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The repeating decimal continues with an infinite number of nines.

In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, for example as 0.9..., or in a variety of other variants such as 0.9, 0.(9), or {{{1}}}) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

Every nonzero, terminating decimal (with infinitely many trailing 0s) has an equal twin representation with infinitely many trailing 9s (for example, 8.32 and 8.31999...). The terminating decimal representation is usually preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases (with a given base's largest digit) or in any similar representation of the real numbers.

The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.


0.999... sections
Intro  [[0.999...?section=Algebraic_proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}|Algebraic proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}]]  [[0.999...?section=Analytic_proofs{{safesubst:#invoke:anchor|main}}|Analytic proofs{{safesubst:#invoke:anchor|main}}]]  [[0.999...?section=Proofs_from_the_construction_of_the_real_numbers{{safesubst:#invoke:anchor|main}}|Proofs from the construction of the real numbers{{safesubst:#invoke:anchor|main}}]]  Generalizations  Applications  Skepticism in education  In popular culture  [[0.999...?section=In_alternative_number_systems{{safesubst:#invoke:anchor|main}}|In alternative number systems{{safesubst:#invoke:anchor|main}}]]  Related questions  See also  Notes  References  Further reading  External links  

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{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }}

The repeating decimal continues with an infinite number of nines.

In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, for example as 0.9..., or in a variety of other variants such as 0.9, 0.(9), or {{{1}}}) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

Every nonzero, terminating decimal (with infinitely many trailing 0s) has an equal twin representation with infinitely many trailing 9s (for example, 8.32 and 8.31999...). The terminating decimal representation is usually preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases (with a given base's largest digit) or in any similar representation of the real numbers.

The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it. Such skepticism is common enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.


0.999... sections
Intro  [[0.999...?section=Algebraic_proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}|Algebraic proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}]]  [[0.999...?section=Analytic_proofs{{safesubst:#invoke:anchor|main}}|Analytic proofs{{safesubst:#invoke:anchor|main}}]]  [[0.999...?section=Proofs_from_the_construction_of_the_real_numbers{{safesubst:#invoke:anchor|main}}|Proofs from the construction of the real numbers{{safesubst:#invoke:anchor|main}}]]  Generalizations  Applications  Skepticism in education  In popular culture  [[0.999...?section=In_alternative_number_systems{{safesubst:#invoke:anchor|main}}|In alternative number systems{{safesubst:#invoke:anchor|main}}]]  Related questions  See also  Notes  References  Further reading  External links  

PREVIOUS: IntroNEXT: [[0.999...?section=Algebraic_proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}|Algebraic proofs{{safesubst:#invoke:anchor|main}}{{safesubst:#invoke:anchor|main}}]]
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