and only if their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see Archimedean property for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

[edit] Proofs

[edit] Algebraic

[edit] Fractions

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using long division, a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.[1]

Another form of this proof multiplies 1/9 = 0.111… by 9.

\begin{align} 0.333\dots &{} = \frac{1}{3} \\ 3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\ 0.999\dots &{} = 1 \end{align}

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

An even easier version of the same proof is based on the following equations:

1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots

Since both equations are valid, by the transitive property, 0.999… must equal 1. Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

[edit] Digit manipulation

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number.

To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.[1] Written as a sequence of equations,

\begin{align} c &= 0.999\ldots \\ 10 c &= 9.999\ldots \\ 10 c - c &= 9.999\ldots - 0.999\ldots \\ 9 c &= 9 \\ c &= 1 \\ 0.999\ldots &= 1 \end{align}

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

[edit] Real analysis

Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form

b_0.b_1b_2b_3b_4b_5\dots

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a positional notation, so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

[edit] Infinite series and sequences

Further information: Decimal representation

Perhaps the most common development of decimal expansions is to define them as sums of infinite series. In general:

b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .

For 0.999… one can apply the powerful convergence theorem concerning geometric series:[2]

If | r | < 1 then ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.

Since 0.999… is such a sum with a common ratio r=\textstyle\frac{1}{10}, the theorem makes short work of the question:

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

This proof (actually, that 10 equals 9.999…) appears as early as 1770 in Leonhard Euler's Elements of Algebra.[3]

Limits: The unit interval, including the base-4 decimal sequence (.3, .33, .333, …) converging to 1.
Limits: The unit interval, including the base-4 decimal sequence (.3, .33, .333, …) converging to 1.

The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebra proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999….[4] A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.[5]

A sequence (x0, x1, x2, …) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:

0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,[6]

The last step — that lim 1/10n = 0 — is often justified by the axiom that the real numbers have the Archimedean property. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".[7] Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1.

[edit] Nested intervals and least upper bounds

Further information: Nested intervals
Nested intervals: in base 3, 1 = 1.000… = 0.222…
Nested intervals: in base 3, 1 = 1.000… = 0.222…

The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b0, b1, b2, b3, …, and one writes

x = b0.b1b2b3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.[8]

One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b0.b1b2b3… is defined to be the unique number contained within all the intervals [b0, b0 + 1], [b0.b1, b0.b1 + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.[9]

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b0.b1b2b3… to be the least upper bound of the set of approximants {b0, b0.b1, b0.b1b2, …}.[10] One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,

The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.[11]

[edit] Real numbers

Further information: Construction of real numbers

Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.[12]

[edit] Dedekind cuts

Further information: Dedekind cut

In the Dedekind cut approach, each real number x is defined as the infinite set of all rational numbers that are less than x.[13] In particular, the real number 1 is the set of all rational numbers that are less than 1.[14] Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers r such that r < 0, or r < 0.9, or r < 0.99, or r is less than some other number of the form \begin{align}1-(\tfrac{1}{10})^n\end{align}.[15] Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number \begin{align}\tfrac{a}{b}<1\end{align}, which implies \begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.[16] The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in Mathematics Magazine, which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.[17] Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."[18] A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "Alternative number systems" below.

[edit] Cauchy sequences

Further information: Cauchy sequence

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between x and y is defined as the absolute value |x − y|, where the absolute value |z| is defined as the maximum of z and −z, thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence (x0, x1, x2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |xm − xn| ≤ δ for all m, n > N. (The distance between terms becomes smaller than any positive rational.)[19]

If (xn) and (yn) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (xn − yn) has the limit 0. Truncations of the decimal number b0.b1b2b3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.[20] Thus in this formalism the task is to show that the sequence of rational numbers

\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right) = \left(1, {1 \over 10}, {1 \over 100}, \dots \right)

has the limit 0. Considering the nth term of the sequence, for n=0,1,2,…, it must therefore be shown that

\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.

This limit is plain;[21] one possible proof is that for ε = a/b > 0 one can take N = b in the definition of the limit of a sequence. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.[16] The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work A comprehensive textbook of classical mathematics: A contemporary interpretation. The book is written specifically to offer a second look at familiar concepts in a contemporary light.[22]

[edit] Generalizations

Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.[23]

Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111… equals 1, and in base 3 (the ternary numeral system) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.[24]

Alternative representations of 1 also occur in non-integer bases. For example, in the golden ratio base, the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for almost all q between 1 and 2, there are uncountably many base-q expansions of 1. On the other hand, there are still uncountably many q (including all natural numbers greater than 1) for which there is only one base-q expansion of 1, other than the trivial 1.000…. This result was first obtained by Paul Erdős, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the Komornik-Loreti constant q = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the Thue-Morse sequence, which does not repeat.[25]

A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example:[26]

Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary point-set topology"; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.[27]

[edit] Applications

One application of 0.999… as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include:

  • 1/7 = 0.142857142857… and 142 + 857 = 999.
  • 1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.

E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.b1b2b3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.[28] Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.[29]

Positions of 1/4, 2/3, and 1 in the Cantor set
Positions of 1/4, 2/3, and 1 in the Cantor set

Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:

  • A point in the unit interval lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.[30]

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.[31] A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.[32]

[edit] 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.[33]
  • 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".[34]
  • 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.[35]
  • Some students regard 0.999… as having a fixed value which is less than 1 by an infinitesimal but non-zero amount.
  • Some students believe that the value of a convergent series is at best an approximation, that 0.\bar{9} \approx 1.

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'".[36]

Of the elementary proofs, multiplying 0.333… = 1⁄3 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.[37] 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… = 1⁄3 using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.[38] Others still are able to prove that 1⁄3 = 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."[39]

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 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.[40]

[edit] 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.[41] The FAQ briefly covers 1⁄3, 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 1⁄3 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.[42]

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 Fools' 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.[43]

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

[edit] 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.[44]

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.

[edit] 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.[45] 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.[46]

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.[47] 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 1/3 by an infinitesimal:

0.333…;…000… does not exist, while
0.333…;…333… = 1/3 exactly.[48]

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 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the surreal number 1/ω, where ω is the first infinite ordinal; the relevant game is LRRRR… or 0.000…2.[49]

[edit] 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 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.[50]

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.[51]

[edit] 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….[52] For an infinite string of 9s including a last 9, one must look elsewhere.

The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.
The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.

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 pn, 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.[53] 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:

\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.[54]

(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.[53]

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"[55] 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.[56]

[edit] Related questions

  • 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.[57]
  • 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;[58] 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.[59]
  • 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.[60] 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).[61][62]

[edit] See also

Wikimedia Commons has media related to:

[edit] Notes

  1. ^ a b cf. with the binary version of the same argument in Silvanus P. Thompson, Calculus made easy, St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.
  2. ^ Rudin p.61, Theorem 3.26; J. Stewart p.706
  3. ^ Euler p.170
  4. ^ Grattan-Guinness p.69; Bonnycastle p.177
  5. ^ For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31
  6. ^ The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) Thomas' Calculus: Early Transcendentals 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).
  7. ^ Davies p.175; Smith and Harrington p.115
  8. ^ Beals p.22; I. Stewart p.34
  9. ^ Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46
  10. ^ Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27
  11. ^ Apostol p.12
  12. ^ The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30
  13. ^ Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number x can be named by giving an infinite set of rationals, namely all the rationals less than x. We will in effect define x to be the set of rationals smaller than x. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"
  14. ^ Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".
  15. ^ Richman p.399
  16. ^ a b J J O'Connor and E F Robertson (October 2005). History topic: The real numbers: Stevin to Hilbert. MacTutor History of Mathematics. Retrieved on 2006-08-30.
  17. ^ Mathematics Magazine:Guidelines for Authors. Mathematical Association of America. Retrieved on 2006-08-23.
  18. ^ Richman pp.398–399
  19. ^ Griffiths & Hilton §24.2 "Sequences" p.386
  20. ^ Griffiths & Hilton pp.388, 393
  21. ^ Griffiths & Hilton pp.395
  22. ^ Griffiths & Hilton pp.viii, 395
  23. ^ Petkovšek p.408
  24. ^ Protter and Morrey p.503; Bartle and Sherbert p.61
  25. ^ Komornik and Loreti p.636
  26. ^ Kempner p.611; Petkovšek p.409
  27. ^ Petkovšek pp.410–411
  28. ^ Leavitt 1984 p.301
  29. ^ Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98
  30. ^ Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.
  31. ^ Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.
  32. ^ Rudin p.50, Pugh p.98
  33. ^ 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."
  34. ^ Tall and Schwarzenberger pp.6–7; Tall 2000 p.221
  35. ^ Tall and Schwarzenberger p.6; Tall 2000 p.221
  36. ^ Tall 2000 p.221
  37. ^ Tall 1976 pp.10–14
  38. ^ Pinto and Tall p.5, Edwards and Ward pp.416–417
  39. ^ Mazur pp.137–141
  40. ^ Dubinsky et al. 261–262
  41. ^ As observed by Richman (p.396). Hans de Vreught (1994). sci.math FAQ: Why is 0.9999… = 1?. Retrieved on 2006-06-29.
  42. ^ Cecil Adams (2003-07-11). An infinite question: Why doesn't .999~ = 1?. The Straight Dope. Chicago Reader. Retrieved on 2006-09-06.
  43. ^ Blizzard Entertainment Announces .999~ (Repeating) = 1. Press Release. Blizzard Entertainment (2004-04-01). Retrieved on 2006-09-03.
  44. ^ Gowers p.60
  45. ^ Berz 439–442
  46. ^ John L. Bell. "An Invitation to Smooth Infinitesimal Analysis" (PDF). Retrieved on 2006-06-29.
  47. ^ For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis.
  48. ^ Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.
  49. ^ Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111…2 follows directly from Berlekamp's Rule, and it is discussed by A. N. Walker (1999). Hackenstrings and the 0.999… ≟ 1 FAQ. Retrieved on 2006-06-29.
  50. ^ Richman pp.397–399
  51. ^ Richman pp.398–400. Rudin (p.23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.
  52. ^ Gardiner p.98; Gowers p.60
  53. ^ a b Fjelstad p.11
  54. ^ Fjelstad pp.14–15
  55. ^ DeSua p.901
  56. ^ DeSua pp.902–903
  57. ^ Wallace p.51, Maor p.17
  58. ^ See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57
  59. ^ Maor p.54
  60. ^ Munkres p.34, Exercise 1(c)
  61. ^ Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics, 2e, W. H. Freeman, 462. ISBN 0-7167-1088-9. 
  62. ^ Floating point types. MSDN C# Language Specification. Retrieved on 2006-08-29.

[edit] References

  • Alligood, Sauer, and Yorke (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, 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. 
  • Berz, Martin (1992). "Automatic differentiation as nonarchimedean analysis". Computer Arithmetic and Enclosure Methods: 439–450, Elsevier. 
  • 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. [1973] (1978). Functions of one complex variable I, 2e, 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. (November 1960). "A system isomorphic to the reals" (restricted access). The American Mathematical Monthly 67: 900–903. 
  • 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. 
  • Edwards, Barbara and Michael Ward (May 2004). "Surprises from mathematics education research: Student (mis)use of mathematical definitions". The American Mathematical Monthly 111: 411–425. 
  • 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 [1770] (1822). in John Hewlett and Francis Horner, English translators.: Elements of Algebra, 3rd English edition, Orme Longman. 
  • Fjelstad, Paul (January 1995). "The repeating integer paradox" (restricted access). The College Mathematics Journal 26: 11–15. doi:10.2307/2687285. 
  • Gardiner, Anthony [1982] (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.; P.J. Hilton (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. London: Van Nostrand Reinhold. ISBN 0-442-02863-6. 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 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. (December 1936). "Anormal Systems of Numeration" (restricted access). The American Mathematical Monthly 43: 610–617. 
  • Komornik, Vilmos; and Paola Loreti (1998). "Unique Developments in Non-Integer Bases" (restricted access). The American Mathematical Monthly 105: 636–639. 
  • Leavitt, W.G. (1967). "A Theorem on Repeating Decimals" (restricted access). The American Mathematical Monthly 74: 669–673. 
  • Leavitt, W.G. (September 1984). "Repeating Decimals" (restricted access). The College Mathematics Journal 15: 299–308. 
  • Lewittes, Joseph (2006). Midy's Theorem for Periodic Decimals. New York Number Theory Workshop on Combinatorial and Additive Number Theory. arXiv.
  • Lightstone, A.H. (March 1972). "Infinitesimals" (restricted access). The American Mathematical Monthly 79: 242–251. 
  • 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. [1975] (2000). Topology, 2e, 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 (May 1990). "Ambiguous Numbers are Dense" (restricted access). American Mathematical Monthly 97: 408–411. 
  • Pinto, Márcia and David Tall (2001). "Following students' development in a traditional university analysis course". PME25: v4: 57–64. 
  • Protter, M.H. and C.B. Morrey (1991). A first course in real analysis, 2e, 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 (December 1999). "Is 0.999… = 1?" (restricted access). Mathematics Magazine 72 (5): 396–400.  Free HTML preprint: Richman, Fred (1999-06-08). Is 0.999… = 1?. Retrieved on 2006-08-23. Note: the journal article contains material and wording not found in the preprint.
  • Robinson, Abraham (1996). Non-standard analysis, Revised edition, Princeton University Press. ISBN 0-691-04490-2. 
  • Rosenlicht, Maxwell (1985). Introduction to Analysis. Dover. ISBN 0-486-65038-3. 
  • Rudin, Walter [1953] (1976). Principles of mathematical analysis, 3e, 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" (restricted access). Mathematics Magazine 51: 90–98. 
  • 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, 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". Mathematics Teaching 82: 44–49. 
  • Tall, David (1976/7). "Conflicts and Catastrophes in the Learning of Mathematics". Mathematical Education for Teaching 2: 2–18. 
  • Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology". Mathematics Education Research Journal 12: 210–230. 
  • von Mangoldt, Dr. Hans (1911). "Reihenzahlen", Einführung in die höhere Mathematik, 1st ed. (in German), Leipzig: Verlag von S. Hirzel. 
  • Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 0-393-00338-8. 

[edit] External links