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{{Short description|Alternative decimal expansion of 1}}
{{pp-pc1}}
{{Use dmy dates|date=December 2012}} {{Use dmy dates|date=May 2024}}
] ]
In ], '''0.999...''' (also written as '''0.{{overline|9}}''', '''0.{{overset|.|9}}''', or '''0.(9)''') denotes the smallest number greater than every ] in the sequence {{nowrap|(0.9, 0.99, 0.999, ...)}}. It can be proved that this number is{{spaces}}]; that is,
: <math>0.999... = 1.</math>
Despite common misconceptions, 0.999... is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent {{em|exactly}} the same number.


An elementary proof is given below that involves only ] and the fact that there is no ] less than all 1/10<sup>{{math|''n''}}</sup>, where {{math|''n''}} is a natural number, a property that results immediately from the ] of the ]s.
In ], the ] '''0.999...''' (sometimes written with more or fewer 9s before the final ], for example as '''0.9...''', or in a variety of other variants such as '''0.<span style="text-decoration: overline;">9</span>''', '''0.(9)''', or {{nowrap|<math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math>}}) denotes a ] that can be shown to be ]. In other words, the symbols "0.999..." and "1" represent the same number. ] of this ] have been formulated with varying degrees of ], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.


There are many other ways of showing this equality, from ] arguments to ] ]. The intuitive arguments are generally based on properties of ]s that are extended without proof to infinite decimals. The proofs are generally based on basic properties of real numbers and methods of ], such as ] and ]s. A question studied in ] is why some people reject this equality.
Every nonzero, terminating decimal (with infinitely many trailing 0s) has an equal twin representation with infinitely many trailing 9s (for example, 8.32 and 8.31999...). The terminating decimal representation is usually preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other ] (with a given base's largest digit) or in any similar representation of the real numbers.


In ], 0.999... can have the same meaning, a different definition, or be undefined. Every nonzero ] has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all ]s that represent the real numbers.
The equality of 0.999... and 1 is closely related to the absence of nonzero ]s in the real number system, the most commonly used system in ]. Some ], such as the ], do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.


== Elementary proof ==
The equality 0.999...&nbsp;=&nbsp;1 has long been accepted by mathematicians and is part of general ]. Nonetheless, some students find it sufficiently ] that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.
]: any point {{math|1=''x''}} before the finish line lies between two of the points {{math|1=''P''<sub>''n''</sub>}} (inclusive).|class=skin-invert-image]]


It is possible to prove the equation {{nowrap|1=0.999... = 1}} using just the mathematical tools of comparison and addition of (finite) ]s, without any reference to more advanced topics such as ] and ]. The proof given ] is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the ], there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so {{nowrap|1=0.999... = 1}}.
==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as ], ], and digit manipulation to build transformations preserving equality from 0.999... to 1. However, these proofs are not entirely rigorous as they don't include a careful analytic definition of 0.999...


=== Intuitive explanation ===
===Fractions and long division{{anchor|Fractions}}===
If one places 0.9, 0.99, 0.999, etc. on the ], one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. For any number <math>x</math> that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than {{tmath|1= x }}. So, it does not make sense to identify 0.999... with any number smaller than 1. Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines. Therefore, 0.999... cannot be identified with any number larger than 1, either. Because 0.999... cannot be bigger than 1 or smaller than 1, it must equal 1 if it is to be any real number at all.{{sfnp|Cheng|2023|p=141}}{{sfnp|Diamond|1955}}
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 {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for {{nowrap|1=0.999... = 1}}. Multiplication of 9 times 1 produces 9 in each digit, so {{nowrap|9 × 0.111...}} equals 0.999... and {{nowrap|9 × {{frac|1|9}}}} equals 1, so {{nowrap|1=0.999... = 1}}:


=== Rigorous proof ===
:<math>
\begin{align}
\frac{1}{9} & = 0.111\dots \\
9 \times \frac{1}{9} & = 9 \times 0.111\dots \\
1 & = 0.999\dots
0.999\dots =1
\end{align}
</math>


Denote by 0.(9)<sub>{{math|''n''}}</sub> the number 0.999...9, with <math> n </math> nines after the decimal point. Thus {{nowrap|1=0.(9)<sub>1</sub> = 0.9}}, {{nowrap|1=0.(9)<sub>2</sub> = 0.99}}, {{nowrap|1=0.(9)<sub>3</sub> = 0.999}}, and so on. One has {{nowrap|1=1 − 0.(9)<sub>1</sub> = 0.1 = {{tmath|1= \textstyle \frac{1}{10} }}}}, {{nowrap|1= 1 − 0.(9)<sub>2</sub> = 0.01 = {{tmath|1= \textstyle \frac{1}{10^2} }}}}, and so on; that is, {{nowrap|1=1 − 0.(9)<sub>{{math|''n''}}</sub> = <math display="inline"> \frac{1}{10^n} </math>}} for every ] {{tmath|1= n }}.
This result is consistent with other ninth fractions, all of which have repeating decimals, such as 3/9 and 8/9. If 0.999... is to be consistent, it must equal 9/9 = 1.
:<math>
\begin{align}
0.333\dots & = \frac{3}{9} \\
0.888\dots & = \frac{8}{9} \\
0.999\dots & = \frac{9}{9} = 1
\end{align}
</math>


Let <math>x</math> be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, {{nowrap|0.(9)<sub>{{math|''n''}}</sub> < <math> x </math> ≤ 1}}, for every {{tmath|1= n }}. By subtracting these inequalities from 1, one gets {{nowrap|0 ≤ 1 − <math> x </math> < {{tmath|1= \textstyle \frac{1}{10^n} }}}}.
===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&nbsp;×&nbsp;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&nbsp;−&nbsp;9&nbsp;=&nbsp;0 for each such digit. The final step uses algebra:


The end of the proof requires that there is no positive number that is less than <math display="inline">\frac{1}{10^n} </math> for all {{tmath|1= n }}. This is one version of the ], which is true for real numbers.{{sfnp|Baldwin|Norton|2012}}{{sfnp|Meier|Smith|2017|loc=§8.2}} This property implies that if {{nowrap|1 − <math> x </math> < {{tmath|1= \textstyle \frac{1}{10^n} }}}} for all {{tmath|n}}, then {{nowrap|1 − <math> x</math>}} can only be equal to 0. So, {{nowrap|1=<math> x </math> = 1}} and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, {{nowrap|1=1 = 0.999...}}.
:<math>
\begin{align}
x &= 0.999\ldots \\
10x &= 9.999\ldots \\
10x &= 9+0.999\ldots \\
10x &= 9 + x\\
9x &= 9\\
x &= 1
\end{align}
</math>


This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into ], such as ]s, with infinitely small numbers (]s) and infinitely large numbers (]s).{{sfnp|Stewart|2009|p=175}}{{sfnp|Propp|2023}} When using such systems, the notation 0.999... is generally not used, as there is no smallest number among the numbers larger than all 0.(9)<sub>{{math|''n''}}</sub>.{{efn|For example, one can show this as follows: if {{math|''x''}} is any number such that {{nowrap|0.(9)<sub>{{math|''n''}}</sub> ≤ {{math|''x'' < 1}}}}, then {{nowrap|0.(9)<sub>{{math|''n''}}−1</sub> ≤ 10{{math|''x''}} − 9 < {{math|''x''}} < 1}}. Thus if {{math|''x''}} has this property for all {{math|''n''}}, the smaller number {{nowrap|10{{math|''x''}} − 9}} does, as well.}}
===Discussion===
Although these proofs demonstrate that 0.999...&nbsp;=&nbsp;1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999...&nbsp;=&nbsp;1 but 0.333...&nbsp;<&nbsp;0.34. 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 ], which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186. William Byers argues that a student who agrees that 0.999...&nbsp;=&nbsp;1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation (Byers pp. 39–41). 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".(p. 396)</ref>


=== Least upper bounds and completeness ===
Once a ] 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.


Part of what this argument shows is that there is a ] of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the ]s of the ] is the ], which states that every bounded sequence has a least upper bound.{{sfnp|Stillwell|1994|p=42}}{{sfnp|Earl|Nicholson|2021|loc="bound"}} This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations.{{sfnp|Rosenlicht|1985|p=27}} The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers has a least upper bound and that this least upper bound is equal to one.{{sfnp|Bauldry|2009|p=47}}
==Analytic proofs{{anchor|Analytic}}==
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 ]. 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''<sub>0</sub> and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots.</math>


== Algebraic arguments <span class="anchor" id="Proofs"></span><span class="anchor" id="Algebraic"></span> ==
It should be noted that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a ], so for example the digit 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.
Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. {{harvtxt|Byers|2007}} discusses the argument that, in elementary school, one is taught that {{nowrap|1=<math display="inline"> \frac{1}{3} </math> = 0.333...}}, so, ignoring all essential subtleties, "multiplying" this identity by 3 gives {{nowrap|1=1 = 0.999...}}. He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the ]; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...{{px2}}." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to&nbsp;1.{{sfnp|Byers|2007|p=39}} {{harvtxt|Richman|1999}} discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.{{sfnp|Richman|1999}}


Byers also presents the following argument.
===Infinite series and sequences===
{{block indent|1=<math>
{{further2|]}}
\begin{align}
x &= 0.999\ldots \\
10x &= 9.999\ldots && \text{by multiplying by }10\\
10x &= 9+0.999\ldots && \text{by splitting off integer part}\\
10x &= 9 + x && \text{by definition of }x\\
9x &= 9 && \text{by subtracting }x\\
x &= 1 && \text{by dividing by }9
\end{align}
</math>}}


Students who did not accept the first argument sometimes accept the second argument, but, in Byers's opinion, still have not resolved the ambiguity, and therefore do not understand the representation of infinite decimals. {{harvtxt|Peressini|Peressini|2007}}, presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and ].{{sfnp|Peressini|Peressini|2007|p=}} {{harvtxt|Baldwin|Norton|2012}}, citing {{harvtxt|Katz|Katz|2010a}}, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.{{sfnmp
Perhaps the most common development of decimal expansions is to define them as sums of ]. In general:
| 1a1 = Baldwin | 1a2 = Norton | 1y = 2012
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>
| 2a1 = Katz | 2a2 = Katz | 2y = 2010a
}} {{harvtxt|Cheng|2023}} concurs, arguing that knowing one can multiply 0.999... by 10 by shifting the decimal point presumes an answer to the deeper question of how one gives a meaning to the expression 0.999... at all.{{sfnp|Cheng|2023|p=136}} The same argument is also given by {{harvtxt|Richman|1999}}, who notes that skeptics may question whether <math> x </math> is ]{{snd}} that is, whether it makes sense to subtract <math> x </math> from both sides.{{sfnp|Richman|1999}} {{Harvtxt|Eisenmann|2008}} similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.{{Sfnp|Eisenmann|2008|p=38}}


== Analytic proofs <span class="anchor" id="Analytic"></span> ==
For 0.999... one can apply the ] theorem concerning ]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
] is the study of the logical underpinnings of ], including the behavior of sequences and series of real numbers.{{sfnp|Tao|2003}} The proofs in this section establish {{nowrap|1=0.999... = 1}} using techniques familiar from real analysis.
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>


=== Infinite series and sequences ===
Since 0.999... is such a sum with a common ratio r = {{frac|1|10}}, the theorem makes short work of the question:
{{further|Decimal representation}}
:<math>0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in ]'s '']''.<ref>Euler p. 170</ref>


A common development of decimal expansions is to define them as sums of ]. In general:
] fraction sequence (.3, .33, .333, ...) converging to 1.]]
<math display="block"> b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>
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 ] 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...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> 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.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>


For 0.999... one can apply the ] theorem concerning ], stating that if {{nowrap|{{tmath|1= \vert r \vert }} < 1}}, then:{{sfnmp
A ] (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ...) has a ] ''x'' if the distance |''x''&nbsp;−&nbsp;''x''<sub>''n''</sub>| becomes arbitrarily small as ''n'' increases. The statement that 0.999...&nbsp;=&nbsp;1 can itself be interpreted and proven as a limit:<ref>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).</ref>
| 1a1 = Rudin | 1y = 1976 | 1p = 61 | 1loc = Theorem 3.26
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>
| 2a1 = Stewart | 2y = 1999 | 2p = 706
}}
<math display="block"> ar + ar^2 + ar^3 + \cdots = \frac{ar}{1-r}.</math>


Since 0.999... is such a sum with <math> a = 9 </math> and common ratio {{tmath|1= \textstyle r = \frac{1}{10} }}, the theorem makes short work of the question:
The last step, that {{frac|1|10<sup>''n''</sup>}} → 0 as ''n'' → ∞, is often justified by the ] of the real numbers. 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".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such ]s are often interpreted by students as implying that 0.999... itself is less than 1.
<math display="block"> 0.999\ldots = 9\left(\tfrac{1}{10}\right) + 9\left({\tfrac{1}{10}}\right)^2 + 9\left({\tfrac{1}{10}}\right)^3 + \cdots = \frac{9\left({\tfrac{1}{10}}\right)}{1-{\tfrac{1}{10}}} = 1.</math>
This proof appears as early as 1770 in ]'s '']''.{{sfnp|Euler|1822|p=170}}


] fraction sequence {{nowrap|1=(.3, .33, .333, ...)}} converging to 1.|class=skin-invert-image]]
===Nested intervals and least upper bounds===
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 ] 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...{{px2}}.{{sfnmp
{{further2|]}}
| 1a1 = Grattan-Guinness | 1y = 1970 | 1p = 69
| 2a1 = Bonnycastle |2y=1806 | 2p = 177
}} 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 several proof-based introductions to calculus or analysis.{{sfnmp
| 1a1 = Stewart | 1y = 1999 | 1p = 706
| 2a1 = Rudin | 2y = 1976 | 2p = 61
| 3a1 = Protter | 3a2 = Morrey | 3y = 1991 | 3p = 213
| 4a1 = Pugh | 4y = 2002 | 4p = 180
| 5a1 = Conway | 5y = 1978 | 5p = 31
}}


A ] {{nowrap|(<math> x_0 </math>, <math> x_1 </math>, <math> x_2 </math>, ...)}} has the value <math> x </math> as its ] if the distance <math> \left\vert x - x_n \right\vert </math> becomes arbitrarily small as <math> n </math> increases. The statement that {{nowrap|1=0.999... = 1}} can itself be interpreted and proven as a limit:{{efn|The limit follows, for example, from {{harvtxt|Rudin|1976}}, p. 57, Theorem 3.20e. For a more direct approach, see also {{harvtxt|Finney|Weir|Giordano|2001}}, section 8.1, example 2(a), example 6(b).}}
]
<math display="block"> 0.999\ldots \ \overset{\underset{\mathrm{def}}{}}{=} \ \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} \ \overset{\underset{\mathrm{def}}{}}{=} \ \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1 - 0 = 1. </math>
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.
The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10<sup>{{math|''-n''}}</sup> approaches 0 as <math> n </math> approaches infinity ({{tmath|1= \infty }}), is often justified by the ] of the real numbers. 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".{{sfnmp
| 1a1 = Davies | 1y = 1846 | 1p = 175
| 2a1 = Smith | 2a2 = Harrington | 2y = 1895 | 2p = 115
}} Such ]s are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.{{sfnp|Tall|2000|p=221}}


=== Nested intervals and least upper bounds ===
If a real number ''x'' is known to lie in the ] (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: , , , and so on up to . The number ''x'' must belong to one of these; if it belongs to then one records the digit "2" and subdivides that interval into , , ..., , . Continuing this process yields an infinite sequence of ], labeled by an infinite sequence of digits ''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, ''b''<sub>3</sub>, ..., and one writes
{{further|Nested intervals}}


]
:<math>x = b_0.b_1b_2b_3 \dots.</math>
The series definition above defines 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 <math> x </math> is known to lie in the ] {{nowrap|}} (that is, 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: {{nowrap|}}, {{nowrap|}}, {{nowrap|}}, and so on up to {{nowrap|}}. The number <math> x </math> must belong to one of these; if it belongs to {{nowrap|}}, then one records the digit "2" and subdivides that interval into {{nowrap|}}, {{nowrap|}}, ..., {{nowrap|}}, {{nowrap|}}. Continuing this process yields an infinite sequence of ], labeled by an infinite sequence of digits {{tmath|1= b_1 }}, {{tmath|1= b_2 }}, {{tmath|1= b_3 }}, ..., and one writes
In this formalism, the identities 1&nbsp;=&nbsp;0.999... and 1&nbsp;=&nbsp;1.000... reflect, respectively, the fact that 1 lies in both and , 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.<ref>Beals p. 22; I. Stewart p. 34</ref>
<math display="block"> x = b_0.b_1b_2b_3 \ldots \,. </math>


In this formalism, the identities {{nowrap|1=1 = 0.999...}} and {{nowrap|1=1 = 1.000...}} reflect, respectively, the fact that 1 lies in both {{nowrap|}}. and {{nowrap|}}, 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.{{sfnmp
One straightforward choice is the ], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their ]. So ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>... is defined to be the unique number contained within all the intervals , , and so on. 0.999... is then the unique real number that lies in all of the intervals , , , and for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>
| 1a1 = Beals | 1y = 2004 | 1p = 22
| 2a1 = Stewart | 2y = 2009 | 2p = 34
}}


One straightforward choice is the ], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their ]. So {{tmath|1= b_1 }}, {{tmath|1= b_2 }}, {{tmath|1= b_3 }}, ... is defined to be the unique number contained within all the intervals {{nowrap|{{bracket|<math> b_0 </math>, <math> b_0 </math> + 1}}}}, {{nowrap|{{bracket|<math> b_0.b_1 </math>, <math> b_0.b_1 </math> + 0.1}}}}, and so on. 0.999... is then the unique real number that lies in all of the intervals {{nowrap|}}, {{nowrap|}}, {{nowrap|}}, and {{nowrap|}} for every finite string of 9s. Since 1 is an element of each of these intervals, {{nowrap|1=0.999... = 1}}.{{sfnmp
The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of ]s or '']''. To directly exploit these objects, one may define ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>... to be the least upper bound of the set of approximants {''b''<sub>0</sub>, ''b''<sub>0</sub>.''b''<sub>1</sub>, ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> 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,
| 1a1 = Bartle | 1a2 = Sherbert | 1y = 1982 | 1pp = 60–62
| 2a1 = Pedrick | 2y = 1994 | 2p = 29
| 3a1 = Sohrab | 3y = 2003 | 3p = 46
}}


The nested intervals theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of ]s or '']''. To directly exploit these objects, one may define {{tmath|1= b_0.b_1 b_2 b_3 }}... to be the least upper bound of the set of approximants {{tmath|1= b_0 }}, {{tmath|1= b_0. b_1 }}, {{tmath|1= b_0. b_1 b_2 }},&nbsp;...{{px2}}.{{sfnmp
<blockquote>
| 1a1 = Apostol | 1y = 1974 | 1pp = 9, 11–12
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.<ref>Apostol p. 12</ref>
| 2a1 = Beals | 2y= 2004 | 2p = 22
</blockquote>
| 3a1 = Rosenlicht | 3y = 1985 | 3p = 27
}} One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying {{nowrap|1=0.999... = 1}} again. ] 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."{{sfnp|Apostol|1974|p=12}}


==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}== == Proofs from the construction of the real numbers <span class="anchor" id="Based on the construction of the real numbers"></span> ==
{{further2|]}} {{further|Construction of the real numbers}}


Some approaches explicitly define real numbers to be certain ], using ]. The ]s&nbsp;– 0, 1, 2, 3, and so on&nbsp;– 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 ]s, and to further extend to ratios, giving the ]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ], so that one number can be compared to another and found to be less than, greater than, or equal to another number. Some approaches explicitly define real numbers to be certain ], using ]. The ]s {{nowrap|{{mset|0, 1, 2, 3, ...}}}} 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 ]s, and to further extend to ratios, giving the ]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division.{{sfnp|Cheng|2023|pp=153–156}}{{sfnp|Conway|2001|pp=25–27}} More subtly, they include ], so that one number can be compared to another and found to be less than, greater than, or equal to another number.{{sfnp|Rudin|1976|pp=3,8}}


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: ]s and ]s. 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.<ref>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</ref> 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: ]s and ]s. Proofs that {{nowrap|1=0.999... = 1}} that directly uses 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 toward 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.{{efn|The historical synthesis is claimed by {{harvtxt|Griffiths|Hilton|1970}}, p.&nbsp;xiv and again by {{harvtxt|Pugh|2002}}, p.&nbsp;10; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see {{harvtxt|Pugh|2002}}, p.&nbsp;17 or {{harvtxt|Rudin|1976}}, p.&nbsp;17. For viewpoints on logic, see {{harvtxt|Pugh|2002}}, p.&nbsp;10, {{harvtxt|Rudin|1976}}, p.ix, or {{harvtxt|Munkres|2000}}, p.&nbsp;30.}}


===Dedekind cuts=== === Dedekind cuts ===
{{further2|]}} {{further|Dedekind cut}}


In the ] approach, each real number ''x'' is defined as the '''] of all rational numbers less than ''x'''''.<ref>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..."</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp. 17–20, Richman p. 399, or Enderton p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1<sup>−</sup>, and 1<sub>''R''</sub>, 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".</ref> 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 In the ] approach, each real number <math> x </math> is defined as the ] of all ]s less than {{tmath|1= x }}.{{efn|{{harvtxt|Enderton|1977}}, p. 113 qualifies this description: "The idea behind Dedekind cuts is that a real number {{math|''x''}} can be named by giving an infinite set of rationals, namely all the rationals less than {{math|''x''}}. We will in effect define {{math|''x''}} to be the set of rationals smaller than {{math|''x''}}. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way&nbsp;..."}} In particular, the real number 1 is the set of all rational numbers that are less than 1.{{efn|{{harvtxt|Rudin|1976}}, pp. 17–20, {{harvtxt|Richman|1999}}, p.&nbsp;399, or {{harvtxt|Enderton|1977}}, p.&nbsp;119. To be precise, Rudin, Richman, and Enderton call this cut 1∗, 1<sub>−</sub>, and 1<sub>R</sub>, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "non-principal Dedekind cut".}} Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers that are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers <math> r </math> such that {{nowrap|<math> r </math> < 0}}, or {{nowrap|<math> r </math> < 0.9}}, or {{nowrap|<math> r </math> < 0.99}}, or <math> r </math> is less than some other number of the form{{sfnp|Richman|1999|p=399}}
<math display="block"> 1-\frac{1}{10^n} = 0.(9)_n = 0.\underbrace{99\ldots 9}_{n\text{ nines}}. </math>
:<math>\begin{align}1-\left(\tfrac{1}{10}\right)^n\end{align}.</math><ref>Richman p. 399</ref>
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
:<math>\begin{align}\tfrac{a}{b}<1,\end{align}</math>
which implies
:<math>\begin{align}\tfrac{a}{b}<1-\left(\tfrac{1}{10}\right)^b\end{align}.</math>
Since 0.999... and 1 contain the same rational numbers, they are the same set: 0.999... = 1.


Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as
The definition of real numbers as Dedekind cuts was first published by ] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |first1 = J. J. | last1 = O'Connor | first2 = E. F. | last2 = Robertson |work=MacTutor History of Mathematics |accessdate=2006-08-30 |archiveurl=http://web.archive.org/web/20070929095431/http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |archivedate=2007-09-29 |deadurl=no |date=October 2005}}</ref>
<math display="block"> \frac{a}{b}<1,</math>
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 '']'',<ref name="MathMag">{{cite web|url=https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=754744&toolAttachmentId=145726|title=Is 0.999... = 1?}}</ref> which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any ] of the rational numbers yields the same results; in particular, he uses ]s, for which the proof is more immediate. He also notes that typically the definitions allow
with <math> b > 0 </math> and {{tmath|1= b > a }}. This implies
{ 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."<ref>Richman pp. 398–399</ref> 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 "]" below.
<math display="block"> 1-\frac{a}{b} = \frac{b-a}{b} \ge \frac{1}{b} > \frac{1}{10^b}, </math>
and thus
<math display="block"> \frac{a}{b} < 1 - \frac{1}{10^b}. </math>


Since
===Cauchy sequences===
<math display="block"> 1-\frac{1}{10^b} = 0.(9)_b < 0.999\ldots, </math>
{{further2|]}}
by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, {{nowrap|1=0.999... = 1}}.


The definition of real numbers as Dedekind cuts was first published by ] in 1872.{{sfnp|O'Connor|Robertson|2005}}
Another approach is to define a real number as the '''limit of a Cauchy sequence of rational numbers'''. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x''&nbsp;−&nbsp;''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 ] property using this distance. That is, in the sequence (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''<sub>''m''</sub>&nbsp;−&nbsp;''x''<sub>''n''</sub>|&nbsp;≤&nbsp;δ for all ''m'', ''n''&nbsp;>&nbsp;''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is {{nowrap|1=0.999 ... = 1}}?" by Fred Richman in '']''.{{sfnp|Richman|1999}} Richman notes that taking Dedekind cuts in any ] of the rational numbers yields the same results; in particular, he uses ]s, for which the proof is more immediate. He also notes that typically the definitions allow {{nowrap|{{mset|<math> x </math> | <math> x </math> < 1}}}} to be a cut but not {{nowrap|{{mset|<math> x </math> | <math> x </math> ≤ 1}}}} (or vice versa).<ref>{{harvtxt|Richman|1999}}, p. 398–399. "Why do that? Precisely to rule out the existence of distinct numbers 0.{{overline|9}} and 1. So we see that in the traditional definition of the real numbers, the equation {{nowrap|1=0.{{overline|9}} = 1}} is built in at the beginning."</ref> 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 <math display="inline"> \frac{1}{3} </math> has no representation; see ''{{slink||Alternative number systems}}'' below.


=== Cauchy sequences ===
If (''x''<sub>''n''</sub>) and (''y''<sub>''n''</sub>) are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''<sub>''n''</sub>&nbsp;−&nbsp;''y''<sub>''n''</sub>) has the limit 0. Truncations of the decimal number ''b''<sub>0</sub>.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp. 388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers
{{further|Cauchy sequence}}
:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>


Another approach is to define a real number as the limit of a ] of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between <math> x </math> and <math> y </math> is defined as the absolute value {{tmath|1= \left\vert x - y \right\vert }}, where the absolute value <math> \left\vert z \right\vert </math> is defined as the maximum of <math> z </math> and {{tmath|1= -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 {{tmath|1= x_0 }}, {{tmath|1= x_1 }}, {{tmath|1= x_2 }},&nbsp;..., a mapping from natural numbers to rationals, for any positive rational <math> \delta </math> there is an <math> N </math> such that <math> \left\vert x_m - x_n \right\vert \le \delta </math> for all {{tmath|1= m, n > N }}; the distance between terms becomes smaller than any positive rational.{{sfnp|Griffiths|Hilton|1970|p=386|loc=§24.2 "Sequences"}}
has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>


If <math> (x_n) </math> and <math> (y_n) </math> are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence <math> (x_n - y_n) </math> has the limit 0. Truncations of the decimal number {{tmath|1= b_0. b_1 b_2 b_3 }}... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.{{sfnp|Griffiths|Hilton|1970|pp=388, 393}} Thus in this formalism the task is to show that the sequence of rational numbers
This limit is plain<ref>Griffiths & Hilton p. 395</ref> if one understands the ]. So again 0.999...&nbsp;=&nbsp;1.
<math display="block">
\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \ldots\right) = \left(1, {1 \over 10}, {1 \over 100}, \ldots \right)
</math>
has a limit 0. Considering the {{tmath|1= n }}th term of the sequence, for {{tmath|1= n \in \mathbb{N} }}, it must therefore be shown that
<math display="block"> \lim_{n\rightarrow\infty}\frac{1}{10^n} = 0. </math>
This can be proved by the ]. So again, {{nowrap|1=0.999... = 1}}.{{sfnp|Griffiths|Hilton|1970|p=395}}


The definition of real numbers as Cauchy sequences was first published separately by ] and ], also in 1872.<ref name="MacTutor2" /> 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.<ref>Griffiths & Hilton pp.viii, 395</ref> The definition of real numbers as Cauchy sequences was first published separately by ] and ], also in 1872.{{sfnp|O'Connor|Robertson|2005}} The above approach to decimal expansions, including the proof that {{nowrap|1=0.999... = 1}}, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''.{{sfnp|Griffiths|Hilton|1970|pp=viii, 395}}


=== Infinite decimal representation === === Infinite decimal representation ===
{{further2|]}}


Commonly in ]' mathematics education, the real numbers are constructed by defining a number using an integer followed by a ] and an infinite sequence written out as a string to represent the ] of any given real number. In this construction, the set of any combination of integers and digits after the decimal point (or radix point in non-base 10 systems) are the set of real numbers. This construction can too be rigorously shown to satisfy all of the ] after defining an ] over the set that '''defines''' 1 =<sub>eq</sub> 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version.<ref>{{cite arXiv |title=A new approach to the real numbers |author=Liangpan Li |deadurl=no |eprint=1101.1800 |class=math.CA |date=March 2011}}</ref> With this construction of the reals, all proofs of the statement 1 = .999... can be viewed as implicitly assuming the equality when any operations are performed on the real numbers. Commonly in ]' mathematics education, the real numbers are constructed by defining a number using an integer followed by a ] and an infinite sequence written out as a string to represent the ] of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the ] after defining an ] over the set that defines {{nowrap|1=1 =<sub>eq</sub> 0.999...}} as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version. In other words, the equality {{nowrap|1=0.999... = 1}} holding true is a necessary condition for strings of digits to behave as real numbers should.{{sfnp|Gowers|2001}}{{sfnp|Li|2011}}


==Generalizations== === Dense order ===
{{further|Dense order}}
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 ].<ref>Petkovšek p. 408</ref>
One of the notions that can resolve the issue is the requirement that real numbers be densely ordered. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal. Therefore, if 0.99999... were to be different from 1, there would have to be another real number in between them but there is none: a single digit cannot be changed in either of the two to obtain such a number.{{sfnp|Artigue|2002|p=212|loc="... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of&nbsp;1)."}}


== Generalizations ==
Second, a comparable theorem applies in each radix or ]. For example, in base 2 (the ]) 0.111... equals 1, and in base 3 (the ]) 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.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>
The result that {{nowrap|1=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&nbsp;0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are ].{{sfnp|Petkovšek|1990|p=408}}{{sfnp|Rosenlicht|1985|p=27}}


Second, a comparable theorem applies in each radix or ]. For example, in base 2 (the ]) 0.111... equals 1, and in base 3 (the ]) 0.222... equals 1. In general, any terminating base <math> b </math> expression has a counterpart with repeated trailing digits equal to {{nowrap|<math> b </math> − 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.{{sfnmp
Alternative representations of 1 also occur in non-integer bases. For example, in the ], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for ] ''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 ], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the ] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the ], which does not repeat.<ref>Komornik and Loreti p. 636</ref>
| 1a1 = Protter | 1a2 = Morrey | 1y = 1991 | 1p = 503
| 2a1 = Bartle | 2a2 = Sherbert | 2y = 1982 | 2p = 61
}}


Alternative representations of 1 also occur in non-integer bases. For example, in the ], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for ] <math> q </math> between 1 and 2, there are uncountably many {{nowrap|base-<math> q </math>}} expansions of 1. In contrast, there are still uncountably many {{tmath|1= q }}, including all natural numbers greater than 1, for which there is only one {{nowrap|1=base-<math> q </math>}} expansion of 1, other than the trivial 1.000...{{px2}}. This result was first obtained by ], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and ] determined the smallest such base, the ] {{nowrap|1=<math> q </math> = 1.787231650...}}{{px2}}. In this base, {{nowrap|1=1 = 0.11010011001011010010110011010011...}}; the digits are given by the ], which does not repeat.{{sfnp|Komornik|Loreti|1998|p=636}}
A more far-reaching generalization addresses ]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the ] system, <sup>1</sup>/<sub>2</sub> = 0.111... = 1.<u>111</u>....
*In the reverse ] (using bases 2!,3!,4!,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....


A more far-reaching generalization addresses ]. They too have multiple representations, and in some sense, the difficulties are even worse. For example:{{sfnmp
===Impossibility of unique representation===
| 1a1 = Kempner | 1y = 1936 | 1p = 611
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 ]. Indeed the following two properties account for the difficulty:
| 2a1 = Petkovšek | 2y = 1990 | 2p = 409
}}
* In the ] system, {{nowrap|1=<math display="inline"> \frac{1}{2} </math> = 0.111... = 1.{{underline|111}}...{{px2}}}}.
* In the reverse ] (using bases 2!, 3!, 4!,&nbsp;... for positions ''after'' the decimal point), {{nowrap|1=1 = 1.000... = 0.1234...}}{{px2}}.


{{harvtxt|Petkovšek|1990}} 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 ]"; it involves viewing sets of positional values as ]s and noticing that their real representations are given by ].{{sfnp|Petkovšek|1990|pp=410–411}}
* If an ] of the ]s is ] 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 ]s 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 ]es (initial substrings) ''p''<sub>1</sub>, ''p''<sub>2</sub> of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''<sub>1</sub>, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''<sub>2</sub>. Then ''L'' has a largest element, starting with ''p''<sub>1</sub> and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''<sub>2</sub> by the smallest symbol in all positions.


== Applications ==
The first point follows from basic properties of the real numbers: ''L'' has a ] and ''R'' has an ], 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 ]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''<sub>1</sub>&nbsp;=&nbsp;"0", ''p''<sub>2</sub>&nbsp;=&nbsp;"1". In fact one need not use the same alphabet for all positions (so that for instance ] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a ] 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 ] map from the collection of strings to an interval of the real numbers cannot be a ]: either some numbers do not correspond to any string, or some of them correspond to more than one string.
One application of 0.999... as a representation of 1 occurs in elementary ]. In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain ]s.{{sfnmp|1a1=Goodwyn|1y=1802|2a1=Dickson|2y=1919|2pp=161}} Examples include:
* <math display="inline"> \frac{1}{7} </math> = 0.{{overline|142857}} and {{nowrap|1=142 + 857 = 999}}.
* <math display="inline"> \frac{1}{73} </math> = 0.{{overline|01369863}} and {{nowrap|1=0136 + 9863 = 9999}}.
E. Midy proved a general result about such fractions, now called ], in 1836. The publication was obscure, and it is unclear whether his proof directly involved 0.999..., but at least one modern proof by William G. Leavitt does. If it can be proved that if a decimal of the form {{tmath|1= 0.b_1 b_2 b_3 }}... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.{{sfnp|Leavitt|1984|p=301}} Investigations in this direction can motivate such concepts as ]s, ], ]s, ] of ] elements, and ].{{sfnmp
| 1a1 = Ginsberg | 1y = 2004 | 1pp = 26–30
| 2a1 = Lewittes | 2y = 2006 | 2pp = 1–3
| 3a1 = Leavitt | 3y = 1967 | 3pp = 669, 673
| 4a1 = Shrader-Frechette | 4y = 1978 | 4pp = 96–98
}}


]]]
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 ]"; it involves viewing sets of positional values as ]s and noticing that their real representations are given by ].<ref>Petkovšek pp. 410–411</ref>
Returning to real analysis, the base-3 analogue {{nowrap|1=0.222... = 1}} plays a key role in the characterization of one of the simplest ]s, the middle-thirds ]: a point in the ] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.


The {{tmath|1= n }}th digit of the representation reflects the position of the point in the {{tmath|1= n }}th stage of the construction. For example, the point <math display="inline"> \frac{2}{3} </math> is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and the left of every deletion thereafter. The point <math display="inline"> \frac{1}{3} </math> is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and the right of every deletion thereafter.{{sfnmp
==Applications==
| 1a1 = Pugh | 1y = 2002 | 1p = 97
One application of 0.999... as a representation of 1 occurs in elementary ]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain ]s. Examples include:
| 2a1 = Alligood | 2a2 = Sauer | 2a3 = Yorke | 2y = 1996 | 2pp = 150–152
*<sup>1</sup>/<sub>7</sub> = 0.142857142857... and 142 + 857 = 999.
| 3a1 = Protter | 3a2 = Morrey | 3y = 1991 | 3p = 507
*<sup>1</sup>/<sub>73</sub> = 0.0136986301369863... and 0136 + 9863 = 9999.
| 4a1 = Pedrick | 4y = 1994 | 4p = 29
E. Midy proved a general result about such fractions, now called '']'', 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 it can be proved that a decimal of the form 0.''b''<sub>1</sub>''b''<sub>2</sub>''b''<sub>3</sub>... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as ]s, ], ]s, ] of ] elements, and ].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>
}}


Repeating nines also turns up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying ] to decimal expansions, of the ] 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.{{efn|{{harvtxt|Maor|1987}}, p. 60 and {{harvtxt|Mankiewicz|2000}}, p. 151 review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. {{harvtxt|Munkres|2000}}, p. 50 mentions the latter method.}} A variant that may be closer to Cantor's original argument uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.{{sfnmp
]]]
| 1a1 = Rudin | 1y = 1976 | 1p = 50
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest ]s, the middle-thirds ]:
| 2a1 = Pugh | 2y = 2002 | 2p = 98
*A point in the ] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.
}}


== Skepticism in education ==
The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point <sup>2</sup>⁄<sub>3</sub> 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 <sup>1</sup>⁄<sub>3</sub> 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.<ref>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.</ref>

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 ] to decimal expansions, of the ] 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.<ref>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.</ref> 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.<ref>Rudin p. 50, Pugh p. 98</ref>

==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 ] concept and disagreements over the nature of ]s. There are many common contributing factors to the confusion: Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the ] concept and disagreements over the nature of ]s. 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 ], which is amplified by the appearance of the seemingly well-understood number 1.<ref>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&nbsp;... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref> * 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 ], which is amplified by the appearance of the seemingly well-understood number 1.{{efn|{{harvtxt|Bunch|1982}}, p.&nbsp;119; {{harvtxt|Tall|Schwarzenberger|1978}}, p.&nbsp;6. The last suggestion is due to {{harvtxt|Burrell|1998}}, p. 28: "Perhaps the most reassuring of all numbers is 1&nbsp;... So it is particularly unsettling when someone tries to pass off 0.9~ as 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".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref> * 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".{{sfnmp
| 1a1 = Tall | 1a2 = Schwarzenberger | 1y = 1978 | 1pp = 6–7
| 2a1 = Tall | 2y = 2000 | 2p = 221
*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.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>
}}
* 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.{{sfnmp
| 1a1 = Tall | 1a2 = Schwarzenberger | 1y = 1978 | 1p = 6
| 2a1 = Tall | 2y = 2000 | 2p = 221
}}


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 ]s to better understand 0.999... 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 ]s to better understand 0.999...; see ''{{slink|#In alternative number systems}}'' below.


Many of these explanations were found by ], 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'".<ref>Tall 2000 p. 221</ref> Many of these explanations were found by ], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered with 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{{'"}}.{{sfnp|Tall|2000|p=221}}


Of the elementary proofs, multiplying 0.333... = <sup>1</sup>⁄<sub>3</sub> 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.<ref>Tall 1976 pp. 10–14</ref> 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... = <sup>1</sup>⁄<sub>3</sub> using a ] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that <sup>1</sup>⁄<sub>3</sub> = 0.333..., but, upon being confronted by the ], insist that "logic" supersedes the mathematical calculations. The elementary argument of multiplying {{nowrap|1=0.333... = <math display="inline"> \frac{1}{3} </math>}} by 3 can convince reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief in the first equation and their disbelief in the second, some students either begin to disbelieve the first equation or simply become frustrated.{{sfnp|Tall|1976|pp=10–14}} 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...{{px2}}. For example, one real analysis student was able to prove that {{nowrap|1=0.333... = <math display= "inline"> \frac{1}{3} </math>}} using a ] definition but then insisted that {{nowrap|0.999... < 1}} based on her earlier understanding of ].{{sfnmp
| 1a1 = Pinto | 1a2 = Tall | 1y = 2001 | 1p = 5
| 2a1 = Edwards | 2a2 = Ward | 2y = 2004 | 2pp = 416–417
}} Others still can prove that {{nowrap|1=<math display="inline"> \frac{1}{3} </math> = 0.333...}}, but, upon being confronted by the ], insist that "logic" supersedes the mathematical calculations.


] 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."<ref>Mazur pp. 137–141</ref> {{harvtxt|Mazur|2005}} 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 {{nowrap|1=9.99... = 10}}, calling it a "wildly imagined infinite growing process".{{sfnp|Mazur|2005|pp=137–141}}


As part of Ed Dubinsky's ] of mathematical learning, he 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 <sup>1</sup>⁄<sub>3</sub> as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' pp. 261–262</ref> As part of the ] of mathematical learning, {{harvtxt|Dubinsky|Weller|McDonald|Brown|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. They also link this mental ability of encapsulation to viewing <math display="inline"> \frac{1}{3} </math> as a number in its own right and to dealing with the set of natural numbers as a whole.{{sfnp|Dubinsky|Weller|McDonald|Brown|2005|pp=261–262}}


== Cultural phenomenon ==
==In popular culture==
With the rise of the ], debates about 0.999... have escaped the classroom and are commonplace on ]s and ]s, including many that nominally have little to do with mathematics. In the newsgroup {{mono|sci.math}}, arguing over 0.999... is described as a "popular sport", and it is one of the questions answered in its ].<ref>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ | first = Hans | last = de Vreught |year=1994 |title=sci.math FAQ: Why is 0.9999... = 1? |accessdate=2006-06-29 |archiveurl=http://web.archive.org/web/20070929122649/http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |archivedate=2007-09-29 |deadurl=no}}</ref> The FAQ briefly covers <sup>1</sup>⁄<sub>3</sub>, multiplication by 10, and limits, and it alludes to Cauchy sequences as well. With the rise of the ], debates about 0.999... have become commonplace on ]s and ]s, including many that nominally have little to do with mathematics. In the newsgroup {{mono|sci.math}} in the 1990s, arguing over 0.999... became a "popular sport", and was one of the questions answered in its ].{{sfnp|Richman|1999|p=396}}{{sfnp|de Vreught|1994}} The FAQ briefly covers {{tmath|1= \textstyle \frac{1}{3} }}, multiplication by 10, and limits, and alludes to Cauchy sequences as well.


A 2003 edition of the general-interest newspaper column '']'' discusses 0.999... via <sup>1</sup>⁄<sub>3</sub> and limits, saying of misconceptions, A 2003 edition of the general-interest newspaper column '']'' discusses 0.999... via <math display="inline"> \frac{1}{3} </math> and limits, saying of misconceptions,
<blockquote> {{blockquote|text=
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. 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.{{sfnp|Adams|2003}}
Nonsense.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |authorlink=Cecil Adams | first = Cecil | last = Adams |work=] |publisher=] |accessdate=2006-09-06 |archiveurl= http://web.archive.org/web/20060815010844/http://www.straightdope.com/columns/030711.html |archivedate= 15 August 2006 |deadurl=no}}</ref>
}}
</blockquote>


A '']'' article reports that the concept of 0.999... is "hotly disputed on websites ranging from '']'' message boards to ] forums".{{sfnp|Ellenberg|2014}}
''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 ]'s ] forums that the company issued a "press release" on ] 2004 that it is 1:
0.999... features also in ]s, such as:{{sfnp|Renteln|Dundes|2005|p=27}}
<blockquote>
{{blockquote|
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.<ref>{{cite press release |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16| archiveurl= http://web.archive.org/web/20091104222830/http://us.blizzard.com/en-us/company/press/pressreleases.html?040401| archivedate= 4 November 2009 }}</ref>
Q: How many mathematicians does it take to ]?{{br}}
</blockquote>
A: 0.999999....
Two proofs are then offered, based on limits and multiplication by 10.
}}


The fact that 0.999... is equal to 1 has been compared to ].{{sfnmp|1a1=Richman |1y=1999 |2a1=Adams |2y=2003 |3a1=Ellenberg |3y=2014}} The runner paradox can be mathematically modeled and then, like 0.999..., resolved using a geometric series. However, it is not clear whether this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.{{sfnmp
0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
| 1a1 = Wallace | 1y = 2003 | 1p = 51
<blockquote>
| 2a1 = Maor | 2y = 1987 | 2p = 17
Q: How many mathematicians does it take to ]?
}}


== In alternative number systems <span class="anchor" id="Alternative number systems"></span> ==
A: 0.999999....
Although the real numbers form an extremely useful ], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and ] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity {{nowrap|1=0.999... = 1}} is a convention as well:
</blockquote>
{{blockquote|
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.{{sfnp|Gowers|2002|p=60}}
}}


=== Infinitesimals ===
==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful ], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and ] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
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.<ref>Gowers p. 60</ref>
</blockquote>
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&nbsp;—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|Infinitesimal}} {{main|Infinitesimal}}


Some proofs that 0.999...&nbsp;=&nbsp;1 rely on the ] of the real numbers: that there are no nonzero ]s. Specifically, the difference 1&nbsp;&nbsp;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. Some proofs that {{nowrap|1=0.999... = 1}} rely on the ] of the real numbers: that there are no nonzero ]s. Specifically, the difference {{nowrap|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 zero, and therefore the two values are the same.

However, there are mathematically coherent ordered ]s, including various alternatives to the real numbers, which are non-Archimedean. ] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> ] developed a decimal expansion for ]s in (0, 1)<sup>∗</sup>.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

:<math>0.d_1d_2d_3 \dots;\dots d_{\infty - 1}d_\infty d_{\infty + 1}\dots,</math>


However, there are mathematically coherent ordered ]s, including various alternatives to the real numbers, which are non-Archimedean. ] provides a number system with a full array of infinitesimals (and their inverses).{{efn|For a full treatment of non-standard numbers, see {{harvtxt|Robinson|1996}}.}} ] developed a decimal expansion for ]s in {{nowrap|(0, 1)<sup>∗</sup>}}. Lightstone shows how to associate each number with a sequence of digits,
indexed by the ] 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 ]. 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.
<math display="block"> 0.d_1d_2d_3 \ldots;\ldots d_{\infty - 1}d_\infty d_{\infty + 1}\ldots,</math>
indexed by the ] numbers. While he does not directly discuss 0.999..., he shows the real number <math display="inline"> \frac{1}{3} </math> is represented by 0.333...;...333..., which is a consequence of the ]. As a consequence the number {{nowrap|1=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.{{sfnp|Lightstone|1972|pp=245–247}}


The standard definition of the number 0.999... is the ] 0.9, 0.99, 0.999, ... A different definition involves what ] refers to as ''ultralimit'', i.e., the equivalence class of this sequence in the ], which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number {{nowrap|1 = ''u''<sub>''H''</sub>=0.999...;...999000...,}} with last digit 9 at infinite ] rank ''H'', satisfies a strict inequality {{nowrap|''u''<sub>''H''</sub> < 1.}} Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be The standard definition of the number 0.999... is the ] 0.9, 0.99, 0.999, ...{{px2}}. A different definition involves an ''ultralimit'', i.e., the equivalence class {{nowrap|}} of this sequence in the ], which is a number that falls short of 1 by an infinitesimal amount.{{sfnp|Tao|2012|pp=156–180}} More generally, the hyperreal number {{nowrap|1=<math> u_H </math> = 0.999...;...999000...}}, with last digit 9 at infinite ] rank {{tmath|1= H }}, satisfies a strict inequality {{tmath|1= u_H < 1 }}. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be{{sfnp|Katz|Katz|2010a}}
:<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}.</math><ref>Katz & Katz 2010</ref> <math display="block"> \underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}. </math>
All such interpretations of "0.999..." are infinitely close to 1. ] 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....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|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.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref> All such interpretations of "0.999..." are infinitely close to 1. ] 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....{{efn|{{harvtxt|Stewart|2009}}, p.&nbsp;175; the full discussion of 0.999... is spread through pp. 172–175.}} Along with {{harvtxt|Katz|Katz|2010b}}, {{harvtxt|Ely|2010}} also questions the assumption that students' ideas about {{nowrap|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.{{sfnmp
| 1a1 = Katz | 1a2 = Katz | 1y = 2010b
Jose Benardete in his book ''Infinity: An essay in metaphysics'' argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:
| 2a1 = Ely | 2y = 2010
:The intelligibility of the continuum has been found—many times over—to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a ''number'' ... though not as a ''real'' number.<ref>{{cite book|first=José Amado |last=Benardete |title=Infinity: An essay in metaphysics |publisher=Clarendon Press |year=1964 |page=279 |url=http://books.google.com/?id=wMgtAAAAMAAJ| accessdate= 27 November 2011 }}</ref>
}}


===Hackenbush=== === Hackenbush ===
] provides alternative reals as well, with infinite Blue-Red ] as one particularly relevant example. In 1974, ] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of ]. For example, the value of the Hackenbush string LRRLRLRL... is 0.010101<sub>2</sub>...&nbsp;=&nbsp;<sup>1</sup>/<sub>3</sub>. However, the value of LRLLL... (corresponding to 0.111...<sub>2</sub>) is infinitesimally less than 1. The difference between the two is the ] <sup>1</sup>/<sub>ω</sub>, where ω is the first ]; the relevant game is LRRRR... or 0.000...<sub>2</sub>.<ref>Berlekamp, Conway, and Guy (pp. 79–80, 307–311) discuss 1 and <sup>1</sup>/<sub>3</sub> and touch on <sup>1</sup>/<sub>ω</sub>. The game for 0.111...<sub>2</sub> follows directly from Berlekamp's Rule.</ref> ] provides a generalized concept of number that encompasses the real numbers and much more besides.{{sfnp|Conway|2001|pp=3–5,12–13,24–27}} For example, in 1974, ] described a correspondence between strings of red and blue segments in ] and binary expansions of real numbers, motivated by the idea of ]. For example, the value of the Hackenbush string LRRLRLRL... is {{nowrap|1=0.010101...<sub>2</sub> = <math display="inline"> \frac{1}{3} </math>.}} However, the value of LRLLL... (corresponding to 0.111...<sub>2</sub>) is infinitesimally less than 1. The difference between the two is the ] <math display="inline"> \frac{1}{\omega} </math>, where <math> \omega </math> is the first ]; the relevant game is LRRRR... or 0.000...<sub>2</sub>.{{efn|{{harvtxt|Berlekamp|Conway|Guy|1982}}, pp. 79–80, 307–311 discuss 1 and {{sfrac|1|3}} and touch on {{sfrac|1|{{math|''ω''}}}}. The game for 0.111...<sub>2</sub> follows directly from Berlekamp's Rule.}}


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...<sub>2</sub>&nbsp;=&nbsp;0.11000...<sub>2</sub>, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRLLL... while the second corresponds to the different path LRLLRRR.... This is 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, {{nowrap|1=0.10111...<sub>2</sub> = 0.11000...<sub>2</sub>}}, which are both equal to {{tmath|1= \textstyle \frac{3}{4} }}, but the first representation corresponds to the binary tree path LRLRLLL..., while the second corresponds to the different path LRLLRRR...{{px2}}.


===Revisiting subtraction=== === Revisiting subtraction ===
Another manner in which the proofs might be undermined is if 1&nbsp;&nbsp;0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include ] ]s, ]s and ]s. Richman considers two such systems, designed so that 0.999... < 1. Another manner in which the proofs might be undermined is if {{nowrap|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 ] ]s, ]s, and ]s. {{harvtxt|Richman|1999}} considers two such systems, designed so that {{nowrap|0.999... < 1}}.{{sfnp|Richman|1999}}


First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the ] and an addition operation, noting that 0.999...&nbsp;<&nbsp;1 simply because 0&nbsp;<&nbsp;1 in the ones place, but for any nonterminating ''x'', one has 0.999...&nbsp;+&nbsp;''x''&nbsp;=&nbsp;1&nbsp;+&nbsp;''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to <sup>1</sup>⁄<sub>3</sub>. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref> First, {{harvtxt|Richman|1999}} defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the ] and an addition operation, noting that {{nowrap|0.999... < 1}} simply because {{nowrap|0 < 1}} in the ones place, but for any nonterminating {{tmath|1= x }}, one has {{nowrap|1=0.999... + <math> x </math> = 1 + {{tmath|1= x }}}}. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to {{tmath|1= \textstyle \frac{1}{3} }}. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.{{sfnp|Richman|1999|pp=397–399}}


In the process of defining multiplication, Richman also defines another system he calls "cut ''D''", which is the set of ]s of decimal fractions. Ordinarily this definition leads to the real numbers, but for a decimal fraction ''d'' he allows both the cut (−∞,&nbsp;''d''&nbsp;) and the "principal cut" (−∞,&nbsp;''d''&nbsp;]. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again 0.999...&nbsp;<&nbsp;1. There are no positive infinitesimals in cut ''D'', but there is "a sort of negative infinitesimal," 0<sup>−</sup>, which has no decimal expansion. He concludes that 0.999...&nbsp;=&nbsp;1&nbsp;+&nbsp;0<sup>−</sup>, while the equation "0.999... + ''x'' = 1" In the process of defining multiplication, Richman also defines another system he calls "cut {{tmath|1= D }}", which is the set of ]s of decimal fractions. Ordinarily, this definition leads to the real numbers, but for a decimal fraction <math> d </math> he allows both the cut {{nowrap|({{tmath|1= -\infty }}, {{tmath|1= d }})}} and the "principal cut" {{nowrap|({{tmath|1= -\infty }}, {{tmath|1= d }}]}}. The result is that the real numbers are "living uneasily together with" the decimal fractions. Again {{nowrap|0.999... < 1}}. There are no positive infinitesimals in cut {{tmath|1= D }}, but there is "a sort of negative infinitesimal", 0<sup>−</sup>, which has no decimal expansion. He concludes that {{nowrap|1=0.999... = 1 + 0<sup>−</sup>}}, while the equation "{{nowrap|1=0.999... + <math> x </math> = 1}}" has no solution.{{efn|{{harvtxt|Richman|1999}}, pp. 398–400. {{harvtxt|Rudin|1976}}, p. 23 assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.}}
has no solution.<ref>Richman pp. 398–400. Rudin (p. 23) assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.</ref>


===''p''-adic numbers=== === ''p''-adic numbers ===
{{main|p-adic number}} {{main|p-adic number}}


When asked about 0.999..., novices often believe there should be a "final 9," believing 1&nbsp;&nbsp;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 final 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 "final 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9. When asked about 0.999..., novices often believe there should be a "final 9", believing {{nowrap|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 final 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 "final 9" in 0.999...{{px2}}.{{sfnmp
| 1a1 = Gardiner | 1y = 2003 | 1p = 98
| 2a1 = Gowers | 2y = 2002 | 2p = 60
}} However, there is a system that contains an infinite string of 9s including a last 9.


] ]


The ]s are an alternative number system of interest in ]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via ]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''p<sup>n</sup>'', than it is to 1. The ''p''-adic numbers form a ] for prime ''p'' and a ] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals. The ] are an alternative number system of interest in ]. Like the real numbers, the {{nowrap|1=<math> p </math>-}}adic numbers can be built from the rational numbers via ]s; the construction uses a different metric in which 0 is closer to {{tmath|1= p }}, and much closer to {{tmath|1= p^n }}, than it is to 1.{{sfnp|Mascari|Miola|1988|p=}} The {{nowrap|1=<math> p </math>-}}adic numbers form a ] for prime <math> p </math> and a ] for other {{tmath|1= p }}, including 10. So arithmetic can be performed in the {{nowrap|1=<math> p </math>-}}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&nbsp;+&nbsp;...999&nbsp;=&nbsp;...000&nbsp;=&nbsp;0, and so ...999&nbsp;=&nbsp;−1.<ref name="Fjelstad11">Fjelstad p. 11</ref> 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: 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: {{nowrap|1=1 + ...999 = ...000 = 0}}, and so {{nowrap|1=...999 = −1}}.{{sfnp|Fjelstad|1995|p=11}} 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:{{sfnp|Fjelstad|1995|pp=14–15}}
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref> <math display="block">\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math>


(Compare with the series ].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999...&nbsp;=&nbsp;1 but was inspired to take the multiply-by-10 proof ] in the opposite direction: if ''x''&nbsp;=&nbsp;...999 then 10''x''&nbsp;=&nbsp; ...990, so 10''x''&nbsp;=&nbsp;''x''&nbsp;&nbsp;9, hence ''x''&nbsp;=&nbsp;−1 again.<ref name="Fjelstad11" /> Compare with the series in the ]. A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that {{nowrap|1=0.999... = 1}} but was inspired to take the multiply-by-10 proof ] in the opposite direction: if {{nowrap|1=<math> x </math> = ...999}}, then {{nowrap|1=10<math> x </math> = ...990}}, so {{nowrap|1=10<math> x </math> = <math> x </math> 9 }}, hence {{nowrap|1=<math> x </math> = −1}} again.{{sfnp|Fjelstad|1995|p=11}}


As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} 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.<ref>DeSua pp. 902–903</ref> As a final extension, since {{nowrap|1=0.999... = 1}} (in the reals) and {{nowrap|1=...999 = −1}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"{{sfnp|DeSua|1960|p=901}} one may add the two equations and arrive at {{nowrap|1=...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 in the ] ] of the ], with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.{{sfnp|DeSua|1960|p=902&ndash;903}}


==Related questions== == See also ==
<!--] should be worked in somewhere and explained, not necessarily here.-->
* ], 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.<ref>Wallace p. 51, Maor p. 17</ref>
* ] 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 ], where the ], i.e. the ], has a "]". Here, it makes sense to define <sup>1</sup>/<sub>0</sub> to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> 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.<ref>Maor p. 54</ref>
* ] 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&nbsp;=&nbsp;0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the ] or ] formats, or floating point numbers as specified by the ]).<ref>{{cite book | last1 = Kroemer | first1 = Herbert | last2 = Kittel | first2 = Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=] C# Language Specification |accessdate=2006-08-29| archiveurl= http://web.archive.org/web/20060824085452/http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp| archivedate= 24 August 2006}}</ref>

==See also==
* ]
* ]
* ]
* ] * ]
* ]

== Notes ==
{{notelist|colwidth=30em}}


==Notes== == References ==
{{reflist|colwidth=30em}} {{reflist|colwidth=30em}}


==References== == Sources ==
{{refbegin|colwidth=30em}} {{refbegin|colwidth=30em}}
* {{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=11 July 2003 |author-link=Cecil Adams |first=Cecil |last=Adams |work=] |publisher=] |access-date=2006-09-06 |archive-url=https://web.archive.org/web/20060815010844/http://www.straightdope.com/columns/030711.html |archive-date=15 August 2006 |url-status=live}}
* {{cite book |author=Alligood, K. T.; Sauer, T. D.; Yorke, J. A. |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
* {{cite book |last1=Alligood |first1=K. T. |last2=Sauer |first2=T. D. |last3=Yorke |first3=J. A. |author-link3=James A. Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=978-0-387-94677-1}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix) *: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}} * {{cite book |last=Apostol |first=Tom M. |author-link=Tom M. Apostol |year=1974 |url=https://archive.org/details/mathematicalanal02edtomm |title=Mathematical Analysis |edition=2e |publisher=Addison-Wesley |isbn=978-0-201-00288-1}}
*: A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic." (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.&nbsp;9–11) *: A transition from calculus to advanced analysis, ''Mathematical analysis'' is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic". (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp.&nbsp;9–11)
* {{cite book |last=Artigue |first=Michèle |editor1-first=Derek |editor1-last=Holton |editor2-first=Michèle |editor2-last=Artigue |editor3-first=Urs |editor3-last=Kirchgräber |editor4-first=Joel |editor4-last=Hillel |editor5-first=Mogens |editor5-last=Niss |editor6-first=Alan |editor6-last=Schoenfeld |title=The Teaching and Learning of Mathematics at University Level |series=New ICMI Study Series |year=2002 |volume=7 |publisher=Springer, Dordrecht |doi=10.1007/0-306-47231-7 |isbn=978-0-306-47231-2}}
* {{cite book |author=Bartle, R. G.; Sherbert, D. R. |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
* {{cite journal |last1=Baldwin |first1=Michael |last2=Norton |first2=Anderson |title=Does 0.999... Really Equal 1? |url=https://eric.ed.gov/?id=EJ961516 |year=2012 |journal=] |volume=21 |issue=2 |pages=58–67}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}} * {{cite book |last1=Bauldry |first1=William C. |year=2009 |url=https://books.google.com/books?id=ab-2vpx0FyYC&pg=PA47 |title=Introduction to Real Analysis: An Educational Approach |publisher=John Wiley & Sons |isbn=978-0-470-37136-7}}
*: This book is intended as introduction to real analysis aimed at upper- undergraduate and graduate-level. (pp.&nbsp;xi-xii)
* {{cite book |year=1982 |title=] |publisher=Academic Press |isbn=0-12-091101-9 |author1 = Berlekamp|first1 = E. R.|last2 = Conway|first2=J. H. |last3=Guy |first3=R. K. |authorlink1=Elwyn Berlekamp |authorlink2=John Horton Conway |authorlink3=Richard k. Guy }}
* {{cite book |last1=Bartle |first1=R. G. |author-link1=Robert G. Bartle |last2=Sherbert |first2=D. R. |year=1982 |url=https://archive.org/details/introductiontore0000bart |title=Introduction to Real Analysis |publisher=Wiley |isbn=978-0-471-05944-8}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |conference=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |id = {{citeseerx|10.1.1.31.3019}} |year=1992}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis". Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite journal|title=Why Does 0.999... = 1?: A Perennial Question and Number Sense|last1=Beswick|first1=Kim|journal=Australian Mathematics Teacher|volume=60|pages=7–9|year=2004|issue=4}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}} * {{cite book |last=Beals |first=Richard |author-link=Richard Beals (mathematician) |url=https://books.google.com/books?id=cXAqJUYqXx0C |title=Analysis: An Introduction |year=2004 |publisher=Cambridge University Press |isbn=978-0-521-60047-7}}
* {{cite book |year=1982 |title=Winning Ways for your Mathematical Plays |publisher=Academic Press |isbn=978-0-12-091101-1 |last1=Berlekamp |first1=E. R. |last2=Conway |first2=J. H. |last3=Guy |first3=R. K. |author-link1=Elwyn Berlekamp |author-link2=John Horton Conway |title-link=Winning Ways for your Mathematical Plays}}
* {{cite book |last=Bonnycastle |first=John |date=1806 |title=An introduction to algebra; with notes and observations: designed for the use of schools and places of public education |url=http://hdl.handle.net/2027/mdp.39015063620382 |edition=First American |location=Philadelphia |hdl=2027/mdp.39015063620382}}
* {{cite book |last=Bunch |first=Bryan H. |url=https://archive.org/details/mathematicalfall0000bunc |title=Mathematical Fallacies and Paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=978-0-442-24905-2}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119) *: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}} * {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=978-0-87779-621-3 |url=https://archive.org/details/merriamwebstersg00burr}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}} * {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton University Press |isbn=978-0-691-12738-5 |url=https://archive.org/details/howmathematician00byer}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}} * {{cite book |last=Cheng |first=Eugenia |author-link=Eugenia Cheng |title=Is Math Real? How Simple Questions Lead Us To Mathematics' Deepest Truths |year=2023 |publisher=Basic Books |isbn=978-1-541-6-01826}}
* {{cite book |last=Conway |first=John B. |author-link=John B. Conway |url=https://archive.org/details/isbn_9781461263142 |title=Functions of One Complex Variable I |edition=2e |publisher=Springer-Verlag |orig-year=1973 |year=1978 |isbn=978-0-387-90328-6}}
*: 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)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175| accessdate= 4 July 2011}} * {{cite book |last=Conway |first=John H. |author-link=John Horton Conway |title=On Numbers and Games |title-link=On Numbers and Games |publisher=A K Peters |year=2001 |edition=2nd |isbn=1-56881-127-6}}
* {{cite book |last=Davies |first=Charles |author-link=Charles Davies (professor) |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=https://archive.org/details/universityarith00davigoog |page= |access-date=4 July 2011}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |jstor=2309468 |journal=The American Mathematical Monthly |volume=67 |pages=900–903 |doi=10.2307/2309468 |issue=9 |date=November 1960}}
* {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |first=Hans |last=de Vreught |year=1994 |title=sci.math FAQ: Why is 0.9999... = 1? |access-date=29 June 2006 |archive-url=https://web.archive.org/web/20070929122649/http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |archive-date=29 September 2007 |url-status=live}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2}}
* {{cite journal |author=Edwards, Barbara; Ward, Michael |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |jstor=4145268| accessdate= 4 July 2011| archiveurl= http://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf| archivedate= 22 July 2011 |issue=5 |date=May 2004}} * {{cite journal |last=DeSua |first=Frank C. |title=A System Isomorphic to the Reals |url=https://archive.org/details/sim_american-mathematical-monthly_1960-11_67_9/page/900 |jstor=2309468 |journal=] |volume=67 |pages=900–903 |doi=10.2307/2309468 |issue=9 |date=November 1960}}
* {{cite journal |last=Diamond |first=Louis E. |title=Irrational Numbers |journal=] |publisher=Mathematical Association of America |year=1955 |volume=29 |number=2 |pages=89–99 |doi=10.2307/3029588 |jstor=3029588}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
* {{cite book |last=Dickson |first=Leonard Eugene |title=History of the Theory of Numbers |volume=1 |publisher=Carnegie Institution of Washington |year=1919}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |url=https://archive.org/details/sim_educational-studies-in-mathematics_2005_60_2/page/253 |journal=] |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2 |s2cid=45937062}}
* {{cite book |last1=Earl |first1=Richard |last2=Nicholson |first2=James |title=The Concise Oxford Dictionary of Mathematics |edition=6th |year=2021 |publisher=Oxford University Press |isbn=978-0-192-58405-2}}
* {{cite journal |last1=Edwards |first1=Barbara |last2=Ward |first2=Michael |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=] |volume=111 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |jstor=4145268 |access-date=4 July 2011 |archive-url=https://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |archive-date=22 July 2011 |issue=5 |date=May 2004 |citeseerx=10.1.1.453.7466}}
* {{cite web |url=http://www.slate.com/blogs/how_not_to_be_wrong/2014/06/06/does_0_999_1_and_are_divergent_series_the_invention_of_the_devil.html |work=] |last=Ellenberg |first=Jordan |title=Does {{nowrap |1=0.999... = 1}}? And Are Divergent Series the Invention of the Devil?|date=6 June 2014|author-link=Jordan Ellenberg|archive-date=8 August 2023|archive-url=https://web.archive.org/web/20230808022739/https://slate.com/human-interest/2014/06/does-0-999-1-and-are-divergent-series-the-invention-of-the-devil.html}}
* {{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |url=https://archive.org/details/sim_journal-for-research-in-mathematics-education_2010-03_41_2/page/117 |journal=] |volume=41 |issue=2 |pages=117–146 |doi=10.5951/jresematheduc.41.2.0117}}
*: 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 {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite book |last=Enderton |first=Herbert B. |author-link=Herbert Enderton |year=1977 |url=https://archive.org/details/elementsofsetthe0000ende |title=Elements of Set Theory |publisher=Elsevier |isbn=978-0-12-238440-0}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii) *: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7| accessdate= 4 July 2011}} * {{cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |orig-year=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |others=John Hewlett and Francis Horner, English translators |publisher=Orme Longman |url=https://archive.org/details/elementsalgebra00lagrgoog |page= |isbn=978-0-387-96014-2 |access-date=4 July 2011}}
* {{cite book |last1=Finney |first1=Ross L. |last2=Weir |first2=Maurice D. |last3=Giordano |first3=Frank R. |year=2001 |title=Thomas' Calculus: Early Transcendentals |edition=10th |publisher=Addison-Wesley |location=New York}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |jstor=2687285 |journal=The College Mathematics Journal |volume=26 |pages=11–15 |doi=10.2307/2687285 |issue=1 |date=January 1995}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}} * {{cite journal |last=Fjelstad |first=Paul |title=The Repeating Integer Paradox |url=https://www.tandfonline.com/doi/abs/10.1080/07468342.1995.11973659 |jstor=2687285 |journal=] |volume=26 |pages=11–15 |doi=10.2307/2687285 |issue=1 |date=January 1995}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}} * {{cite book |last=Gardiner |first=Anthony |author-link=Tony Gardiner |url=https://books.google.com/books?id=NiDCYJ8vrGQC |title=Understanding Infinity: The Mathematics of Infinite Processes |orig-year=1982 |year=2003 |publisher=Dover |isbn=978-0-486-42538-2}}
* {{Cite journal |title=Midy's (nearly) secret theorem – an extension after 165 years |journal=] |volume=35 |issue=1 |year=2004 |first=Brian |last=Ginsberg |pages=26–30 |doi=10.1080/07468342.2004.11922047 |url=https://www.tandfonline.com/doi/abs/10.1080/07468342.2004.11922047}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{Cite journal |first=H. |last=Goodwyn |year=1802 |journal=] |series=New Series |volume=1 |pages=314–316 |title=Curious properties of prime Numbers, taken as the Divisors of unity. By a Correspondent |url=https://archive.org/details/journalofnatural01lond/page/314/mode/2up}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
* {{cite web |last=Gowers |first=Timothy |author-link=William Timothy Gowers |url=https://www.dpmms.cam.ac.uk/~wtg10/decimals.html |title=What is so wrong with thinking of real numbers as infinite decimals? |website=Department of Pure Mathematics and Mathematical Statistics |publisher=Cambridge University |access-date=2024-10-03 |year=2001}}
* {{cite book |last=Gowers |first=Timothy |author-link=William Timothy Gowers |url=https://books.google.com/books?id=DBxSM7TIq48C |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford University Press |isbn=978-0-19-285361-5}}
* {{cite book |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |year=1970 |title=The Development of the Foundations of Mathematical Analysis from Euler to Riemann |publisher=MIT Press |isbn=978-0-262-07034-8 |url=https://archive.org/details/developmentoffo00ivor}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. |author-link2=Peter Hilton |title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation |url=https://archive.org/details/comprehensivetex0000grif |url-access=registration |year=1970 |publisher=Van Nostrand Reinhold |location=London |isbn=978-0-442-02863-3 |id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for ]-area ] mathematics teachers. The course was intended to convey a university-level perspective on ], 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 ], which is not reproduced here. (pp. vii, xiv) *: This book grew out of a course for ]-area ] mathematics teachers. The course was intended to convey a university-level perspective on ], 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 ], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/| accessdate= 4 July 2011| archiveurl= http://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/| archivedate= 20 July 2011 |deadurl=no |bibcode=2010arXiv1007.3018U |arxiv=1007.3018 }} * {{cite journal |last1=Katz |first1=Karin Usadi |last2=Katz |first2=Mikhail G. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |doi=10.54870/1551-3440.1381 |url=http://www.math.umt.edu/TMME/vol7no1/ |access-date=4 July 2011 |archive-url=https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/ |archive-date=20 July 2011 |url-status=dead |bibcode=2010arXiv1007.3018U |arxiv=1007.3018 |s2cid=11544878}}
* {{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |author2-link=Mikhail Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |bibcode=2010arXiv1003.1501K |issue=3 |s2cid=115168622}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |jstor=2300532 |journal=The American Mathematical Monthly |volume=43 |pages=610–617 |doi=10.2307/2300532 |issue=10 |date=December 1936}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |jstor=2589246 |journal=The American Mathematical Monthly |volume=105 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }} * {{cite journal |last=Kempner |first=Aubrey J. |author-link=Aubrey Kempner |title=Anormal Systems of Numeration |jstor=2300532 |journal=] |volume=43 |pages=610–617 |doi=10.2307/2300532 |issue=10 |date=December 1936}}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=The American Mathematical Monthly |volume=74 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }} * {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |author-link2=Paola Loreti |jstor=2589246 |journal=] |volume=105 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7}}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |jstor=2686394 |journal=The College Mathematics Journal |volume=15 |pages=299–308 |doi=10.2307/2686394 |issue=4 |date=September 1984}} * {{cite arXiv |last=Li |first=Liangpan |title=A new approach to the real numbers |eprint=1101.1800 |class=math.CA |date=March 2011}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=The American Mathematical Monthly |volume=79 |pages=242–251 |doi=10.2307/2316619 |issue=3 |date=March 1972}} * {{cite journal |last=Leavitt |first=William G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=] |volume=74 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 |url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1047&context=mathfacpub}}
* {{cite journal |last=Leavitt |first=William G. |title=Repeating Decimals |url=https://archive.org/details/sim_college-mathematics-journal_1984-09_15_4/page/299 |jstor=2686394 |journal=] |volume=15 |pages=299–308 |doi=10.2307/2686394 |issue=4 |date=September 1984}}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
* {{cite arXiv |eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006}}
* {{cite journal |last=Lightstone |first=Albert H. |author-link=A. H. Lightstone |title=Infinitesimals |url=https://archive.org/details/sim_american-mathematical-monthly_1972-03_79_3/page/242 |jstor=2316619 |journal=] |volume=79 |pages=242–251 |doi=10.2307/2316619 |issue=3 |date=March 1972}}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |url=https://archive.org/details/storyofmathemati0000mank_k4e8 |title=The Story of Mathematics |publisher=Cassell |isbn=978-0-304-35473-3}}
*: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.&nbsp;8) *: Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p.&nbsp;8)
* {{cite book |last1=Mascari |first1=Gianfranco |last2=Miola |first2=Alfonso |year=1988 |editor-last1=Beth |editor-first1=Thomas |editor-last2=Clausen |editor-first2=Michael |title=Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra |contribution=On the integration of numeric and algebraic computations |doi=10.1007/BFb0039172 |isbn=978-3-540-39133-3}}
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
* {{cite book |last=Maor |first=Eli |author-link=Eli Maor |title=To Infinity and Beyond: A Cultural History of the Infinite |url=https://archive.org/details/toinfinitybeyond0000maor |url-access=registration |year=1987 |publisher=Birkhäuser |isbn=978-3-7643-3325-6}}
*: 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) *: 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)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}} * {{cite book |last=Mazur |first=Joseph |author-link=Joseph Mazur |url=https://books.google.com/books?id=7KSxBwAAQBAJ |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=978-0-13-147994-4}}
* {{cite book |last1=Meier |first1=John |last2=Smith |first2=Derek |title=Exploring Mathematics: An Engaging Introduction to Proof |publisher=Cambridge University Press |year=2017 |isbn=978-1-107-12898-9}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
* {{cite book |last=Munkres |first=James R. |author-link=James Munkres |title=Topology |year=2000 |orig-year=1975 |edition=2e |publisher=Prentice-Hall |isbn=978-0-13-181629-9}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.&nbsp;30)
*: 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's treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.&nbsp;30)
* {{cite book |last=Núñez |first=Rafael |chapter=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |title=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | isbn=978-0-387-25717-4| accessdate= 4 July 2011| archiveurl= http://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html| archivedate= 18 July 2011}}
* {{Cite journal |first1=Maria Angeles |last1=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999...=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf |access-date=4 July 2011 |ref=none}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{MacTutor|class=HistTopics|title=The real numbers: Stevin to Hilbert|date=October 2005}}{{sfn whitelist|CITEREFO'ConnorRobertson2005}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |jstor=2324393 |journal=] |volume=97 |pages=408–411 |doi=10.2307/2324393 |issue=5 |date=May 1990}} * {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |url=https://archive.org/details/firstcourseinana0000pedr |url-access=registration |year=1994 |publisher=Springer |isbn=978-0-387-94108-0}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor1-first=Bart |editor1-last=van Kerkhove |editor2-first=Jean Paul |editor2-last=van Bendegem |editor-link2=Jean Paul Van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |url=https://archive.org/details/perspectivesonma0000unse_f3x1 |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite book |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=PME25: Following students' development in a traditional university analysis course |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf| archivedate= 30 May 2009}}
* {{cite journal |last=Petkovšek |first=Marko |author-link=Marko Petkovšek |title=Ambiguous Numbers are Dense |url=https://archive.org/details/sim_american-mathematical-monthly_1990-05_97_5/page/408 |jstor=2324393 |journal=] |volume=97 |pages=408–411 |doi=10.2307/2324393 |issue=5 |date=May 1990}}
* {{cite book |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7 |author1 = Protter|first1 = M. H.|last2 = Morrey, Jr.|first2=Charles B.|authorlink2=Charles B. Morrey }}
* {{cite book |last1=Pinto |first1=Márcia |last2=Tall |first2=David O. |author-link2=David Tall |title=PME25: Following students' development in a traditional university analysis course |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |archive-date=30 May 2009}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus." (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nondecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.&nbsp;56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.&nbsp;503–507)
* {{cite web |last=Propp |first=James |author-link=Jim Propp |title=The One About .999... |website=Mathematical Enchantments |url=https://mathenchant.wordpress.com/2015/09/17/the-one-about-999/ |date=17 September 2015 |access-date=24 May 2024}}
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
* {{cite web |last=Propp |first=James |author-link=Jim Propp |title=Denominators and Doppelgängers |website=Mathematical Enchantments |url=https://mathenchant.wordpress.com/2023/01/17/denominators-and-doppelgangers/ |date=17 January 2023 |access-date=16 April 2024}}
* {{cite book |year=1991 |edition=2e |url=https://archive.org/details/firstcourseinrea0000prot |title=A First Course in Real Analysis |publisher=Springer |isbn=978-0-387-97437-8 |last1=Protter |first1=Murray H. |author-link1=Murray H. Protter |last2=Morrey |first2=Charles B. Jr. |author-link2=Charles B. Morrey}}
*: This book aims to "present a theoretical foundation of analysis that is suitable for students who have completed a standard course in calculus". (p. vii) At the end of Chapter 2, the authors assume as an axiom for the real numbers that bounded, nondecreasing sequences converge, later proving the nested intervals theorem and the least upper bound property. (pp.&nbsp;56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp.&nbsp;503–507)
* {{cite book |last=Pugh |first=Charles Chapman |author-link=Charles C. Pugh |title=Real Mathematical Analysis |year=2002 |publisher=Springer-Verlag |isbn=978-0-387-95297-0 |url=https://archive.org/details/realmathematical00char}}
*: While assuming familiarity with the rational numbers, Pugh introduces ]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.&nbsp;10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book. *: While assuming familiarity with the rational numbers, Pugh introduces ]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.&nbsp;10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf| archivedate= 25 February 2009 |date=January 2005}} * {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Alan |author-link2=Alan Dundes |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=] |volume=52 |issue=1 |pages=24–34 |url=https://www.ams.org/notices/200501/fea-dundes.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf |archive-date=25 February 2009 |date=January 2005}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |title=Is 0.999... = 1? |jstor=2690798 |journal=] volume=72 |volume=72 |issue=5 |pages=396–400 |date=December 1999}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred |last=Richman |title=Is 0.999... = 1? |accessdate=2006-08-23| archiveurl= http://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm| archivedate= 2 September 2006 |deadurl=no|date=June 1999}} Note: the journal article contains material and wording not found in the preprint. * {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |title=Is 0.999... = 1? |url=https://archive.org/details/sim_mathematics-magazine_1999-12_72_5/page/396 |jstor=2690798 |journal=] |volume=72 |issue=5 |pages=396–400 |date=December 1999}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred |last=Richman |title=Is 0.999... = 1? |access-date=23 August 2006 |archive-url=https://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm |archive-date=2 September 2006 |url-status=dead |date=June 1999 |ref=none}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}} * {{cite book |last=Robinson |first=Abraham |jstor=j.ctt1cx3vb6 |author-link=Abraham Robinson |title=Non-standard Analysis |year=1996 |edition=Revised |publisher=Princeton University Press |isbn=978-0-691-04490-3}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=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.&nbsp;27–31) as infinite decimals with 0.999...&nbsp;=&nbsp;1 as part of the definition. * {{cite book |last=Rosenlicht |first=Maxwell |author-link=Maxwell Rosenlicht |year=1985 |url=https://archive.org/details/introductiontoan0000rose |title=Introduction to Analysis |publisher=Dover |isbn=978-0-486-65038-8}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp.&nbsp;27–31) as infinite decimals with 0.999...&nbsp;=&nbsp;1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}} * {{cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of Mathematical Analysis |url=https://archive.org/details/principlesofmath00rudi |url-access=registration |edition=3e |year=1976 |orig-year=1953 |publisher=McGraw-Hill |isbn=978-0-07-054235-8}}
*: 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) *: 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)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |pages=90–98 |date=March 1978}} * {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=] |volume=51 |issue=2 |pages=90–98 |date=March 1978}}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7| accessdate= 4 July 2011}} * {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=https://archive.org/details/arithmeticforsc02smitgoog |page= |isbn=978-0-665-54808-6 |access-date=4 July 2011}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}} * {{cite book |last=Sohrab |first=Houshang |url=https://books.google.com/books?id=gBPI_oYZoMMC |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=978-0-8176-4211-2}}
* {{cite book |last=Stewart |first=Ian |author-link=Ian Stewart (mathematician) |url=https://archive.org/details/professorstewart0000stew_i8r6 |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite journal | last1=Starbird | first1=M. | last2=Starbird | first2=T. | title= Required Redundancy in the Representation of Reals| volume=114 | pages=769–774|journal=Proceedings of the American Mathematical Society|jstor=2159403|publisher=AMS | issue=3 | doi=10.1090/S0002-9939-1992-1086343-5 |date=March 1992}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}} * {{cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=978-0-534-36298-0 |url=https://archive.org/details/calculusearlytra00stew}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=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. *: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{citation |first=John |last=Stillwell |author-link=John Stillwell |title=Elements of Algebra: Geometry, Numbers, Equations |url=https://books.google.com/books?id=jWgPAQAAMAAJ |year=1994 |publisher=Springer |isbn=9783540942900}}
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf| archivedate= 30 May 2009}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1977 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf| archivedate= 26 March 2009}} * {{cite journal |last1=Tall |first1=David |author-link1=David Tall |last2=Schwarzenberger |first2=R. L. E. |author-link2=Rolph Ludwig Edward Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |archive-date=30 May 2009}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03| archiveurl= http://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf| archivedate= 30 May 2009 |doi=10.1007/BF03217085|bibcode=2000MEdRJ..12..196T }} * {{cite journal |last=Tall |first=David O. |author-link=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |archive-date=26 March 2009}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development in Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |access-date=3 May 2009 |archive-url=https://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |archive-date=30 May 2009 |doi=10.1007/BF03217085 |bibcode=2000MEdRJ..12..196T |s2cid=143438975}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}} * {{cite book |last=Tao |first=Terence |author-link=Terence Tao |title=Higher order Fourier analysis |publisher=American Mathematical Society |year=2012 |url=https://terrytao.files.wordpress.com/2011/03/higher-book.pdf}}
* {{cite web |last=Tao |first=Terence |author-link=Terence Tao |title=Math 131AH: Week 1 |url=https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf |website=Honors Analysis |publisher=UCLA Mathematics |year=2003 |access-date=23 May 2024}}
* {{cite book |last=Wallace |first=David Foster |author-link=David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=978-0-393-00338-3 |url=https://archive.org/details/everythingmore00davi}}
*{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is not true that 0.999 . . . < 1? |url=http://teaching.math.rs/vol/tm1114.pdf |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=38 |ref={{harvid|Eisenmann|2008}}}}
{{refend}} {{refend}}


==Further reading== === Further reading ===
{{refbegin|colwidth=30em}} {{refbegin|colwidth=30em}}
* {{cite journal |title=Why Does 0.999... = 1?: A Perennial Question and Number Sense |url=https://eric.ed.gov/?id=EJ717818 |last1=Beswick |first1=Kim |journal=Australian Mathematics Teacher |volume=60 |pages=7–9 |year=2004 |issue=4 |ref=none}}
*{{Cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409|bibcode=1987JSP....47..409B }}
* {{cite journal |journal=] |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409–438 |bibcode=1987JSP....47..409B |s2cid=120281766 |ref=none}}
*{{Cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |date=March 1997}}
*{{Cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |first1=J. B.|last1=Calvert|first2=E. R.|last2=Tuttle|first3=Michael S.|last3=Martin|first4=Peter|last4=Warren |journal=The History Teacher |volume=14 |issue=2 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |date=February 1981}} * {{cite journal |title=81.15 A Case of Conflict |url=https://archive.org/details/sim_mathematical-gazette_1997-03_81_490/page/109 |first=Bob |last=Burn |journal=] |volume=81 |issue=490 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |date=March 1997 |s2cid=187823601 |ref=none}}
*{{Cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |pages=13–15, 36 |jstor=40248503 |date=November 2005}} * {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |url=https://archive.org/details/sim_history-teacher_1981-02_14_2/page/167 |first1=J. B. |last1=Calvert |first2=E. R. |last2=Tuttle |first3=Michael S. |last3=Martin |first4=Peter |last4=Warren |journal=The History Teacher |volume=14 |issue=2 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |date=February 1981 |ref=none}}
*{{Cite journal |title=Rational Approximations to π |first1=K. Y.|last1=Choong|first2=D. E.|last2=Daykin|first3=C. R.|last3=Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |date=April 1971}} * {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)1/3 |journal=] |volume=25 |issue=3 |pages=13–15, 36 |jstor=40248503 |date=November 2005 |ref=none}}
* {{cite journal |title=Rational Approximations to π |url=https://archive.org/details/sim_mathematics-of-computation_1971-04_25_114/page/387 |first1=K. Y. |last1=Choong |first2=D. E. |last2=Daykin |first3=C. R. |last3=Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |date=April 1971 |ref=none}}
*{{Cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student's understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22}}
* {{cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student's understanding and use of mathematical definitions in real analysis |editor1-last=Dossey |editor1-first=J. |editor2-last=Swafford |editor2-first=J.O. |editor3-last=Parmentier |editor3-first=M. |editor4-last=Dossey |editor4-first=A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22 |ref=none}}
*{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf| accessdate= 4 July 2011}}
*{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}} * {{cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf |access-date=4 July 2011 |ref=none}}
* {{cite journal |last1=Ferrini-Mundy |first1=J. |last2=Graham |first2=K. |year=1994 |title=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes: Research Issues in Undergraduate Mathematics Learning |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |ref=none}}
*: 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 {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |url=https://archive.org/details/sim_mathematical-gazette_1985-06_69_448/page/77 |first=Tony |last=Gardiner |journal=] |volume=69 |issue=448 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |date=June 1985 |s2cid=125222118 |ref=none}}
*{{Cite journal |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |title=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes: Research issues in undergraduate mathematics learning |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky }}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |url=https://archive.org/details/sim_mathematical-gazette_1988-12_72_462/page/276 |first=John |last=Monaghan |journal=] |volume=72 |issue=462 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |date=December 1988 |s2cid=125825964 |ref=none}}
* {{cite arXiv | eprint=0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 | class=math.NT }}
* {{cite book |last=Núñez |first=Rafael |author-link=Rafael E. Núñez |chapter=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |title=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |chapter-url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html |isbn=978-0-387-25717-4 |access-date=4 July 2011 |archive-url=https://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html |archive-date=18 July 2011 |ref=none}}
*{{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |issue=3}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=] |volume=55 |issue=1–3 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132 |date=March 2004 |s2cid=120453706 |ref=none}}
*{{Cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |date=June 1985}}
*{{Cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |date=December 1988}} * {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |url=https://archive.org/details/sim_american-mathematical-monthly_1996-02_103_2/page/107 |first=James T. |last=Sandefur |journal=] |volume=103 |issue=2 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103 |date=February 1996 |ref=none}}
*{{Cite journal |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999...=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf| accessdate= 4 July 2011}} * {{cite journal |last=Sierpińska |first=Anna |author-link=Anna Sierpińska |title=Humanities students and epistemological obstacles related to limits |journal=] |volume=18 |issue=4 |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354 |date=November 1987 |s2cid=144880659 |ref=none}}
*{{Cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132 |date=March 2004}} * {{cite journal |last1=Starbird |first1=Michael |author-link1=Michael Starbird |last2=Starbird |first2=Thomas |title=Required Redundancy in the Representation of Reals |volume=114 |pages=769–774 |journal=Proceedings of the American Mathematical Society |jstor=2159403 |issue=3 |doi=10.1090/S0002-9939-1992-1086343-5 |date=March 1992 |doi-access=free |ref=none}}
*{{Cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103 |date=February 1996}} * {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=] |volume=31 |issue=3 |pages=258–276 |jstor=749807 |doi=10.2307/749807 |date=May 2000 |ref=none}}
*{{Cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354 |date=November 1987}} * {{cite journal |first=David O. |last=Tall |author-link=David Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6 |s2cid=14289039 |ref=none}}
*{{Cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |pages=258–276 |jstor=749807 |doi=10.2307/749807 |date=May 2000}} * {{cite journal |first=David O. |last=Tall |author-link=David Tall |title=Intuitions of infinity |url=https://archive.org/details/sim_mathematics-in-school_1981-05_10_3/page/30 |journal=Mathematics in School |date=May 1981 |volume=10 |issue=3 |pages=30–33 |jstor=30214290 |ref=none}}
*{{Cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
*{{Cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |date=May 1981 |volume=10|issue=3|pages=30–33|jstor=30214290}}
{{refend}} {{refend}}


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Latest revision as of 14:07, 12 December 2024

Alternative decimal expansion of 1

Stylistic impression of the number 0.9999..., representing the digit 9 repeating infinitely

In mathematics, 0.999... (also written as 0.9, 0..9, or 0.(9)) denotes the smallest number greater than every number in the sequence (0.9, 0.99, 0.999, ...). It can be proved that this number is 1; that is,

0.999... = 1. {\displaystyle 0.999...=1.}

Despite common misconceptions, 0.999... is not "almost exactly 1" or "very, very nearly but not quite 1"; rather, "0.999..." and "1" represent exactly the same number.

An elementary proof is given below that involves only elementary arithmetic and the fact that there is no positive real number less than all 1/10, where n is a natural number, a property that results immediately from the Archimedean property of the real numbers.

There are many other ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The intuitive arguments are generally based on properties of finite decimals that are extended without proof to infinite decimals. The proofs are generally based on basic properties of real numbers and methods of calculus, such as series and limits. A question studied in mathematics education is why some people reject this equality.

In other number systems, 0.999... can have the same meaning, a different definition, or be undefined. Every nonzero terminating decimal has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all positional numeral systems that represent the real numbers.

Elementary proof

The Archimedean property: any point x before the finish line lies between two of the points Pn (inclusive).

It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series and limits. The proof given below is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line, there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1.

Intuitive explanation

If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1. For any number x {\displaystyle x} that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than ⁠ x {\displaystyle x} ⁠. So, it does not make sense to identify 0.999... with any number smaller than 1. Meanwhile, every number larger than 1 will be larger than any decimal of the form 0.999...9 for any finite number of nines. Therefore, 0.999... cannot be identified with any number larger than 1, either. Because 0.999... cannot be bigger than 1 or smaller than 1, it must equal 1 if it is to be any real number at all.

Rigorous proof

Denote by 0.(9)n the number 0.999...9, with n {\displaystyle n} nines after the decimal point. Thus 0.(9)1 = 0.9, 0.(9)2 = 0.99, 0.(9)3 = 0.999, and so on. One has 1 − 0.(9)1 = 0.1 = ⁠ 1 10 {\displaystyle \textstyle {\frac {1}{10}}} ⁠, 1 − 0.(9)2 = 0.01 = ⁠ 1 10 2 {\displaystyle \textstyle {\frac {1}{10^{2}}}} ⁠, and so on; that is, 1 − 0.(9)n = 1 10 n {\textstyle {\frac {1}{10^{n}}}} for every natural number n {\displaystyle n} ⁠.

Let x {\displaystyle x} be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, 0.(9)n < x {\displaystyle x} ≤ 1, for every ⁠ n {\displaystyle n} ⁠. By subtracting these inequalities from 1, one gets 0 ≤ 1 − x {\displaystyle x} < ⁠ 1 10 n {\displaystyle \textstyle {\frac {1}{10^{n}}}} ⁠.

The end of the proof requires that there is no positive number that is less than 1 10 n {\textstyle {\frac {1}{10^{n}}}} for all ⁠ n {\displaystyle n} ⁠. This is one version of the Archimedean property, which is true for real numbers. This property implies that if 1 − x {\displaystyle x} < ⁠ 1 10 n {\displaystyle \textstyle {\frac {1}{10^{n}}}} ⁠ for all ⁠ n {\displaystyle n} ⁠, then 1 − x {\displaystyle x} can only be equal to 0. So, x {\displaystyle x} = 1 and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, 1 = 0.999....

This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, the notation 0.999... is generally not used, as there is no smallest number among the numbers larger than all 0.(9)n.

Least upper bounds and completeness

Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: the smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers has a least upper bound and that this least upper bound is equal to one.

Algebraic arguments

Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. Byers (2007) discusses the argument that, in elementary school, one is taught that 1 3 {\textstyle {\frac {1}{3}}} = 0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1 = 0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...‍." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. Richman (1999) discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.

Byers also presents the following argument.

x = 0.999 10 x = 9.999 by multiplying by  10 10 x = 9 + 0.999 by splitting off integer part 10 x = 9 + x by definition of  x 9 x = 9 by subtracting  x x = 1 by dividing by  9 {\displaystyle {\begin{aligned}x&=0.999\ldots \\10x&=9.999\ldots &&{\text{by multiplying by }}10\\10x&=9+0.999\ldots &&{\text{by splitting off integer part}}\\10x&=9+x&&{\text{by definition of }}x\\9x&=9&&{\text{by subtracting }}x\\x&=1&&{\text{by dividing by }}9\end{aligned}}}

Students who did not accept the first argument sometimes accept the second argument, but, in Byers's opinion, still have not resolved the ambiguity, and therefore do not understand the representation of infinite decimals. Peressini & Peressini (2007), presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. Baldwin & Norton (2012), citing Katz & Katz (2010a), also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature. Cheng (2023) concurs, arguing that knowing one can multiply 0.999... by 10 by shifting the decimal point presumes an answer to the deeper question of how one gives a meaning to the expression 0.999... at all. The same argument is also given by Richman (1999), who notes that skeptics may question whether x {\displaystyle x} is cancellable – that is, whether it makes sense to subtract x {\displaystyle x} from both sides. Eisenmann (2008) similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.

Analytic proofs

Real analysis is the study of the logical underpinnings of calculus, including the behavior of sequences and series of real numbers. The proofs in this section establish 0.999... = 1 using techniques familiar from real analysis.

Infinite series and sequences

Further information: Decimal representation

A common development of decimal expansions is to define them as sums of infinite series. In general: b 0 . b 1 b 2 b 3 b 4 = b 0 + b 1 ( 1 10 ) + b 2 ( 1 10 ) 2 + b 3 ( 1 10 ) 3 + b 4 ( 1 10 ) 4 + . {\displaystyle b_{0}.b_{1}b_{2}b_{3}b_{4}\ldots =b_{0}+b_{1}\left({\tfrac {1}{10}}\right)+b_{2}\left({\tfrac {1}{10}}\right)^{2}+b_{3}\left({\tfrac {1}{10}}\right)^{3}+b_{4}\left({\tfrac {1}{10}}\right)^{4}+\cdots .}

For 0.999... one can apply the convergence theorem concerning geometric series, stating that if ⁠ | r | {\displaystyle \vert r\vert } ⁠ < 1, then: a r + a r 2 + a r 3 + = a r 1 r . {\displaystyle ar+ar^{2}+ar^{3}+\cdots ={\frac {ar}{1-r}}.}

Since 0.999... is such a sum with a = 9 {\displaystyle a=9} and common ratio ⁠ r = 1 10 {\displaystyle \textstyle r={\frac {1}{10}}} ⁠, the theorem makes short work of the question: 0.999 = 9 ( 1 10 ) + 9 ( 1 10 ) 2 + 9 ( 1 10 ) 3 + = 9 ( 1 10 ) 1 1 10 = 1. {\displaystyle 0.999\ldots =9\left({\tfrac {1}{10}}\right)+9\left({\tfrac {1}{10}}\right)^{2}+9\left({\tfrac {1}{10}}\right)^{3}+\cdots ={\frac {9\left({\tfrac {1}{10}}\right)}{1-{\tfrac {1}{10}}}}=1.} This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra.

Limits: The unit interval, including the base-4 fraction 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 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 maneuver on 0.999...‍. 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 several proof-based introductions to calculus or analysis.

A sequence ( x 0 {\displaystyle x_{0}} , x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , ...) has the value x {\displaystyle x} as its limit if the distance | x x n | {\displaystyle \left\vert x-x_{n}\right\vert } becomes arbitrarily small as n {\displaystyle n} increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit: 0.999   = d e f   lim n 0. 99 9 n   = d e f   lim n k = 1 n 9 10 k = lim n ( 1 1 10 n ) = 1 lim n 1 10 n = 1 0 = 1. {\displaystyle 0.999\ldots \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }0.\underbrace {99\ldots 9} _{n}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {9}{10^{k}}}=\lim _{n\to \infty }\left(1-{\frac {1}{10^{n}}}\right)=1-\lim _{n\to \infty }{\frac {1}{10^{n}}}=1-0=1.} The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10 approaches 0 as n {\displaystyle n} approaches infinity (⁠ {\displaystyle \infty } ⁠), is often justified by the Archimedean property of the real numbers. 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". Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.

Nested intervals and least upper bounds

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

The series definition above defines 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 {\displaystyle x} is known to lie in the closed interval (that is, 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: , , , and so on up to . The number x {\displaystyle x} must belong to one of these; if it belongs to , then one records the digit "2" and subdivides that interval into , , ..., , . Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits ⁠ b 1 {\displaystyle b_{1}} ⁠, ⁠ b 2 {\displaystyle b_{2}} ⁠, ⁠ b 3 {\displaystyle b_{3}} ⁠, ..., and one writes x = b 0 . b 1 b 2 b 3 . {\displaystyle x=b_{0}.b_{1}b_{2}b_{3}\ldots \,.}

In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both . and , 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.

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 1 {\displaystyle b_{1}} ⁠, ⁠ b 2 {\displaystyle b_{2}} ⁠, ⁠ b 3 {\displaystyle b_{3}} ⁠, ... is defined to be the unique number contained within all the intervals [ b 0 {\displaystyle b_{0}} , b 0 {\displaystyle b_{0}} + 1], [ b 0 . b 1 {\displaystyle b_{0}.b_{1}} , b 0 . b 1 {\displaystyle b_{0}.b_{1}} + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals , , , and for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.

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 0 . b 1 b 2 b 3 {\displaystyle b_{0}.b_{1}b_{2}b_{3}} ⁠... to be the least upper bound of the set of approximants ⁠ b 0 {\displaystyle b_{0}} ⁠, ⁠ b 0 . b 1 {\displaystyle b_{0}.b_{1}} ⁠, ⁠ b 0 . b 1 b 2 {\displaystyle b_{0}.b_{1}b_{2}} ⁠, ...‍. One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes, "the fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."

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, ...} 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.

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 that directly uses 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 toward 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.

Dedekind cuts

Further information: Dedekind cut

In the Dedekind cut approach, each real number x {\displaystyle x} is defined as the infinite set of all rational numbers less than ⁠ x {\displaystyle x} ⁠. In particular, the real number 1 is the set of all rational numbers that are less than 1. Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers that are less than some stage of the expansion. So the real number 0.999... is the set of rational numbers r {\displaystyle r} such that r {\displaystyle r} < 0, or r {\displaystyle r} < 0.9, or r {\displaystyle r} < 0.99, or r {\displaystyle r} is less than some other number of the form 1 1 10 n = 0. ( 9 ) n = 0. 99 9 n  nines . {\displaystyle 1-{\frac {1}{10^{n}}}=0.(9)_{n}=0.\underbrace {99\ldots 9} _{n{\text{ nines}}}.}

Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as a b < 1 , {\displaystyle {\frac {a}{b}}<1,} with b > 0 {\displaystyle b>0} and ⁠ b > a {\displaystyle b>a} ⁠. This implies 1 a b = b a b 1 b > 1 10 b , {\displaystyle 1-{\frac {a}{b}}={\frac {b-a}{b}}\geq {\frac {1}{b}}>{\frac {1}{10^{b}}},} and thus a b < 1 1 10 b . {\displaystyle {\frac {a}{b}}<1-{\frac {1}{10^{b}}}.}

Since 1 1 10 b = 0. ( 9 ) b < 0.999 , {\displaystyle 1-{\frac {1}{10^{b}}}=0.(9)_{b}<0.999\ldots ,} by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1.

The definition of real numbers as Dedekind cuts was first published by Richard Dedekind in 1872. 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. 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 {\displaystyle x} | x {\displaystyle x} < 1} to be a cut but not { x {\displaystyle x} | x {\displaystyle x} ≤ 1} (or vice versa). 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 {\textstyle {\frac {1}{3}}} has no representation; see § Alternative number systems below.

Cauchy sequences

Further information: Cauchy sequence

Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between x {\displaystyle x} and y {\displaystyle y} is defined as the absolute value ⁠ | x y | {\displaystyle \left\vert x-y\right\vert } ⁠, where the absolute value | z | {\displaystyle \left\vert z\right\vert } is defined as the maximum of z {\displaystyle z} and ⁠ z {\displaystyle -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 0 {\displaystyle x_{0}} ⁠, ⁠ x 1 {\displaystyle x_{1}} ⁠, ⁠ x 2 {\displaystyle x_{2}} ⁠, ..., a mapping from natural numbers to rationals, for any positive rational δ {\displaystyle \delta } there is an N {\displaystyle N} such that | x m x n | δ {\displaystyle \left\vert x_{m}-x_{n}\right\vert \leq \delta } for all ⁠ m , n > N {\displaystyle m,n>N} ⁠; the distance between terms becomes smaller than any positive rational.

If ( x n ) {\displaystyle (x_{n})} and ( y n ) {\displaystyle (y_{n})} are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence ( x n y n ) {\displaystyle (x_{n}-y_{n})} has the limit 0. Truncations of the decimal number ⁠ b 0 . b 1 b 2 b 3 {\displaystyle b_{0}.b_{1}b_{2}b_{3}} ⁠... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers ( 1 0 , 1 9 10 , 1 99 100 , ) = ( 1 , 1 10 , 1 100 , ) {\displaystyle \left(1-0,1-{9 \over 10},1-{99 \over 100},\ldots \right)=\left(1,{1 \over 10},{1 \over 100},\ldots \right)} has a limit 0. Considering the ⁠ n {\displaystyle n} ⁠th term of the sequence, for ⁠ n N {\displaystyle n\in \mathbb {N} } ⁠, it must therefore be shown that lim n 1 10 n = 0. {\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{10^{n}}}=0.} This can be proved by the definition of a limit. 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. 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.

Infinite decimal representation

Commonly in secondary schools' mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 =eq 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version. In other words, the equality 0.999... = 1 holding true is a necessary condition for strings of digits to behave as real numbers should.

Dense order

Further information: Dense order

One of the notions that can resolve the issue is the requirement that real numbers be densely ordered. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal. Therefore, if 0.99999... were to be different from 1, there would have to be another real number in between them but there is none: a single digit cannot be changed in either of the two to obtain such a number.

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.

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. In general, any terminating base b {\displaystyle b} expression has a counterpart with repeated trailing digits equal to b {\displaystyle b} − 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.

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 {\displaystyle q} between 1 and 2, there are uncountably many base- q {\displaystyle q} expansions of 1. In contrast, there are still uncountably many ⁠ q {\displaystyle q} ⁠, including all natural numbers greater than 1, for which there is only one base- q {\displaystyle 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 {\displaystyle q} = 1.787231650...‍. In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the Thue–Morse sequence, which does not repeat.

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:

  • In the balanced ternary system, 1 2 {\textstyle {\frac {1}{2}}} = 0.111... = 1.111...‍.
  • In the reverse factorial number system (using bases 2!, 3!, 4!, ... for positions after the decimal point), 1 = 1.000... = 0.1234...‍.

Petkovšek (1990) 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.

Applications

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

  • 1 7 {\textstyle {\frac {1}{7}}} = 0.142857 and 142 + 857 = 999.
  • 1 73 {\textstyle {\frac {1}{73}}} = 0.01369863 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 whether his proof directly involved 0.999..., but at least one modern proof by William G. Leavitt does. If it can be proved that if a decimal of the form ⁠ 0. b 1 b 2 b 3 {\displaystyle 0.b_{1}b_{2}b_{3}} ⁠... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.

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 the 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 ⁠ n {\displaystyle n} ⁠th digit of the representation reflects the position of the point in the ⁠ n {\displaystyle n} ⁠th stage of the construction. For example, the point 2 3 {\textstyle {\frac {2}{3}}} is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and the left of every deletion thereafter. The point 1 3 {\textstyle {\frac {1}{3}}} is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and the right of every deletion thereafter.

Repeating nines also turns 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. A variant that may be closer to Cantor's original argument uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.

Skepticism in education

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

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

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...; see § In alternative number systems below.

Many of these explanations were found by David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered with 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'".

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

Mazur (2005) tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator", and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process".

As part of the APOS Theory of mathematical learning, Dubinsky et al. (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. They also link this mental ability of encapsulation to viewing 1 3 {\textstyle {\frac {1}{3}}} as a number in its own right and to dealing with the set of natural numbers as a whole.

Cultural phenomenon

With the rise of the Internet, debates about 0.999... have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup sci.math in the 1990s, arguing over 0.999... became a "popular sport", and was one of the questions answered in its FAQ. The FAQ briefly covers ⁠ 1 3 {\displaystyle \textstyle {\frac {1}{3}}} ⁠, multiplication by 10, and limits, and alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column The Straight Dope discusses 0.999... via 1 3 {\textstyle {\frac {1}{3}}} and limits, saying of misconceptions,

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

A Slate article reports that the concept of 0.999... is "hotly disputed on websites ranging from World of Warcraft message boards to Ayn Rand forums". 0.999... features also in mathematical jokes, such as:

Q: How many mathematicians does it take to screw in a lightbulb?
A: 0.999999....

The fact that 0.999... is equal to 1 has been compared to Zeno's paradox of the runner. The runner paradox can be mathematically modeled and then, like 0.999..., resolved using a geometric series. However, it is not clear whether this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.

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.

Infinitesimals

Main article: Infinitesimal

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 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. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses). A. H. Lightstone developed a decimal expansion for hyperreal numbers in (0, 1). Lightstone shows how to associate each number with a sequence of digits, 0. d 1 d 2 d 3 ; d 1 d d + 1 , {\displaystyle 0.d_{1}d_{2}d_{3}\ldots ;\ldots d_{\infty -1}d_{\infty }d_{\infty +1}\ldots ,} indexed by the hypernatural numbers. While he does not directly discuss 0.999..., he shows the real number 1 3 {\textstyle {\frac {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.

The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ...‍. A different definition involves an ultralimit, i.e., the equivalence class of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount. More generally, the hyperreal number u H {\displaystyle u_{H}} = 0.999...;...999000..., with last digit 9 at infinite hypernatural rank ⁠ H {\displaystyle H} ⁠, satisfies a strict inequality ⁠ u H < 1 {\displaystyle u_{H}<1} ⁠. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be 0. 999 H = 1 1 10 H . {\displaystyle {\underset {H}{0.\underbrace {999\ldots } }}\;=1\;-\;{\frac {1}{10^{H}}}.} 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.... Along with Katz & Katz (2010b), Ely (2010) 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.

Hackenbush

Combinatorial game theory provides a generalized concept of number that encompasses the real numbers and much more besides. For example, in 1974, Elwyn Berlekamp described a correspondence between strings of red and blue segments in Hackenbush 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...2 = 1 3 {\textstyle {\frac {1}{3}}} . However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the surreal number 1 ω {\textstyle {\frac {1}{\omega }}} , where ω {\displaystyle \omega } is the first infinite ordinal; the relevant game is LRRRR... or 0.000...2.

This is 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...2 = 0.11000...2, which are both equal to ⁠ 3 4 {\displaystyle \textstyle {\frac {3}{4}}} ⁠, but the first representation corresponds to the binary tree path LRLRLLL..., while the second corresponds to the different path LRLLRRR...‍.

Revisiting 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 (1999) considers two such systems, designed so that 0.999... < 1.

First, Richman (1999) 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 {\displaystyle x} ⁠, one has 0.999... + x {\displaystyle x} = 1 + ⁠ x {\displaystyle x} ⁠. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to ⁠ 1 3 {\displaystyle \textstyle {\frac {1}{3}}} ⁠. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.

In the process of defining multiplication, Richman also defines another system he calls "cut ⁠ D {\displaystyle 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 {\displaystyle d} he allows both the cut (⁠ {\displaystyle -\infty } ⁠, ⁠ d {\displaystyle d} ⁠) and the "principal cut" (⁠ {\displaystyle -\infty } ⁠, ⁠ d {\displaystyle 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 {\displaystyle 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 {\displaystyle x} = 1" has no solution.

p-adic numbers

Main article: p-adic number

When asked about 0.999..., novices often believe there should be a "final 9", believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the final 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 "final 9" in 0.999...‍. However, there is a system that contains an infinite string of 9s including a last 9.

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

The p {\displaystyle p} -adic numbers are an alternative number system of interest in number theory. Like the real numbers, the p {\displaystyle 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 {\displaystyle p} ⁠, and much closer to ⁠ p n {\displaystyle p^{n}} ⁠, than it is to 1. The p {\displaystyle p} -adic numbers form a field for prime p {\displaystyle p} and a ring for other ⁠ p {\displaystyle p} ⁠, including 10. So arithmetic can be performed in the p {\displaystyle p} -adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1. Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula: 999 = 9 + 9 ( 10 ) + 9 ( 10 ) 2 + 9 ( 10 ) 3 + = 9 1 10 = 1. {\displaystyle \ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\cdots ={\frac {9}{1-10}}=-1.}

Compare with the series in the section 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 {\displaystyle x} = ...999, then 10 x {\displaystyle x} = ...990, so 10 x {\displaystyle x} = x {\displaystyle x} − 9, hence x {\displaystyle x} = −1 again.

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

See also

Notes

  1. For example, one can show this as follows: if x is any number such that 0.(9)nx < 1, then 0.(9)n−1 ≤ 10x − 9 < x < 1. Thus if x has this property for all n, the smaller number 10x − 9 does, as well.
  2. The limit follows, for example, from Rudin (1976), p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir & Giordano (2001), section 8.1, example 2(a), example 6(b).
  3. The historical synthesis is claimed by Griffiths & Hilton (1970), p. xiv and again by Pugh (2002), p. 10; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh (2002), p. 17 or Rudin (1976), p. 17. For viewpoints on logic, see Pugh (2002), p. 10, Rudin (1976), p.ix, or Munkres (2000), p. 30.
  4. Enderton (1977), 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 ..."
  5. Rudin (1976), pp. 17–20, Richman (1999), p. 399, or Enderton (1977), p. 119. To be precise, Rudin, Richman, and Enderton call this cut 1∗, 1, and 1R, respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "non-principal Dedekind cut".
  6. Maor (1987), p. 60 and Mankiewicz (2000), p. 151 review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (2000), p. 50 mentions the latter method.
  7. Bunch (1982), p. 119; Tall & Schwarzenberger (1978), p. 6. The last suggestion is due to Burrell (1998), 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."
  8. For a full treatment of non-standard numbers, see Robinson (1996).
  9. Stewart (2009), p. 175; the full discussion of 0.999... is spread through pp. 172–175.
  10. Berlekamp, Conway & Guy (1982), pp. 79–80, 307–311 discuss 1 and ⁠1/3⁠ and touch on ⁠1/ω⁠. The game for 0.111...2 follows directly from Berlekamp's Rule.
  11. Richman (1999), pp. 398–400. Rudin (1976), p. 23 assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.

References

  1. Cheng (2023), p. 141.
  2. Diamond (1955).
  3. Baldwin & Norton (2012).
  4. Meier & Smith (2017), §8.2.
  5. Stewart (2009), p. 175.
  6. Propp (2023).
  7. Stillwell (1994), p. 42.
  8. Earl & Nicholson (2021), "bound".
  9. ^ Rosenlicht (1985), p. 27.
  10. Bauldry (2009), p. 47.
  11. Byers (2007), p. 39.
  12. ^ Richman (1999).
  13. Peressini & Peressini (2007), p. 186.
  14. Baldwin & Norton (2012); Katz & Katz (2010a).
  15. Cheng (2023), p. 136.
  16. Eisenmann (2008), p. 38.
  17. Tao (2003).
  18. Rudin (1976), p. 61, Theorem 3.26; Stewart (1999), p. 706.
  19. Euler (1822), p. 170.
  20. Grattan-Guinness (1970), p. 69; Bonnycastle (1806), p. 177.
  21. Stewart (1999), p. 706; Rudin (1976), p. 61; Protter & Morrey (1991), p. 213; Pugh (2002), p. 180; Conway (1978), p. 31.
  22. Davies (1846), p. 175; Smith & Harrington (1895), p. 115.
  23. ^ Tall (2000), p. 221.
  24. Beals (2004), p. 22; Stewart (2009), p. 34.
  25. Bartle & Sherbert (1982), pp. 60–62; Pedrick (1994), p. 29; Sohrab (2003), p. 46.
  26. Apostol (1974), pp. 9, 11–12; Beals (2004), p. 22; Rosenlicht (1985), p. 27.
  27. Apostol (1974), p. 12.
  28. Cheng (2023), pp. 153–156.
  29. Conway (2001), pp. 25–27.
  30. Rudin (1976), pp. 3, 8.
  31. Richman (1999), p. 399.
  32. ^ O'Connor & Robertson (2005).
  33. Richman (1999), p. 398–399. "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."
  34. Griffiths & Hilton (1970), p. 386, §24.2 "Sequences".
  35. Griffiths & Hilton (1970), pp. 388, 393.
  36. Griffiths & Hilton (1970), p. 395.
  37. Griffiths & Hilton (1970), pp. viii, 395.
  38. Gowers (2001).
  39. Li (2011).
  40. Artigue (2002), p. 212, "... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1).".
  41. Petkovšek (1990), p. 408.
  42. Protter & Morrey (1991), p. 503; Bartle & Sherbert (1982), p. 61.
  43. Komornik & Loreti (1998), p. 636.
  44. Kempner (1936), p. 611; Petkovšek (1990), p. 409.
  45. Petkovšek (1990), pp. 410–411.
  46. Goodwyn (1802); Dickson (1919), pp. 161.
  47. Leavitt (1984), p. 301.
  48. Ginsberg (2004), pp. 26–30; Lewittes (2006), pp. 1–3; Leavitt (1967), pp. 669, 673; Shrader-Frechette (1978), pp. 96–98.
  49. Pugh (2002), p. 97; Alligood, Sauer & Yorke (1996), pp. 150–152; Protter & Morrey (1991), p. 507; Pedrick (1994), p. 29.
  50. Rudin (1976), p. 50; Pugh (2002), p. 98.
  51. Tall & Schwarzenberger (1978), pp. 6–7; Tall (2000), p. 221.
  52. Tall & Schwarzenberger (1978), p. 6; Tall (2000), p. 221.
  53. Tall (1976), pp. 10–14.
  54. Pinto & Tall (2001), p. 5; Edwards & Ward (2004), pp. 416–417.
  55. Mazur (2005), pp. 137–141.
  56. Dubinsky et al. (2005), pp. 261–262.
  57. Richman (1999), p. 396.
  58. de Vreught (1994).
  59. Adams (2003).
  60. Ellenberg (2014).
  61. Renteln & Dundes (2005), p. 27.
  62. Richman (1999); Adams (2003); Ellenberg (2014).
  63. Wallace (2003), p. 51; Maor (1987), p. 17.
  64. Gowers (2002), p. 60.
  65. Lightstone (1972), pp. 245–247.
  66. Tao (2012), pp. 156–180.
  67. Katz & Katz (2010a).
  68. Katz & Katz (2010b); Ely (2010).
  69. Conway (2001), pp. 3–5, 12–13, 24–27.
  70. Richman (1999), pp. 397–399.
  71. Gardiner (2003), p. 98; Gowers (2002), p. 60.
  72. Mascari & Miola (1988), p. 83–84.
  73. ^ Fjelstad (1995), p. 11.
  74. Fjelstad (1995), pp. 14–15.
  75. DeSua (1960), p. 901.
  76. DeSua (1960), p. 902–903.

Sources

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