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0.999...

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In <a href="Mathematics" title="Mathematics">mathematics</a>, 0.999... (also written as 0.9, 0..9, or 0.(9)) is a <a href="Repeating_decimal" title="Repeating decimal">repeating decimal</a> that is an alternative way of writing the number <a href="1" title="1">1</a>. Following the standard rules for representing <a href="Real_number" title="Real number">real numbers</a> in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is 1; that is,

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<annotation encoding="application/x-tex">{\displaystyle 0.999\ldots =1.}</annotation>

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</math><img src="./fd615f072319989ad7676875cba2af8073a13c3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.315ex; height:2.176ex;" alt="{\displaystyle 0.999\ldots =1.}" loading="lazy"></dd></dl>

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.

There are many ways of showing this equality, from <a href="Mathematical_intuition" class="mw-redirect" title="Mathematical intuition">intuitive</a> arguments to <a href="Mathematical_rigor" class="mw-redirect" title="Mathematical rigor">mathematically rigorous</a> <a href="Mathematical_proof" title="Mathematical proof">proofs</a>. The intuitive arguments are generally based on properties of <a href="Finite_decimal" class="mw-redirect" title="Finite decimal">finite decimals</a> that are extended without proof to infinite decimals. An elementary but rigorous proof is given below that involves only <a href="Elementary_arithmetic" title="Elementary arithmetic">elementary arithmetic</a> and the <a href="Archimedean_property" title="Archimedean property">Archimedean property</a>: for each real number, there is a <a href="Natural_number" title="Natural number">natural number</a> that is greater (for example, by rounding up). Other proofs are generally based on basic properties of real numbers and methods of <a href="Calculus" title="Calculus">calculus</a>, such as <a href="Series_(mathematics)" title="Series (mathematics)">series</a> and <a href="Limit_(mathematics)" title="Limit (mathematics)">limits</a>. A question studied in <a href="Mathematics_education" title="Mathematics education">mathematics education</a> is why some people reject this equality.

In <a href="#Alternative_number_systems">other number systems</a>, 0.999... can have the same meaning, a different definition, or be undefined. Every nonzero <a href="Terminating_decimal" class="mw-redirect" title="Terminating decimal">terminating decimal</a> has two equal representations (for example, 8.32000... and 8.31999...). Having values with multiple representations is a feature of all <a href="Positional_numeral_system" class="mw-redirect" title="Positional numeral system">positional numeral systems</a> that represent the real numbers.

<div class="mw-heading mw-heading2"><h2 id="Elementary_proof">Elementary proof</h2></div>

It is possible to prove the equation 0.999... = 1 using just the mathematical tools of comparison and addition of (finite) <a href="Decimal_number" class="mw-redirect" title="Decimal number">decimal numbers</a>, without any reference to more advanced topics. The proof given <a href="#Rigorous_proof">below</a> is a direct formalization of <a class="mw-selflink-fragment" href="#Intuitive_explanation">the intuitive fact</a> that, if one draws 0.9, 0.99, 0.999, etc. on the <a href="Number_line" title="Number line">number line</a>, 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.

<div class="mw-heading mw-heading3"><h3 id="Intuitive_explanation">Intuitive explanation</h3></div>

If one places 0.9, 0.99, 0.999, etc. on the <a href="Number_line" title="Number line">number line</a>, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> that is less than 1, the sequence 0.9, 0.99, 0.999, and so on will eventually reach a number larger than ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠. 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.<a href="#cite_note-FOOTNOTECheng2023141-1">[1]</a><a href="#cite_note-FOOTNOTEDiamond1955-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Rigorous_proof">Rigorous proof</h3></div>

Denote by 0.(9)n the number 0.999...9, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> 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 = ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{10}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{10}}}</annotation>

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</math><img src="./99b2a631124308e27e35e78162e901bae97fe317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{10}}}" loading="lazy">⁠, 1 − 0.(9)2 = 0.01 = ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{10^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{10^{2}}}}</annotation>

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</math><img src="./ace5d7d92fb7545e53b9c5dd413372165039a4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.312ex; height:4.009ex;" alt="{\displaystyle \textstyle {\frac {1}{10^{2}}}}" loading="lazy">⁠, and so on; that is, 1 − 0.(9)n = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{10^{n}}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{10^{n}}}}</annotation>

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</math><img src="./40e51ce7ea098f1e70bdb126defedc781d1fceee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.445ex; height:3.676ex;" alt="{\textstyle {\frac {1}{10^{n}}}}" loading="lazy"> for every <a href="Natural_number" title="Natural number">natural number</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> be a number not greater than 1 and greater than 0.9, 0.99, 0.999, etc.; that is, 0.(9)n < <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ≤ 1, for every ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠. By subtracting these inequalities from 1, one gets 0 ≤ 1 − <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> < ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{10^{n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{10^{n}}}}</annotation>

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</math><img src="./a4bc2c4239d90067ca41658b86dcedab6eb04c02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.445ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{10^{n}}}}" loading="lazy">⁠.

The end of the proof requires that there is no positive number that is less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{10^{n}}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{10^{n}}}}</annotation>

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</math><img src="./40e51ce7ea098f1e70bdb126defedc781d1fceee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.445ex; height:3.676ex;" alt="{\textstyle {\frac {1}{10^{n}}}}" loading="lazy"> for all ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠. This follows from the <a href="Archimedean_property" title="Archimedean property">Archimedean property</a>, which can be expressed as, "for every real number, there is a natural number that is greater". By computing the <a href="Multiplicative_inverse" title="Multiplicative inverse">reciprocal</a>, this implies that for every positive real number, there are natural numbers whose reciprocals are smaller. Therefore, for any positive real number, there must be some ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠ such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{10^{n}}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{10^{n}}}}</annotation>

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</math><img src="./40e51ce7ea098f1e70bdb126defedc781d1fceee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.445ex; height:3.676ex;" alt="{\textstyle {\frac {1}{10^{n}}}}" loading="lazy"> is smaller.<a href="#cite_note-FOOTNOTEBaldwinNorton2012-3">[3]</a><a href="#cite_note-FOOTNOTEMeierSmith2017§8.2-4">[4]</a> This property implies that if 1 − <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> < ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{10^{n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{10^{n}}}}</annotation>

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</math><img src="./a4bc2c4239d90067ca41658b86dcedab6eb04c02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.445ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{10^{n}}}}" loading="lazy">⁠ for all ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠, then 1 − <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> can only be equal to 0. So, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = 1 and 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc. That is, 1 = 0.999..., as claimed.<a href="#cite_note-FOOTNOTEStewartTall201538–39-5">[5]</a>

This proof relies on the Archimedean property of rational and real numbers. Real numbers may be enlarged into <a href="Number_systems" class="mw-redirect" title="Number systems">number systems</a>, such as <a href="Hyperreal_number" title="Hyperreal number">hyperreal numbers</a>, with infinitely small numbers (<a href="Infinitesimal" title="Infinitesimal">infinitesimals</a>) and infinitely large numbers (<a href="Infinite_number" class="mw-redirect" title="Infinite number">infinite numbers</a>).<a href="#cite_note-FOOTNOTEStewart2009175-6">[6]</a><a href="#cite_note-FOOTNOTEPropp2023-7">[7]</a> 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.<a href="#cite_note-8">[a]</a>

<div class="mw-heading mw-heading3"><h3 id="Least_upper_bounds_and_completeness">Least upper bounds and completeness</h3></div>

Part of what this argument shows is that there is a <a href="Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bound</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 <a href="Axiom" title="Axiom">axioms</a> of the <a href="Real_number_system" class="mw-redirect" title="Real number system">real number system</a> is the <a href="Completeness_axiom" class="mw-redirect" title="Completeness axiom">completeness axiom</a>, which states that every bounded sequence has a least upper bound.<a href="#cite_note-FOOTNOTEStillwell199442-9">[8]</a><a href="#cite_note-FOOTNOTEEarlNicholson2021"bound"-10">[9]</a> 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.<a href="#cite_note-FOOTNOTERosenlicht198527-11">[10]</a> 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.<a href="#cite_note-FOOTNOTEBauldry200947-12">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Algebraic_arguments">Algebraic arguments </h2></div>

Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. <a href="#CITEREFByers2007">Byers (2007)</a> discusses the argument that, in elementary school, one is taught that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> = 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 <a href="Equals_sign" title="Equals sign">equals sign</a>; 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.<a href="#cite_note-FOOTNOTEByers200739-13">[12]</a> <a href="#CITEREFRichman1999">Richman (1999)</a> 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.<a href="#cite_note-FOOTNOTERichman1999-14">[13]</a>

Byers also presents the following argument.

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</style><div class="block-indent"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<mtext>by multiplying by </mtext>

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<mtext>by splitting off integer part</mtext>

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<mtext>by definition of </mtext>

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<mtext>by subtracting </mtext>

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<mtext>by dividing by </mtext>

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./68d42bec21541c7d4d9cfebc62fb72467528b8a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:51.656ex; height:18.009ex;" alt="{\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}}}" loading="lazy"></div>

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. <a href="#CITEREFPeressiniPeressini2007">Peressini & Peressini (2007)</a>, 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 <a href="Completeness_axiom" class="mw-redirect" title="Completeness axiom">completeness</a>.<a href="#cite_note-FOOTNOTEPeressiniPeressini2007[httpsarchiveorgdetailsperspectivesonma0000unse_f3x1page186mode1upqequalityviewtheater_186]-15">[14]</a> <a href="#CITEREFBaldwinNorton2012">Baldwin & Norton (2012)</a>, citing <a href="#CITEREFKatzKatz2010a">Katz & Katz (2010a)</a>, also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.<a href="#cite_note-FOOTNOTEBaldwinNorton2012KatzKatz2010a-16">[15]</a> <a href="#CITEREFCheng2023">Cheng (2023)</a> 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.<a href="#cite_note-FOOTNOTECheng2023136-17">[16]</a> The same argument is also given by <a href="#CITEREFRichman1999">Richman (1999)</a>, who notes that skeptics may question whether <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is <a href="Cancelling_out" title="Cancelling out">cancellable</a> – that is, whether it makes sense to subtract <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> from both sides.<a href="#cite_note-FOOTNOTERichman1999-14">[13]</a> <a href="#CITEREFEisenmann2008">Eisenmann (2008)</a> similarly argues that both the multiplication and subtraction which removes the infinite decimal require further justification.<a href="#cite_note-FOOTNOTEEisenmann200838-18">[17]</a>

<div class="mw-heading mw-heading2"><h2 id="Analytic_proofs">Analytic proofs </h2></div>

<a href="Real_analysis" title="Real analysis">Real analysis</a> is the study of the logical underpinnings of <a href="Calculus" title="Calculus">calculus</a>, including the behavior of sequences and series of real numbers.<a href="#cite_note-FOOTNOTETao2003-19">[18]</a> The proofs in this section establish 0.999... = 1 using techniques familiar from real analysis.

<div class="mw-heading mw-heading3"><h3 id="Infinite_series_and_sequences">Infinite series and sequences</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Decimal_representation" title="Decimal representation">Decimal representation</a></div>

A common development of decimal expansions is to define them as <a href="Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a>. In general:

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<annotation encoding="application/x-tex">{\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 .}</annotation>

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For 0.999... one can apply the <a href="Convergent_series" title="Convergent series">convergence</a> theorem concerning <a href="Geometric_series" title="Geometric series">geometric series</a>, stating that if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert r\vert }">

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<annotation encoding="application/x-tex">{\displaystyle \vert r\vert }</annotation>

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</math><img src="./e22ec28b9f23ee22ea5531e2ead8db7d44d75ee2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.342ex; height:2.843ex;" alt="{\displaystyle \vert r\vert }" loading="lazy">⁠ < 1, then:<a href="#cite_note-FOOTNOTERudin197661Theorem_3.26Stewart1999706-20">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle ar+ar^{2}+ar^{3}+\cdots ={\frac {ar}{1-r}}.}</annotation>

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Since 0.999... is such a sum with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=9}">

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<annotation encoding="application/x-tex">{\displaystyle a=9}</annotation>

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</math><img src="./b0f292aed6e12d795abb12b74fd0a1992410a188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=9}" loading="lazy"> and common ratio ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r={\frac {1}{10}}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle r={\frac {1}{10}}}</annotation>

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</math><img src="./1cd93721ced1bbda03bfe7d6efc0bfe83cd6e7ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.627ex; height:3.676ex;" alt="{\displaystyle \textstyle r={\frac {1}{10}}}" loading="lazy">⁠, the theorem makes short work of the question:

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<annotation encoding="application/x-tex">{\displaystyle 0.999\ldots =0+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.}</annotation>

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This proof appears as early as 1770 in <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>'s <a href="Elements_of_Algebra" title="Elements of Algebra">Elements of Algebra</a>.<a href="#cite_note-FOOTNOTEEuler1822170-21">[20]</a>

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 <a href="#Algebraic_arguments">algebraic proof</a> 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 href="#cite_note-FOOTNOTEGrattan-Guinness197069Bonnycastle1806177-22">[21]</a> 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 href="#cite_note-FOOTNOTEStewart1999706Rudin197661ProtterMorrey1991213Pugh2002180Conway197831-23">[22]</a>

A <a href="Sequence" title="Sequence">sequence</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>

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</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

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</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy">, ...) has the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> as its <a href="Limit_of_a_sequence" title="Limit of a sequence">limit</a> if the distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert x-x_{n}\right\vert }">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<a href="#cite_note-24">[b]</a>

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<annotation encoding="application/x-tex">{\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.}</annotation>

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The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that 10-n approaches 0 as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> approaches infinity (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

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<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

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</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy">⁠), is often justified by the <a href="Archimedean_property" title="Archimedean property">Archimedean property</a> 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".<a href="#cite_note-FOOTNOTEDavies1846175SmithHarrington1895115-25">[23]</a> Such <a href="Heuristic" title="Heuristic">heuristics</a> are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.<a href="#cite_note-FOOTNOTETall2000221-26">[24]</a>

<div class="mw-heading mw-heading3"><h3 id="Nested_intervals_and_least_upper_bounds">Nested intervals and least upper bounds</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Nested_intervals" title="Nested intervals">Nested intervals</a></div>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is known to lie in the <a href="Closed_interval" class="mw-redirect" title="Closed interval">closed interval</a> [0, 10] (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: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both [0, 1]. and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<a href="#cite_note-FOOTNOTEBeals200422Stewart200934-27">[25]</a>

One straightforward choice is the <a href="Nested_intervals_theorem" class="mw-redirect" title="Nested intervals theorem">nested intervals theorem</a>, which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>. So ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math><img src="./9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math><img src="./9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy"> + 1], [<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}}</annotation>

</semantics>

</math><img src="./772eb3ddc18e240e57f7e8e7a46a363dc252e30c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.138ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}}</annotation>

</semantics>

</math><img src="./772eb3ddc18e240e57f7e8e7a46a363dc252e30c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.138ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}}" loading="lazy"> + 0.1], and so on. 0.999... is then the unique real number that lies in all of the intervals [0, 1], [0.9, 1], [0.99, 1], and [0.99...9, 1] for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.<a href="#cite_note-FOOTNOTEBartleSherbert198260–62Pedrick199429Sohrab200346-28">[26]</a>

The nested intervals theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of <a href="Least_upper_bound" class="mw-redirect" title="Least upper bound">least upper bounds</a> or <a href="Suprema" class="mw-redirect" title="Suprema">suprema</a>. To directly exploit these objects, one may define ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}b_{2}b_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}b_{2}b_{3}}</annotation>

</semantics>

</math><img src="./4464f24330322ec7e629e036c34ed2ae045d3af7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.241ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}b_{2}b_{3}}" loading="lazy">⁠... to be the least upper bound of the set of approximants ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}}</annotation>

</semantics>

</math><img src="./9e425056f502ca07b103ffbf6ac4720e0f8a01f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0}}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}}</annotation>

</semantics>

</math><img src="./772eb3ddc18e240e57f7e8e7a46a363dc252e30c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.138ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}b_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}b_{2}}</annotation>

</semantics>

</math><img src="./64b897ef71b116da6589b22e7f7b233b689ba2c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.189ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}b_{2}}" loading="lazy">⁠, ...‍.<a href="#cite_note-FOOTNOTEApostol19749,_11–12Beals200422Rosenlicht198527-29">[27]</a> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. <a href="Tom_Apostol" class="mw-redirect" title="Tom Apostol">Tom Apostol</a> 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."<a href="#cite_note-FOOTNOTEApostol197412-30">[28]</a>

<div class="mw-heading mw-heading2"><h2 id="Proofs_from_the_construction_of_the_real_numbers">Proofs from the construction of the real numbers </h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">Construction of the real numbers</a></div>

Some approaches explicitly define real numbers to be certain <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">structures built upon the rational numbers</a>, using <a href="Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>. The <a href="Natural_number" title="Natural number">natural numbers</a> {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 <a href="Integer" title="Integer">integers</a>, and to further extend to ratios, giving the <a href="Rational_number" title="Rational number">rational numbers</a>. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division.<a href="#cite_note-FOOTNOTECheng2023153–156-31">[29]</a><a href="#cite_note-FOOTNOTEConway200125–27-32">[30]</a> More subtly, they include <a href="Order_theory" title="Order theory">ordering</a>, so that one number can be compared to another and found to be less than, greater than, or equal to another number.<a href="#cite_note-FOOTNOTERudin19763,_8-33">[31]</a>

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: <a href="Dedekind_cut" title="Dedekind cut">Dedekind cuts</a> and <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>. 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.<a href="#cite_note-34">[c]</a>

<div class="mw-heading mw-heading3"><h3 id="Dedekind_cuts">Dedekind cuts</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Dedekind_cut" title="Dedekind cut">Dedekind cut</a></div>

In the <a href="Dedekind_cut" title="Dedekind cut">Dedekind cut</a> approach, each real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is defined as the <a href="Infinite_set" title="Infinite set">infinite set</a> of all <a href="Rational_number" title="Rational number">rational numbers</a> less than ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠.<a href="#cite_note-35">[d]</a> In particular, the real number 1 is the set of all rational numbers that are less than 1.<a href="#cite_note-36">[e]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> < 0, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> < 0.9, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> < 0.99, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> is less than some other number of the form<a href="#cite_note-FOOTNOTERichman1999399-37">[32]</a>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</munder>

</mrow>

<mtext> nines</mtext>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{10^{n}}}=0.(9)_{n}=0.\underbrace {99\ldots 9} _{n{\text{ nines}}}.}</annotation>

</semantics>

</math>

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

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}<1,}</annotation>

</semantics>

</math>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b>0}</annotation>

</semantics>

</math><img src="./94436473a90bd55191a79c59474cb5456dcbec00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b>0}" loading="lazy"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b>a}</annotation>

</semantics>

</math><img src="./5b1cfc86ca957eea4f09d683db2412a173f6f404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b>a}" loading="lazy">⁠. This implies

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-{\frac {a}{b}}={\frac {b-a}{b}}\geq {\frac {1}{b}}>{\frac {1}{10^{b}}},}</annotation>

</semantics>

</math>

and thus

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}<1-{\frac {1}{10^{b}}}.}</annotation>

</semantics>

</math>

Since

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-{\frac {1}{10^{b}}}=0.(9)_{b}<0.999\ldots ,}</annotation>

</semantics>

</math>

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 <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> in 1872.<a href="#cite_note-FOOTNOTEO'ConnorRobertson2005-38">[33]</a>

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 <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>.<a href="#cite_note-FOOTNOTERichman1999-14">[13]</a> Richman notes that taking Dedekind cuts in any <a href="Dense_subset" class="mw-redirect" title="Dense subset">dense subset</a> of the rational numbers yields the same results; in particular, he uses <a href="Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fractions</a>, for which the proof is more immediate. He also notes that typically the definitions allow {<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> | <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> < 1} to be a cut but not {<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> | <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> ≤ 1} (or vice versa).<a href="#cite_note-39">[34]</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> has no representation; see <a href="#Alternative_number_systems">§ Alternative number systems</a> below.

<div class="mw-heading mw-heading3"><h3 id="Cauchy_sequences">Cauchy sequences</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></div>

Another approach is to define a real number as the limit of a <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a> of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> is defined as the absolute value ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert x-y\right\vert }">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert x-y\right\vert }</annotation>

</semantics>

</math><img src="./0b1ec1c8668b6fd0b64d1fb704446951036b2857.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.619ex; height:2.843ex;" alt="{\displaystyle \left\vert x-y\right\vert }" loading="lazy">⁠, where the absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert z\right\vert }">

<mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert z\right\vert }</annotation>

</semantics>

</math><img src="./e09d238ddf8dd3a23b8355dccaf1b6fe1b39a68a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.382ex; height:2.843ex;" alt="{\displaystyle \left\vert z\right\vert }" loading="lazy"> is defined as the maximum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -z}</annotation>

</semantics>

</math><img src="./78c4571a4a3ceb7e7e55712372835ebe65d20f3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.896ex; height:2.176ex;" alt="{\displaystyle -z}" loading="lazy">⁠, 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 ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>

</semantics>

</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

</semantics>

</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy">⁠, ..., a mapping from natural numbers to rationals, for any positive rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> there is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert x_{m}-x_{n}\right\vert \leq \delta }">

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\vert x_{m}-x_{n}\right\vert \leq \delta }</annotation>

</semantics>

</math><img src="./7ae00366b54f37dcb47605888803142066caa749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.834ex; height:2.843ex;" alt="{\displaystyle \left\vert x_{m}-x_{n}\right\vert \leq \delta }" loading="lazy"> for all ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n>N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n>N}</annotation>

</semantics>

</math><img src="./bfa46859a96c4fc2561c5a9488980723c4673282.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.631ex; height:2.509ex;" alt="{\displaystyle m,n>N}" loading="lazy">⁠; the distance between terms becomes smaller than any positive rational.<a href="#cite_note-FOOTNOTEGriffithsHilton1970386§24.2_"Sequences"-40">[35]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n})}</annotation>

</semantics>

</math><img src="./012f44968fa86fe5e3827e9957d957b08f2d9e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.357ex; height:2.843ex;" alt="{\displaystyle (x_{n})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{n})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{n})}</annotation>

</semantics>

</math><img src="./81e3bac0852409ba19a3346beca1b87fe9f5aac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.167ex; height:2.843ex;" alt="{\displaystyle (y_{n})}" loading="lazy"> are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{n}-y_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{n}-y_{n})}</annotation>

</semantics>

</math><img src="./36d3cf718d0a091528df9d29031b01943d077744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.556ex; height:2.843ex;" alt="{\displaystyle (x_{n}-y_{n})}" loading="lazy"> has the limit 0. Truncations of the decimal number ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}.b_{1}b_{2}b_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}.b_{1}b_{2}b_{3}}</annotation>

</semantics>

</math><img src="./4464f24330322ec7e629e036c34ed2ae045d3af7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.241ex; height:2.509ex;" alt="{\displaystyle b_{0}.b_{1}b_{2}b_{3}}" loading="lazy">⁠... generate a sequence of rationals, which is Cauchy; this is taken to define the real value of the number.<a href="#cite_note-FOOTNOTEGriffithsHilton1970388,_393-41">[36]</a> Thus in this formalism the task is to show that the sequence of rational numbers

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(1-0,1-{9 \over 10},1-{99 \over 100},\ldots \right)=\left(1,{1 \over 10},{1 \over 100},\ldots \right)}</annotation>

</semantics>

</math>

has a limit 0. Considering the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠th term of the sequence, for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">⁠, it must therefore be shown that

</mrow>

</munder>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty }{\frac {1}{10^{n}}}=0.}</annotation>

</semantics>

</math>

This can be proved by the <a href="Limit_of_a_sequence#Formal_Definition" title="Limit of a sequence">definition of a limit</a>. So again, 0.999... = 1.<a href="#cite_note-FOOTNOTEGriffithsHilton1970395-42">[37]</a>

The definition of real numbers as Cauchy sequences was first published separately by <a href="Eduard_Heine" title="Eduard Heine">Eduard Heine</a> and <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, also in 1872.<a href="#cite_note-FOOTNOTEO'ConnorRobertson2005-38">[33]</a> 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.<a href="#cite_note-FOOTNOTEGriffithsHilton1970viii,_395-43">[38]</a>

<div class="mw-heading mw-heading3"><h3 id="Infinite_decimal_representation">Infinite decimal representation</h3></div>

Commonly in <a href="Secondary_schools" class="mw-redirect" title="Secondary schools">secondary schools</a>' mathematics education, the real numbers are constructed by defining a number using an integer followed by a <a href="Radix_point" class="mw-redirect" title="Radix point">radix point</a> and an infinite sequence written out as a string to represent the <a href="Fractional_part" title="Fractional part">fractional part</a> 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 <a href="Real_number#Axiomatic_approach" title="Real number">real axioms</a> after defining an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> 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.<a href="#cite_note-FOOTNOTEGowers2001-44">[39]</a><a href="#cite_note-FOOTNOTELi2011-45">[40]</a>

<div class="mw-heading mw-heading3"><h3 id="Dense_order">Dense order</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Dense_order" title="Dense order">Dense order</a></div>

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.<sup id="cite_ref-FOOTNOTEArtigue2002212"..._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)."_46-0" class="reference"><a href="#cite_note-FOOTNOTEArtigue2002212"..._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)."-46">[41]</a>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

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 <a href="Dense_set" title="Dense set">dense</a>.<a href="#cite_note-FOOTNOTEPetkovšek1990408-47">[42]</a><a href="#cite_note-FOOTNOTERosenlicht198527-11">[10]</a>

Second, a comparable theorem applies in each <a href="Radix" title="Radix">radix</a> (base). For example, in base 2 (the <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numeral system</a>) 0.111... equals 1, and in base 3 (the <a href="Ternary_numeral_system" title="Ternary numeral system">ternary numeral system</a>) 0.222... equals 1. In general, any terminating base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> expression has a counterpart with repeated trailing digits equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> − 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.<a href="#cite_note-FOOTNOTEProtterMorrey1991503BartleSherbert198261-48">[43]</a>

Alternative representations of 1 also occur in non-integer bases. For example, in the <a href="Golden_ratio_base" title="Golden ratio base">golden ratio base</a>, the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for <a href="Almost_all" title="Almost all">almost all</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> between 1 and 2, there are uncountably many base-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> expansions of 1. In contrast, there are still uncountably many ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">⁠, including all natural numbers greater than 1, for which there is only one base-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> expansion of 1, other than the trivial 1.000...‍. This result was first obtained by <a href="Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a>, Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and <a href="Paola_Loreti" title="Paola Loreti">Paola Loreti</a> determined the smallest such base, the <a href="Komornik%E2%80%93Loreti_constant" title="Komornik–Loreti constant">Komornik–Loreti constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> = 1.787231650...‍. In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the <a href="Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a>, which does not repeat.<a href="#cite_note-FOOTNOTEKomornikLoreti1998636-49">[44]</a>

A more far-reaching generalization addresses <a href="Non-standard_positional_numeral_systems" title="Non-standard positional numeral systems">the most general positional numeral systems</a>. They too have multiple representations, and in some sense, the difficulties are even worse. For example:<a href="#cite_note-FOOTNOTEKempner1936611Petkovšek1990409-50">[45]</a>

<ul><li>In the <a href="Balanced_ternary" title="Balanced ternary">balanced ternary</a> system, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./990274ef9b8937f955b175041b7cd0d5a2d482ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}}" loading="lazy"> = 0.111... = 1.111...‍.</li>

<li>In the reverse <a href="Factorial_number_system" title="Factorial number system">factorial number system</a> (using bases 2!, 3!, 4!, ... for positions after the decimal point), 1 = 1.000... = 0.1234...‍.</li></ul>

<a href="#CITEREFPetkovšek1990">Petkovšek (1990)</a> 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 <a href="Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a>"; it involves viewing sets of positional values as <a href="Stone_space" title="Stone space">Stone spaces</a> and noticing that their real representations are given by <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous functions</a>.<a href="#cite_note-FOOTNOTEPetkovšek1990410–411-51">[46]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

One application of 0.999... as a representation of 1 occurs in elementary <a href="Number_theory" title="Number theory">number theory</a>. In 1802, H. Goodwyn published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain <a href="Prime_number" title="Prime number">prime numbers</a>.<a href="#cite_note-FOOTNOTEGoodwyn1802Dickson1919161-52">[47]</a> Examples include:

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{7}}}</annotation>

</semantics>

</math><img src="./1b8a7a43dffc86577e36d1bcb444aaca8e159d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{7}}}" loading="lazy"> = 0.142857 and 142 + 857 = 999.</li>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{73}}}</annotation>

</semantics>

</math><img src="./d18709097eafa796f452ae0cbea2c8245d535f2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.676ex;" alt="{\textstyle {\frac {1}{73}}}" loading="lazy"> = 0.01369863 and 0136 + 9863 = 9999.</li></ul>

E. Midy proved a general result about such fractions, now called <a href="Midy's_theorem" title="Midy's theorem">Midy's theorem</a>, 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 ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.b_{1}b_{2}b_{3}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0.b_{1}b_{2}b_{3}}</annotation>

</semantics>

</math><img src="./fc3aac660f3e635b721231f41b8ace1c89cf0772.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.965ex; height:2.509ex;" alt="{\displaystyle 0.b_{1}b_{2}b_{3}}" loading="lazy">⁠... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<a href="#cite_note-FOOTNOTELeavitt1984301-53">[48]</a> Investigations in this direction can motivate such concepts as <a href="Greatest_common_divisor" title="Greatest common divisor">greatest common divisors</a>, <a href="Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>, <a href="Fermat_prime" class="mw-redirect" title="Fermat prime">Fermat primes</a>, <a href="Order_(group_theory)" title="Order (group theory)">order</a> of <a href="Group_(mathematics)" title="Group (mathematics)">group</a> elements, and <a href="Quadratic_reciprocity" title="Quadratic reciprocity">quadratic reciprocity</a>.<a href="#cite_note-FOOTNOTEGinsberg200426–30Lewittes20061–3Leavitt1967669,_673Shrader-Frechette197896–98-54">[49]</a>

Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in the characterization of one of the simplest <a href="Fractal" title="Fractal">fractals</a>, the middle-thirds <a href="Cantor_set" title="Cantor set">Cantor set</a>: a point in the <a href="Unit_interval" title="Unit interval">unit interval</a> lies in the Cantor set <a href="If_and_only_if" title="If and only if">if and only if</a> it can be represented in ternary using only the digits 0 and 2.

The ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠th digit of the representation reflects the position of the point in the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">⁠th stage of the construction. For example, the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {2}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {2}{3}}}</annotation>

</semantics>

</math><img src="./ab0e30d9a795c6a0635f89ec69e4ef7ed13a0d14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {2}{3}}}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> 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.<a href="#cite_note-FOOTNOTEPugh200297AlligoodSauerYorke1996150–152ProtterMorrey1991507Pedrick199429-55">[50]</a>

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 <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">his 1891 diagonal argument</a> to decimal expansions, of the <a href="Uncountability" class="mw-redirect" title="Uncountability">uncountability</a> 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 href="#cite_note-56">[f]</a> 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.<a href="#cite_note-FOOTNOTERudin197650Pugh200298-57">[51]</a>

<div class="mw-heading mw-heading2"><h2 id="Skepticism_in_education">Skepticism in education</h2></div>

Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit</a> concept and disagreements over the nature of <a href="Infinitesimal" title="Infinitesimal">infinitesimals</a>. There are many common contributing factors to the confusion:

<ul><li>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 <a href="Paradox" title="Paradox">paradox</a>, which is amplified by the appearance of the seemingly well-understood number 1.<a href="#cite_note-58">[g]</a></li>

<li>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".<a href="#cite_note-FOOTNOTETallSchwarzenberger19786–7Tall2000221-59">[52]</a></li>

<li>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.<a href="#cite_note-FOOTNOTETallSchwarzenberger19786Tall2000221-60">[53]</a></li></ul>

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 <a href="Counterexample" title="Counterexample">counterexamples</a> to better understand 0.999...; see <a href="#In_alternative_number_systems">§ In alternative number systems</a> below.

Many of these explanations were found by <a href="David_Tall" title="David Tall">David Tall</a>, 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'".<a href="#cite_note-FOOTNOTETall2000221-26">[24]</a>

The elementary argument of multiplying 0.333... = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> 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.<a href="#cite_note-FOOTNOTETall197610–14-61">[54]</a> 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... = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> using a <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a> definition but then insisted that 0.999... < 1 based on her earlier understanding of <a href="Long_division" title="Long division">long division</a>.<a href="#cite_note-FOOTNOTEPintoTall20015EdwardsWard2004416–417-62">[55]</a> Others still can prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> = 0.333..., but, upon being confronted by the <a href="#Algebraic_arguments">fractional proof</a>, insist that "logic" supersedes the mathematical calculations.

<a href="#CITEREFMazur2005">Mazur (2005)</a> 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".<a href="#cite_note-FOOTNOTEMazur2005137–141-63">[56]</a>

As part of the <a href="APOS_Theory" title="APOS Theory">APOS Theory</a> of mathematical learning, <a href="#CITEREFDubinskyWellerMcDonaldBrown2005">Dubinsky et al. (2005)</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> as a number in its own right and to dealing with the set of natural numbers as a whole.<a href="#cite_note-FOOTNOTEDubinskyWellerMcDonaldBrown2005261–262-64">[57]</a>

<div class="mw-heading mw-heading2"><h2 id="Cultural_phenomenon">Cultural phenomenon</h2></div>

With the rise of the <a href="Internet" title="Internet">Internet</a>, debates about 0.999... have become commonplace on <a href="Newsgroup" class="mw-redirect" title="Newsgroup">newsgroups</a> and <a href="Message_board" class="mw-redirect" title="Message board">message boards</a>, including many that nominally have little to do with mathematics. In the newsgroup <style data-mw-deduplicate="TemplateStyles:r886049734">

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</style>sci.math in the 1990s, arguing over 0.999... became a "popular sport", and was one of the questions answered in its <a href="FAQ" title="FAQ">FAQ</a>.<a href="#cite_note-FOOTNOTERichman1999396-65">[58]</a><a href="#cite_note-FOOTNOTEde_Vreught1994-66">[59]</a> The FAQ briefly covers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{3}}}">

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./4cd0a7660677cda599a22c28518d7cea18e4e302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{3}}}" loading="lazy">⁠, multiplication by 10, and limits, and alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column <a href="The_Straight_Dope" title="The Straight Dope">The Straight Dope</a> discusses 0.999... via <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> and limits, saying of misconceptions,

/* start https://en.wikipedia.org/ */

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</style><blockquote class="templatequote">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 href="#cite_note-FOOTNOTEAdams2003-67">[60]</a></blockquote>

A <a href="Slate_(magazine)" title="Slate (magazine)">Slate</a> article reports that the concept of 0.999... is "hotly disputed on websites ranging from <a href="World_of_Warcraft" title="World of Warcraft">World of Warcraft</a> message boards to <a href="Ayn_Rand" title="Ayn Rand">Ayn Rand</a> forums".<a href="#cite_note-FOOTNOTEEllenberg2014-68">[61]</a>

0.999... features also in <a href="Mathematical_joke" title="Mathematical joke">mathematical jokes</a>, such as:<a href="#cite_note-FOOTNOTERentelnDundes200527-69">[62]</a>

Q: How many mathematicians does it take to <a href="Lightbulb_joke" title="Lightbulb joke">screw in a lightbulb</a>?

A: 0.999999....

</blockquote>

The fact that 0.999... is equal to 1 has been compared to <a href="Zeno's_paradoxes#Dichotomy_paradox" title="Zeno's paradoxes">Zeno's paradox of the runner</a>.<a href="#cite_note-FOOTNOTERichman1999Adams2003Ellenberg2014-70">[63]</a> 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.<a href="#cite_note-FOOTNOTEWallace200351Maor198717-71">[64]</a>

<div class="mw-heading mw-heading2"><h2 id="In_alternative_number_systems">In alternative number systems </h2></div>

Although the real numbers form an extremely useful <a href="Number_system" class="mw-redirect" title="Number system">number system</a>, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and <a href="Timothy_Gowers" title="Timothy Gowers">Timothy Gowers</a> 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.<a href="#cite_note-FOOTNOTEGowers200260-72">[65]</a>

</blockquote>

<div class="mw-heading mw-heading3"><h3 id="Infinitesimals">Infinitesimals</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinitesimal" title="Infinitesimal">Infinitesimal</a></div>

Some proofs that 0.999... = 1 rely on the <a href="Archimedean_property" title="Archimedean property">Archimedean property</a> of the real numbers: that there are no nonzero <a href="Infinitesimal" title="Infinitesimal">infinitesimals</a>. 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 <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a>, including various alternatives to the real numbers, which are non-Archimedean. <a href="Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">Non-standard analysis</a> provides a number system with a full array of infinitesimals (and their inverses).<a href="#cite_note-73">[h]</a> <a href="A._H._Lightstone" title="A. H. Lightstone">A. H. Lightstone</a> developed a decimal expansion for <a href="Hyperreal_number" title="Hyperreal number">hyperreal numbers</a> in (0, 1)∗. Lightstone shows how to associate each number with a sequence of digits,

<msub>

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<annotation encoding="application/x-tex">{\displaystyle 0.d_{1}d_{2}d_{3}\ldots ;\ldots d_{\infty -1}d_{\infty }d_{\infty +1}\ldots ,}</annotation>

</semantics>

</math>

indexed by the <a href="Hypernatural" class="mw-redirect" title="Hypernatural">hypernatural</a> numbers. While he does not directly discuss 0.999..., he shows the real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

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</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy"> is represented by 0.333...;...333..., which is a consequence of the <a href="Transfer_principle" title="Transfer principle">transfer principle</a>. 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.<a href="#cite_note-FOOTNOTELightstone1972245–247-74">[66]</a>

The standard definition of the number 0.999... is the <a href="Limit_of_a_sequence" title="Limit of a sequence">limit of the sequence</a> 0.9, 0.99, 0.999, ...‍. A different definition involves an ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the <a href="Ultrapower_construction" class="mw-redirect" title="Ultrapower construction">ultrapower construction</a>, which is a number that falls short of 1 by an infinitesimal amount.<a href="#cite_note-FOOTNOTETao2012156–180-75">[67]</a> More generally, the hyperreal number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{H}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{H}}</annotation>

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</math><img src="./1c505b6e6e7dc565e9ba59f62a2d5d71f89e5b7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.021ex; height:2.009ex;" alt="{\displaystyle u_{H}}" loading="lazy"> = 0.999...;...999000..., with last digit 9 at infinite <a href="Hypernatural" class="mw-redirect" title="Hypernatural">hypernatural</a> rank ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">⁠, satisfies a strict inequality ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{H}<1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{H}<1}</annotation>

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</math><img src="./b72f9aa70d678e58b398c85b77c85ae88694256b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.282ex; height:2.509ex;" alt="{\displaystyle u_{H}<1}" loading="lazy">⁠. Accordingly, an alternative interpretation for "zero followed by infinitely many 9s" could be<a href="#cite_note-FOOTNOTEKatzKatz2010a-76">[68]</a>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\underset {H}{0.\underbrace {999\ldots } }}\;=1\;-\;{\frac {1}{10^{H}}}.}</annotation>

</semantics>

</math>

All such interpretations of "0.999..." are infinitely close to 1. <a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> 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....<a href="#cite_note-77">[i]</a> Along with <a href="#CITEREFKatzKatz2010b">Katz & Katz (2010b)</a>, <a href="#CITEREFEly2010">Ely (2010)</a> 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.<a href="#cite_note-FOOTNOTEKatzKatz2010bEly2010-78">[69]</a>

<div class="mw-heading mw-heading3"><h3 id="Hackenbush">Hackenbush</h3></div>

<a href="Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a> provides a generalized concept of number that encompasses the real numbers and much more besides.<a href="#cite_note-FOOTNOTEConway20013–5,_12–13,_24–27-79">[70]</a> For example, in 1974, <a href="Elwyn_Berlekamp" title="Elwyn Berlekamp">Elwyn Berlekamp</a> described a correspondence between strings of red and blue segments in <a href="Hackenbush" title="Hackenbush">Hackenbush</a> and binary expansions of real numbers, motivated by the idea of <a href="Data_compression" title="Data compression">data compression</a>. For example, the value of the Hackenbush string LRRLRLRL... is 0.010101...2 = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./26ea1944c3f21fcd8a37aa679a3bce05d5cd6e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\textstyle {\frac {1}{3}}}" loading="lazy">. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the <a href="Surreal_number" title="Surreal number">surreal number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\omega }}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {1}{\omega }}}</annotation>

</semantics>

</math><img src="./211234b9efc822082477fa14025f9cc6eb052d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.858ex; height:3.343ex;" alt="{\textstyle {\frac {1}{\omega }}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> is the first <a href="Ordinal_number" title="Ordinal number">infinite ordinal</a>; the relevant game is LRRRR... or 0.000...2.<a href="#cite_note-80">[j]</a>

This is true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding <a href="Binary_tree" title="Binary tree">binary tree</a> paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {3}{4}}}">

<mfrac>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {3}{4}}}</annotation>

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</math><img src="./37cc8ead04380ed5ef601fb13c3b8ddb16c94e8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle \textstyle {\frac {3}{4}}}" loading="lazy">⁠, but the first representation corresponds to the binary tree path LRLRLLL..., while the second corresponds to the different path LRLLRRR...‍.

<div class="mw-heading mw-heading3"><h3 id="Revisiting_subtraction">Revisiting subtraction</h3></div>

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 <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="Semigroup" title="Semigroup">semigroups</a>, <a href="Commutative_monoid" class="mw-redirect" title="Commutative monoid">commutative monoids</a>, and <a href="Semiring" title="Semiring">semirings</a>. <a href="#CITEREFRichman1999">Richman (1999)</a> considers two such systems, designed so that 0.999... < 1.<a href="#cite_note-FOOTNOTERichman1999-14">[13]</a>

First, <a href="#CITEREFRichman1999">Richman (1999)</a> defines a nonnegative decimal number to be a literal decimal expansion. He defines the <a href="Lexicographical_order" class="mw-redirect" title="Lexicographical order">lexicographical order</a> and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠, one has 0.999... + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = 1 + ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠. So one peculiarity of the decimal numbers is that addition cannot always be canceled; another is that no decimal number corresponds to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {1}{3}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {1}{3}}}</annotation>

</semantics>

</math><img src="./4cd0a7660677cda599a22c28518d7cea18e4e302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.658ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {1}{3}}}" loading="lazy">⁠. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<a href="#cite_note-FOOTNOTERichman1999397–399-81">[71]</a>

In the process of defining multiplication, Richman also defines another system he calls "cut ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">⁠", which is the set of <a href="Dedekind_cut" title="Dedekind cut">Dedekind cuts</a> of decimal fractions. Ordinarily, this definition leads to the real numbers, but for a decimal fraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> he allows both the cut (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

</semantics>

</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">⁠) and the "principal cut" (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

</semantics>

</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">⁠]. 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 ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">⁠, 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... + <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = 1" has no solution.<a href="#cite_note-82">[k]</a>

<div class="mw-heading mw-heading3"><h3 id="p-adic_numbers">p-adic numbers</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="P-adic_number" title="P-adic number">p-adic number</a></div>

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...‍.<a href="#cite_note-FOOTNOTEGardiner200398Gowers200260-83">[72]</a> However, there is a system that contains an infinite string of 9s including a last 9.

The <a href="P-adic_number" title="P-adic number"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-adic numbers</a> are an alternative number system of interest in <a href="Number_theory" title="Number theory">number theory</a>. Like the real numbers, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-adic numbers can be built from the rational numbers via <a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a>; the construction uses a different metric in which 0 is closer to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">⁠, and much closer to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{n}}</annotation>

</semantics>

</math><img src="./c6a7a7e74ae90ab94f01e1629177758fb68b423b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.477ex; height:2.676ex;" alt="{\displaystyle p^{n}}" loading="lazy">⁠, than it is to 1.<a href="#cite_note-FOOTNOTEMascariMiola1988[httpsbooksgooglecombooksidYVKzPSseyu4CpgPA83_83&ndash;84]-84">[73]</a> The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-adic numbers form a <a href="Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a> for prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> for other ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">⁠, including 10. So arithmetic can be performed in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-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.<a href="#cite_note-FOOTNOTEFjelstad199511-85">[74]</a> 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:<a href="#cite_note-FOOTNOTEFjelstad199514–15-86">[75]</a>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ldots 999=9+9(10)+9(10)^{2}+9(10)^{3}+\cdots ={\frac {9}{1-10}}=-1.}</annotation>

</semantics>

</math>

Compare with the series in the <a href="#Infinite_series_and_sequences">section above</a>. 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 <a href="#Algebraic_arguments">above</a> in the opposite direction: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = ...999, then 10<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = ...990, so 10<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> − 9, hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> = −1 again.<a href="#cite_note-FOOTNOTEFjelstad199511-85">[74]</a>

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"<a href="#cite_note-FOOTNOTEDeSua1960901-87">[76]</a> 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 <a href="Doubly_infinite" class="mw-redirect" title="Doubly infinite">doubly infinite</a> <a href="Positional_notation" title="Positional notation">decimal expansion</a> of the <a href="Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)">10-adic solenoid</a>, with eventually repeating left ends to represent the real numbers and eventually repeating right ends to represent the 10-adic numbers.<a href="#cite_note-FOOTNOTEDeSua1960902&ndash;903-88">[77]</a>

<ul><li><a href="Finitism" title="Finitism">Finitism</a></li>

<li><a href="Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li>

<li><a href="Dichotomy_paradox" class="mw-redirect" title="Dichotomy paradox">Dichotomy paradox</a>, a paradox formed based on not intuitively understanding infinite sequences</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> For example, one can show this as follows: if x is any number such that 0.(9)n ≤ x < 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.

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> The limit follows, for example, from <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 57, Theorem 3.20e. For a more direct approach, see also <a href="#CITEREFFinneyWeirGiordano2001">Finney, Weir & Giordano (2001)</a>, section 8.1, example 2(a), example 6(b).

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> The historical synthesis is claimed by <a href="#CITEREFGriffithsHilton1970">Griffiths & Hilton (1970)</a>, p. xiv and again by <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 10; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 17 or <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 17. For viewpoints on logic, see <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 10, <a href="#CITEREFRudin1976">Rudin (1976)</a>, p.ix, or <a href="#CITEREFMunkres2000">Munkres (2000)</a>, p. 30.

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFEnderton1977">Enderton (1977)</a>, 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 ..."

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <a href="#CITEREFRudin1976">Rudin (1976)</a>, pp. 17–20, <a href="#CITEREFRichman1999">Richman (1999)</a>, p. 399, or <a href="#CITEREFEnderton1977">Enderton (1977)</a>, 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".

</li>

<li id="cite_note-56"><a href="#cite_ref-56">^</a> <a href="#CITEREFMaor1987">Maor (1987)</a>, p. 60 and <a href="#CITEREFMankiewicz2000">Mankiewicz (2000)</a>, p. 151 review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. <a href="#CITEREFMunkres2000">Munkres (2000)</a>, p. 50 mentions the latter method.

</li>

<li id="cite_note-58"><a href="#cite_ref-58">^</a> <a href="#CITEREFBunch1982">Bunch (1982)</a>, p. 119; <a href="#CITEREFTallSchwarzenberger1978">Tall & Schwarzenberger (1978)</a>, p. 6. The last suggestion is due to <a href="#CITEREFBurrell1998">Burrell (1998)</a>, 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."

</li>

<li id="cite_note-73"><a href="#cite_ref-73">^</a> For a full treatment of non-standard numbers, see <a href="#CITEREFRobinson1996">Robinson (1996)</a>.

</li>

<li id="cite_note-77"><a href="#cite_ref-77">^</a> <a href="#CITEREFStewart2009">Stewart (2009)</a>, p. 175; the full discussion of 0.999... is spread through pp. 172–175.

</li>

<li id="cite_note-80"><a href="#cite_ref-80">^</a> <a href="#CITEREFBerlekampConwayGuy1982">Berlekamp, Conway & Guy (1982)</a>, 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.

</li>

<li id="cite_note-82"><a href="#cite_ref-82">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>, pp. 398–400. <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 23 assigns this alternative construction (but over the rationals) as the last exercise of Chapter 1.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-FOOTNOTECheng2023141-1"><a href="#cite_ref-FOOTNOTECheng2023141_1-0">^</a> <a href="#CITEREFCheng2023">Cheng (2023)</a>, p. 141.

</li>

<li id="cite_note-FOOTNOTEDiamond1955-2"><a href="#cite_ref-FOOTNOTEDiamond1955_2-0">^</a> <a href="#CITEREFDiamond1955">Diamond (1955)</a>.

</li>

<li id="cite_note-FOOTNOTEBaldwinNorton2012-3"><a href="#cite_ref-FOOTNOTEBaldwinNorton2012_3-0">^</a> <a href="#CITEREFBaldwinNorton2012">Baldwin & Norton (2012)</a>.

</li>

<li id="cite_note-FOOTNOTEMeierSmith2017§8.2-4"><a href="#cite_ref-FOOTNOTEMeierSmith2017§8.2_4-0">^</a> <a href="#CITEREFMeierSmith2017">Meier & Smith (2017)</a>, §8.2.

</li>

<li id="cite_note-FOOTNOTEStewartTall201538–39-5"><a href="#cite_ref-FOOTNOTEStewartTall201538–39_5-0">^</a> <a href="#CITEREFStewartTall2015">Stewart & Tall (2015)</a>, pp. 38–39.

</li>

<li id="cite_note-FOOTNOTEStewart2009175-6"><a href="#cite_ref-FOOTNOTEStewart2009175_6-0">^</a> <a href="#CITEREFStewart2009">Stewart (2009)</a>, p. 175.

</li>

<li id="cite_note-FOOTNOTEPropp2023-7"><a href="#cite_ref-FOOTNOTEPropp2023_7-0">^</a> <a href="#CITEREFPropp2023">Propp (2023)</a>.

</li>

<li id="cite_note-FOOTNOTEStillwell199442-9"><a href="#cite_ref-FOOTNOTEStillwell199442_9-0">^</a> <a href="#CITEREFStillwell1994">Stillwell (1994)</a>, p. 42.

</li>

<li id="cite_note-FOOTNOTEEarlNicholson2021"bound"-10"><a href="#cite_ref-FOOTNOTEEarlNicholson2021"bound"_10-0">^</a> <a href="#CITEREFEarlNicholson2021">Earl & Nicholson (2021)</a>, "bound".

</li>

<li id="cite_note-FOOTNOTERosenlicht198527-11">^ <a href="#cite_ref-FOOTNOTERosenlicht198527_11-0">a</a> <a href="#cite_ref-FOOTNOTERosenlicht198527_11-1">b</a> <a href="#CITEREFRosenlicht1985">Rosenlicht (1985)</a>, p. 27.

</li>

<li id="cite_note-FOOTNOTEBauldry200947-12"><a href="#cite_ref-FOOTNOTEBauldry200947_12-0">^</a> <a href="#CITEREFBauldry2009">Bauldry (2009)</a>, p. 47.

</li>

<li id="cite_note-FOOTNOTEByers200739-13"><a href="#cite_ref-FOOTNOTEByers200739_13-0">^</a> <a href="#CITEREFByers2007">Byers (2007)</a>, p. 39.

</li>

<li id="cite_note-FOOTNOTERichman1999-14">^ <a href="#cite_ref-FOOTNOTERichman1999_14-0">a</a> <a href="#cite_ref-FOOTNOTERichman1999_14-1">b</a> <a href="#cite_ref-FOOTNOTERichman1999_14-2">c</a> <a href="#cite_ref-FOOTNOTERichman1999_14-3">d</a> <a href="#CITEREFRichman1999">Richman (1999)</a>.

</li>

<li id="cite_note-FOOTNOTEPeressiniPeressini2007[httpsarchiveorgdetailsperspectivesonma0000unse_f3x1page186mode1upqequalityviewtheater_186]-15"><a href="#cite_ref-FOOTNOTEPeressiniPeressini2007[httpsarchiveorgdetailsperspectivesonma0000unse_f3x1page186mode1upqequalityviewtheater_186]_15-0">^</a> <a href="#CITEREFPeressiniPeressini2007">Peressini & Peressini (2007)</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/perspectivesonma0000unse_f3x1/page/186/mode/1up?q=equality&view=theater">186</a>.

</li>

<li id="cite_note-FOOTNOTEBaldwinNorton2012KatzKatz2010a-16"><a href="#cite_ref-FOOTNOTEBaldwinNorton2012KatzKatz2010a_16-0">^</a> <a href="#CITEREFBaldwinNorton2012">Baldwin & Norton (2012)</a>; <a href="#CITEREFKatzKatz2010a">Katz & Katz (2010a)</a>.

</li>

<li id="cite_note-FOOTNOTECheng2023136-17"><a href="#cite_ref-FOOTNOTECheng2023136_17-0">^</a> <a href="#CITEREFCheng2023">Cheng (2023)</a>, p. 136.

</li>

<li id="cite_note-FOOTNOTEEisenmann200838-18"><a href="#cite_ref-FOOTNOTEEisenmann200838_18-0">^</a> <a href="#CITEREFEisenmann2008">Eisenmann (2008)</a>, p. 38.

</li>

<li id="cite_note-FOOTNOTETao2003-19"><a href="#cite_ref-FOOTNOTETao2003_19-0">^</a> <a href="#CITEREFTao2003">Tao (2003)</a>.

</li>

<li id="cite_note-FOOTNOTERudin197661Theorem_3.26Stewart1999706-20"><a href="#cite_ref-FOOTNOTERudin197661Theorem_3.26Stewart1999706_20-0">^</a> <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 61, Theorem 3.26; <a href="#CITEREFStewart1999">Stewart (1999)</a>, p. 706.

</li>

<li id="cite_note-FOOTNOTEEuler1822170-21"><a href="#cite_ref-FOOTNOTEEuler1822170_21-0">^</a> <a href="#CITEREFEuler1822">Euler (1822)</a>, p. 170.

</li>

<li id="cite_note-FOOTNOTEGrattan-Guinness197069Bonnycastle1806177-22"><a href="#cite_ref-FOOTNOTEGrattan-Guinness197069Bonnycastle1806177_22-0">^</a> <a href="#CITEREFGrattan-Guinness1970">Grattan-Guinness (1970)</a>, p. 69; <a href="#CITEREFBonnycastle1806">Bonnycastle (1806)</a>, p. 177.

</li>

<li id="cite_note-FOOTNOTEStewart1999706Rudin197661ProtterMorrey1991213Pugh2002180Conway197831-23"><a href="#cite_ref-FOOTNOTEStewart1999706Rudin197661ProtterMorrey1991213Pugh2002180Conway197831_23-0">^</a> <a href="#CITEREFStewart1999">Stewart (1999)</a>, p. 706; <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 61; <a href="#CITEREFProtterMorrey1991">Protter & Morrey (1991)</a>, p. 213; <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 180; <a href="#CITEREFConway1978">Conway (1978)</a>, p. 31.

</li>

<li id="cite_note-FOOTNOTEDavies1846175SmithHarrington1895115-25"><a href="#cite_ref-FOOTNOTEDavies1846175SmithHarrington1895115_25-0">^</a> <a href="#CITEREFDavies1846">Davies (1846)</a>, p. 175; <a href="#CITEREFSmithHarrington1895">Smith & Harrington (1895)</a>, p. 115.

</li>

<li id="cite_note-FOOTNOTETall2000221-26">^ <a href="#cite_ref-FOOTNOTETall2000221_26-0">a</a> <a href="#cite_ref-FOOTNOTETall2000221_26-1">b</a> <a href="#CITEREFTall2000">Tall (2000)</a>, p. 221.

</li>

<li id="cite_note-FOOTNOTEBeals200422Stewart200934-27"><a href="#cite_ref-FOOTNOTEBeals200422Stewart200934_27-0">^</a> <a href="#CITEREFBeals2004">Beals (2004)</a>, p. 22; <a href="#CITEREFStewart2009">Stewart (2009)</a>, p. 34.

</li>

<li id="cite_note-FOOTNOTEBartleSherbert198260–62Pedrick199429Sohrab200346-28"><a href="#cite_ref-FOOTNOTEBartleSherbert198260–62Pedrick199429Sohrab200346_28-0">^</a> <a href="#CITEREFBartleSherbert1982">Bartle & Sherbert (1982)</a>, pp. 60–62; <a href="#CITEREFPedrick1994">Pedrick (1994)</a>, p. 29; <a href="#CITEREFSohrab2003">Sohrab (2003)</a>, p. 46.

</li>

<li id="cite_note-FOOTNOTEApostol19749,_11–12Beals200422Rosenlicht198527-29"><a href="#cite_ref-FOOTNOTEApostol19749,_11–12Beals200422Rosenlicht198527_29-0">^</a> <a href="#CITEREFApostol1974">Apostol (1974)</a>, pp. 9, 11–12; <a href="#CITEREFBeals2004">Beals (2004)</a>, p. 22; <a href="#CITEREFRosenlicht1985">Rosenlicht (1985)</a>, p. 27.

</li>

<li id="cite_note-FOOTNOTEApostol197412-30"><a href="#cite_ref-FOOTNOTEApostol197412_30-0">^</a> <a href="#CITEREFApostol1974">Apostol (1974)</a>, p. 12.

</li>

<li id="cite_note-FOOTNOTECheng2023153–156-31"><a href="#cite_ref-FOOTNOTECheng2023153–156_31-0">^</a> <a href="#CITEREFCheng2023">Cheng (2023)</a>, pp. 153–156.

</li>

<li id="cite_note-FOOTNOTEConway200125–27-32"><a href="#cite_ref-FOOTNOTEConway200125–27_32-0">^</a> <a href="#CITEREFConway2001">Conway (2001)</a>, pp. 25–27.

</li>

<li id="cite_note-FOOTNOTERudin19763,_8-33"><a href="#cite_ref-FOOTNOTERudin19763,_8_33-0">^</a> <a href="#CITEREFRudin1976">Rudin (1976)</a>, pp. 3, 8.

</li>

<li id="cite_note-FOOTNOTERichman1999399-37"><a href="#cite_ref-FOOTNOTERichman1999399_37-0">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>, p. 399.

</li>

<li id="cite_note-FOOTNOTEO'ConnorRobertson2005-38">^ <a href="#cite_ref-FOOTNOTEO'ConnorRobertson2005_38-0">a</a> <a href="#cite_ref-FOOTNOTEO'ConnorRobertson2005_38-1">b</a> <a href="#CITEREFO'ConnorRobertson2005">O'Connor & Robertson (2005)</a>.

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>, 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."

</li>

<li id="cite_note-FOOTNOTEGriffithsHilton1970386§24.2_"Sequences"-40"><a href="#cite_ref-FOOTNOTEGriffithsHilton1970386§24.2_"Sequences"_40-0">^</a> <a href="#CITEREFGriffithsHilton1970">Griffiths & Hilton (1970)</a>, p. 386, §24.2 "Sequences".

</li>

<li id="cite_note-FOOTNOTEGriffithsHilton1970388,_393-41"><a href="#cite_ref-FOOTNOTEGriffithsHilton1970388,_393_41-0">^</a> <a href="#CITEREFGriffithsHilton1970">Griffiths & Hilton (1970)</a>, pp. 388, 393.

</li>

<li id="cite_note-FOOTNOTEGriffithsHilton1970395-42"><a href="#cite_ref-FOOTNOTEGriffithsHilton1970395_42-0">^</a> <a href="#CITEREFGriffithsHilton1970">Griffiths & Hilton (1970)</a>, p. 395.

</li>

<li id="cite_note-FOOTNOTEGriffithsHilton1970viii,_395-43"><a href="#cite_ref-FOOTNOTEGriffithsHilton1970viii,_395_43-0">^</a> <a href="#CITEREFGriffithsHilton1970">Griffiths & Hilton (1970)</a>, pp. viii, 395.

</li>

<li id="cite_note-FOOTNOTEGowers2001-44"><a href="#cite_ref-FOOTNOTEGowers2001_44-0">^</a> <a href="#CITEREFGowers2001">Gowers (2001)</a>.

</li>

<li id="cite_note-FOOTNOTELi2011-45"><a href="#cite_ref-FOOTNOTELi2011_45-0">^</a> <a href="#CITEREFLi2011">Li (2011)</a>.

</li>

<li id="cite_note-FOOTNOTEArtigue2002212"..._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)."-46"><a href="#cite_ref-FOOTNOTEArtigue2002212"..._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)."_46-0">^</a> <a href="#CITEREFArtigue2002">Artigue (2002)</a>, 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).".

</li>

<li id="cite_note-FOOTNOTEPetkovšek1990408-47"><a href="#cite_ref-FOOTNOTEPetkovšek1990408_47-0">^</a> <a href="#CITEREFPetkovšek1990">Petkovšek (1990)</a>, p. 408.

</li>

<li id="cite_note-FOOTNOTEProtterMorrey1991503BartleSherbert198261-48"><a href="#cite_ref-FOOTNOTEProtterMorrey1991503BartleSherbert198261_48-0">^</a> <a href="#CITEREFProtterMorrey1991">Protter & Morrey (1991)</a>, p. 503; <a href="#CITEREFBartleSherbert1982">Bartle & Sherbert (1982)</a>, p. 61.

</li>

<li id="cite_note-FOOTNOTEKomornikLoreti1998636-49"><a href="#cite_ref-FOOTNOTEKomornikLoreti1998636_49-0">^</a> <a href="#CITEREFKomornikLoreti1998">Komornik & Loreti (1998)</a>, p. 636.

</li>

<li id="cite_note-FOOTNOTEKempner1936611Petkovšek1990409-50"><a href="#cite_ref-FOOTNOTEKempner1936611Petkovšek1990409_50-0">^</a> <a href="#CITEREFKempner1936">Kempner (1936)</a>, p. 611; <a href="#CITEREFPetkovšek1990">Petkovšek (1990)</a>, p. 409.

</li>

<li id="cite_note-FOOTNOTEPetkovšek1990410–411-51"><a href="#cite_ref-FOOTNOTEPetkovšek1990410–411_51-0">^</a> <a href="#CITEREFPetkovšek1990">Petkovšek (1990)</a>, pp. 410–411.

</li>

<li id="cite_note-FOOTNOTEGoodwyn1802Dickson1919161-52"><a href="#cite_ref-FOOTNOTEGoodwyn1802Dickson1919161_52-0">^</a> <a href="#CITEREFGoodwyn1802">Goodwyn (1802)</a>; <a href="#CITEREFDickson1919">Dickson (1919)</a>, pp. 161.

</li>

<li id="cite_note-FOOTNOTELeavitt1984301-53"><a href="#cite_ref-FOOTNOTELeavitt1984301_53-0">^</a> <a href="#CITEREFLeavitt1984">Leavitt (1984)</a>, p. 301.

</li>

<li id="cite_note-FOOTNOTEGinsberg200426–30Lewittes20061–3Leavitt1967669,_673Shrader-Frechette197896–98-54"><a href="#cite_ref-FOOTNOTEGinsberg200426–30Lewittes20061–3Leavitt1967669,_673Shrader-Frechette197896–98_54-0">^</a> <a href="#CITEREFGinsberg2004">Ginsberg (2004)</a>, pp. 26–30; <a href="#CITEREFLewittes2006">Lewittes (2006)</a>, pp. 1–3; <a href="#CITEREFLeavitt1967">Leavitt (1967)</a>, pp. 669, 673; <a href="#CITEREFShrader-Frechette1978">Shrader-Frechette (1978)</a>, pp. 96–98.

</li>

<li id="cite_note-FOOTNOTEPugh200297AlligoodSauerYorke1996150–152ProtterMorrey1991507Pedrick199429-55"><a href="#cite_ref-FOOTNOTEPugh200297AlligoodSauerYorke1996150–152ProtterMorrey1991507Pedrick199429_55-0">^</a> <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 97; <a href="#CITEREFAlligoodSauerYorke1996">Alligood, Sauer & Yorke (1996)</a>, pp. 150–152; <a href="#CITEREFProtterMorrey1991">Protter & Morrey (1991)</a>, p. 507; <a href="#CITEREFPedrick1994">Pedrick (1994)</a>, p. 29.

</li>

<li id="cite_note-FOOTNOTERudin197650Pugh200298-57"><a href="#cite_ref-FOOTNOTERudin197650Pugh200298_57-0">^</a> <a href="#CITEREFRudin1976">Rudin (1976)</a>, p. 50; <a href="#CITEREFPugh2002">Pugh (2002)</a>, p. 98.

</li>

<li id="cite_note-FOOTNOTETallSchwarzenberger19786–7Tall2000221-59"><a href="#cite_ref-FOOTNOTETallSchwarzenberger19786–7Tall2000221_59-0">^</a> <a href="#CITEREFTallSchwarzenberger1978">Tall & Schwarzenberger (1978)</a>, pp. 6–7; <a href="#CITEREFTall2000">Tall (2000)</a>, p. 221.

</li>

<li id="cite_note-FOOTNOTETallSchwarzenberger19786Tall2000221-60"><a href="#cite_ref-FOOTNOTETallSchwarzenberger19786Tall2000221_60-0">^</a> <a href="#CITEREFTallSchwarzenberger1978">Tall & Schwarzenberger (1978)</a>, p. 6; <a href="#CITEREFTall2000">Tall (2000)</a>, p. 221.

</li>

<li id="cite_note-FOOTNOTETall197610–14-61"><a href="#cite_ref-FOOTNOTETall197610–14_61-0">^</a> <a href="#CITEREFTall1976">Tall (1976)</a>, pp. 10–14.

</li>

<li id="cite_note-FOOTNOTEPintoTall20015EdwardsWard2004416–417-62"><a href="#cite_ref-FOOTNOTEPintoTall20015EdwardsWard2004416–417_62-0">^</a> <a href="#CITEREFPintoTall2001">Pinto & Tall (2001)</a>, p. 5; <a href="#CITEREFEdwardsWard2004">Edwards & Ward (2004)</a>, pp. 416–417.

</li>

<li id="cite_note-FOOTNOTEMazur2005137–141-63"><a href="#cite_ref-FOOTNOTEMazur2005137–141_63-0">^</a> <a href="#CITEREFMazur2005">Mazur (2005)</a>, pp. 137–141.

</li>

<li id="cite_note-FOOTNOTEDubinskyWellerMcDonaldBrown2005261–262-64"><a href="#cite_ref-FOOTNOTEDubinskyWellerMcDonaldBrown2005261–262_64-0">^</a> <a href="#CITEREFDubinskyWellerMcDonaldBrown2005">Dubinsky et al. (2005)</a>, pp. 261–262.

</li>

<li id="cite_note-FOOTNOTERichman1999396-65"><a href="#cite_ref-FOOTNOTERichman1999396_65-0">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>, p. 396.

</li>

<li id="cite_note-FOOTNOTEde_Vreught1994-66"><a href="#cite_ref-FOOTNOTEde_Vreught1994_66-0">^</a> <a href="#CITEREFde_Vreught1994">de Vreught (1994)</a>.

</li>

<li id="cite_note-FOOTNOTEAdams2003-67"><a href="#cite_ref-FOOTNOTEAdams2003_67-0">^</a> <a href="#CITEREFAdams2003">Adams (2003)</a>.

</li>

<li id="cite_note-FOOTNOTEEllenberg2014-68"><a href="#cite_ref-FOOTNOTEEllenberg2014_68-0">^</a> <a href="#CITEREFEllenberg2014">Ellenberg (2014)</a>.

</li>

<li id="cite_note-FOOTNOTERentelnDundes200527-69"><a href="#cite_ref-FOOTNOTERentelnDundes200527_69-0">^</a> <a href="#CITEREFRentelnDundes2005">Renteln & Dundes (2005)</a>, p. 27.

</li>

<li id="cite_note-FOOTNOTERichman1999Adams2003Ellenberg2014-70"><a href="#cite_ref-FOOTNOTERichman1999Adams2003Ellenberg2014_70-0">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>; <a href="#CITEREFAdams2003">Adams (2003)</a>; <a href="#CITEREFEllenberg2014">Ellenberg (2014)</a>.

</li>

<li id="cite_note-FOOTNOTEWallace200351Maor198717-71"><a href="#cite_ref-FOOTNOTEWallace200351Maor198717_71-0">^</a> <a href="#CITEREFWallace2003">Wallace (2003)</a>, p. 51; <a href="#CITEREFMaor1987">Maor (1987)</a>, p. 17.

</li>

<li id="cite_note-FOOTNOTEGowers200260-72"><a href="#cite_ref-FOOTNOTEGowers200260_72-0">^</a> <a href="#CITEREFGowers2002">Gowers (2002)</a>, p. 60.

</li>

<li id="cite_note-FOOTNOTELightstone1972245–247-74"><a href="#cite_ref-FOOTNOTELightstone1972245–247_74-0">^</a> <a href="#CITEREFLightstone1972">Lightstone (1972)</a>, pp. 245–247.

</li>

<li id="cite_note-FOOTNOTETao2012156–180-75"><a href="#cite_ref-FOOTNOTETao2012156–180_75-0">^</a> <a href="#CITEREFTao2012">Tao (2012)</a>, pp. 156–180.

</li>

<li id="cite_note-FOOTNOTEKatzKatz2010a-76"><a href="#cite_ref-FOOTNOTEKatzKatz2010a_76-0">^</a> <a href="#CITEREFKatzKatz2010a">Katz & Katz (2010a)</a>.

</li>

<li id="cite_note-FOOTNOTEKatzKatz2010bEly2010-78"><a href="#cite_ref-FOOTNOTEKatzKatz2010bEly2010_78-0">^</a> <a href="#CITEREFKatzKatz2010b">Katz & Katz (2010b)</a>; <a href="#CITEREFEly2010">Ely (2010)</a>.

</li>

<li id="cite_note-FOOTNOTEConway20013–5,_12–13,_24–27-79"><a href="#cite_ref-FOOTNOTEConway20013–5,_12–13,_24–27_79-0">^</a> <a href="#CITEREFConway2001">Conway (2001)</a>, pp. 3–5, 12–13, 24–27.

</li>

<li id="cite_note-FOOTNOTERichman1999397–399-81"><a href="#cite_ref-FOOTNOTERichman1999397–399_81-0">^</a> <a href="#CITEREFRichman1999">Richman (1999)</a>, pp. 397–399.

</li>

<li id="cite_note-FOOTNOTEGardiner200398Gowers200260-83"><a href="#cite_ref-FOOTNOTEGardiner200398Gowers200260_83-0">^</a> <a href="#CITEREFGardiner2003">Gardiner (2003)</a>, p. 98; <a href="#CITEREFGowers2002">Gowers (2002)</a>, p. 60.

</li>

<li id="cite_note-FOOTNOTEMascariMiola1988[httpsbooksgooglecombooksidYVKzPSseyu4CpgPA83_83&ndash;84]-84"><a href="#cite_ref-FOOTNOTEMascariMiola1988[httpsbooksgooglecombooksidYVKzPSseyu4CpgPA83_83&ndash;84]_84-0">^</a> <a href="#CITEREFMascariMiola1988">Mascari & Miola (1988)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YVKzPSseyu4C&pg=PA83">83–84</a>.

</li>

<li id="cite_note-FOOTNOTEFjelstad199511-85">^ <a href="#cite_ref-FOOTNOTEFjelstad199511_85-0">a</a> <a href="#cite_ref-FOOTNOTEFjelstad199511_85-1">b</a> <a href="#CITEREFFjelstad1995">Fjelstad (1995)</a>, p. 11.

</li>

<li id="cite_note-FOOTNOTEFjelstad199514–15-86"><a href="#cite_ref-FOOTNOTEFjelstad199514–15_86-0">^</a> <a href="#CITEREFFjelstad1995">Fjelstad (1995)</a>, pp. 14–15.

</li>

<li id="cite_note-FOOTNOTEDeSua1960901-87"><a href="#cite_ref-FOOTNOTEDeSua1960901_87-0">^</a> <a href="#CITEREFDeSua1960">DeSua (1960)</a>, p. 901.

</li>

<li id="cite_note-FOOTNOTEDeSua1960902&ndash;903-88"><a href="#cite_ref-FOOTNOTEDeSua1960902&ndash;903_88-0">^</a> <a href="#CITEREFDeSua1960">DeSua (1960)</a>, p. 902–903.

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2></div>

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</style><cite id="CITEREFAdams2003" class="citation web cs1"><a href="Cecil_Adams" title="Cecil Adams">Adams, Cecil</a> (11 July 2003). <a rel="nofollow" class="external text" href="http://www.straightdope.com/columns/030711.html">"An infinite question: Why doesn't .999~ = 1?"</a>. <a href="The_Straight_Dope" title="The Straight Dope">The Straight Dope</a>. <a href="Chicago_Reader" title="Chicago Reader">Chicago Reader</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060815010844/http://www.straightdope.com/columns/030711.html">Archived</a> from the original on 15 August 2006. Retrieved 6 September 2006.</cite></li>

<li><cite id="CITEREFAlligoodSauerYorke1996" class="citation book cs1">Alligood, K. T.; Sauer, T. D.; <a href="James_A._Yorke" title="James A. Yorke">Yorke, J. A.</a> (1996). "4.1 Cantor Sets". Chaos: An introduction to dynamical systems. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94677-1</bdi>.</cite>

<dl><dd>This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)</dd></dl></li>

<li><cite id="CITEREFApostol1974" class="citation book cs1"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalanal02edtomm">Mathematical Analysis</a> (2e ed.). Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-00288-1</bdi>.</cite>

<dl><dd>A transition from calculus to advanced analysis, Mathematical analysis is intended to be "honest, rigorous, up to date, and, at the same time, not too pedantic". (pref.) Apostol's development of the real numbers uses the least upper bound axiom and introduces infinite decimals two pages later. (pp. 9–11)</dd></dl></li>

<li><cite id="CITEREFArtigue2002" class="citation book cs1">Artigue, Michèle (2002). Holton, Derek; Artigue, Michèle; Kirchgräber, Urs; Hillel, Joel; Niss, Mogens; Schoenfeld, Alan (eds.). The Teaching and Learning of Mathematics at University Level. New ICMI Study Series. Vol. 7. Springer, Dordrecht. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F0-306-47231-7">10.1007/0-306-47231-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-306-47231-2</bdi>.</cite></li>

<li><cite id="CITEREFBaldwinNorton2012" class="citation journal cs1">Baldwin, Michael; Norton, Anderson (2012). <a rel="nofollow" class="external text" href="https://eric.ed.gov/?id=EJ961516">"Does 0.999... Really Equal 1?"</a>. <a href="The_Mathematics_Educator" title="The Mathematics Educator">The Mathematics Educator</a>. 21 (2): 58–67.</cite></li>

<li><cite id="CITEREFBauldry2009" class="citation book cs1">Bauldry, William C. (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ab-2vpx0FyYC&pg=PA47">Introduction to Real Analysis: An Educational Approach</a>. John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-37136-7</bdi>.</cite>

<dl><dd>This book is intended as introduction to real analysis aimed at upper- undergraduate and graduate-level. (pp. xi-xii)</dd></dl></li>

<li><cite id="CITEREFBartleSherbert1982" class="citation book cs1"><a href="Robert_G._Bartle" title="Robert G. Bartle">Bartle, R. G.</a>; Sherbert, D. R. (1982). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontore0000bart">Introduction to Real Analysis</a>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-05944-8</bdi>.</cite>

<dl><dd>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)</dd></dl></li>

<li><cite id="CITEREFBeals2004" class="citation book cs1"><a href="Richard_Beals_(mathematician)" title="Richard Beals (mathematician)">Beals, Richard</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cXAqJUYqXx0C">Analysis: An Introduction</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-60047-7</bdi>.</cite></li>

<li><cite id="CITEREFBerlekampConwayGuy1982" class="citation book cs1"><a href="Elwyn_Berlekamp" title="Elwyn Berlekamp">Berlekamp, E. R.</a>; <a href="John_Horton_Conway" title="John Horton Conway">Conway, J. H.</a>; Guy, R. K. (1982). <a href="Winning_Ways_for_your_Mathematical_Plays" class="mw-redirect" title="Winning Ways for your Mathematical Plays">Winning Ways for your Mathematical Plays</a>. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-091101-1</bdi>.</cite></li>

<li><cite id="CITEREFBonnycastle1806" class="citation book cs1">Bonnycastle, John (1806). <a rel="nofollow" class="external text" href="http://hdl.handle.net/2027/mdp.39015063620382">An introduction to algebra; with notes and observations: designed for the use of schools and places of public education</a> (First American ed.). Philadelphia. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.39015063620382">2027/mdp.39015063620382</a>.</cite></li>

<li><cite id="CITEREFBunch1982" class="citation book cs1">Bunch, Bryan H. (1982). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalfall0000bunc">Mathematical Fallacies and Paradoxes</a>. Van Nostrand Reinhold. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-442-24905-2</bdi>.</cite>

<dl><dd>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)</dd></dl></li>

<li><cite id="CITEREFBurrell1998" class="citation book cs1">Burrell, Brian (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/merriamwebstersg00burr">Merriam-Webster's Guide to Everyday Math: A Home and Business Reference</a>. Merriam-Webster. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-87779-621-3</bdi>.</cite></li>

<li><cite id="CITEREFByers2007" class="citation book cs1">Byers, William (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/howmathematician00byer">How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics</a>. Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-12738-5</bdi>.</cite></li>

<li><cite id="CITEREFCheng2023" class="citation book cs1"><a href="Eugenia_Cheng" title="Eugenia Cheng">Cheng, Eugenia</a> (2023). Is Math Real? How Simple Questions Lead Us To Mathematics' Deepest Truths. Basic Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-541-6-01826</bdi>.</cite></li>

<li><cite id="CITEREFConway1978" class="citation book cs1"><a href="John_B._Conway" title="John B. Conway">Conway, John B.</a> (1978) [1973]. <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9781461263142">Functions of One Complex Variable I</a> (2e ed.). Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90328-6</bdi>.</cite></li>

<li><cite id="CITEREFConway2001" class="citation book cs1"><a href="John_Horton_Conway" title="John Horton Conway">Conway, John H.</a> (2001). <a href="On_Numbers_and_Games" title="On Numbers and Games">On Numbers and Games</a> (2nd ed.). A K Peters. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-56881-127-6</bdi>.</cite></li>

<li><cite id="CITEREFDavies1846" class="citation book cs1"><a href="Charles_Davies_(professor)" title="Charles Davies (professor)">Davies, Charles</a> (1846). <a rel="nofollow" class="external text" href="https://archive.org/details/universityarith00davigoog">The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications</a>. A.S. Barnes. p. <a rel="nofollow" class="external text" href="https://archive.org/details/universityarith00davigoog/page/n181">175</a>. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFde_Vreught1994" class="citation web cs1">de Vreught, Hans (1994). <a rel="nofollow" class="external text" href="http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/">"sci.math FAQ: Why is 0.9999... = 1?"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070929122649/http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/">Archived</a> from the original on 29 September 2007. Retrieved 29 June 2006.</cite></li>

<li><cite id="CITEREFDeSua1960" class="citation journal cs1">DeSua, Frank C. (November 1960). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1960-11_67_9/page/900">"A System Isomorphic to the Reals"</a>. <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 67 (9): 900–903. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2309468">10.2307/2309468</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2309468">2309468</a>.</cite></li>

<li><cite id="CITEREFDiamond1955" class="citation journal cs1">Diamond, Louis E. (1955). "Irrational Numbers". <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>. 29 (2). Mathematical Association of America: 89–99. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3029588">10.2307/3029588</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3029588">3029588</a>.</cite></li>

<li><cite id="CITEREFDickson1919" class="citation book cs1">Dickson, Leonard Eugene (1919). History of the Theory of Numbers. Vol. 1. Carnegie Institution of Washington.</cite></li>

<li><cite id="CITEREFDubinskyWellerMcDonaldBrown2005" class="citation journal cs1">Dubinsky, Ed; Weller, Kirk; McDonald, Michael; Brown, Anne (2005). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_educational-studies-in-mathematics_2005_60_2/page/253">"Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2"</a>. <a href="Educational_Studies_in_Mathematics" title="Educational Studies in Mathematics">Educational Studies in Mathematics</a>. 60 (2): 253–266. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10649-005-0473-0">10.1007/s10649-005-0473-0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:45937062">45937062</a>.</cite></li>

<li><cite id="CITEREFEarlNicholson2021" class="citation book cs1">Earl, Richard; Nicholson, James (2021). The Concise Oxford Dictionary of Mathematics (6th ed.). Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-192-58405-2</bdi>.</cite></li>

<li><cite id="CITEREFEdwardsWard2004" class="citation journal cs1">Edwards, Barbara; Ward, Michael (May 2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110722153906/http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf">"Surprises from mathematics education research: Student (mis)use of mathematical definitions"</a> (PDF). <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 111 (5): 411–425. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.453.7466">10.1.1.453.7466</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F4145268">10.2307/4145268</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/4145268">4145268</a>. Archived from <a rel="nofollow" class="external text" href="http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf">the original</a> (PDF) on 22 July 2011. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFEllenberg2014" class="citation web cs1"><a href="Jordan_Ellenberg" title="Jordan Ellenberg">Ellenberg, Jordan</a> (6 June 2014). <a rel="nofollow" class="external text" href="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">"Does 0.999... = 1? And Are Divergent Series the Invention of the Devil?"</a>. <a href="Slate_(magazine)" title="Slate (magazine)">Slate</a>. Archived from <a rel="nofollow" class="external text" href="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">the original</a> on 8 August 2023.</cite></li>

<li><cite id="CITEREFEly2010" class="citation journal cs1">Ely, Robert (2010). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_journal-for-research-in-mathematics-education_2010-03_41_2/page/117">"Nonstandard student conceptions about infinitesimals"</a>. <a href="Journal_for_Research_in_Mathematics_Education" title="Journal for Research in Mathematics Education">Journal for Research in Mathematics Education</a>. 41 (2): 117–146. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5951%2Fjresematheduc.41.2.0117">10.5951/jresematheduc.41.2.0117</a>.</cite>

<dl><dd>This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for 0.999... falling short of 1 by an infinitesimal 0.000...1.</dd></dl></li>

<li><cite id="CITEREFEnderton1977" class="citation book cs1"><a href="Herbert_Enderton" title="Herbert Enderton">Enderton, Herbert B.</a> (1977). <a rel="nofollow" class="external text" href="https://archive.org/details/elementsofsetthe0000ende">Elements of Set Theory</a>. Elsevier. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-238440-0</bdi>.</cite>

<dl><dd>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)</dd></dl></li>

<li><cite id="CITEREFEuler1822" class="citation book cs1"><a href="Leonhard_Euler" title="Leonhard Euler">Euler, Leonhard</a> (1822) [1770]. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsalgebra00lagrgoog">Elements of Algebra</a>. John Hewlett and Francis Horner, English translators (3rd English ed.). Orme Longman. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsalgebra00lagrgoog/page/n205">170</a>. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFFinneyWeirGiordano2001" class="citation book cs1">Finney, Ross L.; Weir, Maurice D.; Giordano, Frank R. (2001). Thomas' Calculus: Early Transcendentals (10th ed.). New York: Addison-Wesley.</cite></li>

<li><cite id="CITEREFFjelstad1995" class="citation journal cs1">Fjelstad, Paul (January 1995). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/07468342.1995.11973659">"The Repeating Integer Paradox"</a>. <a href="The_College_Mathematics_Journal" title="The College Mathematics Journal">The College Mathematics Journal</a>. 26 (1): 11–15. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2687285">10.2307/2687285</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2687285">2687285</a>.</cite></li>

<li><cite id="CITEREFGardiner2003" class="citation book cs1"><a href="Tony_Gardiner" title="Tony Gardiner">Gardiner, Anthony</a> (2003) [1982]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NiDCYJ8vrGQC">Understanding Infinity: The Mathematics of Infinite Processes</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-42538-2</bdi>.</cite></li>

<li><cite id="CITEREFGinsberg2004" class="citation journal cs1">Ginsberg, Brian (2004). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/07468342.2004.11922047">"Midy's (nearly) secret theorem – an extension after 165 years"</a>. <a href="The_College_Mathematics_Journal" title="The College Mathematics Journal">The College Mathematics Journal</a>. 35 (1): 26–30. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F07468342.2004.11922047">10.1080/07468342.2004.11922047</a>.</cite></li>

<li><cite id="CITEREFGoodwyn1802" class="citation journal cs1">Goodwyn, H. (1802). <a rel="nofollow" class="external text" href="https://archive.org/details/journalofnatural01lond/page/314/mode/2up">"Curious properties of prime Numbers, taken as the Divisors of unity. By a Correspondent"</a>. <a href="Journal_of_Natural_Philosophy%2C_Chemistry%2C_and_the_Arts" title="Journal of Natural Philosophy, Chemistry, and the Arts">Journal of Natural Philosophy, Chemistry, and the Arts</a>. New Series. 1: 314–316.</cite></li>

<li><cite id="CITEREFGowers2001" class="citation web cs1"><a href="William_Timothy_Gowers" class="mw-redirect" title="William Timothy Gowers">Gowers, Timothy</a> (2001). <a rel="nofollow" class="external text" href="https://www.dpmms.cam.ac.uk/~wtg10/decimals.html">"What is so wrong with thinking of real numbers as infinite decimals?"</a>. Department of Pure Mathematics and Mathematical Statistics. Cambridge University. Retrieved 3 October 2024.</cite></li>

<li><cite id="CITEREFGowers2002" class="citation book cs1"><a href="William_Timothy_Gowers" class="mw-redirect" title="William Timothy Gowers">Gowers, Timothy</a> (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DBxSM7TIq48C">Mathematics: A Very Short Introduction</a>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-285361-5</bdi>.</cite></li>

<li><cite id="CITEREFGrattan-Guinness1970" class="citation book cs1"><a href="Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, Ivor</a> (1970). <a rel="nofollow" class="external text" href="https://archive.org/details/developmentoffo00ivor">The Development of the Foundations of Mathematical Analysis from Euler to Riemann</a>. MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-262-07034-8</bdi>.</cite></li>

<li><cite id="CITEREFGriffithsHilton1970" class="citation book cs1">Griffiths, H. B.; <a href="Peter_Hilton" title="Peter Hilton">Hilton, P. J.</a> (1970). <a rel="nofollow" class="external text" href="https://archive.org/details/comprehensivetex0000grif">A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation</a>. London: Van Nostrand Reinhold. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-442-02863-3</bdi>. <a href="LCC_(identifier)" class="mw-redirect" title="LCC (identifier)">LCC</a> <a rel="nofollow" class="external text" href="https://catalog.loc.gov/vwebv/search?searchCode=CALL%2B&searchArg=QA37.2+G75&searchType=1&recCount=25">QA37.2 G75</a>.</cite>

<dl><dd>This book grew out of a course for <a href="Birmingham" title="Birmingham">Birmingham</a>-area <a href="Grammar_school" title="Grammar school">grammar school</a> mathematics teachers. The course was intended to convey a university-level perspective on <a href="Mathematics_education" title="Mathematics education">school mathematics</a>, 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 <a href="Ideal_theory" title="Ideal theory">ideal theory</a>, which is not reproduced here. (pp. vii, xiv)</dd></dl></li>

<li><cite id="CITEREFKatzKatz2010a" class="citation journal cs1">Katz, Karin Usadi; <a href="Mikhail_Katz" title="Mikhail Katz">Katz, Mikhail G.</a> (2010a). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/">"When is .999... less than 1?"</a>. The Montana Mathematics Enthusiast. 7 (1): 3–30. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1007.3018">1007.3018</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010arXiv1007.3018U">2010arXiv1007.3018U</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.54870%2F1551-3440.1381">10.54870/1551-3440.1381</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11544878">11544878</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.umt.edu/TMME/vol7no1/">the original</a> on 20 July 2011. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFKatzKatz2010b" class="citation journal cs1">Katz, Karin Usadi; <a href="Mikhail_Katz" title="Mikhail Katz">Katz, Mikhail G.</a> (2010b). "Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era". <a href="Educational_Studies_in_Mathematics" title="Educational Studies in Mathematics">Educational Studies in Mathematics</a>. 74 (3): 259. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1003.1501">1003.1501</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010arXiv1003.1501K">2010arXiv1003.1501K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10649-010-9239-4">10.1007/s10649-010-9239-4</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115168622">115168622</a>.</cite></li>

<li><cite id="CITEREFKempner1936" class="citation journal cs1"><a href="Aubrey_Kempner" class="mw-redirect" title="Aubrey Kempner">Kempner, Aubrey J.</a> (December 1936). "Anormal Systems of Numeration". <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 43 (10): 610–617. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2300532">10.2307/2300532</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2300532">2300532</a>.</cite></li>

<li><cite id="CITEREFKomornikLoreti1998" class="citation journal cs1">Komornik, Vilmos; <a href="Paola_Loreti" title="Paola Loreti">Loreti, Paola</a> (1998). "Unique Developments in Non-Integer Bases". <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 105 (7): 636–639. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2589246">10.2307/2589246</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2589246">2589246</a>.</cite></li>

<li><cite id="CITEREFLi2011" class="citation arxiv cs1">Li, Liangpan (March 2011). "A new approach to the real numbers". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.1800">1101.1800</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CA">math.CA</a>].</cite></li>

<li><cite id="CITEREFLeavitt1967" class="citation journal cs1">Leavitt, William G. (1967). <a rel="nofollow" class="external text" href="http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1047&context=mathfacpub">"A Theorem on Repeating Decimals"</a>. <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 74 (6): 669–673. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2314251">10.2307/2314251</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2314251">2314251</a>.</cite></li>

<li><cite id="CITEREFLeavitt1984" class="citation journal cs1">Leavitt, William G. (September 1984). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_college-mathematics-journal_1984-09_15_4/page/299">"Repeating Decimals"</a>. <a href="The_College_Mathematics_Journal" title="The College Mathematics Journal">The College Mathematics Journal</a>. 15 (4): 299–308. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2686394">10.2307/2686394</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2686394">2686394</a>.</cite></li>

<li><cite id="CITEREFLewittes2006" class="citation arxiv cs1">Lewittes, Joseph (2006). "Midy's Theorem for Periodic Decimals". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.NT/0605182">math.NT/0605182</a>.</cite></li>

<li><cite id="CITEREFLightstone1972" class="citation journal cs1"><a href="A._H._Lightstone" title="A. H. Lightstone">Lightstone, Albert H.</a> (March 1972). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1972-03_79_3/page/242">"Infinitesimals"</a>. <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 79 (3): 242–251. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2316619">10.2307/2316619</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2316619">2316619</a>.</cite></li>

<li><cite id="CITEREFMankiewicz2000" class="citation book cs1">Mankiewicz, Richard (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/storyofmathemati0000mank_k4e8">The Story of Mathematics</a>. Cassell. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-304-35473-3</bdi>.</cite>

<dl><dd>Mankiewicz seeks to represent "the history of mathematics in an accessible style" by combining visual and qualitative aspects of mathematics, mathematicians' writings, and historical sketches. (p. 8)</dd></dl></li>

<li><cite id="CITEREFMascariMiola1988" class="citation book cs1">Mascari, Gianfranco; Miola, Alfonso (1988). "On the integration of numeric and algebraic computations". In Beth, Thomas; Clausen, Michael (eds.). Applicable Algebra, Error-Correcting Codes, Combinatorics and Computer Algebra. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0039172">10.1007/BFb0039172</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-39133-3</bdi>.</cite></li>

<li><cite id="CITEREFMaor1987" class="citation book cs1"><a href="Eli_Maor" title="Eli Maor">Maor, Eli</a> (1987). <a rel="nofollow" class="external text" href="https://archive.org/details/toinfinitybeyond0000maor">To Infinity and Beyond: A Cultural History of the Infinite</a>. Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-3325-6</bdi>.</cite>

<dl><dd>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)</dd></dl></li>

<li><cite id="CITEREFMazur2005" class="citation book cs1"><a href="Joseph_Mazur" title="Joseph Mazur">Mazur, Joseph</a> (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7KSxBwAAQBAJ">Euclid in the Rainforest: Discovering Universal Truths in Logic and Math</a>. New York: Pi Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-147994-4</bdi>.</cite></li>

<li><cite id="CITEREFMeierSmith2017" class="citation book cs1">Meier, John; Smith, Derek (2017). Exploring Mathematics: An Engaging Introduction to Proof. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-12898-9</bdi>.</cite></li>

<li><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="James_Munkres" title="James Munkres">Munkres, James R.</a> (2000) [1975]. Topology (2e ed.). Prentice-Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-181629-9</bdi>.</cite>

<dl><dd>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. 30)</dd></dl></li>

<li><cite class="citation journal cs1">Navarro, Maria Angeles; Carreras, Pedro Pérez (2010). <a rel="nofollow" class="external text" href="http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf">"A Socratic methodological proposal for the study of the equality 0.999...=1"</a> (PDF). The Teaching of Mathematics. 13 (1): 17–34. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFO'ConnorRobertson2005" class="citation cs2">O'Connor, John J.; <a href="Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a> (October 2005), <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/{{{id}}}.html">"The real numbers: Stevin to Hilbert"</a>, <a href="MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite></li>

<li><cite id="CITEREFPedrick1994" class="citation book cs1">Pedrick, George (1994). <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinana0000pedr">A First Course in Analysis</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94108-0</bdi>.</cite></li>

<li><cite id="CITEREFPeressiniPeressini2007" class="citation book cs1">Peressini, Anthony; Peressini, Dominic (2007). "Philosophy of Mathematics and Mathematics Education". In van Kerkhove, Bart; <a href="Jean_Paul_Van_Bendegem" title="Jean Paul Van Bendegem">van Bendegem, Jean Paul</a> (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/perspectivesonma0000unse_f3x1">Perspectives on Mathematical Practices</a>. Logic, Epistemology, and the Unity of Science. Vol. 5. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-5033-6</bdi>.</cite></li>

<li><cite id="CITEREFPetkovšek1990" class="citation journal cs1"><a href="Marko_Petkov%C5%A1ek" title="Marko Petkovšek">Petkovšek, Marko</a> (May 1990). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1990-05_97_5/page/408">"Ambiguous Numbers are Dense"</a>. <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 97 (5): 408–411. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2324393">10.2307/2324393</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2324393">2324393</a>.</cite></li>

<li><cite id="CITEREFPintoTall2001" class="citation book cs1">Pinto, Márcia; <a href="David_Tall" title="David Tall">Tall, David O.</a> (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090530043127/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf">PME25: Following students' development in a traditional university analysis course</a> (PDF). pp. v4: 57–64. Archived from <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf">the original</a> (PDF) on 30 May 2009. Retrieved 3 May 2009.</cite></li>

<li><cite id="CITEREFPropp2015" class="citation web cs1"><a href="Jim_Propp" title="Jim Propp">Propp, James</a> (17 September 2015). <a rel="nofollow" class="external text" href="https://mathenchant.wordpress.com/2015/09/17/the-one-about-999/">"The One About .999..."</a> Mathematical Enchantments. Retrieved 24 May 2024.</cite></li>

<li><cite id="CITEREFPropp2023" class="citation web cs1"><a href="Jim_Propp" title="Jim Propp">Propp, James</a> (17 January 2023). <a rel="nofollow" class="external text" href="https://mathenchant.wordpress.com/2023/01/17/denominators-and-doppelgangers/">"Denominators and Doppelgängers"</a>. Mathematical Enchantments. Retrieved 16 April 2024.</cite></li>

<li><cite id="CITEREFProtterMorrey1991" class="citation book cs1"><a href="Murray_H._Protter" title="Murray H. Protter">Protter, Murray H.</a>; <a href="Charles_B._Morrey" class="mw-redirect" title="Charles B. Morrey">Morrey, Charles B. Jr.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/firstcourseinrea0000prot">A First Course in Real Analysis</a> (2e ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-97437-8</bdi>.</cite>

<dl><dd>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. 56–64) Decimal expansions appear in Appendix 3, "Expansions of real numbers in any base". (pp. 503–507)</dd></dl></li>

<li><cite id="CITEREFPugh2002" class="citation book cs1"><a href="Charles_C._Pugh" title="Charles C. Pugh">Pugh, Charles Chapman</a> (2002). <a rel="nofollow" class="external text" href="https://archive.org/details/realmathematical00char">Real Mathematical Analysis</a>. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95297-0</bdi>.</cite>

<dl><dd>While assuming familiarity with the rational numbers, Pugh introduces <a href="Dedekind_cut" title="Dedekind cut">Dedekind cuts</a> as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.</dd></dl></li>

<li><cite id="CITEREFRentelnDundes2005" class="citation journal cs1">Renteln, Paul; <a href="Alan_Dundes" title="Alan Dundes">Dundes, Alan</a> (January 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090225124532/http://www.ams.org/notices/200501/fea-dundes.pdf">"Foolproof: A Sampling of Mathematical Folk Humor"</a> (PDF). <a href="Notices_of_the_AMS" class="mw-redirect" title="Notices of the AMS">Notices of the AMS</a>. 52 (1): 24–34. Archived from <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200501/fea-dundes.pdf">the original</a> (PDF) on 25 February 2009. Retrieved 3 May 2009.</cite></li>

<li><cite id="CITEREFRichman1999" class="citation journal cs1">Richman, Fred (December 1999). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-magazine_1999-12_72_5/page/396">"Is 0.999... = 1?"</a>. <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>. 72 (5): 396–400. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690798">10.2307/2690798</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690798">2690798</a>.</cite> Free HTML preprint: <cite class="citation web cs1">Richman, Fred (June 1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060902040839/http://www.math.fau.edu/Richman/HTML/999.htm">"Is 0.999... = 1?"</a>. Archived from <a rel="nofollow" class="external text" href="http://www.math.fau.edu/Richman/HTML/999.htm">the original</a> on 2 September 2006. Retrieved 23 August 2006.</cite> Note: the journal article contains material and wording not found in the preprint.</li>

<li><cite id="CITEREFRobinson1996" class="citation book cs1"><a href="Abraham_Robinson" title="Abraham Robinson">Robinson, Abraham</a> (1996). Non-standard Analysis (Revised ed.). Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-04490-3</bdi>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/j.ctt1cx3vb6">j.ctt1cx3vb6</a>.</cite></li>

<li><cite id="CITEREFRosenlicht1985" class="citation book cs1"><a href="Maxwell_Rosenlicht" title="Maxwell Rosenlicht">Rosenlicht, Maxwell</a> (1985). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoan0000rose">Introduction to Analysis</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-65038-8</bdi>.</cite> This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.</li>

<li><cite id="CITEREFRudin1976" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976) [1953]. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofmath00rudi">Principles of Mathematical Analysis</a> (3e ed.). McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054235-8</bdi>.</cite>

<dl><dd>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)</dd></dl></li>

<li><cite id="CITEREFShrader-Frechette1978" class="citation journal cs1">Shrader-Frechette, Maurice (March 1978). "Complementary Rational Numbers". <a href="Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>. 51 (2): 90–98. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690144">10.2307/2690144</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690144">2690144</a>.</cite></li>

<li><cite id="CITEREFSmithHarrington1895" class="citation book cs1">Smith, Charles; Harrington, Charles (1895). <a rel="nofollow" class="external text" href="https://archive.org/details/arithmeticforsc02smitgoog">Arithmetic for Schools</a>. Macmillan. p. <a rel="nofollow" class="external text" href="https://archive.org/details/arithmeticforsc02smitgoog/page/n129">115</a>. Retrieved 4 July 2011.</cite></li>

<li><cite id="CITEREFSohrab2003" class="citation book cs1">Sohrab, Houshang (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=gBPI_oYZoMMC">Basic Real Analysis</a>. Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4211-2</bdi>.</cite></li>

<li><cite id="CITEREFStewart2009" class="citation book cs1"><a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/professorstewart0000stew_i8r6">Professor Stewart's Hoard of Mathematical Treasures</a>. Profile Books. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84668-292-6</bdi>.</cite></li>

<li><cite id="CITEREFStewartTall2015" class="citation book cs1"><a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a>; <a href="David_Tall" title="David Tall">Tall, David</a> (2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xSN-BwAAQBAJ&pg=PA38">The Foundations of Mathematics</a> (2nd ed.). Oxford University Press. pp. 38–39. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-870644-1</bdi>.</cite></li>

<li><cite id="CITEREFStewart1999" class="citation book cs1"><a href="James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (1999). <a rel="nofollow" class="external text" href="https://archive.org/details/calculusearlytra00stew">Calculus: Early transcendentals</a> (4e ed.). Brooks/Cole. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-36298-0</bdi>.</cite>

<dl><dd>This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.</dd></dl></li>

<li><cite id="CITEREFStillwell1994" class="citation cs2"><a href="John_Stillwell" title="John Stillwell">Stillwell, John</a> (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jWgPAQAAMAAJ">Elements of Algebra: Geometry, Numbers, Equations</a>, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783540942900</bdi></cite></li>

<li><cite id="CITEREFTallSchwarzenberger1978" class="citation journal cs1"><a href="David_Tall" title="David Tall">Tall, David</a>; <a href="Rolph_Ludwig_Edward_Schwarzenberger" title="Rolph Ludwig Edward Schwarzenberger">Schwarzenberger, R. L. E.</a> (1978). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090530043040/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf">"Conflicts in the Learning of Real Numbers and Limits"</a> (PDF). Mathematics Teaching. 82: 44–49. Archived from <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf">the original</a> (PDF) on 30 May 2009. Retrieved 3 May 2009.</cite></li>

<li><cite id="CITEREFTall1976" class="citation journal cs1"><a href="David_Tall" title="David Tall">Tall, David O.</a> (1976). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090326052901/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf">"Conflicts and Catastrophes in the Learning of Mathematics"</a> (PDF). Mathematical Education for Teaching. 2 (4): 2–18. Archived from <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf">the original</a> (PDF) on 26 March 2009. Retrieved 3 May 2009.</cite></li>

<li><cite id="CITEREFTall2000" class="citation journal cs1">Tall, David (2000). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090530043111/http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf">"Cognitive Development in Advanced Mathematics Using Technology"</a> (PDF). Mathematics Education Research Journal. 12 (3): 210–230. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000MEdRJ..12..196T">2000MEdRJ..12..196T</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03217085">10.1007/BF03217085</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:143438975">143438975</a>. Archived from <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf">the original</a> (PDF) on 30 May 2009. Retrieved 3 May 2009.</cite></li>

<li><cite id="CITEREFTao2012" class="citation book cs1"><a href="Terence_Tao" title="Terence Tao">Tao, Terence</a> (2012). <a rel="nofollow" class="external text" href="https://terrytao.files.wordpress.com/2011/03/higher-book.pdf">Higher order Fourier analysis</a> (PDF). American Mathematical Society.</cite></li>

<li><cite id="CITEREFTao2003" class="citation web cs1"><a href="Terence_Tao" title="Terence Tao">Tao, Terence</a> (2003). <a rel="nofollow" class="external text" href="https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf">"Math 131AH: Week 1"</a> (PDF). Honors Analysis. UCLA Mathematics. Retrieved 23 May 2024.</cite></li>

<li><cite id="CITEREFWallace2003" class="citation book cs1"><a href="David_Foster_Wallace" title="David Foster Wallace">Wallace, David Foster</a> (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/everythingmore00davi">Everything and more: a compact history of infinity</a>. Norton. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-393-00338-3</bdi>.</cite></li>

<li><cite id="CITEREFEisenmann2008" class="citation journal cs1">Eisenmann, Petr (2008). <a rel="nofollow" class="external text" href="http://teaching.math.rs/vol/tm1114.pdf">"Why is not true that 0.999 . . . < 1?"</a> (PDF). The Teaching of Mathematics. 11 (1): 38.</cite></li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3></div>

<ul><li><cite class="citation journal cs1">Beswick, Kim (2004). <a rel="nofollow" class="external text" href="https://eric.ed.gov/?id=EJ717818">"Why Does 0.999... = 1?: A Perennial Question and Number Sense"</a>. Australian Mathematics Teacher. 60 (4): 7–9.</cite></li>

<li><cite class="citation journal cs1">Burkov, S. E. (1987). "One-dimensional model of the quasicrystalline alloy". <a href="Journal_of_Statistical_Physics" title="Journal of Statistical Physics">Journal of Statistical Physics</a>. 47 (3/4): 409–438. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987JSP....47..409B">1987JSP....47..409B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01007518">10.1007/BF01007518</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120281766">120281766</a>.</cite></li>

<li><cite class="citation journal cs1">Burn, Bob (March 1997). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematical-gazette_1997-03_81_490/page/109">"81.15 A Case of Conflict"</a>. <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 81 (490): 109–112. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3618786">10.2307/3618786</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3618786">3618786</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:187823601">187823601</a>.</cite></li>

<li><cite class="citation journal cs1">Calvert, J. B.; Tuttle, E. R.; Martin, Michael S.; Warren, Peter (February 1981). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_history-teacher_1981-02_14_2/page/167">"The Age of Newton: An Intensive Interdisciplinary Course"</a>. The History Teacher. 14 (2): 167–190. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F493261">10.2307/493261</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/493261">493261</a>.</cite></li>

<li><cite class="citation journal cs1">Choi, Younggi; Do, Jonghoon (November 2005). "Equality Involved in 0.999... and (-8)1/3". <a href="For_the_Learning_of_Mathematics" title="For the Learning of Mathematics">For the Learning of Mathematics</a>. 25 (3): 13–15, 36. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40248503">40248503</a>.</cite></li>

<li><cite class="citation journal cs1">Choong, K. Y.; Daykin, D. E.; Rathbone, C. R. (April 1971). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-of-computation_1971-04_25_114/page/387">"Rational Approximations to π"</a>. Mathematics of Computation. 25 (114): 387–392. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2004936">10.2307/2004936</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2004936">2004936</a>.</cite></li>

<li><cite class="citation book cs1">Edwards, B. (1997). "An undergraduate student's understanding and use of mathematical definitions in real analysis". In Dossey, J.; Swafford, J.O.; Parmentier, M.; Dossey, A.E. (eds.). Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Vol. 1. Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education. pp. 17–22.</cite></li>

<li><cite class="citation journal cs1">Eisenmann, Petr (2008). <a rel="nofollow" class="external text" href="http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf">"Why is it not true that 0.999... < 1?"</a> (PDF). The Teaching of Mathematics. 11 (1): 35–40. Retrieved 4 July 2011.</cite></li>

<li><cite class="citation journal cs1">Ferrini-Mundy, J.; Graham, K. (1994). Kaput, J.; Dubinsky, E. (eds.). "Research in calculus learning: Understanding of limits, derivatives and integrals". MAA Notes: Research Issues in Undergraduate Mathematics Learning. 33: 31–45.</cite></li>

<li><cite class="citation journal cs1">Gardiner, Tony (June 1985). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematical-gazette_1985-06_69_448/page/77">"Infinite processes in elementary mathematics: How much should we tell the children?"</a>. <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 69 (448): 77–87. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3616921">10.2307/3616921</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3616921">3616921</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125222118">125222118</a>.</cite></li>

<li><cite class="citation journal cs1">Monaghan, John (December 1988). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematical-gazette_1988-12_72_462/page/276">"Real Mathematics: One Aspect of the Future of A-Level"</a>. <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 72 (462): 276–281. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3619940">10.2307/3619940</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3619940">3619940</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:125825964">125825964</a>.</cite></li>

<li><cite class="citation book cs1"><a href="Rafael_E._N%C3%BA%C3%B1ez" title="Rafael E. Núñez">Núñez, Rafael</a> (2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110718014351/http://www.cogsci.ucsd.edu/~nunez/web/publications.html">"Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics"</a>. 18 Unconventional Essays on the Nature of Mathematics. Springer. pp. 160–181. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-25717-4</bdi>. Archived from <a rel="nofollow" class="external text" href="http://www.cogsci.ucsd.edu/~nunez/web/publications.html">the original</a> on 18 July 2011. Retrieved 4 July 2011.</cite></li>

<li><cite class="citation journal cs1">Przenioslo, Malgorzata (March 2004). "Images of the limit of function formed in the course of mathematical studies at the university". <a href="Educational_Studies_in_Mathematics" title="Educational Studies in Mathematics">Educational Studies in Mathematics</a>. 55 (1–3): 103–132. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AEDUC.0000017667.70982.05">10.1023/B:EDUC.0000017667.70982.05</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120453706">120453706</a>.</cite></li>

<li><cite class="citation journal cs1">Sandefur, James T. (February 1996). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_american-mathematical-monthly_1996-02_103_2/page/107">"Using Self-Similarity to Find Length, Area, and Dimension"</a>. <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>. 103 (2): 107–120. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2975103">10.2307/2975103</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2975103">2975103</a>.</cite></li>

<li><cite class="citation journal cs1"><a href="Anna_Sierpi%C5%84ska" title="Anna Sierpińska">Sierpińska, Anna</a> (November 1987). "Humanities students and epistemological obstacles related to limits". <a href="Educational_Studies_in_Mathematics" title="Educational Studies in Mathematics">Educational Studies in Mathematics</a>. 18 (4): 371–396. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00240986">10.1007/BF00240986</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3482354">3482354</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144880659">144880659</a>.</cite></li>

<li><cite class="citation journal cs1"><a href="Michael_Starbird" title="Michael Starbird">Starbird, Michael</a>; Starbird, Thomas (March 1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1992-1086343-5">"Required Redundancy in the Representation of Reals"</a>. Proceedings of the American Mathematical Society. 114 (3): 769–774. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9939-1992-1086343-5">10.1090/S0002-9939-1992-1086343-5</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2159403">2159403</a>.</cite></li>

<li><cite class="citation journal cs1">Szydlik, Jennifer Earles (May 2000). "Mathematical Beliefs and Conceptual Understanding of the Limit of a Function". <a href="Journal_for_Research_in_Mathematics_Education" title="Journal for Research in Mathematics Education">Journal for Research in Mathematics Education</a>. 31 (3): 258–276. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F749807">10.2307/749807</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/749807">749807</a>.</cite></li>

<li><cite class="citation journal cs1"><a href="David_Tall" title="David Tall">Tall, David O.</a> (2009). "Dynamic mathematics and the blending of knowledge structures in the calculus". ZDM Mathematics Education. 41 (4): 481–492. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11858-009-0192-6">10.1007/s11858-009-0192-6</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14289039">14289039</a>.</cite></li>

<li><cite class="citation journal cs1"><a href="David_Tall" title="David Tall">Tall, David O.</a> (May 1981). <a rel="nofollow" class="external text" href="https://archive.org/details/sim_mathematics-in-school_1981-05_10_3/page/30">"Intuitions of infinity"</a>. Mathematics in School. 10 (3): 30–33. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30214290">30214290</a>.</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:0.999%E2%80%A6" class="extiw external" title="commons:Category:0.999…">0.999…</a>.</div></div>

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<li><a rel="nofollow" class="external text" href="http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html">Proof of the equality based on arithmetic</a> from Math Central</li>

<li><a href="David_Tall" title="David Tall">David Tall</a>'s <a rel="nofollow" class="external text" href="http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html">research on mathematics cognition</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304055237/https://www.dpmms.cam.ac.uk/~wtg10/decimals.html">What is so wrong with thinking of real numbers as infinite decimals?</a></li>

<li><a rel="nofollow" class="external text" href="http://at.metamath.org/mpegif/0.999....html">Theorem 0.999...</a> on <a href="Metamath" title="Metamath">Metamath</a></li></ul>

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<li><a href="Absolute_difference" title="Absolute difference">Absolute difference</a></li>

<li><a href="Cantor_set" title="Cantor set">Cantor set</a></li>

<li><a href="Cantor%E2%80%93Dedekind_axiom" title="Cantor–Dedekind axiom">Cantor–Dedekind axiom</a></li>

<li><a href="Completeness_of_the_real_numbers" title="Completeness of the real numbers">Completeness</a></li>

<li><a href="Construction_of_the_real_numbers" title="Construction of the real numbers">Construction</a></li>

<li><a href="Decidability_of_first-order_theories_of_the_real_numbers" title="Decidability of first-order theories of the real numbers">Decidability of first-order theories</a></li>

<li><a href="Extended_real_number_line" title="Extended real number line">Extended real number line</a></li>

<li><a href="Gregory_number" title="Gregory number">Gregory number</a></li>

<li><a href="Irrational_number" title="Irrational number">Irrational number</a></li>

<li><a href="Normal_number" title="Normal number">Normal number</a></li>

<li><a href="Rational_number" title="Rational number">Rational number</a></li>

<li><a href="Rational_zeta_series" title="Rational zeta series">Rational zeta series</a></li>

<li><a href="Real_coordinate_space" title="Real coordinate space">Real coordinate space</a></li>

<li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski axiomatization</a></li>

<li><a href="Vitali_set" title="Vitali set">Vitali set</a></li></ul>

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