In mathematics, the recurring decimal 0.999… , which is also written as or , denotes a real number equal to 1. In other words, "0.999…" represents the same number as the symbol "1". The equality has long been accepted by professional mathematicians and taught in textbooks. Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience. Image File history File links Download high-resolution version (2554x398, 73 KB) A view of 0. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... ‹ The template below is being considered for deletion. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Look up Rigour in Wiktionary, the free dictionary. ...

In the last few decades, researchers of mathematics education have studied the reception of this equality among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique decimal expansion, that nonzero infinitesimal quantities should exist, or that the expansion of 0.999… eventually terminates. Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ... In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ... A numeral is a symbol or group of symbols that represents a number. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

The non-uniqueness of such expansions is not limited to the decimal system. The same phenomenon occurs in integer bases other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. For reasons of simplicity, the terminating decimal is almost always the preferred representation, further contributing to the misconception that it is the only representation. In fact, once infinite expansions are allowed, all positional numeral systems contain an infinity of ambiguous numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations. These various identities have been applied to better understand patterns in the decimal expansions of fractions and the structure of a simple fractal, the Cantor set. They also occur in a classic investigation of the infinitude of the entire set of real numbers. The integers are commonly denoted by the above symbol. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... The use of non-integer numbers as radix, or bases, in a positional numbering system. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base (or radix) of that numeral system. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

Number systems in which 0.999… is strictly less than 1 can be constructed, but only outside the standard real number system which is used in elementary mathematics. For the socioeconomic meaning, see social inequality. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Introduction

0.999… is a number written in the decimal numeral system, and some of the simplest proofs that 0.999… = 1 rely on the convenient arithmetic properties of this system. Most of decimal arithmetic — addition, subtraction, multiplication, division, and comparison — uses manipulations at the digit level that are much the same as those for integers. As with integers, any two finite decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1. For other uses, see Decimal (disambiguation). ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... This article is about inequalities in mathematics. ... The integers are commonly denoted by the above symbol. ...

The meaning of "…" (ellipsis) in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some finite portion is left unstated or otherwise omitted. When used to specify a recurring decimal, "…" means that some infinite portion is left unstated. In particular, 0.999… indicates the limit of the sequence (0.9,0.99,0.999,0.9999,…) (or, equivalently, the sum of all terms of the form 9 × 0.1^k for integers k=1 to infinity). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1. This article is about the punctuation symbol. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... For other senses of this word, see sequence (disambiguation). ...

There are many proofs that 0.999… = 1. Before demonstrating this using algebraic methods, consider that two real numbers are identical if and only if their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a closed interval defined by the above sequence and the triangle inequality). Thus the difference is 0 and the numbers are identical. This also explains why 0.333… = ¹⁄₃, etc. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using fractions, ¹⁄₃ = ²⁄₆. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on). For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ...

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: The limit of a sequence is one of the oldest concepts in mathematical analysis. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

• 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.^[1]
• Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".^[2]
• 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.^[3]
• Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount.
• Some students believe that the value of a convergent series is an approximation, not the actual value.

These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999…. Look up paradox in Wiktionary, the free dictionary. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ... In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ...

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".^[4]

Of the elementary proofs, multiplying 0.333… = ¹⁄₃ by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.^[5] Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = ¹⁄₃ using a supremum definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.^[6] 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. In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ...

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."^[7] Joseph Mazur (born in the Bronx in 1942) is a professor of Mathematics at Marlboro College, in Marlboro, Vermont. ...

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky et al. also link this mental ability of encapsulation to viewing ¹⁄₃ as a number in its own right and to dealing with the set of natural numbers as a whole.^[8]

Proofs

Algebra

Fractions

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using long division, a simple division of integers like ¹⁄₃ becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × ¹⁄₃ equals 1, so .^[9] In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

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

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

Since both equations are valid, by the transitive property, 0.999… must equal 1. Similarly, ³/₃ = 1, and ³/₃ = 0.999… So, 0.999… must equal 1. In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c. ...

Digit manipulation

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

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

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it can be proven by investigating the fundamental relationship between decimals and the numbers they represent. For finite decimals, this process relies only on the arithmetic of real numbers. To prove that the manipulations also work for infinite decimals, one needs the methods of real analysis. Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

Real analysis

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

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. A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ...

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: It has been suggested that this article or section be merged with decimal. ... In mathematics, a series is a sum of a sequence of terms. ...

For 0.999… one can apply the powerful convergence theorem concerning infinite geometric series:^[10] In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, as a geometric series; see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

If | r | < 1 then

Since 0.999… is such a sum with a common ratio , the theorem makes short work of the question:

This proof (actually, that 10 equals 9.999…) appears as early as 1770 in Leonhard Euler's Elements of Algebra.^[11] Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Elements of Algebra is a mathematics textbook by the famous mathematician Leonhard Euler, originally published circa 1765. ...

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

The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebra proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999….^[12] A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.^[13] Image File history File links Base4_333. ... Image File history File links Base4_333. ...

A sequence (x₀, x₁, x₂, …) has a limit x if the distance |x − x_n| becomes arbitrarily small as n increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit: The limit of a sequence is one of the oldest concepts in mathematical analysis. ...

^[14]

The last step — that lim ¹/_10ⁿ = 0 — is often justified by the axiom that the real numbers have the Archimedean property. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook The University Arithmetic explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 Arithmetic for Schools says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".^[15] Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1. In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... Look up Heuristic in Wiktionary, the free dictionary. ...

Nested intervals and least upper bounds

Further information: Nested intervals

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

The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it. In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of In for all n. ... Image File history File links 999_Intervals_C.svg‎ File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): 0. ... Image File history File links 999_Intervals_C.svg‎ File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): 0. ...

If a real number x is known to lie in the closed interval [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number x must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b₀, b₁, b₂, b₃, …, and one writes In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of In for all n. ...

x = b₀.b₁b₂b₃…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.^[16]

One straightforward choice is the nested intervals theorem, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their intersection. So b₀.b₁b₂b₃… is defined to be the unique number contained within all the intervals [b₀, b₀ + 1], [b₀.b₁, b₀.b₁ + 0.1], and so on. 0.999… is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99…9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999… = 1.^[17] In mathematics, a sequence of nested intervals is understood as a collection of sets of real numbers In such that each set In is an interval of the real line, for n = 1, 2, 3, ... , and that further In + 1 is a subset of In for all n. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or suprema. To directly exploit these objects, one may define b₀.b₁b₂b₃… to be the least upper bound of the set of approximants {b₀, b₀.b₁, b₀.b₁b₂, …}.^[18] One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes, In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ...

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.^[19]

Real numbers

Main article: Construction of real numbers

Other approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found less than, greater than, or equal. In mathematics, there are a number of ways of defining the real number system as an ordered field. ... In mathematics, there are a number of ways of defining the real number system as an ordered field. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The integers are commonly denoted by the above symbol. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...

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.^[20]

Dedekind cuts

Further information: Dedekind cut

In the Dedekind cut approach, each real number x is the infinite set of all rational numbers that are less than x.^[21] In particular, the real number 1 is the set of all rational numbers that are less than 1.^[22] 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 .^[23] 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. In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards... In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards...

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872.^[24] 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.^[25] Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."^[26] A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "Different answers from alternative number systems" below. Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Mathematics Magazine is a a bimonthly publication of the Mathematical Association of America. ... In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if, intuitively, any point in X can be well-approximated by points in A. Formally, A is dense in X if for any point x in X, any neighborhood of... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and &#8722; (minus...

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 are Cauchy using this distance. That is, in the sequence (x₀, x₁, x₂, …), a mapping from natural numbers to rationals, for any positive rational δ there is an N such that |x_m − x_n| ≤ δ for all m, n > N. (The distance between terms becomes arbitrarily small.)^[27] In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...

If (x_n) and (y_n) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (x_n − y_n) has the limit 0. Truncations of the decimal number b₀.b₁b₂b₃… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.^[28] Thus in this formalism the task is to show that the sequence of 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;^[29] 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 limit of a sequence is one of the oldest concepts in mathematical analysis. ...

The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872.^[24] 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.^[30] Heinrich Eduard Heine (March 15, 1821–October 21, 1881) was a German mathematician. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ...

Generalizations

Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.^[31]

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.^[32] In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Ternary or trinary is the base-3 numeral system. ...

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, q = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the Thue-Morse sequence, which does not repeat.^[33] Golden ratio base refers to the use of the golden ratio, the irrational number ≈1. ... In mathematics, the phrase almost all has a number of specialised uses. ... Paul ErdÅ‘s, also ErdÅ‘s Pál, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... In mathematics and its applications, the Thue-Morse sequence, or Prouhet-Thue-Morse sequence, is a certain binary sequence whose initial segments alternate (in a certain sense). ...

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:^[34] Non-standard positional numeral systems are numeral systems that may be denoted positional systems, but that deviate in one way or another from the following description of standard positional system: In a standard positional numeral system, the base b is a positive integer, and b different glyphs are used to...

• In the balanced ternary system, ¹/₂ = 0.111… = 1.111….
• In the factoradic system, 1 = 1.000… = 0.1234….

Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary point-set topology"; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.^[35] Balanced ternary is a non-standard positional numeral system, useful for comparison logic. ... The factorial based radix or factoradic is a factorial based mixed radix numeral scheme: radix: 5! 4! 3! 2! 1! decimal: 120 24 6 2 1 In this numbering system, the rightmost digit may be 0 or 1, the next 0, 1, or 2, and so on. ... In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. ... In mathematics, Stones representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Boolean spaces, i. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...

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: Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

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

E. Midy proved a general result about such fractions, now called Midy's Theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.b₁b₂b₃… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.^[36] Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.^[37] In mathematics, Midys theorem, named after French mathematician E. Midy[1], is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a recurring decimal expansion with an even period. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ...

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

Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set: Image File history File links Cantor_base_3. ... Image File history File links Cantor_base_3. ... The boundary of the Mandelbrot set is a famous example of a fractal. ... The Cantor set, introduced by German mathematician Georg Cantor, is a remarkable construction involving only the real numbers between zero and one. ...

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

The nth digit of the representation reflects the position of the point in the nth stage of the construction. For example, the point ²⁄₃ is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point ¹⁄₃ is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.^[38] In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying his 1891 diagonal argument to decimal expansions, of the uncountability of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.^[39] 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.^[40] Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In mathematics, an uncountable set is a set which is not countable. ...

In popular culture

With the rise of the Internet, debates about 0.999… have escaped the classroom and are commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its FAQ.^[41] The FAQ briefly covers ⅓, multiplication by 10, and limits, and it alludes to Cauchy sequences as well. A newsgroup is a repository usually within the Usenet system, for messages posted from many users at different locations. ... This article or section does not cite its references or sources. ... FAQ is an abbreviation for Frequently Asked Question(s). The term refers to listed questions and answers, all supposed to be frequently asked in some context, and pertaining to a particular topic. ...

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999… via ⅓ and limits, saying of misconceptions, A Columnist is a journalist who produces a specific form of writing for publication called a column. ... Cecil Adams is the pen name of the author of The Straight Dope since 1973, a popular question and answer column published in The Chicago Reader, syndicated in thirty newspapers in the United States and Canada, and available online. ...

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

Nonsense.^[42]

The Straight Dope cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company's president, Mike Morhaime, announced at a press conference on April 1, 2004 that it is 1: Blizzard Entertainment, a division of Vivendi Games, is an American computer game developer and publisher headquartered in Irvine, California. ... Battle. ... Mike Morhaime is the president, and one of the founders of, Blizzard Entertainment, a video game developer located in Irvine, California and currently owned by the VU Games group of Vivendi Universal. ... A joint press conference by U.S. President George W. Bush and British Prime Minister Tony Blair at the White House. ... is the 91st day of the year (92nd in leap years) in the Gregorian calendar. ... Year 2004 (MMIV) was a leap year starting on Thursday of the Gregorian calendar. ...

We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.^[43]

Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

Different answers from alternative number systems

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

However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.^[44]

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

Infinitesimals

Main article: Infinitesimal

Some proofs that 0.999… = 1 rely on the Archimedean property of the standard real numbers: there are no nonzero infinitesimals. There are mathematically coherent ordered algebraic structures, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the dual numbers include a new infinitesimal element ε, analogous to the imaginary unit i in the complex numbers except that ε² = 0. The resulting structure is useful in automatic differentiation. The dual numbers can be given a lexicographic order, in which case the multiples of ε become non-Archimedean elements.^[45] Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... A variety of dualities in mathematics are listed at duality (mathematics). ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics and computer algebra, automatic differentiation, or AD, sometimes alternatively called algorithmic differentiation, is a method to numerically evaluate the derivative of a function specified by a computer program. ... In mathematics, the lexicographical order, or dictionary order, is a natural order structure of the cartesian product of two ordered sets. ...

Another way to construct alternatives to standard reals is to use topos theory and alternative logics rather than set theory and classical logic (which is a special case). For example, smooth infinitesimal analysis has infinitesimals with no reciprocals.^[46] In mathematics, a topos (plural topoi or toposes) is a type of category that behaves like the category of sheaves of sets on a topological space. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... The reciprocal function: y = 1/x. ...

Non-standard analysis is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to calculus.^[47] A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of ¹/₃ by an infinitesimal: Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Calculus (from Latin, pebble or little stone) is a branch of mathematics that includes the study of limits, derivatives, integrals, and infinite series, and constitutes a major part of modern university education. ...

0.333…;…000… does not exist, while

0.333…;…333… = ¹/₃ exactly.^[48]

Combinatorial game theory provides alternative reals as well, with infinite Blue-Red Hackenbush as one particularly relevant example. In 1974, Elwyn Berlekamp described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of data compression. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = ¹/₃. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the surreal number ¹/_ω, where ω is the first infinite ordinal; the relevant game is LRRRR… or 0.000….^[49] Mathematicians playing Konane at a Combinatorial game theory workshop (for technical content, see external link) This article is on the theory of combinatorial games. ... Hackenbush is a two-player partisan mathematical game that consists of several colored line segments connected to the ground. ... Elwyn Ralph Berlekamp (born September 6, 1940 in Dover, Ohio, United States of America) is a professor of mathematics at the University of California, Berkeley. ... “Source coding” redirects here. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...

First, Richman defines a nonnegative decimal number to be a literal decimal expansion. He defines the lexicographical order and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999… + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to ¹⁄₃. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.^[50] In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ...

In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction d he allows both the cut (−∞, d ) and the "principal cut" (−∞, d ]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999… < 1. There are no positive infinitesimals in cut D, but there is "a sort of negative infinitesimal," 0⁻, which has no decimal expansion. He concludes that 0.999… = 1 + 0⁻, while the equation "0.999… + x = 1" has no solution.^[51]

p-adic numbers

Main article: p-adic number

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….^[52] For an infinite string of 9s including a last 9, one must look elsewhere. In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ...

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

The p-adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p-adic numbers can be built from the rational numbers via Cauchy sequences; the construction uses a different metric in which 0 is closer to p, and much closer to pⁿ, than it is to 1 . The p-adic numbers form a field for prime p and a ring for other p, including 10. So arithmetic can be performed in the p-adics, and there are no infinitesimals. Image File history File links 4adic_333. ... Image File history File links 4adic_333. ... In mathematics, the p-adic number systems were first described by Kurt Hensel in 1897. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.^[53] Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

^[54]

(Compare with the series above.) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof above in the opposite direction: if x = …999 then 10x =  …990, so 10x = x − 9, hence x = −1 again.^[53]

As a final extension, since 0.999… = 1 (in the reals) and …999 = −1 (in the 10-adics), then by "blind faith and unabashed juggling of symbols"^[55] one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.^[56]

Related questions

• Zeno's paradoxes, particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.^[57]
• Division by zero occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as complex analysis, where the extended complex plane, i.e. the Riemann sphere, has point "infinity". Here, it makes sense to define ¹/₀ to be infinity;^[58] and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.^[59]
• Negative zero is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.^[60] Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the sign and magnitude or one's complement formats, or floating point numbers as specified by the IEEE floating-point standard).^[61][62]

“Arrow paradox” redirects here. ... In mathematics, a division is called a division by zero if the divisor is zero. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... A rendering of the Riemann Sphere. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a − sign. ... In mathematics, signed numbers in some arbitrary base is done in the usual way, by prefixing it with a - sign. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ...

Notes

Silvanus Phillips Thompson (June 19, 1851 – June 12, 1916). ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 242nd day of the year (243rd in leap years) in the Gregorian calendar. ... The Mathematical Association of America (MAA) is a professional society that focuses on undergraduate mathematics education. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 235th day of the year (236th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 180th day of the year (181st in leap years) in the Gregorian calendar. ... Cecil Adams is a name, generally assumed to be a pseudonym, which designates the unknown author or authors of The Straight Dope, a popular question and answer column published in The Chicago Reader since 1973. ... Cecil Adams is the pen name of the author of The Straight Dope since 1973, a popular question and answer column published in The Chicago Reader, syndicated in thirty newspapers in the United States and Canada, and available online. ... The Chicago Reader is an alternative newsweekly in Chicago, Illinois. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 249th day of the year (250th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 246th day of the year (247th in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 180th day of the year (181st in leap years) in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 180th day of the year (181st in leap years) in the Gregorian calendar. ... The Microsoft Developer Network (MSDN) is the portion of Microsoft responsible for managing the firms relationship with developers. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 241st day of the year (242nd in leap years) in the Gregorian calendar. ...

References

Elwyn Ralph Berlekamp (born September 6, 1940 in Dover, Ohio, United States of America) is a professor of mathematics at the University of California, Berkeley. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Richard Kenneth Guy (born 1916) is a Professor Emeritus in the Department of Mathematics at the University of Calgary. ... Winning Ways for your Mathematical Plays (ISBN 1568811306) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. ... John B. Conway is a mathematician at University of Tennessee. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... William Timothy Gowers (born November 20, 1963, Wiltshire, United Kingdom) is a British mathematician. ... Library of Congress reading room The Library of Congress Classification (LCC) is a system of library classification developed by the Library of Congress. ... Birmingham (pron. ... A grammar school is a school that may, depending on regional usage as exemplified below, provide either secondary education or, a much less common usage, primary education (also known as elementary). Grammar schools trace their origins back to medieval Europe, as schools in which university preparatory subjects, such as Latin... Mathematics education is a term that refers both to the practice of teaching and learning mathematics, as well as to a field of scholarly research on this practice. ... In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. ... arXiv (pronounced archive, as if the X were the Greek letter χ) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and quantitative biology which can be accessed via the Internet. ... December 2004 issue of the American Mathematical Monthly. ... Mathematics Magazine is a a bimonthly publication of the Mathematical Association of America. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 235th day of the year (236th in leap years) in the Gregorian calendar. ... Abraham Robinson Abraham Robinson (October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ... Walter Rudin Walter Rudin is an American mathematician, formerly a professor of mathematics at the University of Wisconsin, Madison. ... Hans Carl Friedrich von Mangoldt (1854-1925) was a German mathematician who contributed to the solution of the prime number theorem. ... David Foster Wallace (born February 21, 1962) is an American novelist, essayist, and short story writer. ...

See also

• Decimal representation
• Infinity
• Limit (mathematics)
• Naive mathematics
• Non-standard analysis
• Real analysis
• Series (mathematics)

It has been suggested that this article or section be merged with decimal. ... The infinity symbol ∞ in several typefaces. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as... Informal mathematics, also called naive mathematics, has historically the been the predominant form of mathematics at most times and in most cultures, and is the subject of many ethno-cultural studies of mathematics. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

External links

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• .999999… = 1? from cut-the-knot
• Why does 0.9999… = 1 ?
• Ask A Scientist: Repeating Decimals
• Proof of the equality based on arithmetic
• Repeating Nines
• Point nine recurring equals one
• David Tall's research on mathematics cognition

Image File history File links 0. ... Image File history File links Sound-icon. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 292nd day of the year (293rd in leap years) in the Gregorian calendar. ... Image File history File links Sound-icon. ... Image File history File links Commons-logo. ... cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ... Image File history File links LinkFA-star. ...