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[[File:999 Perspective.svg|300px|thumb]]
'''0.999...''', also written as 0.{{overline|9}} or 0.{{overset|.|9}}, and read as "0 point 9 repeating", is one of the ways the number [[1 (number)|1]] can be written. Even though it does not have any ones in it, it is still equal in value to 1.
== About ==
0.{{overline|9}} is a [[repeating decimal]], which means the nines after the decimal point keep going forever. No matter how far you look, there will still be more nines.

It is hard for many people to understand why 0.{{overline|9}} is the same as 1. There are many [[mathematical proof|proofs]] that show why they are the same number, but many of these proofs are very complex.<ref name="Katz">{{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |doi=10.54870/1551-3440.1381 |s2cid=11544878 |url=http://www.math.umt.edu/TMME/vol7no1/ |access-date=2010-10-17 |archive-date=2011-07-20 |archive-url=https://web.archive.org/web/20110720095125/http://www.math.umt.edu/TMME/vol7no1/ |url-status=dead }}</ref>
== Examples ==
One simple way of showing that 0.{{overline|9}} and 1 are the same thing is to [[Division (mathematics)|divide]] them both by the number [[3 (number)|3]]. When 0.{{overline|9}} is divided by 3, the answer is 0.{{overline|3}}. When 1 is divided by 3, the answer is also 0.{{overline|3}}.

:<math>\frac{0.999\ldots}{3}=0.333\ldots</math>
:<math>\frac13=0.333\ldots</math>

If 0.{{overline|9}} and 1 divided by 3 are the same, then they must be the same even without the division. 

Another way of thinking about it is if <math>1/3=0.333\ldots</math> and <math>2/3=0.666\ldots</math>, then <math>3/3=0.999\ldots</math> therefore, as <math>3/3=1</math>, 0.999... must also equal 1. There are many other ways of showing this.<ref name="Katz" />

Another simple way of proving that 0.{{overline|9}} = 1 is by accepting the simple fact that if two numbers are different, there must be at least one number between them. For example, a number between 1 and 0.9 is 0.95, and a number between 0.99 and 1 is 0.995. Since 0.{{overline|9}} has an [[Infinity|infinite]] number of 9s, there cannot be another number after the "last" 9, because there is no last 9. Therefore, there is no number between 0.{{overline|9}} and 1. Therefore, they are equal.

A more mathematical proof can be found with the following [[equation]]:

<math>x=0.999\ldots</math>

<math>10x=9.999\ldots</math>

<math>10x-x=9.999\ldots-x</math>

<math>9x=9.999\ldots-0.999\ldots=9</math>

<math>x=9/9</math>

<math>x=1</math>

<math>0.999\ldots=1</math>
== In popular culture ==
As the [[Internet]] developed, arguments about 0.{{overline|9}} are often on [[newsgroup]]s and [[message board]]s. Even newsgroups and message boards that do not have much to do with math argue about this. In the newsgroup sci.math, arguing about 0.{{overline|9}} is a "popular sport".<ref name=sci.mathfaq/> It is also one of the questions in its [[FAQ]].<ref name=sci.mathfaq>As observed by Richman (p. 396). {{cite web |url=http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0.999eq1/ |author=Hans de Vreught | year=1994 | title=sci.math FAQ: Why is 0.9999... = 1? |access-date=2006-06-29}}</ref>

== 0.999... Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:base4 333.svg|A depiction of the unit interval (black line), with the numbers labeled in base 4: 0, 0.1, 0.2, 0.3, 0.33, 0.333., and 0.3333… = 1/3. Created by Melchoir

File:999 Intervals C.svg|An illustration of the membership of 1 in two sequences of nested intervals corresponding to the base-3 equality 1 = 1.000... = 0.222...

File:Cantor base 3.svg|Positions of {{sfrac|1|4}}, {{sfrac|2|3}}, and 1 in the [[Cantor set]]

File:4adic 333.svg|The 4-adic integers (black points), including the sequence {{nowrap|1=(3, 33, 333, ...)}} converging to −1. The 10-adic analogue is {{nowrap|1=...999 = −1}}.
</gallery>
== Related pages ==
* [[Limit (mathematics)]]
== References ==
{{reflist}}
== Other websites ==
* [http://mathforum.org/dr.math/faq/faq.0.9999.html Why does 0.9999... = 1?]
* [http://www.newton.dep.anl.gov/askasci/math99/math99167.htm Ask A Scientist: Repeating Decimals] {{Webarchive|url=https://web.archive.org/web/20150226225642/http://newton.dep.anl.gov/askasci/math99/math99167.htm |date=2015-02-26 }}
* [https://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

[[Category:Mathematics]]


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