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Articles 0.999... 1 1 (number) 20 Portal:Mathematics 24 Signed zero 29 Integer 32 Real number 36 References Article Sources and Contributors 44 Image Sources, Licenses and Contributors 46 Article Licenses License 48 0.999... 1 0.999...

In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the only representation. The non-terminating form is more convenient for understanding the decimal expansions of certain fractions and, in base three, for the structure of the ternary Cantor set, a simple fractal. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any positional numeral system for the real numbers contains infinitely many numbers with multiple representations. The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks to students. In the last few decades, researchers of mathematics education have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an appeal to authority from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimal real numbers. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be numbers whose difference from 1 is less than any rational number. Other systems, known as the p-adic numbers, have a different form of "decimal expansions" which behave quite differently than expansions of real numbers. Although the real numbers are the most common object of study in the field of mathematical analysis, the hyperreals and p-adics both have applications in that area.

Algebraic proofs Algebraic proofs showing that 0.999... represents the number 1 use concepts such as fractions, long division, and digit manipulation to build transformations preserving equality from 0.999... to 1.

Fractions and long division 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⁄ becomes a recurring decimal, 0.111..., in which the digits repeat 9 without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × 1⁄ equals 1, so 0.999... = 1: 9 0.999... 2

Another form of this proof multiplies 1⁄ = 0.333... by 3. 3

Digit manipulation When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

Discussion Although these proofs demonstrate that 0.999... = 1, the extent to which they explain the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.[1] William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.[2] Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".[3] Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.

Analytic proofs 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 b and one can neglect negatives, so a decimal expansion has the form 0

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. 0.999... 3

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:

For 0.999... one can apply the convergence theorem concerning geometric series:[4]

If then

Since 0.999... is such a sum with a common ratio r = 1⁄ , the theorem makes short work of the question: 10

This proof (actually, that 10 equals 9.999...) appears as early as 1770 in Leonhard Euler's Elements of Algebra.[5] 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 algebraic 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 [6] maneuver on 0.999... A 19th-century reaction against such Limits: The unit interval, including the base-4 fraction liberal summation methods resulted in the definition that still sequence (.3, .33, .333, ...) converging to 1. 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.[7]

A sequence (x , x , x , ...) has a limit x if the distance |x − x | becomes arbitrarily small as n increases. The statement 0 1 2 n that 0.999... = 1 can itself be interpreted and proven as a limit:[8]

The last step, that 1⁄ n → 0 as n → ∞, is often justified by the axiom that the real numbers have the Archimedean 10 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".[9] Such heuristics are often interpreted by students as implying that 0.999... itself is less than 1. 0.999... 4

Nested intervals and least upper bounds Further information: Nested intervals 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) 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 Nested intervals: in base 3, 1 = 1.000... = 0.222... digits b , b , b , b , ..., and one writes 0 1 2 3

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, 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.[10] 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 b .b b b ... is defined to be the unique number contained within all the intervals [b , b + 1], [b .b , b .b + 0.1], 0 1 2 3 0 0 0 1 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.[11] 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 b .b b b ... to be the 0 1 2 3 least upper bound of the set of approximants {b , b .b , b .b b , ...}.[12] One can then show that this definition (or 0 0 1 0 1 2 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.[13]

Proofs from the construction of the real numbers Further information: Construction of the 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 to be less than, greater than, or equal to another number. 0.999... 5

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.[14]

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.[15] In particular, the real number 1 is the set of all rational numbers that are less than 1.[16] 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

.[17] 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

which implies

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.[18] 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.[19] 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. He also notes that typically the definitions allow { x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."[20] A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "Alternative number systems" below.

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 (x , x , x , ...), a mapping from natural numbers to 0 1 2 rationals, for any positive rational δ there is an N such that |x − x | ≤ δ for all m, n > N. (The distance between terms m n becomes smaller than any positive rational.)[21] If (x ) and (y ) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (x − y ) n n n n has the limit 0. Truncations of the decimal number b .b b b ... generate a sequence of rationals which is Cauchy; 0 1 2 3 this is taken to define the real value of the number.[22] Thus in this formalism the task is to show that the sequence of 0.999... 6

rational numbers

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

This limit is plain;[23] 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.[18] 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.[24]

Generalizations The result that 0.999... = 1 generalizes readily 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.[25] 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.[26] 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.[27] 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:[28] • In the balanced ternary system, 1/ = 0.111... = 1.111.... 2 • In the reverse factorial number system (using bases 2,3,4,... for positions after the decimal point), 1 = 1.000... = 0.1234....

Impossibility of unique representation That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered lexicographically. Indeed the following two properties account for the difficulty: • If an interval of the real numbers is partitioned into two non-empty parts L, R, such that every element of L is (strictly) less than every element of R, then either L contains a largest element or R contains a smallest element, but not both. • The collection of infinite strings of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts L, R, such that every element of L is less than every element of R, while L contains a largest element and R contains a smallest element. Indeed it suffices to take two finite prefixes (initial substrings) p , p of elements from the collection such that they differ only in their final symbol, for which 1 2 0.999... 7

symbol they have successive values, and take for L the set of all strings in the collection whose corresponding prefix is at most p , and for R the remainder, the strings in the collection whose corresponding prefix is at least p . 1 2 Then L has a largest element, starting with p and choosing the largest available symbol in all following positions, 1 while R has a smallest element obtained by following p by the smallest symbol in all positions. 2 The first point follows from basic properties of the real numbers: L has a supremum and R has an infimum, which are easily seen to be equal; being a real number it either lies in R or in L, but not both since L and R are supposed to be disjoint. The second point generalizes the 0.999.../1.000... pair obtained for p = "0", p = "1". In fact one need 1 2 not use the same alphabet for all positions (so that for instance mixed radix systems can be included) or consider the full collection of possible strings; the only important points are that at each position a finite set of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an order preserving map from the collection of strings to an interval of the real numbers cannot be a bijection: either some numbers do not correspond to any string, or some of them correspond to more than one string. Marko Petkovšek has proven that for any positional 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.[29]

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/ = 0.142857142857... and 142 + 857 = 999. 7 • 1/ = 0.0136986301369863... and 0136 + 9863 = 9999. 73 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.b b b ... is a positive integer, then it must be 0.999..., which is then the 1 2 3 source of the 9s in the theorem.[30] Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.[31] 0.999... 8

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⁄ is given the 3 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⁄ is 3 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.[32]

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 Positions of 1/ , 2/ , and 1 in the Cantor set decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... 4 3 A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.[33] 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.[34]

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.[35] • 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".[36] • 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.[37] 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 O. 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'".[38] Of the elementary proofs, multiplying 0.333... = 1⁄ by 3 is apparently a successful strategy for convincing reluctant 3 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.[39] 0.999... 9

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⁄ using a supremum definition, but then 3 insisted that 0.999... < 1 based on her earlier understanding of long division.[40] 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."[41] 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⁄ as a number in its own right and to dealing with the set of 3 natural numbers as a whole.[42]

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.[43] The FAQ briefly covers 1⁄ , multiplication by 10, and limits, and it alludes to Cauchy sequences as well. 3 A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999... via 1⁄ and limits, 3 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.[44] 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.[45] Two proofs are then offered, based on limits and multiplication by 10. 0.999... features also in mathematical folklore, specifically in the following joke:[46] Q: How many mathematicians does it take to screw in a lightbulb? A: 0.999999.... 0.999... 10

In alternative number systems Although the real numbers form an extremely useful number system, the decision to interpret the notation "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.[47] 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 Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same. However, there are mathematically coherent ordered algebraic structures, including various alternatives to the real numbers, which are non-Archimedean. 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.[48] 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. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses).[49] A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1)∗.[50] Lightstone shows how to associate to each number a sequence of digits,

indexed by the hypernatural numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the transfer principle. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number. At the same time, the hyperreal number u =0.999...;...999000..., with last digit 9 at infinite hypernatural rank H, H satisfies a strict inequality u < 1. Accordingly, Karin Katz and Mikhail Katz have proposed an alternative H evaluation of "0.999...":

[51]

All such interpretations of "0.999..." are infinitely close to 1. Ian Stewart characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....[52] Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about 0.999... < 1 are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.[53] [54] 0.999... 11

Hackenbush 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 ... = 1/ . However, the value of LRLLL... (corresponding to 0.111... ) is 2 3 2 infinitesimally less than 1. The difference between the two is the surreal number 1/ , where ω is the first infinite ω ordinal; the relevant game is LRRRR... or 0.000... .[55] 2 This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111... = 0.11000... , which are both equal to 3⁄ , 2 2 4 but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

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⁄ . After defining multiplication, the 3 decimal numbers form a positive, totally ordered, commutative semiring.[56] 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.[57]

p-adic numbers When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they 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....[58] However, there is a system that contains an infinite string of 9s including a last 9. 0.999... 12

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.[59] Another derivation

uses a geometric series. The infinite series implied by "...999" does The 4-adic integers (black points), including the not converge in the real numbers, but it converges in the 10-adics, sequence (3, 33, 333, ...) converging to −1. The 10-adic and so one can re-use the familiar formula: analogue is ...999 = −1.

[60]

(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.[59] 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"[61] 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.[62]

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.[63] • 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/ to be infinity;[64] and, in fact, the results are profound and applicable to many problems 0 in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.[65] • 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.[66] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary 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).[67] [68] 0.999... 13

Notes [1] This argument is found in Peressini and Peressini p. 186 [2] Byers pp. 39–41 [3] Richman p. 396 [4] Rudin p. 61, Theorem 3.26; J. Stewart p. 706 [5] Euler p. 170 [6] Grattan-Guinness p. 69; Bonnycastle p. 177 [7] For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31 [8] 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). [9] Davies p. 175; Smith and Harrington p. 115 [10] Beals p. 22; I. Stewart p. 34 [11] Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46 [12] Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27 [13] Apostol p. 12 [14] 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 [15] 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..." [16] Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1 , R 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". [17] Richman p. 399

[18] J J O'Connor and E F Robertson (October 2005). "History topic: The real numbers: Stevin to Hilbert" (http:/ / www-gap. dcs. st-and. ac. uk/

~history/ PrintHT/ Real_numbers_2. html). MacTutor History of Mathematics. . Retrieved 2006-08-30. [19] Richman [20] Richman pp. 398–399 [21] Griffiths & Hilton §24.2 "Sequences" p. 386 [22] Griffiths & Hilton pp. 388, 393 [23] Griffiths & Hilton p. 395 [24] Griffiths & Hilton pp.viii, 395 [25] Petkovšek p. 408 [26] Protter and Morrey p. 503; Bartle and Sherbert p. 61 [27] Komornik and Loreti p. 636 [28] Kempner p. 611; Petkovšek p. 409 [29] Petkovšek pp. 410–411 [30] Leavitt 1984 p. 301 [31] Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98 [32] 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. [33] 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. [34] Rudin p. 50, Pugh p. 98 [35] 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." [36] Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221 [37] Tall and Schwarzenberger p. 6; Tall 2000 p. 221 [38] Tall 2000 p. 221 [39] Tall 1976 pp. 10–14 [40] Pinto and Tall p. 5, Edwards and Ward pp. 416–417 [41] Mazur pp. 137–141 [42] Dubinsky et al. 261–262

[43] As observed by Richman (p. 396). Hans de Vreught (1994). "sci.math FAQ: Why is 0.9999... = 1?" (http:/ / www. faqs. org/ faqs/

sci-math-faq/ specialnumbers/ 0. 999eq1/ ). . Retrieved 2006-06-29.

[44] Cecil Adams (2003-07-11). "An infinite question: Why doesn't .999~ = 1?" (http:/ / www. straightdope. com/ columns/ 030711. html). The Straight Dope. Chicago Reader. . Retrieved 2006-09-06. 0.999... 14

[45] "Blizzard Entertainment Announces .999~ (Repeating) = 1" (http:/ / us. blizzard. com/ en-us/ company/ press/ pressreleases. html?040401). Press Release. Blizzard Entertainment. 2004-04-01. . Retrieved 2009-11-16. [46] Renteln and Dundes, p. 27 [47] Gowers p. 60 [48] Berz 439–442 [49] For a full treatment of non-standard numbers see for example Robinson's Non-standard Analysis. [50] Lightstone pp. 245–247 [51] Katz & Katz 2010 [52] Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175. [53] Katz & Katz (2010b) [54] R. Ely (2010) [55] Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and 1/ and touch on 1/ . The game for 0.111... follows directly from 3 ω 2 Berlekamp's Rule. [56] Richman pp. 397–399 [57] Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1. [58] Gardiner p. 98; Gowers p. 60 [59] Fjelstad p. 11 [60] Fjelstad pp. 14–15 [61] DeSua p. 901 [62] DeSua pp. 902–903 [63] Wallace p. 51, Maor p. 17 [64] See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57 [65] Maor p. 54 [66] Munkres p. 34, Exercise 1(c) [67] Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2e ed.). W. H. Freeman. p. 462. ISBN 0-7167-1088-9.

[68] "Floating point types" (http:/ / msdn. microsoft. com/ library/ en-us/ csspec/ html/ vclrfcsharpspec_4_1_6. asp). MSDN C# Language Specification. . Retrieved 2006-08-29.

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• Berz, Martin (1992). "Automatic differentiation as nonarchimedean analysis" (http:/ / citeseerx. ist. psu. edu/

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• Burrell, Brian (1998). Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster. ISBN 0-87779-621-1. • Byers, William (2007). How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics. Princeton UP. ISBN 0-691-12738-7. • Conway, John B. (1978) [1973]. 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

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(http:/ / books. google. com/ ?id=X8yv0sj4_1YC& pg=PA170) (3rd English ed.). Orme Longman. ISBN 0-387-96014-7. • Fjelstad, Paul (January 1995). "The repeating integer paradox". The College Mathematics Journal 26 (1): 11–15. doi:10.2307/2687285. JSTOR 2687285. • Gardiner, Anthony (2003) [1982]. Understanding Infinity: The Mathematics of Infinite Processes. Dover. ISBN 0-486-42538-X. • Gowers, Timothy (2002). Mathematics: A Very Short Introduction. Oxford UP. ISBN 0-19-285361-9. • Grattan-Guinness, Ivor (1970). The development of the foundations of mathematical analysis from Euler to Riemann. MIT Press. ISBN 0-262-07034-0. • Griffiths, H. B.; Hilton, P. J. (1970). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. 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)

• Katz, K.; Katz, M. (2010a). "When is .999... less than 1?" (http:/ / www. math. umt. edu/ TMME/ vol7no1/ ). The Montana Mathematics Enthusiast 7 (1): 3–30. • Kempner, A. J. (December 1936). "Anormal Systems of Numeration". The American Mathematical Monthly 43 (10): 610–617. doi:10.2307/2300532. JSTOR 2300532. • Komornik, Vilmos; Loreti, Paola (1998). "Unique Developments in Non-Integer Bases". The American Mathematical Monthly 105 (7): 636–639. doi:10.2307/2589246. JSTOR 2589246. 0.999... 16

• Leavitt, W. G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly 74 (6): 669–673. doi:10.2307/2314251. JSTOR 2314251. • Leavitt, W. G. (September 1984). "Repeating Decimals". The College Mathematics Journal 15 (4): 299–308. doi:10.2307/2686394. JSTOR 2686394. • Lightstone, A. H. (March 1972). "Infinitesimals". The American Mathematical Monthly 79 (3): 242–251. doi:10.2307/2316619. JSTOR 2316619. • Mankiewicz, Richard (2000). The story of mathematics. Cassell. ISBN 0-304-35473-2. Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8) • Maor, Eli (1987). To infinity and beyond: a cultural history of the infinite. Birkhäuser. ISBN 3-7643-3325-1. A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii) • Mazur, Joseph (2005). Euclid in the Rainforest: Discovering Universal Truths in Logic and Math. Pearson: Pi Press. ISBN 0-13-147994-6. • Munkres, James R. (2000) [1975]. 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) • Núñez, Rafael (2006). "Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied

Cognitive Foundations of Mathematics" (http:/ / www. cogsci. ucsd. edu/ ~nunez/ web/ publications. html). 18 Unconventional Essays on the Nature of Mathematics. Springer. pp. 160–181. ISBN 978-0-387-25717-4. • Pedrick, George (1994). A First Course in Analysis. Springer. ISBN 0-387-94108-8. • Peressini, Anthony; Peressini, Dominic (2007). "Philosophy of Mathematics and Mathematics Education". In Bart van Kerkhove, Jean Paul van Bendegem. Perspectives on Mathematical Practices. Logic, Epistemology, and the Unity of Science. 5. Springer. ISBN 978-1-4020-5033-6. • Petkovšek, Marko (May 1990). "Ambiguous Numbers are Dense". American Mathematical Monthly 97 (5): 408–411. doi:10.2307/2324393. JSTOR 2324393. • Pinto, Márcia; Tall, David (2001). "Following students' development in a traditional university analysis course"

(http:/ / www. warwick. ac. uk/ staff/ David. Tall/ pdfs/ dot2001j-pme25-pinto-tall. pdf). PME25. pp. v4: 57–64. Retrieved 2009-05-03. • 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.

• Renteln, Paul; Dundes, Allan (January 2005). "Foolproof: A Sampling of Mathematical Folk Humor" (http:/ /

www. ams. org/ notices/ 200501/ fea-dundes. pdf). Notices of the AMS 52 (1): 24–34. Retrieved 2009-05-03. 0.999... 17

• Richman, Fred (December 1999). "Is 0.999... = 1?". Mathematics Magazine 72 (5): 396–400. doi:10.2307/2690798. JSTOR 2690798. Free HTML preprint: Richman, Fred (1999-06-08). "Is 0.999... = 1?"

(http:/ / www. math. fau. edu/ Richman/ HTML/ 999. htm). Retrieved 2006-08-23. Note: the journal article contains material and wording not found in the preprint. • Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2. • Rosenlicht, Maxwell (1985). Introduction to Analysis. Dover. ISBN 0-486-65038-3. This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition. • Rudin, Walter (1976) [1953]. 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 (March 1978). "Complementary Rational Numbers". Mathematics Magazine 51 (2): 90–98. doi:10.2307/2690144. JSTOR 2690144.

• Smith, Charles; Harrington, Charles (1895). Arithmetic for Schools (http:/ / books. google. com/

books?vid=LCCN02029670& pg=PA115). Macmillan. ISBN 0-665-54808-7. • 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, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. ISBN 978-1-84668-292-6. • 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.

• Tall, D. O.; Schwarzenberger, R. L. E. (1978). "Conflicts in the Learning of Real Numbers and Limits" (http:/ /

www. warwick. ac. uk/ staff/ David. Tall/ pdfs/ dot1978c-with-rolph. pdf). Mathematics Teaching 82: 44–49. Retrieved 2009-05-03.

• Tall, David (1976/7). "Conflicts and Catastrophes in the Learning of Mathematics" (http:/ / www. warwick. ac.

uk/ staff/ David. Tall/ pdfs/ dot1976a-confl-catastrophy. pdf). Mathematical Education for Teaching 2 (4): 2–18. Retrieved 2009-05-03.

• Tall, David (2000). "Cognitive Development In Advanced Mathematics Using Technology" (http:/ / www.

warwick. ac. uk/ staff/ David. Tall/ pdfs/ dot2001b-merj-amt. pdf). Mathematics Education Research Journal 12 (3): 210–230. Retrieved 2009-05-03. • von Mangoldt, Dr. Hans (1911). "Reihenzahlen" (in German). Einführung in die höhere Mathematik (1st ed.). Leipzig: Verlag von S. Hirzel. • Wallace, David Foster (2003). Everything and more: a compact history of infinity. Norton. ISBN 0-393-00338-8. 0.999... 18

Further reading • Burkov, S. E. (1987), "One-dimensional model of the quasicrystalline alloy", Journal of Statistical Physics 47 (3/4): 409, doi:10.1007/BF01007518 • Burn, Bob (March 1997), "81.15 A Case of Conflict", The Mathematical Gazette 81 (490): 109–112, doi:10.2307/3618786, JSTOR 3618786 • J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren (February 1981), "The Age of Newton: An Intensive Interdisciplinary Course", The History Teacher 14 (2): 167–190, doi:10.2307/493261, JSTOR 493261 • Choi, Younggi; Do, Jonghoon (November 2005), "Equality Involved in 0.999... and (-8)⅓", For the Learning of Mathematics 25 (3): 13–15, 36, JSTOR 40248503 • K. Y. Choong, D. E. Daykin, C. R. Rathbone (April 1971), "Rational Approximations to π", Mathematics of Computation 25 (114): 387–392, doi:10.2307/2004936, JSTOR 2004936 • Edwards, B. (1997), "An undergraduate student’s understanding and use of mathematical definitions in real analysis", in Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E., Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 1, Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education, pp. 17–22

• Eisenmann, Petr (2008), "Why is it not true that 0.999... < 1?" (http:/ / elib. mi. sanu. ac. rs/ files/ journals/ tm/ 20/

tm1114. pdf), The Teaching of Mathematics 11 (1): 35–40 • Ely, Robert (2010), "Nonstandard student conceptions about infinitesimals", Journal for Research in Mathematics Education 41 (2): 117–146 This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for 0.999... falling short of 1 by an infinitesimal 0.000...1. • Ferrini-Mundy, J.; Graham, K. (1994), Research in calculus learning: Understanding of limits, derivatives and integrals, in Kaput, J.; Dubinsky, E., "Research issues in undergraduate mathematics learning", MAA Notes 33: 31–45 • Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". arXiv:math.NT/0605182. • Katz, Karin Usadi; Katz, Mikhail G. (2010b), "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era", Educational Studies in Mathematics 74: 259, arXiv:1003.1501, doi:10.1007/s10649-010-9239-4 • Gardiner, Tony (June 1985), "Infinite processes in elementary mathematics: How much should we tell the children?", The Mathematical Gazette 69 (448): 77–87, doi:10.2307/3616921, JSTOR 3616921 • Monaghan, John (December 1988), "Real Mathematics: One Aspect of the Future of A-Level", The Mathematical Gazette 72 (462): 276–281, doi:10.2307/3619940, JSTOR 3619940 • Navarro, Maria Angeles; Carreras, Pedro Pérez (2010), "A Socratic methodological proposal for the study of the

equality 0.999…=1" (http:/ / elib. mi. sanu. ac. rs/ files/ journals/ tm/ 24/ tm1312. pdf), The Teaching of Mathematics 13 (1): 17–34 • Przenioslo, Malgorzata (March 2004), "Images of the limit of function formed in the course of mathematical studies at the university", Educational Studies in Mathematics 55 (1–3): 103–132, doi:10.1023/B:EDUC.0000017667.70982.05 • Sandefur, James T. (February 1996), "Using Self-Similarity to Find Length, Area, and Dimension", The American Mathematical Monthly 103 (2): 107–120, doi:10.2307/2975103, JSTOR 2975103 • Sierpińska, Anna (November 1987), "Humanities students and epistemological obstacles related to limits", Educational Studies in Mathematics 18 (4): 371–396, doi:10.1007/BF00240986, JSTOR 3482354 • Szydlik, Jennifer Earles (May 2000), "Mathematical Beliefs and Conceptual Understanding of the Limit of a Function", Journal for Research in Mathematics Education 31 (3): 258–276, doi:10.2307/749807, JSTOR 749807 • Tall, David O. (2009), "Dynamic mathematics and the blending of knowledge structures in the calculus", ZDM Mathematics Education 41 (4): 481–492, doi:10.1007/s11858-009-0192-6 0.999... 19

• Tall, David O. (May 1981), "Intuitions of infinity", Mathematics in School: 30–33

External links

• .999999... = 1? (http:/ / www. cut-the-knot. org/ arithmetic/ 999999. shtml) from cut-the-knot

• Why does 0.9999... = 1 ? (http:/ / mathforum. org/ dr. math/ faq/ faq. 0. 9999. html)

• Ask A Scientist: Repeating Decimals (http:/ / www. newton. dep. anl. gov/ askasci/ math99/ math99167. htm)

• Proof of the equality based on arithmetic (http:/ / mathcentral. uregina. ca/ QQ/ database/ QQ. 09. 00/ joan2. html)

• Repeating Nines (http:/ / descmath. com/ diag/ nines. html)

• Point nine recurring equals one (http:/ / qntm. org/ pointnine)

• David Tall's research on mathematics cognition (http:/ / www. warwick. ac. uk/ staff/ David. Tall/ themes/

limits-infinity. html)

• What is so wrong with thinking of real numbers as infinite decimals? (http:/ / www. dpmms. cam. ac. uk/ ~wtg10/

decimals. html)

• Theorem 0.999... (http:/ / us. metamath. org/ mpegif/ 0. 999. . . . html) on Metamath

• A Friendly Chat About Whether 0.999... = 1 (http:/ / betterexplained. com/ articles/

a-friendly-chat-about-whether-0-999-1/ ) 1 (number) 20 1 (number)

1

−1 0 1 2 3 4 5 6 7 8 9 → List of numbers — Integers 0 10 20 30 40 50 60 70 80 90 →

Cardinal 1 one

Ordinal 1st first

Numeral system unary

Factorization

Divisors 1

Greek numeral α'

Roman numeral I

Roman numeral (Unicode) Ⅰ, ⅰ کی - ١ Persian ١ Arabic Ge'ez ፩ Bengali ১ Chinese numeral 一，弌，壹

Korean 일, 하나 Devanāgarī १ Telugu ೧ Tamil ௧ Kannada ೧ Hebrew (alef) א Khmer ១

Thai ๑

prefixes mono- /haplo- (from Greek) uni- (from Latin)

Binary 1

Octal 1

Duodecimal 1

Hexadecimal 1

1 (one; /ˈwʌn/ or UK /ˈwɒn/) is a number, numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement. For example, a line segment of "unit length" is a line segment of length 1. 1 (number) 21

As a number One, sometimes referred to as unity, is the integer before two and after zero. One is the first non-zero number in the natural numbers as well as the first odd number in the natural numbers. Any number multiplied by one is the number, as one is the identity for multiplication. As a result, one is its own factorial, its own square, its own cube, and so on. One is also the empty product, as any number multiplied by one is itself, which produces the same result as multiplying by no numbers at all.

As a digit

The glyph used today in the Western world to represent the number 1, a vertical line, often with a serif at the top and sometimes a short horizontal line at the bottom, traces its roots back to the Indians, who wrote 1 as a horizontal line, much like the Chinese character 一. The Gupta wrote it as a curved line, and the Nagari sometimes added a small circle on the left (rotated a quarter turn to the right, this 9-look-alike became the present day numeral 1 in the Gujarati and Punjabi scripts). The Nepali also rotated it to the right but kept the circle small.[1] This eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral . In some European countries (e.g., Germany), the little serif at the top is sometimes extended into a long upstroke, sometimes as long as the vertical line, which can lead to confusion with the glyph for seven in other countries. Where the 1 is written with a long upstroke, the number 7 has a horizontal stroke through the vertical line.

While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, the character usually is of x-height, as, for example, in

. 1 (number) 22

Many older typewriters do not have a separate symbol for 1 and use the lowercase l instead. It is possible to find cases when the uppercase J is used, while it may be for decorative purposes.

Mathematics

Mathematically, 1 is • in arithmetic (algebra) and calculus, the natural number that follows 0 and precedes 2 and the multiplicative identity of the integers, real numbers and complex numbers; • more generally, in abstract algebra, the multiplicative identity

("unity"), usually of a ring. The 24-hour tower clock in Venice, using J as a One cannot be used as the base of a positional numeral system; symbol for 1 sometimes tallying is referred to as "base 1", since only one mark (the tally) is needed, but this is not a positional notation. The logarithms base 1 are undefined, since the function 1x always equals 1 and so has no unique inverse. In the real-number system, 1 can be represented in two ways as a recurring decimal: as 1.000... and as 0.999... (q.v.). Formalizations of the natural numbers have their own representations of 1: • in the Peano axioms, 1 is the successor of 0; • in Principia Mathematica, 1 is defined as the set of all singletons (sets with one element); • in the Von Neumann cardinal assignment of natural numbers, 1 is defined as the set {0}. In a multiplicative group or monoid, the identity element is sometimes denoted 1, but e (from the German Einheit, "unity") is more traditional. However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are general fields. One is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few. In many mathematical and engineering equations, numeric values are typically normalized to fall within the unit interval from 0 to 1, where 1 usually represents the maximum possible value in the range of parameters. Because of the multiplicative identity, if f(x) is a multiplicative function, then f(1) must equal 1. It is also the first and second numbers in the Fibonacci sequence and is the first number in many other mathematical sequences. As a matter of convention, Sloane's early Handbook of Integer Sequences added an initial 1 to any sequence that did not already have it and considered these initial 1's in its lexicographic ordering. Sloane's later Encyclopedia of Integer Sequences and its Web counterpart, the On-Line Encyclopedia of Integer Sequences, ignore initial ones in their lexicographic ordering of sequences, because such initial ones often correspond to trivial cases. One is neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g., 4 = 22 = (-1)6×123×22). The definition of a field requires that 1 must not be equal to 0. Thus, there are no fields of characteristic 1. Nevertheless, abstract algebra can consider the field with one element, which is not a singleton and is not a set at all. One is the only positive integer divisible by exactly one positive integer (whereas prime numbers are divisible by exactly two positive integers, composite numbers are divisible by more than two positive integers, and zero is divisible by all positive integers). One was formerly considered prime by some mathematicians, using the definition that a prime is divisible only by one and itself. However, this complicates the fundamental theorem of arithmetic, so 1 (number) 23

modern definitions exclude units. The last professional mathematician to publicly label 1 a prime number was Henri Lebesgue in 1899. One is one of three possible values of the Möbius function: it takes the value one for square-free integers with an even number of distinct prime factors. One is the only odd number in the range of Euler's totient function φ(x), in the cases x = 1 and x = 2. One is the only 1-perfect number (see multiply perfect number). By definition, 1 is the magnitude or absolute value of a unit vector and a unit matrix (more usually called an identity matrix). Note that the term unit matrix is sometimes used to mean something quite different. By definition, 1 is the probability of an event that is almost certain to occur. One is the most common leading digit in many sets of data, a consequence of Benford's law. The ancient Egyptians represented all fractions (with the exception of 2/3 and 3/4) in terms of sums of fractions with numerator 1 and distinct denominators. For example, . Such representations are popularly known as

Egyptian Fractions or Unit Fractions. The Generating Function that has all coefficients 1 is given by

.

This power series converges and has finite value if and only if, .

Table of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000

Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 0.5 0.25 0.2 0.125 0.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

In technology

• The resin identification code used in recycling to identify polyethylene terephthalate • Used in binary code 1 (number) 24

In science • The atomic number of hydrogen

Notes [1] Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 392, Fig. 24.61

Portal:Mathematics

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Mathematics (books) | History of mathematics | Mathematicians | Awards | Education | Institutes and societies | Literature | Notation | Theorems | Proofs | Unsolved problems • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? Portal:Mathematics 26

• ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined? • ...that outstanding mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?

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ARTICLE INDEX: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z 0-9

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Science: History of science Philosophy of science Systems science Mathematics Biology Chemistry Physics Earth sciences Technology What are portals? · List of portals · Featured portals Signed zero 29 Signed zero

Signed zero is zero with an associated sign. In ordinary arithmetic, −0 = +0 = 0. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero). This occurs in some signed number representations for integers, and in most floating point number representations. The number 0 is usually encoded as +0, but can be represented by either +0 or −0. The IEEE 754 standard for floating point arithmetic (presently used by most computers and programming languages that support floating point numbers) requires both +0 and −0. The zeroes can be considered as a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞, division by zero is only undefined for ±0/±0 and ±∞/±∞. Negatively signed zero echoes the mathematical analysis concept of approaching 0 from below as a one-sided limit, which may be denoted by x → 0−, x → 0−, or x → ↑0. The notation "−0" may be used informally to denote a small negative number that has been rounded to zero. The concept of negative zero also has some theoretical applications in statistical mechanics and other disciplines. It is claimed that the inclusion of signed zero in IEEE 754 makes it much easier to achieve numerical accuracy in some critical problems,[1] in particular when computing with complex elementary functions.[2] On the other hand, the concept of signed zero runs contrary to the general assumption made in most mathematical fields (and in most mathematics courses) that negative zero is the same thing as zero. Representations that allow negative zero can be a source of errors in programs, as software developers do not realize (or may forget) that, while the two zero representations behave as equal under numeric comparisons, they are different bit patterns and yield different results in some operations.

Representations The widely used two's complement encoding does not allow a negative zero. In a 1+7-bit sign-and-magnitude representation for integers, negative zero is represented by the bit string 1000 0000. In an 8-bit one's complement representation, negative zero is represented by the bit string 1111 1111. In both encodings, positive zero is represented by 0000 0000. In IEEE 754 binary floating point numbers, zero values are represented by the biased exponent and significand both being zero. Negative zero has the sign bit set to one. One may obtain Negative zero by IEEE 754 representation in binary32 negative zero as the result of certain computations, for instance as the result of arithmetic underflow on a negative number, or −1.0*0.0, or simply as −0.0.

In IEEE 754 and in IBM's General Decimal Arithmetic encoding specification, a floating-point representation that uses decimal arithmetic, negative zero is represented by an exponent being any valid exponent in the range for the encoding, the true significand being zero, and the sign bit being one. Signed zero 30

Properties and handling The IEEE 754 floating point standard specifies the behavior of positive zero and negative zero under various operations. The outcome may depend on the current IEEE rounding mode settings.

Arithmetic Multiplication and division follow their usual rules for combining signs:

• (for different from 0)

• • Addition and subtraction are handled specially if the values could cancel: • • • • (for any finite , −0 when rounding toward negative) Because of negative zero (and only because of it), the statements z = -(x - y) and z = (-x) - (-y), for floating-point variables x, y, and z, cannot be optimized to z = y - x. Some other special rules: • [3]

• (follows the sign rule for division)

• (for non-zero , follows the sign rule for division)

• (Not a Number or interrupt for indeterminate form)

•

Division of a non-zero number by zero sets the divide by zero flag, and an operation producing a NaN sets the invalid operation flag. An exception handler is called if enabled for the corresponding flag.

Comparisons According to the IEEE 754 standard, negative zero and positive zero should compare as equal with the usual (numerical) comparison operators, like the == operators of C and Java. In those languages, special programming tricks may be needed to distinguish the two values: • casting the number to an integer type, so as to compare the bit patterns; • using the IEEE 754 copysign() function to copy the sign of the zero to some non-zero number; • taking the reciprocal of the zero to obtain either 1/(+0) = +∞ or 1/(−0) = −∞. However, some programming languages may provide alternative comparison operators that do distinguish the two zeros. This is the case, for example, of the equals method in Java's Double class. [4] Signed zero 31

Scientific uses Informally, one may use the notation "−0" for a negative value that was rounded to zero. This notation may be useful when a negative sign is significant; for example, when tabulating Celsius temperatures, where a negative sign means below freezing. In statistical mechanics, one sometimes uses negative temperatures to describe systems with population inversion, which can be considered to have a temperature greater than positive infinity, because the coefficient of energy in the population distribution function is −1/Temperature. In this context, a temperature of −0 is a (theoretical) temperature larger than any other negative temperature, corresponding to the (theoretical) maximum conceivable extent of population inversion, the opposite extreme to +0.[5]

References [1] William Kahan, "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit", in The State of the Art in

Numerical Analysis (http:/ / portal. acm. org/ citation. cfm?id=59678) (eds. Iserles and Powell), Clarendon Press, Oxford, 1987.

[2] William Kahan, Derivatives in the Complex z-plane (http:/ / www. cs. berkeley. edu/ ~wkahan/ Math185/ Derivative. pdf), p10.

[3] Cowlishaw, Mike (7 April 2009). "Decimal Arithmetic: Arithmetic operations – square-root" (http:/ / speleotrove. com/ decimal/ daops. html#refsqrt). speleotrove.com (IBM Corporation). . Retrieved 7 December 2010.

[4] http:/ / java. sun. com/ javase/ 6/ docs/ api/ java/ lang/ Double. html#equals%28java. lang. Object%29 [5] Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 462. ISBN 0-7167-1088-9.

• "Floating point types" (http:/ / msdn. microsoft. com/ library/ en-us/ csspec/ html/ vclrfcsharpspec_4_1_6. asp). MSDN C# Language Specification. Retrieved 15 October 2005.

• "Division operator" (http:/ / msdn. microsoft. com/ library/ en-us/ csspec/ html/ vclrfcsharpspec_7_7_2. asp). MSDN C# Language Specification. Retrieved 15 October 2005.

• Thomas Wang (March 2000). Java Floating-Point Number Intricacies (http:/ / www. concentric. net/ ~Ttwang/

tech/ javafloat. htm). September 2000.

• Mike Colishaw (28 July 2008). "Decimal Arithmetic Specification, version 1.68" (http:/ / speleotrove. com/

decimal/ decarith. html). Retrieved 2008-08-14. — a decimal floating point specification that includes negative zero

Further reading

• Michael Ingrassia. "Fortran 95 SIGN Change" (http:/ / developers. sun. com/ prodtech/ cc/ articles/ sign. html). Sun Developer Network. Retrieved October 15, 2005. — the changes in the Fortran SIGN function in Fortran 95 to accommodate negative zero

• "JScript data types" (http:/ / msdn. microsoft. com/ library/ en-us/ script56/ html/ js56jscondatatype. asp). MSDN JScript. Retrieved October 16, 2005. — JScript's floating point type has negative zero by definition

• "A look at the floating-point support of the Java virtual machine" (http:/ / www. javaworld. com/ javaworld/

jw-10-1996/ jw-10-hood. html). Javaworld. Retrieved October 16, 2005. — representation of negative zero in the Java virtual machine

• Bruce Dawson. Comparing floating point numbers (http:/ / www. cygnus-software. com/ papers/ comparingfloats/

comparingfloats. htm). — how to handle negative zero when comparing floating-point numbers

• John Walker. "Minus Zero" (http:/ / www. fourmilab. ch/ documents/ univac/ minuszero. html). UNIVAC Memories. Retrieved October 17, 2005. — One's complement numbers on the UNIVAC 1100 family computers. Integer 32 Integer

The integers (from the Latin integer, literally "untouched", hence "whole": the word entire comes from the same origin, but via French[1] ) are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {..., −2, −1, 0, 1, 2, ...}. For example, 21, 4, and −2048 are integers; 9.75 and 5½ are not integers.

Symbol often used to denote the set of integers

The set of all integers is often denoted by a boldface Z (or blackboard bold , Unicode U+2124 ℤ), which stands [2] for Zahlen (German for numbers, pronounced German pronunciation: [ˈtsaːlən]). The set is the finite set of integers modulo n (for example, ). The integers (with addition as operation) form the smallest group containing the additive monoid of the natural numbers. Like the natural numbers, the integers form a countably infinite set. In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as Integers can be thought of as discrete, equally spaced points on an infinitely long rational integers to distinguish them from number line. Natural Integers (purple) and Negative Integers (red). the more broadly defined algebraic integers (but with "rational" meaning "quotient of integers", this attempt at precision suffers from circularity).

Algebraic properties Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). The following lists some of the basic properties of addition and multiplication for any integers a, b and c.

Addition Multiplication

Closure: a + b is an integer a × b is an integer

Associativity: a + (b + c) = (a + b) + c a × (b × c) = (a × b) × c

Commutativity: a + b = b + a a × b = b × a

Existence of an identity element: a + 0 = a a × 1 = a

Existence of inverse elements: a + (−a) = 0 An inverse element usually does not exist at all.

Distributivity: a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c)

No zero divisors: If a × b = 0, then a = 0 or b = 0 (or both)

In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... + 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense Integer 33

that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. Adding the last property says that Z is an integral domain. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties Z is a totally ordered set without upper or lower bound. The ordering of Z is given by: ... −3 < −2 < −1 < 0 < 1 < 2 < 3 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: 1. if a < b and c < d, then a + c < b + d 2. if a < b and 0 < c, then ac < bc. It follows that Z together with the above ordering is an ordered ring. The integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition. Integer 34

Construction

The integers can be formally constructed as the equivalence classes of ordered pairs of natural numbers (a, b). The intuition is that (a, b) stands for the result of subtracting b from a. To confirm our expectation that 1 − 2 and 4 − 5 denote the same number, we define an equivalence relation ~ on these pairs with the following rule:

alt=Representation of equivalence classes for the numbers -5 to 5 |Red Points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line. |upright=2

precisely when

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; denoting by [(a,b)] the equivalence class having (a,b) as a member, one has:

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

Hence subtraction can be defined as the addition of the additive inverse:

The standard ordering on the integers is given by: iff It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes. Every equivalence class has a unique member that is of the form (n,0) or (0,n) (or both at once). The natural number n is identified with the class [(n,0)] (in other words the natural numbers are embedded into the integers by map sending n to [(n,0)]), and the class [(0,n)] is denoted −n (this covers all remaining classes, and gives the class [(0,0)] a second time since −0 = 0. Thus, [(a,b)] is denoted by

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity. This notation recovers the familiar representation of the integers as {... −3,−2,−1, 0, 1, 2, 3, ...}. Some examples are: Integer 35

Integers in computing An integer is often a primitive datatype in computer languages. However, integer datatypes can only represent a subset of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation datatypes (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). Variable-length representations of integers, such as bignums, can store any integer that fits in the computer's memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).

Cardinality The cardinality of the set of integers is equal to (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from Z to N. If N = {0, 1, 2, ...} then consider the function:

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,0) (1,1) (2,3) (3,5) ... } If N = {1,2,3,...} then consider the function:

{ ... (-4,8) (-3,6) (-2,4) (-1,2) (0,1) (1,3) (2,5) (3,7) ... } If the domain is restricted to Z then each and every member of Z has one and only one corresponding member of N and by the definition of cardinal equality the two sets have equal cardinality. Integer 36

Notes

[1] Evans, Nick (1995). "A-Quantifiers and Scope" (http:/ / books. google. com/ ?id=NlQL97qBSZkC). In Bach, Emmon W. Quantification in Natural Languages. Dordrecht, The Netherlands; Boston, MA: Kluwer Academic Publishers. pp. 262. ISBN 0792333527.

[2] Miller, Jeff (2010-08-29). "Earliest Uses of Symbols of Number Theory" (http:/ / jeff560. tripod. com/ nth. html). . Retrieved 2010-09-20.

References • Bell, E. T., Men of Mathematics. New York: Simon and Schuster, 1986. (Hardcover; ISBN 0-671-46400-0)/(Paperback; ISBN 0-671-62818-6) • Herstein, I. N., Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1. • Mac Lane, Saunders, and Garrett Birkhoff; Algebra, American Mathematical Society; 3rd edition (April 1999). ISBN 0-8218-1646-2.

• Weisstein, Eric W., " Integer (http:/ / mathworld. wolfram. com/ Integer. html)" from MathWorld.

External links

• The Positive Integers - divisor tables and numeral representation tools (http:/ / www. positiveintegers. org)

• On-Line Encyclopedia of Integer Sequences (http:/ / www. research. att. com/ ~njas/ sequences/ ) cf OEIS This article incorporates material from Integer on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 (an integer), 4/3 (a rational number that is not an integer), 8.6 (a rational number given by a finite decimal representation), √2 (the square root of two, an irrational number) and π (3.1415926535..., a transcendental number). Real numbers A symbol of the set of real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be Real numbers can be thought of as points on an infinitely long number line. determined by a possibly infinite decimal representation (such as that of π above), where the consecutive digits indicate into which tenth of an interval given by the previous digits the real number belongs to. The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.

These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realization that a better definition was needed — was one of the most important developments of 19th century mathematics. The currently standard axiomatic definition is that real numbers form the unique complete totally ordered field (R,+,·,<), up to isomorphism,[1] whereas popular constructive definitions of real numbers include declaring them as equivalence Real number 37

classes of Cauchy sequences of rational numbers, Dedekind cuts, or certain infinite "decimal representations", together with precise interpretations for the arithmetic operations and the order relation. These definitions are equivalent in the realm of classical mathematics.

Basic properties A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823122147… The ellipsis (three dots) indicate that there would still be more digits to come. More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound.

In physics In the physical sciences, most physical constants such as the universal gravitational constant, and physical variables, such as position, mass, speed, and electric charge, are modeled using real numbers. In fact, the fundamental physical theories such as classical mechanics, electromagnetism, quantum mechanics, general relativity and the standard model are described using mathematical structures, typically smooth manifolds or Hilbert spaces, that are based on the real numbers although actual measurements of physical quantities are of finite accuracy and precision. In some recent developments of theoretical physics stemming from the holographic principle, the Universe is seen fundamentally as an information store, essentially zeroes and ones, organized in much less geometrical fashion and manifesting itself as space-time and particle fields only on a more superficial level. This approach removes the real number system from its foundational role in physics and even prohibits the existence of infinite precision real numbers in the physical universe by considerations based on the Bekenstein bound.[2]

In computation Computer arithmetic cannot directly operate on real numbers, but only on a finite subset of rational numbers, limited by the number of bits used to store them, whether as floating-point numbers or arbitrary precision numbers. However, computer algebra systems can operate on irrational quantities exactly by manipulating formulas (such as , , or ) rather than their rational or decimal approximation;[3] however, it is not in

general possible to determine whether two such expressions are equal (the Constant problem). A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms,[4] but an uncountable number of reals, almost all real numbers fail to be computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. If computers could use unlimited-precision real numbers (real computation), then one could solve NP-complete problems, and even #P-complete problems in polynomial time, answering affirmatively the P = NP problem. Real number 38

Notation Mathematicians use the symbol R (or alternatively, , the letter "R" in blackboard bold, Unicode ℝ – U+211D) to represent the set of all real numbers (as this set is naturally endowed with a structure of field, the expression field of the real numbers is more frequently used than set of all real numbers). The notation Rn refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter. For example, a value from R3 consists of three real numbers and specifies the coordinates of a point in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example real matrix, real polynomial and real Lie algebra. As a substantive, the term is used almost strictly in reference to the real numbers themselves (e.g., The "set of all reals").

History Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic "Sulba Sutras" ("The rules of chords") in, ca. 600 BC, include what may be the first 'use' of irrational numbers. The concept of irrationality was implicitly accepted by early Indian mathematicians since Manava (c. 750–690 BC), who were aware that the square roots of certain numbers such as 2 and 61 could not be exactly determined.[5] Around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. The Middle Ages saw the acceptance of zero, negative, integral and fractional numbers, first by Indian and Chinese mathematicians, and then by Arabic mathematicians, who were also the first to treat irrational numbers as algebraic objects,[6] which was made possible by the development of algebra. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers.[7] The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850–930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.[8] In the 16th century, Simon Stevin created the basis for modern decimal notation, and insisted that there is no difference between rational and irrational numbers in this regard. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 18th and 19th centuries there was much work on irrational and transcendental numbers. Johann Heinrich Lambert (1761) gave the first flawed proof that π cannot be rational; Adrien-Marie Legendre (1794) completed the proof, and showed that π is not the square root of a rational number. Paolo Ruffini (1799) and Niels Henrik Abel (1842) both constructed proofs of Abel–Ruffini theorem: that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. Évariste Galois (1832) developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic equation, and then established existence of transcendental numbers, the proof being subsequently displaced by Georg Cantor (1873). Charles Hermite (1873) first proved that e is transcendental, and Ferdinand von Lindemann (1882), showed that π is transcendental. Lindemann's proof was much simplified by Weierstrass (1885), still further by David Hilbert (1893), and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. In 1874 he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, which he published in 1891. See Cantor's first uncountability proof. Real number 39

Definition The real number system can be defined axiomatically up to an isomorphism, which is described below. There are also many ways to construct "the" real number system, for example, starting from natural numbers, then defining rational numbers algebraically, and finally defining real numbers as equivalence classes of their Cauchy sequences or as Dedekind cuts, which are certain subsets of rational numbers. Another possibility is to start from some rigorous axiomatization of Euclidean geometry (Hilbert, Tarski etc.) and then define the real number system geometrically. From the structuralist point of view all these constructions are on equal footing.

Axiomatic approach Let R denote the set of all real numbers. Then: • The set R is a field, meaning that addition and multiplication are defined and have the usual properties. • The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z: • if x ≥ y then x + z ≥ y + z; • if x ≥ 0 and y ≥ 0 then xy ≥ 0. • The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind-complete ordered fields R and R , there exists a unique field isomorphism from R to R , allowing us to think of them as 1 2 1 2 essentially the same mathematical object. For another axiomatization of R, see Tarski's axiomatization of the reals.

Construction from the rational numbers The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,...} converges to a unique real number. For details and other constructions of real numbers, see construction of the real numbers.

Properties

Completeness The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (x ) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly n depending on ε) such that the distance |x − x | is less than ε for all n and m that are both greater than N. In other n m words, a sequence is a Cauchy sequence if its elements x eventually come and remain arbitrarily close to each other. n A sequence (x ) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that n the distance |x − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its n elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: Every Cauchy sequence of real numbers is convergent to a real number. Real number 40

That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...), where each term adds a digit of the decimal expansion of the positive square root of 2, is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the positive square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function

converges to a real number because for every x the sums

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if the limit is not known in advance.

"The complete ordered field" The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (in this case, the additive group of the field) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind-complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield. Real number 41

Advanced properties The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. In fact, the cardinality of the reals equals that of the set of subsets (i.e., the power set) of the natural numbers, and Cantor's diagonal argument states that the latter set's cardinality is strictly bigger than the cardinality of N. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology – in the order topology as intervals, in the metric topology as epsilon-balls. The Dedekind cuts construction uses the order topology presentation, while the Cauchy sequences construction uses the metric topology presentation. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, connected, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, although no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one real root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

Generalizations and extensions The real numbers can be generalized and extended in several different directions: • The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field. • The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, not even an additive group, but it still has a total order; moreover, it is a complete lattice. • The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, not even an additive group. However, it allows division of a non-zero element by zero. It is not ordered anymore. • The long real line pastes together ℵ * + ℵ copies of the real line plus a single point (here ℵ * denotes the 1 1 1 reversed ordering of ℵ ) to create an ordered set that is "locally" identical to the real numbers, but somehow 1 longer; for instance, there is an order-preserving embedding of ℵ in the long real line but not in the real numbers. 1 The long real line is the largest ordered set that is complete and locally Archimedean. As with the previous two examples, this set is no longer a field or additive group. Real number 42

• Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. • Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.

"Reals" in set theory In set theory, specifically descriptive set theory, the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

Real numbers and logic The real numbers are most often formalized using the Zermelo-Fraenkel axiomatization of set theory, but some mathematicians study the real numbers with other logical foundations of mathematics. In particular, the real numbers are also studied in reverse mathematics and in constructive mathematics.[9] Abraham Robinson's theory of nonstandard or hyperreal numbers extends the set of the real numbers by infinitesimal numbers, which allows to built infinitesimal calculus in a way which is closer to the usual intuition of the notion of limit. Edward Nelson's internal set theory is a non Zermelo-Fraenkel set theory which allows to consider the non standard real numbers as elements of the set of the reals (and not of an extension of it as in Robinson's theory). The continuum hypothesis posits that the cardinality of the set of the real numbers is , i.e. the smallest infinite cardinal number after , the cardinality of the integers. Paul Cohen has proved in 1963 that it is an axiom which is independent of the other axioms of set theory, that is one may choose either the continuum hypothesis or its negation as an axiom of set theory, without making it contradictory.

Notes [1] More precisely, given two complete totally ordered fields, there is a unique isomorphism between them. This implies that the identity is the unique field automorphism of the reals which is compatible with the ordering.

[2] Scott Aaronson, NP-complete Problems and Physical Reality (http:/ / arxiv. org/ abs/ quant-ph/ 0502072), ACM SIGACT News, Vol. 36, No. 1. (March 2005), pp. 30–52. [3] Cohen, Joel S. (2002), Computer algebra and symbolic computation: elementary algorithms, 1, A K Peters, Ltd., p. 32, ISBN 9781568811581

[4] James L. Hein, Discrete Structures, Logic, and Computability (http:/ / books. google. com/ books?id=vmlcc2IH9dEC), 3rd edition (Jones and Bartlett Publishers: Sudbury, MA), section 14.1.1 (2010). [5] T. K. Puttaswamy, "The Accomplishments of Ancient Indian Mathematicians", pp. 410–1, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1402002602

[6] O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?" (http:/ / www-history. mcs. st-andrews. ac. uk/

HistTopics/ Arabic_mathematics. html), MacTutor History of Mathematics archive, University of St Andrews, . [7] Matvievskaya, Galina (1987), "The Theory of Quadratic Irrationals in Medieval Oriental Mathematics", Annals of the New York Academy of Sciences 500: 253–277 [254], doi:10.1111/j.1749-6632.1987.tb37206.x [8] Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1402002602 [9] Bishop, Errett; Bridges, Douglas (1985), Constructive analysis, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 279, Berlin, New York: Springer-Verlag, ISBN 978-3-540-15066-4, chapter 2. Real number 43

References • Georg Cantor, 1874, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, volume 77, pages 258–262. • Robert Katz, 1964, Axiomatic Analysis, D. C. Heath and Company. • Edmund Landau, 2001, ISBN 0-8218-2693-X, Foundations of Analysis, American Mathematical Society. • Howie, John M., Real Analysis, Springer, 2005, ISBN 1-85233-314-6

External links

• The real numbers: Pythagoras to Stevin (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/

Real_numbers_1. html)

• The real numbers: Stevin to Hilbert (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/

Real_numbers_2. html)

• The real numbers: Attempts to understand (http:/ / www-groups. dcs. st-and. ac. uk/ ~history/ HistTopics/

Real_numbers_3. html)

• What are the "real numbers," really? (http:/ / www. math. vanderbilt. edu/ ~schectex/ courses/ thereals/ ) Article Sources and Contributors 44 Article Sources and Contributors

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Integer Source: http://en.wikipedia.org/w/index.php?oldid=440204917 Contributors: -Midorihana-, .:Ajvol:., 16@r, 28bytes, 5 albert square, 9258fahsflkh917fas, ALostIguana, Ace ETP, Ahoerstemeier, Airplaneman, Aitias, Akanemoto, Alansohn, Aldaron, Alejo2083, Altenmann, Amyx231, Analoguedragon, Andre Engels, Andres, AndyZ, Antandrus, Apparition11, Arammozuob, Arnejohs, Arthena, Arvindn, AxelBoldt, Az1568, BarretBonden, Bendono, Bjankuloski06en, Blehfu, Blockofwood, Bobo192, Bongwarrior, BrianH123, Brianjd, Brion VIBBER, Bryan Derksen, Camw, Can't sleep, clown will eat me, Capricorn42, Charles Matthews, Chris 73, Christian List, Christopher Parham, Chun-hian, Cikicdragan, Ciphergoth, Clader2, Computer97, Conversion script, Corvus cornix, Cxw, Cybercobra, DB.Gerry, DJ1AM, DMacks, Daniel Arteaga, Darkmeerkat, Darth Panda, Davexia, David Eppstein, DavidCBryant, Davidw1985, Dbenbenn, Deconstructhis, Demmy100, DerHexer, Deryck Chan, Digby Tantrum, Dirtylittlesecerts, Dkusic, Drake Redcrest, Dreadstar, Dreish, Dricherby, Dryke, Dwheeler, Dysprosia, Edgar181, El C, Elassint, Elroch, Epbr123, Eric119, Esanchez7587, Extransit, Faradayplank, Fayimora, Fieldday-sunday, Finell, Fnerchei, Fredrik, FrozenMan, Fæ, GT1345, Gaia Octavia Agrippa, Gaius Cornelius, Gakusha, Gauss, Georgia guy, Gfoley4, Gggh, Giftlite, Graham87, GreekHouse, GromXXVII, Grover cleveland, Guardian72, Hairhorn, Harland1, Henrygb, Hippietrail, Hornlitz, Hut 8.5, Igoldste, Imasleepviking, InvisibleK, Iridescent, Isaac909, Ivan Shmakov, J.delanoy, JSpung, Ja 62, JackSchmidt, Jake Nelson, Jarmiz, Jauhienij, Jfire, Jh12, Jiddisch, Jim.belk, Jj137, Jleedev, Johnferrer, Jojit fb, Jonkerz, Josh Parris, Jshadias, Julian Mendez, Jumbuck, KSmrq, Kadri, Kahriman, Kanags, Karadimos, Katzmik, Kawehi 65, Keilana, King of Hearts, Kingpin13, KnowledgeOfSelf, Knutux, Koeplinger, Ksero, Kuru, Kwamikagami, L Kensington, La Pianista, Lambiam, LayZeeDK, Leonard Vertighel, Leszek Jańczuk, Liempt, Lights, Linas, LizardJr8, Lolimakethingsfun, Loren.wilton, Lowellian, Lukeelms, Lupin, MacMed, Macy, Marc van Leeuwen, Markus Kuhn, MattGiuca, Mb5576, McSly, Mendaliv, Mentifisto, Mephistophelian, Merlion444, Metatron's Cube, Mets501, Michael Hardy, Michael Slone, Mike Segal, Miquonranger03, Moeron, Msh210, NSR, Nakon, NawlinWiki, Nikai, Notinlist, Obradovic Goran, Odie5533, Oleg Alexandrov, Olegalexandrov, OmegaMan, Omerks, Oneiros, Optikos, Oxymoron83, P0mbal, Panoramix, ParisianBlade, Paul August, Pb30, Philip Trueman, PhilippWeissenbacher, Pinethicket, Pleasantville, PookeyMaster, Pretzelpaws, Pufferfish101, Quaeler, Quota, Qxz, RJFJR, RJaguar3, Radon210, Raja Hussain, Rholton, Rich Farmbrough, Riitoken, Rob Hooft, Rumping, SJP, SLSB, Saeed, Salix alba, Salsa Shark, Saric, Sasquatch, Sbowers3, Sceptre, Scgtrp, Seaphoto, Seba5618, Seresin, Shadowjams, Sillychiva593, Sin Harvest, Snigbrook, Spindled, Splintax, Stapler9124, Stephenb, Stevertigo, Sticky Parkin, Stuart Presnell, Stwalkerster, SunCreator, SuperMidget, T.J.V., TakuyaMurata, Tanyakh, Tardis, Tbhotch, TeleComNasSprVen, The Flying Spaghetti Monster, The Thing That Should Not Be, TheRingess, Theda, Timothy Clemans, Tobias Bergemann, TomdFr, Triwbe, Trovatore, Troy 07, Ttwo, Unyoyega, Vegard, Vsmith, Wafulz, Waterfox, Wikipe-tan, Wikipelli, Wimt, Wknight94, Wolfrock, Wyatt915, X!, XJamRastafire, Xantharius, Youssefsan, Zack, Zarcillo, ZooFari, Zundark, 710 anonymous edits

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