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French Quarter (New Orleans)

2017-01-29T18:49:42Z

Loadmaster: /* Landmarks and attractions */ Added image :File:New-Orleans-Street-Artist-1988-205.jpg

{{About|the French Quarter in New Orleans|other cities with "French Quarter" areas}}
{{Redirect|Vieux Carré|the play|Vieux Carré (play)|the cocktail|Carousel Piano Bar & Lounge}}

{{Geobox
| Settlement

| name = French Quarter
| native_name =
| other_name = Vieux Carré
| other_name1 =
| category = [[New Orleans neighborhoods|New Orleans Neighborhood]]

| image = ChrisLitherlandFrenchQuarter2011.jpg
| image_caption = The French Quarter, looking north with Mississippi River to the right

| flag =
| flag_size =
| symbol =
| symbol_size =

| etymology_type =
| etymology =
| nickname =
| motto =

| country = United States
| state = Louisiana
| region_type = City
| region = [[New Orleans]]
| district_type = Planning District
| district = District 1, French Quarter/CBD

| area_imperial = 0.66
| area_land_imperial = 0.49
| area_water_imperial = 0.17
| area_water_percentage = auto
| area_percentage_round = 2
| area_round = 1
| location =
| lat_d = 29
| lat_m = 57
| lat_s = 31
| lat_NS = N
| long_d = 90
| long_m = 03
| long_s = 54
| long_EW = W
| elevation_imperial = 3
| elevation_round = 1

| population_as_of = 2010
| population = 3888
| population_density_imperial = 7935

| established_type = Mayor-Council<ref>[http://www.cityofno.com/pg-1-9.aspx City Of New Orleans : City Charter]</ref>
| established =
| mayor = [[Mitch Landrieu]]

| timezone = [[North American Central Time Zone|CST]]
| utc_offset = -6
| timezone_DST = [[North American Central Time Zone|CDT]]
| utc_offset_DST = -5
| postal_code = 70116 - 70130
| postal_code_type = ZIP Codes
| area_code = [[Area code 504|504]]
| area_code_type =
| code2_type =
| code2 =

| free_type =
| free =
| free1_type =
| free1 =

| map =
| map_size =
| map_caption = Location of the French Quarter and [[New Orleans Central Business District|Central Business District]] in New Orleans
| map_locator =
| map_locator_x =
| map_locator_y =

| website =
}}

The '''French Quarter''', also known as the '''Vieux Carré''', is the oldest [[New Orleans neighborhoods|neighborhood]] in the city of [[New Orleans]]. After New Orleans (''La Nouvelle-Orléans'' in French) was founded in 1718 by [[Jean-Baptiste Le Moyne de Bienville]], the city developed around the ''Vieux Carré'' ("Old Square" in English), a central square. The district is more commonly called the French Quarter today, or simply "The Quarter," related to changes in the city with American immigration after the Louisiana Purchase.<ref>[http://www.inetours.com/New_Orleans/French_Quarter_History.html New Orleans French Quarter History, Architecture and Pictures]</ref> Most of the extant historic buildings were constructed either in the late 18th century, during the city's period of Spanish rule, or were built during the first half of the 19th century, after U.S. annexation and statehood.

The district as a whole has been designated as a [[National Historic Landmark]], with numerous contributing buildings that are separately deemed significant. It is a prime tourist destination in the city, as well as attracting local residents. Because of its distance from areas where the levee was breached during [[Hurricane Katrina]] in 2005 as well as the strength and height of the nearest Mississippi River Levees in contrast to other levees along the canals and lakefront,<ref>http://www.nola.com/katrina/graphics/flashflood.swf</ref> it suffered relatively light damage from floodwater as compared to other areas of the city and the greater region.

==Geography==
The French Quarter is located at {{Coord|29|57|31|N|90|03|54|W|type:city}}<ref name="GR1">{{cite web|url=http://www.census.gov/geo/www/gazetteer/gazette.html|publisher=[[United States Census Bureau]]|accessdate=2011-04-23|date=2011-02-12|title=US Gazetteer files: 2010, 2000, and 1990}}</ref> and has an elevation of {{convert|1|ft|1}}.<ref name="GR3">{{cite web|url=http://geonames.usgs.gov|accessdate=2008-01-31|title=US Board on Geographic Names|publisher=[[United States Geological Survey]]|date=2007-10-25}}</ref> According to the [[United States Census Bureau]], the district has a total area of {{convert|0.66|sqmi|1}}. {{convert|0.49|sqmi|1}} of which is land and {{convert|0.17|sqmi|1}} (25.76%) of which is water.

===Boundaries===
The most common definition of the French Quarter includes all the land stretching along the [[Mississippi River]] from [[Canal Street, New Orleans|Canal Street]] to [[Esplanade Avenue, New Orleans|Esplanade Avenue]] (13 blocks) and inland to [[Rampart Street|North Rampart Street]] (seven to nine blocks). It equals an area of 78 square blocks. Some definitions, such as city zoning laws, exclude the properties facing Canal Street, which had already been redeveloped by the time architectural preservation was considered, and the section between Decatur Street and the river, much of which had long served industrial and warehousing functions.

Any alteration to structures in the remaining blocks is subject to review by the Vieux Carré Commission, which determines whether the proposal is appropriate for the historic character of the district. Its boundaries as defined by the City Planning Commission are: Esplanade Avenue to the north, the [[Mississippi River]] to the east, Canal Street, [[Decatur Street (New Orleans)|Decatur Street]] and Iberville Street to the south and the [[Basin Street]], St. Louis Street and North Rampart Street to the west.<ref>{{cite web|url=http://gnocdc.org/orleans/1/48/index.html|title=French Quarter Neighborhood|author=Greater New Orleans Community Data Center|accessdate=2008-06-21}}</ref>

The National Historic Landmark district is stated to be 85 square blocks.<ref name="nhlsum"/><ref name="nrhpinv2"/> The Quarter is subdistrict of the French Quarter/CBD Area.

===Adjacent neighborhoods===
{{further|Neighborhoods in New Orleans}}
* [[Faubourg Marigny]] (east)
* [[Mississippi River]] (south)
* [[New Orleans Central Business District|Central Business District]] (west)
* [[Iberville Projects|Iberville]] (north)
* [[Tremé]] (north)

==Demographics==
As of the [[census]] of 2000, there were 4,176 people, 2,908 households, and 509 families residing in the neighborhood.<ref name="French Quarter Neighborhood">{{cite web|title=French Quarter Neighborhood|url=http://gnocdc.org/NeighborhoodData/1/FrenchQuarter/index.html|publisher=Greater New Orleans Community Data Center|accessdate=5 January 2012}}</ref> The [[population density]] was 8,523 /mi² (3,212 /km²).

As of the [[census]] of 2010, there were 3,813 people, 2,635 households, and 549 families residing in the neighborhood.<ref name="French Quarter Neighborhood"/>

==History==
{{Infobox NRHP
| name = Vieux Carre Historic District
| nrhp_type = nhld
| image = UpperChartersNOLA.jpg
| caption = French Quarter: Upper Chartres street looking towards [[Jackson Square, New Orleans, Louisiana|Jackson Square]] and the spires of [[St. Louis Cathedral, New Orleans|St. Louis Cathedral]].
| location = [[New Orleans]], [[Louisiana]]
| locmapin = Louisiana
| area =
| built = 1734
| architect = Multiple
| architecture =
| designated_nrhp_type = December 21, 1965<ref name="nhlsum">{{cite web | url=http://tps.cr.nps.gov/nhl/detail.cfm?ResourceId=258&ResourceType=District | title=Vieux Carre Historic District | accessdate=2008-01-31 | work=National Historic Landmark summary listing | publisher=National Park Service}}</ref>
| added = October 15, 1966<ref name="nris">{{NRISref|2007a}}</ref>
| refnum = 66000377
| nocat = yes
}}

Many of the buildings date from before 1803, when New Orleans was acquired by the [[United States]] in the Louisiana Purchase, although some 19th-century and early 20th-century buildings were added to the area. Since the 1920s, the historic buildings have been protected by law and cannot be demolished; and any renovations or new construction in the neighborhood must be carried out in accordance with city regulations, preserving the historic architectural style.

Most of the French Quarter's architecture was built during the late 18th century and the period of [[Louisiana (New Spain)|Spanish rule]] over the city, which is reflected in the architecture of the neighborhood. The [[Great New Orleans Fire (1788)]] and another great fire in 1794 destroyed most of the Quarter's old French colonial architecture, leaving the colony's new Spanish overlords to rebuild it according to more modern tastes. Their strict new fire codes mandated that all structures be physically adjacent and close to the curb to create a firewall. The old French peaked roofs were replaced with flat tiled ones, and wooden siding was banned in favor of fire-resistant [[stucco]], painted in the pastel hues fashionable at the time. As a result, colorful walls and roofs and elaborately decorated ironwork balconies and galleries, from the late 18th and the early 19th centuries, abound. (In southeast Louisiana, a distinction is made between "balconies", which are self-supporting and attached to the side of the building, and "galleries," which are supported from the ground by poles or columns.)

When [[English language|Anglophone]] Americans began to move in after the [[Louisiana Purchase]], they mostly built on available land upriver, across modern-day [[Canal Street, New Orleans|Canal Street]]. This thoroughfare became the meeting place of two cultures, one [[Francophone]] [[Louisiana Creole people|Creole]] and the other Anglophone American. (Local landowners had retained architect and surveyor [[Barthelemy Lafon]] to subdivide their property to create an American suburb). The [[Central reservation|median]] of the wide boulevard became a place where the two contentious cultures could meet and do business in both French and English. As such, it became known as the "neutral ground", and this name is used for medians in the New Orleans area.

Even before the Civil War, French Creoles had become a minority in the French Quarter.<ref name="ellis1">{{cite book | last=Ellis | first=Scott S. | title=Madame Vieux Carré: the French Quarter in the Twentieth Century | publisher=University of Mississippi | year=2010 | isbn=978-1-60473-358-7 | page=7}}</ref> In the late 19th century the Quarter became a less fashionable part of town, and many immigrants from southern [[Italy]] and [[Ireland]] settled there. In 1905, the Italian consul estimated that one-third to one-half of the Quarter’s population were Italian-born or second generation Italian-Americans. Irish immigrants also settled heavily in the Esplanade area, which was called the "Irish Channel".<ref name="ellis2">''Madame Vieux Carré'', p. 11</ref>
[[File:New-orleans10.jpg|275px|thumb|left|Elaborate ironwork galleries on the corner of Royal and St. Peter streets]]

[[File:French Quarter-874.JPG|thumb|right|300px|The balconies and windows are an example of late 18th-century [[Spanish architecture#Spanish Colonial architecture|Spanish architecture]] built after the Great Fires of [[Great New Orleans Fire (1788)|1788]] and [[Great New Orleans Fire (1794)|1794]].]]
In 1917, the closure of [[Storyville, New Orleans|Storyville]] sent much of the vice formerly concentrated therein back into the French Quarter, which "for most of the remaining French Creole families . . was the last straw, and they began to move uptown."<ref name="ellis3">''Madame Vieux Carré'', p. 20-21</ref> This, combined with the loss of the [[French Opera House]] two years later, provided a bookend to the era of French Creole culture in the Quarter.<ref name="ellis4">''Madame Vieux Carré'', p. 21</ref> Many of the remaining French Creoles moved to the University area.<ref>''New Orleans 1900 to 1920'' by Mary Lou Widmer. Pelican Publishing: 2007. ISBN 1-58980-401-5 pg 23</ref>

In the early 20th century, the Quarter's cheap rents and air of decay attracted a [[Bohemianism|bohemian]] artistic community, a trend which became pronounced in the 1920s. Many of these new inhabitants were active in the first preservation efforts in the Quarter, which began around that time.<ref name="ellis5">''Madame Vieux Carré'', p. 24</ref> As a result, the Vieux Carré Commission (VCC) was established in 1925. Although initially only an advisory body, a 1936 referendum to amend the Louisiana constitution afforded it a measure of regulatory power. It began to exercise more power in the 1940s to preserve and protect the district.<ref name="ellis6">''Madame Vieux Carré'', p. 43</ref>

Meanwhile, [[World War II]] brought thousands of servicemen and war workers to New Orleans as well as to the surrounding region's military bases and shipyards. Many of these sojourners paid visits to the Vieux Carré. Although nightlife and vice had already begun to coalesce on [[Bourbon Street]] in the two decades following the closure of Storyville, the war produced a larger, more permanent presence of exotic, risqué, and often raucous entertainment on what became the city's most famous strip. Years of repeated crackdowns on vice in Bourbon Street clubs, which took on new urgency under Mayor [[deLesseps Story Morrison]], reached a crescendo with District Attorney [[Jim Garrison]]'s raids in 1962, but Bourbon Street's clubs were soon back in business.<ref>Souther, J. Mark. "New Orleans on Parade: Tourism and the Transformation of the Crescent City." Baton Rouge: Louisiana State University Press, 2013. pp. 41-50.</ref>

The plan to construct an elevated Riverfront Expressway between the [[Mississippi River]] levee and the French Quarter consumed the attention of Vieux Carré preservationists through much of the 1960s. On December 21, 1965, the "Vieux Carre Historic District" was designated a [[National Historic Landmark]].<ref name="nhlsum"/><ref name="nrhpinv2">{{Cite journal | url={{NHLS url|id=66000377}} | title=National Register of Historic Places Inventory-Nomination: Vieux Carré Historic District | date=February 1975 | author=Patricia Heintzelman | publisher=National Park Service | postscript=}}</ref> After waging a decade-long battle against the [[Vieux Carré Riverfront Expressway]] that utilized the newly passed [[National Historic Preservation Act of 1966]], preservationists and their allies forced the issue into federal court, eventually producing the cancellation of the freeway plan in 1969.<ref>Souther, "New Orleans on Parade," pp. 66-71</ref>

The victory was important for the preservation of the French Quarter, but it was hardly the only challenge. Throughout the 1960s, new hotels opened regularly, often replacing large sections of the French Quarter. The VCC approved these structures as long as their designers adhered to prevailing exterior styles. Detractors, fearing that the Vieux Carré's charm might be compromised by the introduction of too many new inns, lobbied successfully for passage in 1969 of a municipal ordinance that forbade new hotels within the district's boundaries. However, the ordinance failed to stop the proliferation of [[timeshare]] condominiums and clandestine [[bed and breakfast]] inns throughout the French Quarter or high-rise hotels just outside its boundaries.<ref>Souther, "New Orleans on Parade," pp. 54-63, 203</ref> In the 1980s, many long-term residents were driven away by rising rents, as property values rose dramatically with expectations of windfalls from the planned [[1984 World's Fair]] site nearby.

More of the neighborhood was developed to support [[tourism]], which is important to the city's economy. But, the French Quarter still combines residential, hotels, guest houses, bars, restaurants and tourist-oriented commercial properties.

===Effect of Hurricane Katrina===
{{main|Effects of Hurricane Katrina on New Orleans}}
As with other parts of the city developed before the late 19th century, and on higher land predating New Orleans' levee systems, the French Quarter remained substantially dry following Hurricane Katrina. Its elevation is five feet (1.5 m) above sea level.<ref>{{webarchive |url=https://web.archive.org/web/20050911082124/http://today.reuters.co.uk/news/newsArticle.aspx?type=topNews&storyID=2005-08-31T161230Z_01_ROB586049_RTRUKOC_0_UK-WEATHER-KATRINA.xml |date=September 11, 2005 |title=Officials rescue Katrina's survivors amid 'chaos' }} By Rick Wilking, Wed Aug 31, 2005, retrieved on 2009-11-27.</ref> Some streets had minor flooding, and several buildings suffered significant wind damage. Most of the major landmarks suffered only minor damage.<ref>[http://www.frenchquarter.com/index.php FrenchQuarter.com: The Essential Guide to New Orleans' Oldest Neighborhood]</ref> In addition, the Quarter largely escaped the looting and violence that occurred after the storm; nearly all of the antique shops and art galleries in the French Quarter, for example, were untouched.<ref>{{cite news| url=http://www.latimes.com/news/nationworld/nation/la-na-rumors27sep27,0,5492806,full.story?coll=la-home-headlines | work=Los Angeles Times | first1=Susannah | last1=Rosenblatt | first2=James | last2=Rainey | title=Katrina Takes a Toll on Truth, News Accuracy – Los Angeles Times | date=September 27, 2005}}</ref>

Mayor [[Ray Nagin]] officially reopened the French Quarter on September 26, 2005 (almost a month after the storm), for business owners to inspect their property and clean up. Within a few weeks, a large selection of French Quarter businesses had reopened. The [[Historic New Orleans Collection]]'s Williams Research Center Annex was the first new construction completed in the French Quarter after Hurricane Katrina.<ref>[http://www.hnoc.org/visit/buildings_williams_add.html THNOC - WRC Addition]</ref>

==Landmarks and attractions==

===Jackson Square===
{{Main|Jackson Square (New Orleans)}}
[[File:Jackson Square New Orleans.JPG|thumb|right|300px|[[Andrew Jackson|Jackson]] [[equestrian statue]] and [[St. Louis Cathedral (New Orleans)|St. Louis Cathedral]] – flanked by [[the Cabildo]] and [[the Presbytere]]]]
'''Jackson Square''' (formerly ''Place d'Armes'' or ''Plaza de Armas'', in French and Spanish, respectively), originally designed by architect and landscaper Louis H. Pilié (officially credited only with the iron fence), is a public, gated park the size of a city block, located at the front of the French Quarter (GPS {{Coord|29.95748|-90.06310|display=inline}}). In the mid-19th century, the square was named after President (formerly General, of [[Battle of New Orleans]] acclaim) [[Andrew Jackson]].

In 1856, city leaders purchased an [[Equestrian sculpture|equestrian statue]] of Jackson from the sculptor Clark Mills. The statue was placed at the center of the square, which was converted to a park from its previous use as a [[military parade]] ground and execution site. (Convicted criminals were sometimes hanged in the square. After the [[1811 German Coast Uprising|slave insurrection of 1811]] during the [[Territory of Orleans|U.S. territorial period]], some of the insurgents were sentenced to death here in [[Orleans Parish]] under a justice system which had not yet been converted to American ideals, and their severed heads were displayed here.)

The square originally overlooked the [[Mississippi River]] across Decatur Street; however, the view was blocked in the 19th century when larger levees were built along the river. The riverfront was long devoted to shipping-related activities at the heart of the [[port]]. The administration of Mayor [[Moon Landrieu]] put in a scenic boardwalk across from Jackson Square; it is known as the "Moon Walk" in his honor. At the end of the 1980s, old wharves and warehouses were demolished to create [[Woldenberg Park]], extending the riverfront promenade up to [[Canal Street, New Orleans|Canal Street]].

On the opposite side of the square from the River are three 18th‑century historic buildings, which were the city's heart in the colonial era. The center of the three is [[St. Louis Cathedral (New Orleans)|St. Louis Cathedral]]. The [[cathedral]] was designated a [[minor basilica]] by [[Pope Paul VI]]. To its left is [[the Cabildo]], the old city hall, now a museum, where the final transfer papers for the [[Louisiana Purchase]] were signed. To the Cathedral's right is [[the Presbytère]], built to match the Cabildo. The Presbytère, originally planned to house the city's [[Roman Catholic]] priests and authorities, was adapted as a courthouse at the start of the 19th century after the Louisiana Purchase, when civilian government was elevated over church authority. In the 20th century it was adapted as a museum.

On each side of the square are the [[Pontalba Buildings]], matching red-brick, one-block-long, four‑story buildings constructed between 1849 and 1851. The ground floors house shops and restaurants; the upper floors are apartments. The buildings were planned as row townhouses; they were not converted to rental apartments until the 1930s (during the [[Great Depression]]).

The buildings were designed and constructed by [[Baroness Micaela Almonester Pontalba]], daughter of Don [[Andres Almonaster y Rojas]], a prominent Spanish philanthropist in [[Louisiana Creole people|Creole]] New Orleans. Micaela Almonaster was born in Louisiana in 1795. Her father died three years later, and she became sole heiress to his fortune and his New Orleans land holdings.

Directly across from Jackson Square is the [[Jax Brewery]] building, the original home of a local [[beer]]. After the company ceased to operate independently, the building was converted for use by retail businesses, including restaurants and specialty shops. In recent years, some retail space has been converted into riverfront [[condominium]]s. Behind the Jax Brewery lies the [[Toulouse Street]] Wharf, the regular pier for the excursion steamboat, ''[[Natchez (boat)#Current Natchez|Natchez]]''.

From the 1920s through the 1980s, Jackson Square became known for attracting [[Painting|painter]]s, young art students, and [[caricaturist]]s.{{Citation needed|date=March 2011}} In the 1990s, the artists were joined by [[tarot card readers]], [[mime artist|mime]]s, fortune tellers, and other street performers.

Live music has been a regular feature of the entire Quarter, including the Square, for more than a century. Formal concerts are also held, although more rarely. Street musicians play for tips.

Diagonally across the square from the Cabildo is [[Café du Monde]], open 24 hours a day except for [[Christmas Day]] and during [[tropical cyclone|hurricane]]s. The historic open-air [[coffeehouse|cafe]] is known for its [[café au lait]] (literally [[coffee]] served with milk) — coffee blended with [[chicory]] — and [[beignet]]s, made and served there continuously since the [[American Civil War|Civil War]] period (1862). It is a custom for anyone visiting for the first time to blow the [[powdered sugar]] off a beignet and make a wish.
<gallery>
File:French Quarter01 New Orleans.JPG|Carriage in French Quarter
File:BourbonStreet.jpg|The ''Rue Bourbon'', or [[Bourbon Street]], was named for the former ruling dynasty of France.
File:French Quarter02 New Orleans.JPG|French Quarter
File:French Quarter03 New Orleans.JPG|French Quarter
File:Musicians Performing.jpg|Musicians performing in the French Quarter
File:New-Orleans-Street-Artist-1988-205.jpg|[[Street artist]] (1988)
Flea Market.jpg|New Orleans Flea Market
File:4 story in French Quarter.JPG|Four-storied building with balconies
File:20150307-LouisianaSupremeCourt.jpg|The [[Louisiana Supreme Court]] Building
</gallery>

===Bourbon Street===
{{Main|Bourbon Street}}
The most well known of the French Quarter streets, Bourbon Street, or Rue Bourbon, is known for its drinking establishments. Most of the bars frequented by tourists are new but the Quarter also has a number of notable bars with interesting histories. The [[Old Absinthe House]] has kept its name even though [[absinthe]] was banned in the U.S. from 1915 to 2007 because it was believed to have toxic qualities.

[[Pat O'Brien's Bar]] is known both for inventing the red [[Hurricane (cocktail)|"hurricane"]] cocktail and for having the first [[Dueling pianos|dueling piano]] bar. Pat O'Brien's is located at 718 St. Peter Street.<ref>{{cite web
| last = Lind
| first = Angus
| url = http://blog.nola.com/anguslind/2008/11/pat_os_turns_75_this_week.html
| title = Home of the 'Hurricane' Pat O'Brien's turns 75 this week
| publisher = nola.com
| accessdate = 2009-06-19
}}</ref>

[[Lafitte's Blacksmith Shop]] is a tavern located on the corner of Bourbon and St. Philip streets. Built sometime before 1772, it is one of the older surviving structures in New Orleans. It is also the oldest bar in all of America that still operates as a bar. According to legend, the structure was once a business owned by the [[Jean Lafitte|Lafitte brothers]], perhaps as a "front" for their smuggling operations at [[Barataria Bay]].

The [[Napoleon House]] bar and restaurant is in the former home of mayor [[Nicholas Girod]]. It was named for an unrealized plot to rescue [[Napoleon]] from his exile in [[Saint Helena]] and bring him to New Orleans.

The original [[Johnny White's]] bar is a favorite of [[motorcycle|motorcycle biker]]s. In 2005 an offshoot called Johnny White's Hole in the Wall, along with [[Molly's at the Market]], drew national media attention as the only businesses in the city to stay open throughout Hurricane Katrina and the weeks after the storm.{{cn|date=April 2016}}

[[Spirits on Bourbon]] was featured on the season three of ''Bar Rescue.'' It has become a staple of Bourbon Street, with its light-up skull cup and Resurrection drink.

The [[Bourbon Pub]] and Oz, both located at the intersection of Bourbon and St. Ann Streets, are the two largest [[homosexual|gay]] clubs in New Orleans. [[Café Lafitte in Exile]], located at the intersection of Bourbon and Dumaine, is the oldest continuously running [[gay bar]] in the [[United States]]. These and other gay establishments sponsor the raucous [[Southern Decadence]] Festival during [[Labor Day]] weekend. This festival is often referred to as New Orleans' Gay Mardi Gras. St. Ann Street is often called "the Lavender Line" or "the Velvet Line" in reference to its being on the edge of the French Quarter's predominately gay district. While gay residents live throughout the French Quarter, that portion northeast of St. Ann Street is generally considered to be the gay district.{{cn|date=April 2016}}

New Orleans and its French Quarter are one of a few places in the [[United States]] where possession and consumption of [[alcoholic beverage|alcohol]] in [[United States open container laws|open containers]] is allowed on the street.<ref>[http://secure.cityofno.com/SystemModules/PrintPage.aspx?portal=2&load=~/PortalModules/ViewPressRelease.ascx&itemid=509 City of New Orleans memo]</ref>

===Restaurants===
The neighborhood contains many restaurants, ranging from formal to casual, patronized by both visitors and locals. Some are well-known landmarks, such as [[Antoine's]] and [[Tujague's]], which have been in business since the 19th century. [[Arnaud's]], [[Galatoire's]], [[Broussard's]], and [[Brennan's]] are also venerable.

Less historic—but also well-known—French Quarter restaurants include those run by chefs [[Paul Prudhomme]] ("K-Paul's"), [[Emeril Lagasse]] ("NOLA"), and [[John Besh]]. Port of Call on [[Esplanade Avenue, New Orleans|Esplanade Avenue]] has been in business for more than 30 years, and is recognized for its popular "Monsoon" drink (their answer to the "Hurricane" at [[Pat O'Brien's Bar]]) as well as for its food.

The Gumbo Shop is another traditional eatery in the Quarter and where casual dress is acceptable. For a take-out lunch, [[Central Grocery]] on [[Decatur Street (New Orleans)|Decatur Street]] is the home of the original [[muffaletta]] [[Italians in New Orleans|Italian]] [[sandwich]].

===Hotels===
{{see also|Canal Street, New Orleans#Hotels}}
Accommodations in the French Quarter range from large international chain hotels, to [[bed and breakfast]]s, to time-share condominiums and small guest houses with only one or two rooms.

The Audubon Cottages are a collection of seven luxuriously-appointed [[Creole cottage]]s, two of which were utilized by [[John James Audubon]] in the early 19th century when he worked in New Orleans for a short time.
The [[Hotel St. Pierre]] is a small hotel also consisting of historic French Quarter houses, with a courtyard patio.

The French Quarter is well known for its traditional-style hotels, such as the Bourbon Orleans, [[Hotel Monteleone]] (family-owned), Royal Sonesta, the Astor, and the [[Omni Royal Orleans]]. These hotels offer prime locations, beautiful views, and/or historic atmosphere.

==See also==
{{Portal|New Orleans}}
* [[Buildings and architecture of New Orleans]]
* [[French Market]]
* [[French Quarter Festival]], early April
* [[Jackson Square (New Orleans)|Jackson Square]]
* [[Louisiana Creole cuisine]]
* [[Satchmo SummerFest]], early August
* [[List of National Historic Landmarks in Louisiana]]
* [[National Register of Historic Places listings in Orleans Parish, Louisiana]]

==References==
{{Reflist|2}}

==External links==
{{Commons category|French Quarter}}
{{Wikivoyage|New_Orleans/French_Quarter|French Quarter}}
* {{webarchive |url=https://web.archive.org/web/*/http://www.new-orleans.la.us/cnoweb/VCC/index.html |date=* |title=Vieux Carré Commission }} (VCC) (Archive) - City of New Orleans
*[http://penelope.uchicago.edu/Thayer/E/Gazetteer/Places/America/United_States/Louisiana/New_Orleans/_Texts/Iron_Lace*.html Harriet Joor: ''The City of Iron Lace'']
*[http://tps.cr.nps.gov/nhl/detail.cfm?ResourceId=258&ResourceType=District National Historic Landmarks Program: Vieux Carré Historic District]
*[http://www.nps.gov/history/NR/twhp/wwwlps/lessons/20vieux/20vieux.htm ''Vieux Carré:A Creole Neighborhood in New Orleans,'' a National Park Service Teaching with Historic Places (TwHP) lesson plan]
*[http://btr360.com/2015/05/28/a-kora-african-bass-harp-plays-in-jackson-square-new-olreans/ A Travel Description: At Jackson Square in the French Quarter]

{{New Orleans District 1}}
{{Registered Historic Places}}
{{New Orleans}}
{{Ethnic enclaves}}

[[Category:French Quarter| ]]
[[Category:Downtown New Orleans]]
[[Category:Neighborhoods in New Orleans]]
[[Category:Louisiana populated places on the Mississippi River]]
[[Category:National Historic Landmarks in Louisiana]]
[[Category:Busking venues]]
[[Category:Villages in Louisiana]]
[[Category:Tourist attractions in New Orleans]]
[[Category:Historic districts on the National Register of Historic Places in Louisiana]]
[[Category:National Register of Historic Places in New Orleans]]

David A. Clarke

2015-09-01T17:21:30Z

Loadmaster: /* Support for self-defense and Second Amendment rights */ Links

{{BLP primary sources|date=April 2014}}
{{Infobox officeholder
|name = David A. Clarke, Jr.
|image = Sheriff Clarke.png
|caption = Sheriff Clarke at the NPS Graduation Ceremony in 2013
|office = 64th [[List of sheriffs of Milwaukee|Sheriff of Milwaukee]]
|term_start = March 2002
|term_end =
|predecessor = [[Leverett F. Baldwin]]
|successor =
|birth_date = {{birth year and age|1956}}
|birth_place = [[Milwaukee, Wisconsin|Milwaukee]], [[Wisconsin]]
|spouse = Julie Clarke
|alma_mater = [[Concordia University Wisconsin]]
|website = {{URL|http://www.thepeoplessheriff.com}}
}}

'''Sheriff David A. Clarke Jr.''' (born 1956) is the 64th Sheriff of [[Milwaukee County]]. In 2002, Clarke was appointed to a vacancy by Governor [[Scott McCallum]], and later elected that same year to his first four-year term. He was re-elected in November 2006, 2010, and 2014, and is currently serving his fourth full term.

== Early life, education, and early career ==
Clarke was born in the City of Milwaukee, attending [[Marquette University High School]]

He went on to earn a degree in Criminal Justice Management from [[Concordia University Wisconsin]], graduating summa cum laude. In 2003, Concordia University named him Wisconsin Alumnus of the Year. His postgraduate work includes graduating from the FBI National Academy and the National Executive Institute in Quantico, Virginia; completing the Program for Senior Executives in State and Local Government at Harvard University’s John F. Kennedy School of Government; Driving Government Performance: Leadership Strategies that Produce Results at Harvard University’s John F. Kennedy School of Government; and studying with Police Chief William Bratton and L.A. County Sheriff Lee Baca.<ref>{{cite web|title=SHERIFF DAVID CLARKE BIO, MILWAUKEE, A GREAT AMERICAN HERO|url=http://www.ministers-best-friend.com/CHRISTIPEDIA-TM--SHERIFF-DAVID-CLARKE-BIO--MILWAUKEE--A-GREAT-AMERICAN-HERO.html|website=http://www.ministers-best-friend.com|accessdate=5 August 2014}}</ref>

His career in law enforcement began in 1978 at the [[Milwaukee Police Department]] (MPD). After 11 years as a patrol officer, Clarke was promoted to Detective, making the Homicide Division less than 1 year later. In 1992, Clarke was again promoted to Lieutenant of Detectives. The next step was becoming Captain of Police for the MPD in 1996. In 1999, Clarke took over the post of Commanding Officer for MPD's Intelligence Division. Clarke then became Milwaukee County Sheriff in 2002, currently holding the same post.<ref name="milwaukee">{{cite web|url=http://county.milwaukee.gov/MeettheSheriff9152.htm|title=Meet the Sheriff|publisher=county.milwaukee.gov|accessdate=2014-03-30}}</ref>

== Sheriff of Milwaukee ==

=== Persona ===
As Sheriff, Clarke has been known for his outspokenness. Clarke frequently appears at public events on horseback wearing a cowboy hat. Among his controversial remarks were his assertions that Milwaukee County Executive Chris Abele had “penis envy” and must have been on heroin when crafting the county budget.<ref name="Lisa Kaiser, 'Is It Time For a New Sheriff in Town?', Shepherd Express (July 23, 2014).">{{cite web|url=http://expressmilwaukee.com/article-permalink-23685.html|title=Is It Time For a New Sheriff in Town?|publisher=Shepherd Express|accessdate=2014-07-30}}</ref> In 2015, at an [[National Rifle Association|NRA]] event, he, according to Miranda Blue of [[Right Wing Watch]], proposed redesigning the [[Great Seal of the United States]] to include a semi-automatic rifle. <ref>{{cite news|last1=Blue|first1=Miranda|title=NRA Speaker Proposes Adding Semi-Automatic Rifle To US Seal|url=http://www.rightwingwatch.org/content/nra-speaker-proposes-adding-semi-automatic-rifle-us-seal|accessdate=12 April 2015|publisher=Right-wing Watch|date=10 April 2015}}</ref>

=== Party affiliation ===
Clarke has been elected three times to the Sheriff's office as a Democrat, despite not belonging to any party. This has spurred criticism from the local Democratic Party.<ref name="Lisa Kaiser, 'Is It Time For a New Sheriff in Town?', Shepherd Express (July 23, 2014)."/>

Clarke explains his choice to run as a Democrat thusly on his website:

"Like me, most people question why the Office of Sheriff is a partisan election. I have never asked a person to vote for me because I run as a Democrat. I ask them to vote for me based on my 35-year commitment to keeping citizens safe. Most voters get it when it comes to public safety. There is no Democrat or Republican way to be a sheriff. The enemy is not the opposing party; the enemy is the criminal."<ref>{{cite news|last1=Bice|first1=Daniel|title=Sheriff David Clarke files for re-election amid talk of other offices|url=http://www.jsonline.com/blogs/news/261361241.html|accessdate=20 August 2014|work=Journal Sentinel|date=31 May 2014}}</ref>

=== Budget cuts and service reductions ===
In response to budget cuts prompted by the Milwaukee County Pension Scandal,<ref>{{cite web|title=Ten Stories That Changed Our Lives: #6 Milwaukee Co. Pension Scandal|url=http://www.620wtmj.com/news/local/79307597.html|website=http://www.620wtmj.com|accessdate=3 August 2014}}</ref> Clarke began eliminating Department units to save money. He eliminated the gun crime unit, drug unit, and witness protection unit. Many of these units duplicate services provided by municipal departments.<ref>{{cite web|title=Reforming Milwaukee County – Response to the Fiscal Crisis A Report by the Greater Milwaukee Committee|url=http://www.gmconline.org/images/stories/PDF_files/gmcmkecntypowerpoint.pdf|website=http://www.gmconline.org|accessdate=3 August 2014}}</ref>

Milwaukee County Executive Chris Abele has consistently cut the Sheriff's budget further. A Milwaukee Journal-Sentinel news story reports that "County Executive Chris Abele's 2014 budget takes direct aim at Sheriff David A. Clarke Jr.'s office, cutting more than $12 million and 69 jobs, shifting park patrols, emergency management, 911 communications and training divisions elsewhere." <ref>{{cite web|title=Abele wants to cut Clarke's budget; sheriff calls exec 'vindictive little man'|url=http://www.jsonline.com/news/abele-wants-5-million-to-end-long-term-care-at-mental-health-complex-b99107268z1-225351372.html|website=http://www.jsonline.com/|accessdate=4 August 2014}}</ref>

=== House of Corrections turnaround ===
In January 2008, before the Milwaukee County House of Corrections was placed under the management of Sheriff Clarke, a National Institute of Corrections audit of the Milwaukee County House of Correction identified 44 areas of concern, including serious security, morale, and management issues, and described the facility, as “dysfunctional.”

In 2009, Sheriff Clarke took over the facility and quickly overcame a $5 million deficit, as well as most of the issues brought up in the audit, including lack of discipline, poor supervision, employee sick use abuse, inmate fights, and excessive and unnecessary overtime use.

Clarke received praise for rapidly correcting the issues.<ref name="Steve Schultze, 'Sheriff cleans House to address audit's critiques', Milwaukee Journal-Sentinel (April 10, 2009)">{{cite web|url=http://www.jsonline.com/news/milwaukee/42833992.html|title=Sheriff cleans House to address audit's critiques|publisher=Milwaukee Journal-Sentinel|accessdate=2014-07-30}}</ref>

=== Support for self-defense and Second Amendment rights ===
In January 2013, Sheriff Clarke was featured on a series of public radio ads that said citizens could no longer rely on the police for timely protection and should arm themselves. Later that month, Clarke appeared on the CNN program ''[[Piers Morgan Live]]'', with Milwaukee Mayor and [[gun control|gun-control]] advocate [[Tom Barrett (politician)|Tom Barrett]], and attacked the notion that citizens could no longer rely on calling [[9-1-1]]. The appearance sent David Clarke into the national spotlight.<ref name="jsonline">{{cite web|url=http://www.jsonline.com/news/milwaukee/clarke-barrett-square-off-over-guns-on-cnn-h98inr1-188955951.html|title=David Clarke, Tom Barrett square off over guns on CNN|publisher=jsonline.com|accessdate=2014-03-30}}</ref>

=== Christian Centurians lawsuit ===
In 2006, Clarke invited members of a Protestant sectarian organization to speak at mandatory roll calls, over the objections of the Deputies' union and members of various faiths. This resulted in a federal lawsuit in the United States District Court for the Eastern District of Wisconsin, which Clarke lost and subsequently appealed to the U.S. Court of Appeals for the Seventh Circuit, who upheld the lower court's ruling in 2009. The sheriff did not seek review in the U.S. Supreme Court.<ref>{{cite web|title=Milwaukee Deputy Sheriffs Association v. Clarke|url=https://www.au.org/our-work/legal/lawsuits/milwaukee-deputy-sheriffs-association-v-clarke|website=https://www.au.org/|accessdate=4 August 2014}}</ref><ref name="Milwaukee Deputy Sheriffs' Association v. Clarke, No. 08-1515">{{cite web|url=http://caselaw.findlaw.com/us-7th-circuit/1499070.html|title=Milwaukee Deputy Sheriffs' Association v. Clarke, No. 08-1515|accessdate=2014-07-30}}</ref>

=== Constitutional Sheriffs Association Award ===
In 2013, Clarke was honored with the Sheriff of the Year Award by the Constitutional Sheriffs and Peace Officers Association. The official statement credited Clarke with, “Demonstrating true leadership and courage...staying true to his oath, true to his badge, and true to the people he has promised to serve and protect.”<ref name="cspoa">{{cite web|url=http://cspoa.org/|title=Constitutional Sheriffs and Peace Officers Association | CSPOA - Constitutional Sheriffs and Peace Officers Association|publisher=cspoa.org|accessdate=2014-03-30}}</ref> Though Clarke lists the award in his autobiography on the Sheriff’s official website,<ref name="Meet the Sheriff">{{cite web|url=http://county.milwaukee.gov/MeettheSheriff9152.htm|title=Meet the Sheriff|accessdate=2014-07-30}}</ref> it has been a source of controversy,<ref name="Lisa Kaiser, 'Is It Time For a New Sheriff in Town?', Shepherd Express (July 23, 2014)."/> because the CSPOA is thought by some to be a fringe group.{{cn|date=July 2015}}

=== Stance on Marijuana ===
In 2015, during a House Judiciary hearing on police reform by Rep. Steve Cohen, Clarke said that he did not believe that marijuana was less destructive to society than methamphetamine or cocaine.<ref>https://www.youtube.com/watch?t=31&v=nEYP5YAPViM</ref>

== Future ==
In January 2014, Sheriff Clarke announced he is considering a run for [[Mayor of Milwaukee]] in 2016.<ref>{{cite web|url=http://fox6now.com/2014/01/31/milwaukee-co-sheriff-david-clarke-considers-2016-run-for-mayor/|title=Milwaukee Co. Sheriff David Clarke considers 2016 run for mayor|work=FOX6Now.com}}</ref>

== Personal ==
David Clarke and his wife live on the northwest side of Milwaukee.

== Electoral history ==

=== Milwaukee County Sheriff Elections (2002-2014) ===

==== 2002 Democratic Primary ====
*'''David Clarke''' - 59%
*Pete Misko - 26%
*Mark Hayes - 15%<ref name="Milw Election Results">{{cite web|title=Election Results|url=http://city.milwaukee.gov/ElectionResults1717.htm#.U-uMf_ldV8E|website=City of Milwaukee|accessdate=13 August 2014}}</ref>

==== 2002 General Election ====
*'''David Clarke''' (D) - 74%
*Ken Bohn (R), 25%<ref name="Milw Election Results" />

==== 2006 Democratic Primary ====
*'''David Clarke''' - 54%
*Vincent Bobot - 46%<ref name="Milw Election Results" />

==== 2006 General Election ====
*'''David Clarke''' (D) - 78%
*Don Holt (R), 21%<ref name="Milw Election Results" />

==== 2010 Democratic Primary ====
*'''David Clarke''' - 53%
*Chris Moews - 47%<ref name="jsonline2">{{cite web|url=http://www.jsonline.com/news/milwaukee/102928759.html|title=Election 2010 | Milwaukee County Sheriff - Clarke defeats Moews in Milwaukee County sheriff's primary|publisher=jsonline.com|accessdate=2014-03-30}}</ref>

==== 2010 General Election ====
*'''David Clarke''' (D) - 74%
*Steven Duckhorn (R) - 25%<ref name="milwaukee2">{{cite web|url=http://county.milwaukee.gov/ElectionResults23729/2010ElectionResults.htm#Sept14|title=2010 Election Results|publisher=county.milwaukee.gov|accessdate=2014-03-30}}</ref>

==== 2014 Democratic Primary ====
*'''David Clarke''' - 52%
*Chris Moews - 48%<ref>{{cite web|title=Wisconsin 2014 fall primary election results|url=http://www.jsonline.com/news/statepolitics/wisconsin-2014-fall-primary-election-results-253583691.html#Milwaukee_Sheriff_Dem_Primary|website=JS Online|accessdate=13 August 2014}}</ref>

==== 2014 General Election ====
*'''David Clarke''' (D) - 79%
*Angela Walker (I) - 21%<ref>{{cite web|title=Wisconsin 2014 fall general election results|url=http://www.jsonline.com/news/statepolitics/wisconsin-2014-fall-general-election-results-272052351.html#Milwaukee_Sheriff|website=http://www.jsonline.com|accessdate=12 December 2014}}</ref>

==References==
{{Reflist}}

== External links ==
*[http://www.thepeoplessheriff.com Campaign website]

{{Persondata
| NAME = Clarke, David A., Jr
| ALTERNATIVE NAMES = David Clarke
| SHORT DESCRIPTION = American sheriff
| DATE OF BIRTH = 1956
| PLACE OF BIRTH = [[Milwaukee]], [[Wisconsin]], [[United States|U.S.]]
| DATE OF DEATH =
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Clarke, David A.}}
[[Category:1956 births]]
[[Category:American gun rights advocates]]
[[Category:Living people]]
[[Category:People from Milwaukee, Wisconsin]]
[[Category:Concordia University Wisconsin alumni]]
[[Category:Wisconsin sheriffs]]
[[Category:African Americans in law enforcement]]

Parkettierung mit Fünfecken

2015-08-18T19:28:59Z

Loadmaster: /* Dual uniform tilings */ Rm quotes, add caps

[[File:15-monohedral_pentagonal_tiling_types.png|thumb|450px|Examples for each of the 15 known monohedral tiling types by convex pentagons, with tiles colored by their [[isohedral figure|''k''-isohedral]] positions.]]
In [[geometry]], a '''pentagonal tiling''' is a [[tessellation|tiling]] of the plane where each individual piece is in the shape of a [[pentagon]].

A [[regular tiling|regular]] [[pentagonal]] [[tiling]] on the [[Euclidean plane]] is impossible because the [[internal angle]] of a [[Pentagon#Regular pentagons|regular pentagon]], 108°, is not a divisor of 360°, the angle measure of a whole [[turn (geometry)|turn]].

Fifteen types of convex pentagons are known to [[prototile|monohedrally]] tile the plane (the most recent type being discovered in 2015), but it is not known whether this list is complete.<ref name="NPR" />

== History ==
{{harvtxt|Reinhardt|1918}} found the five pentagonal tilings that are "tile transitive", meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the [[automorphism group]] [[group action|acts transitively]] on the tiles). {{harvtxt|Kershner|1968}} found three more types of pentagonal tiles, the tilings of which are never tile transitive; he claimed incorrectly that this was the complete list of pentagons that can tile the plane. In 1975 Richard E. James III found a 9th type, after reading about Kershner's results in [[Martin Gardner]]'s "Mathematical Games" column in ''[[Scientific American]]'' magazine of July 1975 (reprinted in {{harvtxt|Gardner|1988}}); James' tiling is monohedral (it only uses one type of tile) but not tile transitive. {{harvtxt|Schattschneider|1978}} described how [[Marjorie Rice]], an amateur mathematician, discovered four new types of [[Tessellation|tessellating]] pentagons in 1976 and 1977. {{harvtxt|Schattschneider|1985}} described a 14th convex pentagon type found by Rolf Stein in 1985. {{harvtxt|Bagina|2011}} showed that there are only 8 edge-to-edge convex types, a result obtained independently by {{harvtxt|Sugimoto|2012}}. [[University of Washington Bothell]] mathematicians Casey Mann, Jennifer McLoud, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a [[computer algorithm]] (paper pending {{As of|2015|August|lc = y}}).<ref>{{cite news |title= Attack on the pentagon results in discovery of new mathematical tile |first=Alex |last=Bellos |newspaper=[[The Guardian]] |date=11 August 2015 |url=http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile}}</ref>

== Non-convex pentagons ==
With pentagons that are not required to be [[convex]], additional types of tiling are possible. An example is the [[sphinx tiling]], an [[aperiodic tiling]] formed by a pentagonal [[rep-tile]] {{harv|Godrèche|1989}}. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a [[parallelogram]] and then tiling the plane by translates of this parallelogram, a pattern that can also be used for certain other shapes of non-convex pentagons.

== Dual uniform tilings ==

There are three different [[isohedral]] pentagonal tilings generated as [[Dual (polyhedron)|duals]] of the [[uniform tiling]]s:

<gallery widths="220">
File:Tiling Dual Semiregular V3-3-3-4-4 Prismatic Pentagonal.svg|[[Prismatic pentagonal tiling]]<hr/>Instance of ''pentagon {{nowrap|type 1}} tiling'' described in 1918<ref name="Reinhardt1918"/>

File:Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg|[[Cairo pentagonal tiling]]<hr/>Instance of ''pentagon {{nowrap|type 4}} tiling'' described in 1918<ref name="Reinhardt1918"/><ref>[http://www.wolframalpha.com/input/?i=pentagon+type+4+tiling&f=PentagonType4Tiling.A_2*pi%2F3&f=PentagonType4Tiling.b_1 Cairo pentagonal tiling generated by a ''pentagon type '''4''' tiling'' query] and [http://www.wolframalpha.com/input/?i=pentagon+type+2+tiling&f=PentagonType2Tiling.A_2*pi%2F3&f=PentagonType2Tiling.B_pi%2F2&f=PentagonType2Tiling.b_1&f=PentagonType2Tiling.e_1 by a ''pentagon type '''2''' tiling'' query] on [http://www.wolframalpha.com wolframalpha.com] (caution: the wolfram definition of ''pentagon type 2 tiling'' does not correspond with ''type 2'' defined by Reinhardt in 1918)</ref>

File:Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg|[[Floret pentagonal tiling]]<hr/>Instance of ''pentagon {{nowrap|type 5}} tiling'' described in 1918 by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]<ref name="Reinhardt1918">{{Citation | last1=Reinhardt | first1=Karl | title=Über die Zerlegung der Ebene in Polygone | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN316479497&DMDID=DMDLOG_0013&LOGID=LOG_0013&PHYSID=PHYS_0083 | publisher=Borna-Leipzig, Druck von Robert Noske, | language=German | series=Dissertation Frankfurt am Main | year=1918 | pages=77-81}} (caution: there is at least one obvious mistake within this paper, i.e. γ+δ angle sum needs to equal π, not 2π for the first two tiling types defined on page 77)</ref>
</gallery>

== Regular pentagonal tilings in non-Euclidean geometry ==
A [[dodecahedron]] can be considered a regular tiling of 12 pentagons on the surface of a [[sphere]], with [[Schläfli symbol]] {5,3}, having 3 pentagons around each vertex.

In the [[hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]], there are tilings of regular pentagons, for instance [[order-4 pentagonal tiling]], with [[Schläfli symbol]] {5,4}, having 4 pentagons around each vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.

{| class=wikitable
![[Sphere]]
!colspan=6|Hyperbolic plane
|-
|[[File:Uniform tiling 532-t0.png|80px]] [[Dodecahedron|{5,3}]]
|[[File:Uniform tiling 54-t0.png|80px]] [[order-4 pentagonal tiling|{5,4}]]
|[[File:Uniform tiling 55-t0.png|80px]] [[order-5 pentagonal tiling|{5,5}]]
|[[File:Uniform tiling 56-t0.png|80px]] [[order-6 pentagonal tiling|{5,6}]]
|[[File:Uniform tiling 57-t0.png|80px]] [[order-7 pentagonal tiling|{5,7}]]
|[[File:Uniform tiling 58-t0.png|80px]] [[order-8 pentagonal tiling|{5,8}]]
|valign=bottom|...{5,∞}
|}

== Irregular hyperbolic plane pentagonal tilings ==

There are an infinite number of dual [[uniform tilings in hyperbolic plane]] with isogonal irregular pentagonal faces. They have [[face configuration]]s as V3.3.''p''.3.''q''.

{| class=wikitable
|+ Order ''p''-''q'' floret pentagonal tiling
|-
![[Order-3 snub heptagonal tiling#Dual tiling|7-3]]
!8-3
!9-3
!...
!5-4
!6-4
!7-4
!...
!5-5
|- valign=bottom align=center
|[[File:Ord7 3 floret penta til.png|120px]] V3.3.3.3.7
|V3.3.3.3.8
|V3.3.3.3.9
|…
|[[File:Order-5-4 floret pentagonal tiling.png|120px]] V3.3.4.3.5
|V3.3.4.3.6
|V3.3.4.3.7
|…
|V3.3.5.3.5
|…
|}

== See also ==
* [[Mosaic]]
* [[Penrose tiling]]

== References ==
{{reflist|refs=
<ref name="NPR">{{cite news |title= With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem |last= Peralta |first= Eyder |publisher= NPR |date= 14 August 2015 |accessdate= 15 August 2015 |url= http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem?utm_source=facebook.com&utm_medium=social&utm_campaign=npr&utm_term=nprnews&utm_content=20150814 |mode= cs2}}</ref>
}}

'''Bibliography'''
{{refbegin|30em}}
* {{Citation | last=Bagina | first=Olga | title=Tiling the plane with congruent equilateral convex pentagons
| url=http://dx.doi.org/10.1016/j.jcta.2003.11.002 | doi=10.1016/j.jcta.2003.11.002 | mr=2046081 | year=2004 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=105 | issue=2 | pages=221–232}}
* {{citation | last=Bagina | first=Olga | script-title=ru:Мозаики из выпуклых пятиугольников |trans_title=Tilings of the plane with convex pentagons | language=Russian | year=2011 | journal=Vestnik |publisher=Kemerovo State University |issn=2078-1768 |volume=4 | issue=48 | pages=63–73 |url=http://www.mathnet.ru/php/seminars.phtml?presentid=4640 |accessdate=29 January 2013}}
* {{citation |last1=Grünbaum |first1=Branko |authorlink=Branko Grünbaum |last2=Shephard |first2=Geoffrey C. |title=Tilings and Patterns |location=New York |publisher=W. H. Freeman and Company |year=1987 |isbn=0-7167-1193-1 |chapter=Tilings by polygons | mr=0857454}}
* {{citation |last=Gardner |first=Martin |authorlink=Martin Gardner |title=Time travel and other mathematical bewilderments |year=1988 |publisher=W. H. Freeman and Company |location=New York |isbn=0-7167-1925-8 |chapter=Tiling with Convex Polygons | mr=0905872 }}
* {{citation
| last = Godrèche | first = C.
| doi = 10.1088/0305-4470/22/24/006
| issue = 24
| journal = Journal of Physics A: Mathematical and General
| mr = 1030678
| pages = L1163–L1166
| title = The sphinx: a limit-periodic tiling of the plane
| volume = 22
| year = 1989}}
* {{Citation | last1=Hirschhorn | first1=M. D. | last2=Hunt | first2=D. C. | title=Equilateral convex pentagons which tile the plane | url=http://dx.doi.org/10.1016/0097-3165(85)90078-0 | doi=10.1016/0097-3165(85)90078-0 | mr=787713 | year=1985 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=39 | issue=1 | pages=1–18}}
* {{Citation | last=Kershner | first=Richard | title=On paving the plane | url=http://www.jstor.org/stable/2314332 | mr=0236822 | year=1968 | journal=[[American Mathematical Monthly]] | issn=0002-9890 | volume=75 | pages=839–844 | doi=10.2307/2314332}}
* {{Citation | last=Reinhardt | first=Karl | title=Über die Zerlegung der Ebene in Polygone | url=http://resolver.sub.uni-goettingen.de/purl?PPN316479497 | publisher=Borna-Leipzig, Druck von Robert Noske, | language=German | series=Dissertation Frankfurt a.M. | year=1918}}
* {{Citation | last=Schattschneider | first=Doris |authorlink=Doris Schattschneider| title=Tiling the plane with congruent pentagons | url=http://www.jstor.org/stable/2689644 | mr=0493766 | year=1978 | journal=[[Mathematics Magazine]] | issn=0025-570X | volume=51 | issue=1 | pages=29–44 | doi=10.2307/2689644}}
* {{citation| last=Schattschneider | first=Doris |authorlink=Doris Schattschneider|title=A new pentagon tiler |journal=Mathematics Magazine |volume=58 |issue=5 |year=1985 |page=308 |id=The cover has a picture of the new tiling}}
* {{citation | last=Sugimoto | first=Teruhisa | mr=3030316 | title=Convex pentagons for edge-to-edge tiling, I. | pages=93-103 | journal=Forma | volume=27 | year=2012 | ref=harv | issue=1 | url=http://www.scipress.org/journals/forma/pdf/2701/27010093.pdf}}
{{refend}}

== External links ==
{{Commons category|Pentagonal tilings}}
*{{MathWorld|title=Pentagon Tiling|urlname=PentagonTiling}}
*[http://demonstrations.wolfram.com/PentagonTilings/ Pentagon Tilings]
*[http://www.mathpuzzle.com/tilepent.html The 14 Pentagons that Tile the Plane]
*[http://www.jaapsch.net/tilings/#pentagon 15 (monohedral) Tilings with a convex pentagonal tile] with k-isohedral colorings

{{Tessellation}}

[[Category:Tessellation]]

Parkettierung mit Fünfecken

2015-08-18T19:17:35Z

Loadmaster: /* top */ Reword

[[File:15-monohedral_pentagonal_tiling_types.png|thumb|450px|Examples for each of the 15 known monohedral tiling types by convex pentagons, with tiles colored by their [[isohedral figure|''k''-isohedral]] positions.]]
In [[geometry]], a '''pentagonal tiling''' is a [[tessellation|tiling]] of the plane where each individual piece is in the shape of a [[pentagon]].

A [[regular tiling|regular]] [[pentagonal]] [[tiling]] on the [[Euclidean plane]] is impossible because the [[internal angle]] of a [[Pentagon#Regular pentagons|regular pentagon]], 108°, is not a divisor of 360°, the angle measure of a whole [[turn (geometry)|turn]].

Fifteen types of convex pentagons are known to [[prototile|monohedrally]] tile the plane (the most recent type being discovered in 2015), but it is not known whether this list is complete.<ref name="NPR" />

== History ==
{{harvtxt|Reinhardt|1918}} found the five pentagonal tilings that are "tile transitive", meaning that the symmetries of the tiling can take any tile to any other tile (more formally, the [[automorphism group]] [[group action|acts transitively]] on the tiles). {{harvtxt|Kershner|1968}} found three more types of pentagonal tiles, the tilings of which are never tile transitive; he claimed incorrectly that this was the complete list of pentagons that can tile the plane. In 1975 Richard E. James III found a 9th type, after reading about Kershner's results in [[Martin Gardner]]'s "Mathematical Games" column in ''[[Scientific American]]'' magazine of July 1975 (reprinted in {{harvtxt|Gardner|1988}}); James' tiling is monohedral (it only uses one type of tile) but not tile transitive. {{harvtxt|Schattschneider|1978}} described how [[Marjorie Rice]], an amateur mathematician, discovered four new types of [[Tessellation|tessellating]] pentagons in 1976 and 1977. {{harvtxt|Schattschneider|1985}} described a 14th convex pentagon type found by Rolf Stein in 1985. {{harvtxt|Bagina|2011}} showed that there are only 8 edge-to-edge convex types, a result obtained independently by {{harvtxt|Sugimoto|2012}}. [[University of Washington Bothell]] mathematicians Casey Mann, Jennifer McLoud, and David Von Derau discovered a 15th monohedral tiling convex pentagon in 2015 using a [[computer algorithm]] (paper pending {{As of|2015|August|lc = y}}).<ref>{{cite news |title= Attack on the pentagon results in discovery of new mathematical tile |first=Alex |last=Bellos |newspaper=[[The Guardian]] |date=11 August 2015 |url=http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile}}</ref>

== Non-convex pentagons ==
With pentagons that are not required to be [[convex]], additional types of tiling are possible. An example is the [[sphinx tiling]], an [[aperiodic tiling]] formed by a pentagonal [[rep-tile]] {{harv|Godrèche|1989}}. The sphinx may also tile the plane periodically, by fitting two sphinx tiles together to form a [[parallelogram]] and then tiling the plane by translates of this parallelogram, a pattern that can also be used for certain other shapes of non-convex pentagons.

== Dual uniform tilings ==

There are three different [[isohedral]] pentagonal tilings generated as [[Dual (polyhedron)|duals]] of the [[uniform tiling]]s:

<gallery widths="220">
File:Tiling Dual Semiregular V3-3-3-4-4 Prismatic Pentagonal.svg|[[Prismatic pentagonal tiling]]<hr/>instance of ''„pentagon type 1 tiling“'' described 1918<ref name="Reinhardt1918"/>

File:Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg|[[Cairo pentagonal tiling]]<hr/>instance of ''„pentagon type 4 tiling“'' described 1918<ref name="Reinhardt1918"/><ref>[http://www.wolframalpha.com/input/?i=pentagon+type+4+tiling&f=PentagonType4Tiling.A_2*pi%2F3&f=PentagonType4Tiling.b_1 Cairo pentagonal tiling generated by a ''pentagon type '''4''' tiling'' query] and [http://www.wolframalpha.com/input/?i=pentagon+type+2+tiling&f=PentagonType2Tiling.A_2*pi%2F3&f=PentagonType2Tiling.B_pi%2F2&f=PentagonType2Tiling.b_1&f=PentagonType2Tiling.e_1 by a ''pentagon type '''2''' tiling'' query] on [http://www.wolframalpha.com wolframalpha.com] (caution: the wolfram definition of ''pentagon type 2 tiling'' does not correspond with ''type 2'' defined by Reinhardt in 1918)</ref>

File:Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg|[[Floret pentagonal tiling]]<hr/>instance of ''„pentagon type 5 tiling“'' described 1918 by [[Karl Reinhardt (mathematician)|Karl Reinhardt]]<ref name="Reinhardt1918">{{Citation | last1=Reinhardt | first1=Karl | title=Über die Zerlegung der Ebene in Polygone | url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN316479497&DMDID=DMDLOG_0013&LOGID=LOG_0013&PHYSID=PHYS_0083 | publisher=Borna-Leipzig, Druck von Robert Noske, | language=German | series=Dissertation Frankfurt am Main | year=1918 | pages=77-81}} (caution: there is at least one obvious mistake within this paper, i.e. γ+δ angle sum needs to equal π, not 2π for the first two tiling types defined on page 77)</ref>
</gallery>

== Regular pentagonal tilings in non-Euclidean geometry ==
A [[dodecahedron]] can be considered a regular tiling of 12 pentagons on the surface of a [[sphere]], with [[Schläfli symbol]] {5,3}, having 3 pentagons around each vertex.

In the [[hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]], there are tilings of regular pentagons, for instance [[order-4 pentagonal tiling]], with [[Schläfli symbol]] {5,4}, having 4 pentagons around each vertex. Higher order regular tilings {5,n} can be constructed on the hyperbolic plane, ending in {5,∞}.

{| class=wikitable
![[Sphere]]
!colspan=6|Hyperbolic plane
|-
|[[File:Uniform tiling 532-t0.png|80px]] [[Dodecahedron|{5,3}]]
|[[File:Uniform tiling 54-t0.png|80px]] [[order-4 pentagonal tiling|{5,4}]]
|[[File:Uniform tiling 55-t0.png|80px]] [[order-5 pentagonal tiling|{5,5}]]
|[[File:Uniform tiling 56-t0.png|80px]] [[order-6 pentagonal tiling|{5,6}]]
|[[File:Uniform tiling 57-t0.png|80px]] [[order-7 pentagonal tiling|{5,7}]]
|[[File:Uniform tiling 58-t0.png|80px]] [[order-8 pentagonal tiling|{5,8}]]
|valign=bottom|...{5,∞}
|}

== Irregular hyperbolic plane pentagonal tilings ==

There are an infinite number of dual [[uniform tilings in hyperbolic plane]] with isogonal irregular pentagonal faces. They have [[face configuration]]s as V3.3.''p''.3.''q''.

{| class=wikitable
|+ Order ''p''-''q'' floret pentagonal tiling
|-
![[Order-3 snub heptagonal tiling#Dual tiling|7-3]]
!8-3
!9-3
!...
!5-4
!6-4
!7-4
!...
!5-5
|- valign=bottom align=center
|[[File:Ord7 3 floret penta til.png|120px]] V3.3.3.3.7
|V3.3.3.3.8
|V3.3.3.3.9
|…
|[[File:Order-5-4 floret pentagonal tiling.png|120px]] V3.3.4.3.5
|V3.3.4.3.6
|V3.3.4.3.7
|…
|V3.3.5.3.5
|…
|}

== See also ==
* [[Mosaic]]
* [[Penrose tiling]]

== References ==
{{reflist|refs=
<ref name="NPR">{{cite news |title= With Discovery, 3 Scientists Chip Away At An Unsolvable Math Problem |last= Peralta |first= Eyder |publisher= NPR |date= 14 August 2015 |accessdate= 15 August 2015 |url= http://www.npr.org/sections/thetwo-way/2015/08/14/432015615/with-discovery-3-scientists-chip-away-at-an-unsolvable-math-problem?utm_source=facebook.com&utm_medium=social&utm_campaign=npr&utm_term=nprnews&utm_content=20150814 |mode= cs2}}</ref>
}}

'''Bibliography'''
{{refbegin|30em}}
* {{Citation | last=Bagina | first=Olga | title=Tiling the plane with congruent equilateral convex pentagons
| url=http://dx.doi.org/10.1016/j.jcta.2003.11.002 | doi=10.1016/j.jcta.2003.11.002 | mr=2046081 | year=2004 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=105 | issue=2 | pages=221–232}}
* {{citation | last=Bagina | first=Olga | script-title=ru:Мозаики из выпуклых пятиугольников |trans_title=Tilings of the plane with convex pentagons | language=Russian | year=2011 | journal=Vestnik |publisher=Kemerovo State University |issn=2078-1768 |volume=4 | issue=48 | pages=63–73 |url=http://www.mathnet.ru/php/seminars.phtml?presentid=4640 |accessdate=29 January 2013}}
* {{citation |last1=Grünbaum |first1=Branko |authorlink=Branko Grünbaum |last2=Shephard |first2=Geoffrey C. |title=Tilings and Patterns |location=New York |publisher=W. H. Freeman and Company |year=1987 |isbn=0-7167-1193-1 |chapter=Tilings by polygons | mr=0857454}}
* {{citation |last=Gardner |first=Martin |authorlink=Martin Gardner |title=Time travel and other mathematical bewilderments |year=1988 |publisher=W. H. Freeman and Company |location=New York |isbn=0-7167-1925-8 |chapter=Tiling with Convex Polygons | mr=0905872 }}
* {{citation
| last = Godrèche | first = C.
| doi = 10.1088/0305-4470/22/24/006
| issue = 24
| journal = Journal of Physics A: Mathematical and General
| mr = 1030678
| pages = L1163–L1166
| title = The sphinx: a limit-periodic tiling of the plane
| volume = 22
| year = 1989}}
* {{Citation | last1=Hirschhorn | first1=M. D. | last2=Hunt | first2=D. C. | title=Equilateral convex pentagons which tile the plane | url=http://dx.doi.org/10.1016/0097-3165(85)90078-0 | doi=10.1016/0097-3165(85)90078-0 | mr=787713 | year=1985 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=39 | issue=1 | pages=1–18}}
* {{Citation | last=Kershner | first=Richard | title=On paving the plane | url=http://www.jstor.org/stable/2314332 | mr=0236822 | year=1968 | journal=[[American Mathematical Monthly]] | issn=0002-9890 | volume=75 | pages=839–844 | doi=10.2307/2314332}}
* {{Citation | last=Reinhardt | first=Karl | title=Über die Zerlegung der Ebene in Polygone | url=http://resolver.sub.uni-goettingen.de/purl?PPN316479497 | publisher=Borna-Leipzig, Druck von Robert Noske, | language=German | series=Dissertation Frankfurt a.M. | year=1918}}
* {{Citation | last=Schattschneider | first=Doris |authorlink=Doris Schattschneider| title=Tiling the plane with congruent pentagons | url=http://www.jstor.org/stable/2689644 | mr=0493766 | year=1978 | journal=[[Mathematics Magazine]] | issn=0025-570X | volume=51 | issue=1 | pages=29–44 | doi=10.2307/2689644}}
* {{citation| last=Schattschneider | first=Doris |authorlink=Doris Schattschneider|title=A new pentagon tiler |journal=Mathematics Magazine |volume=58 |issue=5 |year=1985 |page=308 |id=The cover has a picture of the new tiling}}
* {{citation | last=Sugimoto | first=Teruhisa | mr=3030316 | title=Convex pentagons for edge-to-edge tiling, I. | pages=93-103 | journal=Forma | volume=27 | year=2012 | ref=harv | issue=1 | url=http://www.scipress.org/journals/forma/pdf/2701/27010093.pdf}}
{{refend}}

== External links ==
{{Commons category|Pentagonal tilings}}
*{{MathWorld|title=Pentagon Tiling|urlname=PentagonTiling}}
*[http://demonstrations.wolfram.com/PentagonTilings/ Pentagon Tilings]
*[http://www.mathpuzzle.com/tilepent.html The 14 Pentagons that Tile the Plane]
*[http://www.jaapsch.net/tilings/#pentagon 15 (monohedral) Tilings with a convex pentagonal tile] with k-isohedral colorings

{{Tessellation}}

[[Category:Tessellation]]

CDC 6600

2015-01-06T17:49:14Z

Loadmaster: /* Description */ Mv lengthy intro para into new subsect

{{more footnotes|date=March 2013}}
[[File:CDC 6600.jc.jpg|thumb|300px|right| The CDC 6600. Behind the system console are two of the "arms" of the plus-sign shaped cabinet with the covers opened. Individual modules can be seen inside. The racks holding the modules are hinged to give access to the racks behind them. Each arm of the machine had up to four such racks. On the right is the cooling system.]]
[[Image:Control Data 6600 mainframe.jpg|thumb|right|300px|A CDC 6600 system [[console]]. The [[computer display|display]]s were driven through software, primarily to provide text display (in a choice of three sizes). It also provided a way to draw simple graphics. Unlike more modern displays, the console was a [[vector display|vector drawing system]] rather than a [[raster display|raster system]]. Lines were drawn by specifying a start and end point. The consoles had a single [[computer font|font]] where each glyph was a series of vectors.]]

The '''CDC 6600''' was the flagship [[mainframe computer|mainframe]] [[supercomputer]] of the [[CDC 6000 series|6000 series]] of computer systems manufactured by [[Control Data Corporation]]. The first CDC 6600 was delivered in 1965 to the [[CERN]] laboratory near [[Geneva]], [[Switzerland]],<ref>[http://timeline.web.cern.ch/the-cdc-6600-arrives-at-cern] CERN Timelines, "The CDC 6600 arrives at CERN"</ref> where it was used to analyse the 2-3 million photographs of bubble-chamber tracks that CERN experiments were producing every year. In 1966 another CDC 6600 was delivered to the Lawrence Radiation Laboratory, part of the University of California at Berkeley, where it was used for the analysis of nuclear events photographed inside the Alvarez bubble chamber.<ref>[http://www.lbl.gov/Science-Articles/Research-Review/Magazine/1981/81fchp6.html] "Bumper Crop", chapter 6 in "Lawrence and His Laboratory", 1981</ref>
The CDC 6600 is generally considered to be the first successful [[supercomputer]], outperforming its fastest predecessor, the [[IBM 7030 Stretch]], by about a factor of three. With performance of about 1 [[FLOPS|megaFLOPS]],<ref>[http://www.princeton.edu/~achaney/tmve/wiki100k/docs/CDC_6600.html]</ref> the CDC 6600 was the world's fastest computer from 1964 to 1969, when it relinquished that status to its successor, the [[CDC 7600]].

A CDC 6600 is on display at the [[Computer History Museum]] in [[Mountain View, California]].

==History and impact==
{{main|Control Data Corporation}}

CDC's first products were based on the machines designed at [[Engineering Research Associates|ERA]], which [[Seymour Cray]] had been asked to update after moving to CDC. After an experimental machine known as the ''Little Character'', they delivered the [[CDC 1604]], one of the first commercial [[Transistor computer|transistor-based computers]], and one of the fastest machines on the market. Management was delighted, and made plans for a new series of machines that were more tailored to business use; they would include instructions for character handling and record keeping for instance. Cray was not interested in such a project, and set himself the goal of producing a new machine that would be 50 times faster than the 1604. When asked to complete a detailed report on plans at one and five years into the future, he wrote back that his five-year goal was "to produce the largest computer in the world", "largest" at that time being synonymous with "fastest", and that his one year plan was "to be one-fifth of the way".

Taking his core team to new offices nearby the original CDC headquarters, they started to experiment with higher quality versions of the "cheap" [[transistor]]s Cray had used in the 1604. After much experimentation, they found that there was simply no way the [[germanium]]-based transistors could be run much faster than those used in the 1604. The "business machine" that management had originally wanted, now forming as the [[CDC 3000]] series, pushed them about as far as they could go. Cray then decided the solution was to work with the then-new [[silicon]]-based transistors from [[Fairchild Semiconductor]], which were just coming onto the market and offered dramatically improved switching performance.

During this period, CDC grew from a startup to a large company and Cray became increasingly frustrated with what he saw as ridiculous management requirements. Things became considerably more tense in 1962 when the new [[CDC 3600]] started to near production quality, and appeared to be exactly what management wanted, when they wanted it. Cray eventually told CDC's CEO, [[William Norris (CEO)|William Norris]] that something had to change, or he would leave the company. Norris felt he was too important to lose, and gave Cray the green light to set up a new lab wherever he wanted.

After a short search, Cray decided to return to his home town of [[Chippewa Falls, WI]], where he purchased a block of land and started up a new lab. Although this process introduced a fairly lengthy delay in the design of his new machine, once in the new lab, without management interference, things started to progress quickly. By this time, the new transistors were becoming quite reliable, and modules built with them tended to work properly on the first try. Working with Jim Thornton, who was the system architect and the 'hidden genius' behind the 6600, the machine soon took form.

More than 100 CDC 6600s were sold over the machine's lifetime. Many of these went to various [[nuclear bomb]]-related labs, and quite a few found their way into university computing labs. Cray immediately turned his attention to its replacement, this time setting a goal of 10 times the performance of the 6600, delivered as the [[CDC 7600]]. The later [[CDC Cyber]] 70 and 170 computers were very similar to the CDC 6600 in overall design and were nearly completely backwards compatible.

==Description==
Typical machines of the era used a single [[Central processing unit|CPU]] to drive the entire system. A typical program would first load data into memory (often using pre-rolled library code), process it, and then write it back out. This required the CPUs to be fairly complex in order to handle the complete set of instructions they would be called on to perform. A complex CPU implied a large CPU, introducing signalling delays while information flowed between the individual modules making it up. These delays set a maximum upper limit on performance, the machine could only operate at a cycle speed that allowed the signals time to arrive at the next module.

Cray took another approach. At the time, CPUs generally ran slower than the [[main memory]] they were attached to. For instance, a processor might take 15 cycles to multiply two numbers, while each memory access took only one or two. This meant there was a significant time where the main memory was idle. It was this idle time that the 6600 exploited.

Instead of trying to make the CPU handle all the tasks, the 6600 CPUs handled arithmetic and logic only. This resulted in a much smaller CPU which could operate at a higher clock speed. Combined with the faster switching speeds of the silicon transistors, the new CPU design easily outperformed everything then available. The new design ran at 10 MHz (100 ns cycle), about ten times faster than other machines on the market. In addition to the clock being faster, the simple processor executed instructions in fewer clock cycles; for instance, the CPU could complete a multiplication in ten cycles.

However, the CPU could only execute a limited number of simple instructions. A typical CPU of the era had a [[Complex instruction set computer|complex instruction set]], which included instructions to handle all the normal "housekeeping" tasks such as memory access and [[input/output]]. Cray instead implemented these instructions in separate, simpler processors dedicated solely to these tasks, leaving the CPU with a much smaller instruction set. (This was the first of what later came to be called [[reduced instruction set computer]] (RISC) design.) By allowing the CPU, peripheral processors (PPs) and I/O to operate in parallel, the design considerably improved the performance of the machine. Under normal conditions a machine with several processors would also cost a great deal more. Key to the 6600's design was to make the I/O processors, known as ''peripheral processors'' (PPs), as simple as possible. The PPs were based on the simple 12-bit [[CDC 160-A]], which ran much slower than the CPU, gathering up data and "squirting" it into main memory at high speed via dedicated hardware.

The 10 PPs were implemented virtually; there was CPU hardware only for a single PP. This CPU hardware was shared and operated on 10 PP register sets which represented each of the 10 PP ''states'' (similar to modern [[temporal multithreading|multithreading]] processors). The PP ''[[barrel processor|register barrel]]'' would "rotate", with each PP register set presented to the "slot" which the actual PP CPU occupied. The shared CPU would execute all or some portion of a PP's instruction whereupon the barrel would "rotate" again, presenting the next PP's register set (state). Multiple "rotations" of the barrel were needed to complete an instruction. A complete barrel "rotation" occurred in 1000 nanoseconds (100 nanoseconds per PP), and an instruction could take from 1 to 5 "rotations" of the barrel to be completed, or more if it was a data transfer instruction.

The basis for the 6600 CPU is what would today be referred to as a [[RISC]] system, one in which the processor is tuned to do instructions which are comparatively simple and have limited and well-defined access to memory. The philosophy of many other machines was toward using instructions which were complicated — for example, a single instruction which would fetch an operand from memory and add it to a value in a register. In the 6600, loading the value from memory would require one instruction, and adding it would require a second one. While slower in theory due to the additional memory accesses, the fact that in well-scheduled code, multiple instructions could be processing in parallel offloaded this expense. This simplification also forced programmers to be very aware of their memory accesses, and therefore code deliberately to reduce them as much as possible.

===Models===
The CDC 6000 series included four basic models, the [[CDC 6400]], the CDC 6500, the CDC 6600, and the CDC 6700. The models of the 6000 series differed only in their CPUs, which were of two kinds, the 6400 CPU and the 6600 CPU. The 6400 CPU had a unified arithmetic unit, rather than discrete [[Execution unit|''functional units'']]. As such, it could not overlap instructions' execution times. For example, in a 6400 CPU, if an add instruction immediately followed a multiply instruction, the add instruction could not be started till the multiply instruction finished, so the net execution time of the two instructions would be the sum of their individual execution times. The 6600 CPU had multiple functional units which could operate simultaneously (i.e., ''in [[Parallel computing|parallel]]''), allowing the CPU to overlap instructions' execution times. For example, a 6600 CPU could begin executing an add instruction in the next CPU cycle following the beginning of a multiply instruction (assuming, of course, that the result of the multiply instruction was not an operand of the add instruction), so the net execution time of the two instructions would simply be the (longer) execution time of the multiply instruction. The 6600 CPU also had an ''instruction stack'', a kind of ''[[instruction cache]]'', which helped increase CPU throughput by reducing the amount of CPU idle time caused by waiting for memory to respond to instruction fetch requests. The two kinds of CPUs were instruction compatible, so that a program that ran on either of the kinds of CPUs would run the same way on the other kind but would run faster on the 6600 CPU. Indeed, all models of the 6000 series were fully inter-compatible. The CDC 6400 had one CPU (a 6400 CPU), the CDC 6500 had two CPUs (both 6400 CPUs), the CDC 6600 had one CPU (a 6600 CPU), and the CDC 6700 had two CPUs (one 6600 CPU and one 6400 CPU).

===Central Processor (CP)===
{| class="infobox" style="font-size:88%"
|-
|style="text-align:center" |''CDC 6x00 registers''
|-
|
{| style="font-size:88%;"
|-
| style="width:10px; text-align:center;"| 59
| style="width:160px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 17
| style="width:70px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 00
| style="width:auto;" | ''(bit position)''
|-
|colspan="6" | '''Operand registers''' ''(60 bits)''
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X0
| style="width:auto; background:white; color:black;"| Register 0
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X1
| style="width:auto; background:white; color:black;"| Register 1
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X2
| style="width:auto; background:white; color:black;"| Register 2
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X3
| style="width:auto; background:white; color:black;"| Register 3
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X4
| style="width:auto; background:white; color:black;"| Register 4
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X5
| style="width:auto; background:white; color:black;"| Register 5
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X6
| style="width:auto; background:white; color:black;"| Register 6
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X7
| style="width:auto; background:white; color:black;"| Register 7
|-
|colspan="6" | '''Address registers''' ''(18 bits)''
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A0
| style="width:auto; background:white; color:black;"| Address 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A1
| style="width:auto; background:white; color:black;"| Address 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A2
| style="width:auto; background:white; color:black;"| Address 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A3
| style="width:auto; background:white; color:black;"| Address 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A4
| style="width:auto; background:white; color:black;"| Address 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A5
| style="width:auto; background:white; color:black;"| Address 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A6
| style="width:auto; background:white; color:black;"| Address 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A7
| style="width:auto; background:white; color:black;"| Address 7
|-
|colspan="6" | '''Increment registers''' ''(18 bits)''
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B0 ''(all bits zero)''
| style="width:auto; background:white; color:black;"| Increment 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B1
| style="width:auto; background:white; color:black;"| Increment 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B2
| style="width:auto; background:white; color:black;"| Increment 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B3
| style="width:auto; background:white; color:black;"| Increment 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B4
| style="width:auto; background:white; color:black;"| Increment 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B5
| style="width:auto; background:white; color:black;"| Increment 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B6
| style="width:auto; background:white; color:black;"| Increment 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B7
| style="width:auto; background:white; color:black;"| Increment 7
|}

|}
The Central Processor (CP) and main memory of the 6400, 6500, and 6600 machines had a 60-bit word length. The Central Processor had eight general purpose [[60-bit]] [[processor register|registers]] X0 through X7, eight [[18-bit]] address registers A0 through A7, and eight 18-bit "increment" registers B0 through B7. B0 was held at zero permanently by the hardware. Many programmers found it useful to set B1 to 1, and similarly treat it as inviolate.

The CP had no instructions for input and output, which are accomplished through Peripheral Processors (below). No opcodes were specifically dedicated to loading or storing memory; this occurred as a side effect of assignment to certain A registers. Setting A1 through A5 loaded the word at that address into X1 through X5 respectively; setting A6 or A7 stored a word from X6 or X7. No side effects were associated with A0. A separate hardware load/store unit, called the ''stunt box'', handled the actual data movement independently of the operation of the instruction stream, allowing other operations to complete while memory was being accessed, which required eight cycles, in the best case.

The 6600 CP included 10 parallel functional units, allowing multiple instructions to be worked on at the same time. Today, this is known as a [[superscalar]] design, but it was unique for its time. Unlike most modern CPU designs, functional units were not pipelined; the functional unit would become busy when an instruction was "issued" to it and would remain busy for the entire time required to execute that instruction. (By contrast, the CDC 7600 introduced pipelining into its functional units.) In the best case, an instruction could be issued to a functional unit every 100{{nbsp}}ns clock cycle. The system read and decoded instructions from memory as fast as possible, generally faster than they could be completed, and fed them off to the units for processing. The units were:
* floating point multiply (two copies)
* floating point divide
* floating point add
* "long" integer add
* incrementers (two copies; performed memory load/store)
* shift
* boolean logic
* branch

Floating-point operations were given pride of place in this [[Computer architecture|architecture]]: the CDC 6600 (and kin) stand virtually alone in being able to execute a 60-bit [[floating point]] multiplication in time comparable to that for a program branch.

Fixed point addition and subtraction of 60-bit numbers were handled in the Long Add Unit, with negative numbers are represented in one's complement notation. Fixed point multiply and divide were performed by converting to and from floating point.<ref>http://ed-thelen.org/comp-hist/CDC-6600-R-M.html#P3-21</ref>

Previously executed instructions were saved in an eight-word [[CPU cache|cache]], called the "stack". In-stack jumps were quicker than out-of-stack jumps because no memory fetch was required. The stack was flushed by an unconditional jump instruction, so unconditional jumps at the ends of loops were conventionally written as conditional jumps that would always succeed.

The system used a 10{{nbsp}}[[Hertz|MHz]] clock, but used a [[clock signal#4-phase clock|four-phase signal]], so the system could at times effectively operate at 40{{nbsp}}MHz. A floating-point multiplication took ten cycles, a division took 29, and the overall performance, taking into account memory delays and other issues, was about 3{{nbsp}}[[MFLOPS]]. Using the best available compilers, late in the machine's history, [[FORTRAN]] programs could expect to maintain about 0.5{{nbsp}}MFLOPS.

===Memory organization===
User programs are restricted to use only a contiguous area of main memory. The portion of memory to which an executing program has access is controlled by the ''RA'' (Relative Address) and ''FL'' (Field Length) registers which are not accessible to the user program. When a user program tries to read or write a word in central memory at address ''a'', the processor will first verify that a is between 0 and FL-1. If it is, the processor accesses the word in central memory at address RA+a. This process is known as base-bound relocation; each user program sees core memory as a contiguous block words with length FL, starting with address 0; in fact the program may be anywhere in the physical memory. Using this technique, each user program can be moved ("relocated") in main memory by the operating system, as long as the RA register reflects its position in memory. A user program which attempts to access memory outside the allowed range (that is, with an address which is not less than FL) will trigger an interrupt, and will be terminated by the operating system. When this happens, the operating system may create a [[core dump]] which records the contents of the program's memory and registers in a file, allowing the developer of the program a means to know what happened. Note the distinction with [[virtual memory]] systems; in this case, the entirety of a process's addressable space must be in core memory, must be contiguous, and its size cannot be larger than the real memory capacity.

All but the first seven [[CDC 6000 series]] machines could be configured with an optional Extended Core Storage (ECS) system. ECS was built from a different variety of core memory than was used in the central memory. This made it economical for it to be both larger and slower. The primary reason was that ECS memory was wired with only two wires per core (contrast with five for central memory). Because it performed very wide transfers, its sequential transfer rate was the same as that of the small core memory. A 6000 CPU could directly perform block memory transfers between a user's program (or operating system) and the ECS unit. Wide data paths were used, so this was a very fast operation. Memory bounds were maintained in a similar manner as central memory — with an RA/FL mechanism maintained by the operating system. ECS could be used for a variety of purposes, including containing user data arrays that were too large for central memory, holding often-used files, swapping, and even as a communication path in a multi-mainframe complex.

===Peripheral Processors (PPs)===
To handle the 'household' tasks, which in other designs, were ascribed to the CPU, Cray included ten other processors, based partly on his earlier computer, the [[CDC 160-A]]. These machines, called Peripheral Processors, or PPs, were full computers in their own right, but were tuned to performing [[Input/output|I/O]] tasks and running the operating system. (Nearly all of the operating system ran on the PP's; thus leaving the full power of the Central Processor available for user programs.) One of the PPs was in overall control of the machine, including control of the program running on the main CPU, while the others would be dedicated to various I/O tasks — similar to [[I/O channel]]s in [[IBM mainframe]]s of the time. When the program needed to perform an I/O operation, it loaded a small program into one of the PPs which did the work. The PP would then inform the CPU via an interrupt, when the task was complete.

Each PP included its own memory of 4096 [[12-bit]] words. This memory served for both for I/O buffering and program storage, but the execution units were shared by 10 PPs, in a configuration called the [[Barrel processor|Barrel and slot]]. This meant that the execution units (the "slot") would execute one instruction cycle from the first PP, then one instruction cycle from the second PP, etc. in a round robin fashion. This was done both to reduce costs, and because access to CP memory required 10 PP clock cycles: when a PP accesses CP memory, the data is available next time the PP receives its slot time.

===Wordlengths, characters===
The central processor had [[60-bit]] words, whilst the peripheral processors had [[12-bit]] words. CDC used the term "byte" to refer to 12-bit entities used by peripheral processors; characters were 6-bit, and central processor instructions were either 15 bits, or 30 bits with a signed 18-bit address field, the latter allowing for a directly addressable memory space of 128K words of central memory (converted to modern terms, with 8-bit bytes, this is 0.94 MB). The signed nature of the address registers limited an individual program to 128K words. (Later CDC 6000-compatible machines could have 256K or more words of central memory, budget permitting, but individual user programs were still limited to 128K words of CM.) Central processor instructions started on a word boundary when they were the target of a jump statement or subroutine return jump instruction, so no-operations were sometimes required to fill out the last 15, 30 or 45 bits of a word.

The 6-bit characters, in an encoding called [[display code]], could be used to store up to 10 characters in a word. They permitted a character set of 64 characters, which is enough for all upper case letters, digits, and some punctuation. Certainly, enough to write [[FORTRAN]], or print financial or scientific reports. There were actually two variations of the [[display code]] character sets in use, 64-character and 63-character. The 64-character set had the disadvantage that two consecutive ':' (colon) characters might be interpreted as the end of a line if they fell at the end of a 10-byte word. A later variant, called [[CDC display code#6/12 display code|6/12 display code]], was also used in the [[CDC Kronos|Kronos]] and [[NOS (software)|NOS]] timesharing systems to allow full use of the [[ASCII]] character set in a manner somewhat compatible with older software.

With no byte addressing instructions at all, code had to be written to pack and shift characters into words. The very large words, and comparatively small amount of memory, meant that programmers would frequently economize on memory by packing data into words at the bit level.

It is interesting to note that due to the large word size, and with 10 characters per word, it was often faster to process words full of characters at a time — rather than unpacking/processing/repacking them. For example, the CDC [[COBOL]] compiler was actually quite good at processing decimal fields using this technique. These sorts of techniques are now commonly used in the 'multi-media' instructions of current processors.

===Physical design===
[[Image:CDCcordwood1.jpg|thumb|right|300px|A CDC 6600 [[Printed circuit board#Cordwood construction| cordwood logic module]] containing 64 silicon transistors. The coaxial connectors are test points. The module is cooled conductively via the front panel. The 6600 model contained nearly 6,000 such modules.<ref>Understanding Computers: Speed and Power 1990, p. 17.</ref>]]

The machine was built in a plus-sign-shaped cabinet with a pump and heat exchanger in the outermost {{convert|18|in|cm|abbr=on}} of each of the four arms. Cooling was done with [[Freon]] circulating within the machine and exchanging heat to an external chilled water supply. Each arm could hold four chassis, each about {{convert|8|in|cm|abbr=on}} thick, hinged near the center, and opening a bit like a book. The intersection of the "plus" was filled with cables which interconnected the chassis. The chassis were numbered from 1 (containing all 10 PPUs and their memories, as well as the 12 rather minimal I/O channels) to 16. The main memory for the CPU was spread over many of the chassis. In a system with only 64K words of main memory, one of the arms of the "plus" was omitted.

The logic of the machine was packaged into modules about {{convert|2.5|in|mm|abbr=on}} square and about {{convert|1|in|cm|abbr=on}} thick. Each module had a connector (30 pins, two vertical rows of 15) on one edge, and six test points on the opposite edge. The module was placed between two aluminum cold plates to remove heat. The module itself consisted of two parallel printed circuit boards, with components mounted either on one of the boards or between the two boards. This provided a very dense package; somewhat difficult to repair, but with good heat removal. It was known as [[Printed circuit board#Cordwood construction|cordwood construction]].

==Operating system and programming==
There was a sore point with the 6600 [[operating system]] support — slipping timelines. The machines originally ran a very simple [[batch processing|job-control]] system known as ''COS'' ([[Chippewa Operating System]]), which was quickly "thrown together" based on the earlier [[CDC 3000]] operating system in order to have something running to test the systems for delivery. However the machines were intended to be delivered with a much more powerful system known as ''SIPROS'' (for Simultaneous Processing Operating System), which was being developed at the company's System Sciences Division in [[Los Angeles]]. Customers were impressed with SIPROS's feature list, and many had SIPROS written into their delivery contracts.

SIPROS turned out to be a major fiasco. Development timelines continued to slip, costing CDC major amounts of profit in the form of delivery delay penalties. After several months of waiting with the machines ready to be shipped, the project was eventually cancelled. The programmers who had worked on COS had little faith in SIPROS and had continued working on improving COS.

[[Operating system development]] then split into two camps. The CDC-sanctioned evolution of COS was undertaken at the [[Sunnyvale, California]] software development lab. Many customers eventually took delivery of their systems with this software, then known as ''[[SCOPE (software)|SCOPE]]'' (Supervisory Control Of Program Execution). (Some Control Data Field Engineers used to refer to SCOPE as ''Sunnyvale's Collection Of Programming Errors''). SCOPE version 1 was, essentially, dis-assembled COS; SCOPE version 2 included new device and file system support; SCOPE version 3 included permanent file support, EI/200 remote batch support, and INTERCOM [[time sharing]] support. SCOPE always had significant reliability and maintainability issues. [[File:CDC 6000 series SCOPE 3.1 building itself.PNG|thumb| CDC 6000 series SCOPE 3.1 building itself while running on Desktop CYBER emulator.]]

The underground evolution of COS took place at the [[Arden Hills, Minnesota]] assembly plant. ''MACE'' ([Greg] Mansfield And [Dave] Cahlander Executive) was written largely by a single programmer in the off-hours when machines were available. Its feature set was essentially the same as COS and SCOPE 1. It retained the earlier COS file system, but made significant advances in code modularity to improve system reliability and adaptiveness to new storage devices. MACE was never an official product, although many customers were able to wrangle a copy from CDC.

The unofficial MACE software was later chosen over the official SCOPE product as the basis of the next CDC operating system, ''[[CDC Kronos|Kronos]]'', named after the [[Chronos|Greek god of time]]. The main marketing reason for its adoption was the development of its TELEX [[time sharing]] feature and its BATCHIO remote batch feature. Kronos continued to use the COS/SCOPE 1 file system with the addition of a permanent file feature.

An attempt to unify the SCOPE and Kronos operating system products produced ''[[NOS (software)|NOS]]'', (Network Operating System). NOS was intended to be the sole operating system for all CDC machines, a fact CDC promoted heavily. Many SCOPE customers remained software-dependent on the SCOPE architecture, so CDC simply renamed it ''[[NOS/BE]]'' (Batch Environment), and were able to claim that everyone was thus running NOS. In practice, it was far easier to modify the Kronos code base to add SCOPE features than the reverse.

The assembly plant environment also produced other operating systems which were never intended for customer use. These included the engineering tools SMM for hardware testing, and KALEIDOSCOPE, for software [[smoke testing (software)|smoke testing]]. Another commonly used tool for CDC Field Engineers during testing was MALET (Maintenance Application Language for Equipment Testing), which was used to stress test components and devices after repairs or servicing by engineers. Testing conditions often used hard disk packs and magnetic tapes which were deliberately marked with errors to determine if the errors would be detected by MALET and the engineer.

==CDC 7600==
The CDC 7600 was originally intended to be fully compatible with the existing 6000-series machines as well. (It started life as the CDC 6800.) But during its design, the designers determined that maintaining complete compatibility with the existing 6000-series machines would limit how much performance improvement they could attain and decided to sacrifice compatibility for performance. While the CDC 7600's CPU was basically instruction compatible with the 6400 and 6600 CPUs, allowing code portability at the [[High-level programming language|high-level language]] source code level, the CDC 7600's hardware, especially that of its Peripheral Processor Units (PPUs), was quite different, and the CDC 7600 required a different operating system. This turned out to be somewhat serendipitous because it allowed the designers to improve on some of the characteristics of the 6000-series design, such as the latter's complete dependence on Peripheral Processors (PPs), particularly the first (called PP0), to control operation of the entire computer system, including the CPU(s). Unlike the 6600 CPU, the CDC 7600's CPU could control its own operation. In fact, the 6000-series machines were [[Retrofitting|retrofitted]] with this capability.

==See also==
* [[History of supercomputing]]

== Notes ==
{{Reflist}}

==References==
* Grishman, Ralph (1974). ''Assembly Language Programming for the Control Data 6000 Series and the Cyber 70 Series''. New York, NY: Algorithmics Press. [http://www.bitsavers.org/pdf/cdc/cyber/books/Grishman_CDC6000AsmLangPgmg.pdf]
* [http://ed-thelen.org/comp-hist/CDC-6600-R-M.html#TOC/ CONTROL DATA 6400/6500/6600 COMPUTER SYSTEMS Reference Manual]
* Thornton, J. (1963). ''Considerations in Computer Design - Leading up to the Control Data 6600'' [http://www.bitsavers.org/pdf/cdc/cyber/cyber_70/thornton_6600_paper.pdf]
* Thornton, J. (1970). ''Design of a Computer—The Control Data 6600''. Glenview, IL: Scott, Foresman and Co. [http://www.bitsavers.org/pdf/cdc/cyber/books/DesignOfAComputer_CDC6600.pdf]
* (1990) ''Understanding Computers: Speed and Power, a TIME LIFE series'' ISBN 0809475863

==External links==
* [http://purl.umn.edu/104327 Neil R. Lincoln with 18 Control Data Corporation (CDC) engineers on computer architecture and design], [[Charles Babbage Institute]], University of Minnesota. Engineers include Robert Moe, Wayne Specker, Dennis Grinna, Tom Rowan, Maurice Hutson, Curt Alexander, Don Pagelkopf, Maris Bergmanis, Dolan Toth, Chuck Hawley, Larry Krueger, Mike Pavlov, Dave Resnick, Howard Krohn, Bill Bhend, Kent Steiner, Raymon Kort, and Neil R. Lincoln. Discussion topics include [[CDC 1604]], CDC 6600, [[CDC 7600]], [[CDC 8600]], [[CDC STAR-100]] and [[Seymour Cray]].
* [http://research.microsoft.com/~gbell/Computer_Structures__Readings_and_Examples/00000509.htm Parallel operation in the Control Data 6600, James Thornton]
* [http://research.microsoft.com/users/gbell/craytalk/sld001.htm Presentation of the CDC 6600 and other machines designed by Seymour Cray] – by C. [[Gordon Bell]] of Microsoft Research (formerly of DEC)
*{{cite web | url = http://www.computerhistory.org/revolution/supercomputers/10/33 | title = CDC 6600’s Five Year Reign | author = | date = 2003 | publisher = Computer History Museum | quote = The 6600 had 400,000 transistors and more than 100 miles of wiring. }} – overview with pictures

{{S-start}}
{{S-ach|rec}}
{{S-bef|before=[[IBM 7030 Stretch]]}}
{{S-ttl
| title = [[Supercomputer|World's most powerful computer]]
| years = 1964 - 1968
}}
{{s-aft|after=[[CDC 7600]]}}
{{S-end}}

{{DEFAULTSORT:Cdc 6600}}
[[Category:1964 introductions]]
[[Category:CDC hardware|6600]]
[[Category:Supercomputers]]
[[Category:Transistorized computers]]
[[Category:Superscalar microprocessors]]

Greg Gutfeld

2014-04-11T20:49:28Z

Loadmaster: ersetzt mit neueren Bild :File:Greg-Gutfeld-3157.jpg

[[Datei:Greg-Gutfeld-3157.jpg|miniatur|Greg Gutfeld (2014)]]
'''Greg Gutfeld''' (* [[12. September]] [[1964]]) ist ein [[Vereinigte Staaten|US-amerikanischer]] [[Journalist]], Fernsehmoderator und Satiriker.

== Leben und Wirken ==
Gutfeld wuchs in [[San Mateo (Kalifornien)|San Mateo]], [[Kalifornien]], auf. Nach dem Besuch der ''Junípero Serra High School'' studierte er an der [[University of California, Berkeley]], die er 1987 mit einem Abschluss in Englisch verließ. Anschließend absolvierte er ein Praktikum bei der Zeitschrift [[The American Spectator]]. Seine erste reguläre Anstellung erhielt er als Redakteur für die Zeitschrift ''Prevention''.

1995 trat Gutfeld in die Redaktion der Zeitschrift ''[[Men’s Health]]'' ein, deren Chefredakteur er 1999 wurde, bis er 2000 durch David Zinczenko ersetzt wurde. Im selben Jahr erhielt er den Posten des Chefredakteurs Zeitschrift ''Stuff'', deren Auflage unter seiner Ägide von 750.000 auf 1,2 Millionen stieg. Von 2004 bis 2006 fungierte Gutfeld schließlich als Chefredakteur der Zeitschrift ''Maxim'' in Großbritannien.

2007 wurde Gutfeld vom Nachrichtensender [[Fox News]] eingestellt, für den er seit dem 5. Februar 2007 das nächtliche Unterhaltungsformat ''Red Eye w/Greg Gutfeld'' präsentiert. Die Sendung, die werktäglich von 3.00 bis 4.00 Uhr Ostküstenzeit ausgestrahlt wird, bietet eine Mischung aus humoristischen Monologen des Gastgebers zu aktuellen Ereignissen sowie zwanglosen Gesprächen zum Tagesgeschehen mit einer Gruppe wechselnder Gäste, wobei die Themenfelder Politik, Gesellschaft und Unterhaltungsindustrie dominieren.

Seit dem 11. Juli 2011 ist Gutfeld außerdem einer der ständigen Co-Modeatoren der politischen Diskussionsrunde ''The Five'', die Fox News täglich um 5.00 Uhr nachmittags Ostküstenzeit ausstrahlt. Diese Sendung, die anstelle der kontroversen Fernseh-Show des Radiotalkers [[Glenn Beck]] ins Programm genommen wurde, ist als Panel aufgebaut, in dem fünf Co-Moderatoren mit unterschiedlichen politischen Standpunkten in einer Tischrunde aktuelle Ereignisse aus den Bereichen Politik und Gesellschaft erörtern.

Gutfeld, der sich selbst als libertär kennzeichnet, ist verheiratet und lebt in New York City.

== Schriften ==
* ''The Official Point System for Keeping Score in the Relationshi Game'', 1997.
* ''The Scorecard at Work: The Official Point System for Keeping Score on the Job'', 1999.
* ''Lessons from the Land of Pork Scratchings'' 2008.
* ''The Bible of Unspeakable Truths'', 2010.

== Weblinks ==
{{Commonscat}}
* {{NNDB Name|488/000345450}}

{{SORTIERUNG:Gutfeld, Greg}}
[[Kategorie:Journalist (Vereinigte Staaten)]]
[[Kategorie:Fernsehmoderator (Vereinigte Staaten)]]
[[Kategorie:US-Amerikaner]]
[[Kategorie:Geboren 1964]]
[[Kategorie:Mann]]

{{Personendaten
|NAME=Gutfeld, Greg
|ALTERNATIVNAMEN=
|KURZBESCHREIBUNG=US-amerikanischer Journalist, Fernsehmoderator und Satiriker
|GEBURTSDATUM=12. September 1964
|GEBURTSORT=
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Flogging a Dead Horse

2014-02-05T00:22:24Z

Loadmaster: /* Lede */ Ersetzt Bild mit retuschierten :Datei:Man sitting on a dead horse (1876 - 1884) (retouched).jpg

[[Datei:Man sitting on a dead horse (1876 - 1884) (retouched).jpg|mini|rechts|Ein Mann in [[Sheboygan]] sitzt etwas unschlüssig auf einem toten Pferd.]]
[[Datei:Reform bill1867punchJohn Tenniel into the dark125.png|mini|rechts|Die britische Parlamentsreform stellte sich keineswegs als tot, sondern als erhebliche Veränderung heraus, [[Punch (Zeitschrift)|Punch]] karikierte sie als Sprung ins Ungewisse.]]

'''Flogging a Dead horse''' (en. ''ein totes Pferd schlagen'') ist eine englischsprachige Redewendung. Sie beschreibt ein Verhalten, das Zeit oder Kraft vergeblich in eine gescheiterte Angelegenheit verwendet.

== Verwendung ==
Der erste prominente Gebrauch des Ausdrucks wird dem englischen Politiker [[John Bright]] zugeschrieben, der damit die spätere [[Reform Act 1867|Britische Parlamentsreform von 1867]] charakterisierte. Bright, einer der bedeutendsten Rhetoriker seiner Generation und ein Vertreter der [[Radikale (Vereinigtes Königreich)|Radikalen]] hatte damit die lange vor sich hindümpelnde englische Wahlrechtsreform bezeichnet. Er versuchte so, die anfänglich wenig interessierten Parlamentsmitglieder zu mehr Engagement zu veranlassen. Die letztlich durchgeführte Reform hatte erhebliche Auswirkungen, da sie die zugelassene Wählerschaft nahezu verdoppelte. Im ''Oxford English Dictionary'' wird ein Artikel von ''The Globe'', einige Jahre später, 1872 als frühester nachweislicher schriftlicher Beleg der Redewendung angegeben.<ref>''The Globe.'' 1. Aug 1872.</ref>

== Mögliche Vorgänger ==
Bereits im 17. Jahrhundert stand der Ausdruck ''dead horse'' für eine im Voraus bezahlte Arbeitsleistung."<ref>Nicker Nicked in Harl. Misc. (Park) II. 110 (1668)</ref> Im voraus bezahlte Arbeit wurde demnach weniger sorgfältig oder vollständig durchgeführt, der Auftraggeber hatte aber wenig Sanktions- oder Motivationsmöglichkeiten. Bei Schiffsbesatzungen, die zumeist für den ersten Monat auf dem Boot im Voraus bezahlt wurden, gab es die Tradition einer Dead-Horse-Feier nach Abschluss dieser Zeit.<ref>Alfred Simmons: ''Old England and New Zealand.'' London 1879, S. 113.</ref>

Das ''tote Pferd schlagen'' war vor Bright nun ungewöhnlich. Bekannt war aber die als etwas gestelzt geltende Redewendung ''to slay the slain'' (die Erschlagenen erschlagen). Sie findet sich bei [[John Dryden]]s ''Alexanders Feast'' und geht auf ein Zitat aus [[Sophokles]]s [[Antigone (Sophokles)|Antigone]] zurück.{{"-en|Nay, allow the claim of the dead; stab not the fallen; what prowess is it to slay the slain anew?|Übersetzung=Weich du dem Toten und verfolge nicht Den, der dahin ist. Welche Kraft ist das, Zu töten Tote? |Monolog des Tiresias<ref>{{cite web|url=http://www.monologuearchive.com/s/sophocles_006.html |title=Antigone: Tiresias' Monologue, Übersetzung nach Hölderlin, Antigonae/Vierter Akt |publisher=Monologue Archive| accessdate=2013-09-21}}</ref>}}

''Slay the slain'' anzuführen, galt als Zeichen von Bildung und Ausdruck eines gehobenen literarischen Stils. Dies findet sich unter anderem bei dem im Rahmen der [[Hippocampus-Debatte]] veröffentlichten Spottgedicht Monkeyana in Punch. Hintergrund war eine öffentlich ausgetragene Debatte im Mai 1861 und weitere Auseinandersetzungen um [[Charles Darwin]]s ''[[Origin of Species]]'' und dessen Verteidigung durch [[Thomas Henry Huxley]].<ref>{{cite web|url=http://aleph0.clarku.edu/huxley/comm/Punch/Monkey.html |title="Monkeyana", ''Punch'', May 1861 |publisher=THE HUXLEY FILE| accessdate=2013-09-21}}</ref>

Brights Zitat fasst damit zwei unterschiedliche Redewendungen aus verschiedenen Sprachebenen in einem einprägsamen Bild zusammen. Olivia Isils Buch zu maritimen Redewendungen in der englischen Alltagssprache nimmt dies im Titel mit auf.{{Zitat-en|When a Loose Cannon Flogs a Dead Horse There's the Devil to Pay: Seafaring Words in Everyday Speech |Übersetzung=Wenn ein wandelndes Pulverfass ein totes Pferd prügelt, kommt das dicke Ende noch zum Schluss - Maritime Ausdrücke in der Alltagssprache|autor=Olivia Isil|ref=<ref>{{cite book |title=When a Loose Cannon Flogs a Dead Horse There's the Devil to Pay: Seafaring Words in Everyday Speech |first=Olivia A. |last=Isil |publisher=International Marine |year=1996 |isbn=0-07-032877-3}}</ref>}}

== Weitere Verwendungen ==
Der israelische Wissenschaftshistoriker [[Joseph Agassi]] verfasste einen Essay unter dem Titel {{Zitat-en|On the Merit of Flogging Dead Horses|Übersetzung=Warum es sich lohnt, tote Pferde zu prügeln|autor=[[Joseph Agassi]]|ref=<ref>Joseph Agassi: ''FOURTH PRELIMINARY ESSAY: ON THE MERIT OF FLOGGING DEAD HORSES.'' In: ''Science and Its History. A Reassessment of the Historiography of Science.'' Springer, 2008, ISBN 978-1-4020-5632-1.</ref>}} Er spricht sich bei wissenschaftlichen Arbeiten dafür aus, gelegentlich doch einst gründlich widerlegte Theorien neu zu untersuchen. Dies sei bei wissenschaftlichem Arbeiten auch nicht zu vermeiden.

Es gibt unter anderem einige Musikalben, die auf die Redewendung anspielen. ''Flogging a Dead Horse'' von den [[Sex Pistols]] stellte bereits nach Auflösung der Band 1980 noch einige erfolgreiche Singles zusammen. ''Beating Dead Horses'' stammt von der Industrial Metal Band 16Volt, ''Beating a Dead Horse to Death... Again'' ist ein 2008 erschienenes Album von Dog Fashion Disco.

Als Buchtitel findet sie sich bei der Biographie und Werkübersicht des Künstlerbrüderpaars [[Jake und Dinos Chapman]].<ref>Jake & Dinos Chapman (Hrsg.): ''Flogging a Dead Horse: The Life and Works of Jake & Dinos Chapman.'' Rizzoli International Publications, New York 2011, ISBN 978-0-8478-3478-5.</ref> Die beiden Provokateure hatten unter anderem Originalaquarelle von Adolf Hitler aufgekauft und unter dem Titel ''If Hitler Had Been a Hippy How Happy Would We Be'' mit Smileys und Sonnenblumen versehen.<ref>Mark Brown: [http://www.theguardian.com/artanddesign/2008/may/30/art ''Hitler gets Chapman treatment as Hell rises from the ashes.''] In: ''The Guardian.'' 30. Mai 2008.</ref>

== Weblinks ==
{{Wiktionary|flog a dead horse|beat a dead horse}}
* [http://www.takeourword.com/TOW207/page1.html "Sensational Etymologies"], TakeOurWord.com
* [http://www.goenglish.com/BeatADeadHorse.asp "Beating A Dead Horse"], GoEnglish.com

== Einzelnachweise ==
<references />

{{SORTIERUNG:Flogging a dead horse}}

[[Kategorie:Redewendung]]

CDC 6600

2013-08-26T22:30:47Z

Loadmaster: /* Central Processor (CP) */ "Instruction pointer" → "Program counter"

{{no footnotes|date=March 2013}}
[[File:CDC 6600.jc.jpg|thumb|300px|right| The CDC 6600. Behind the system console are two of the "arms" of the plus-sign shaped cabinet with the covers opened. Individual modules can be seen inside. The racks holding the modules are hinged to give access to the racks behind them. Each arm of the machine had up to four such racks. On the right is the cooling system.]]
[[Image:Control Data 6600 mainframe.jpg|thumb|right|300px|A CDC 6600 system console. The displays were driven through software, primarily to provide text display (in a choice of three sizes). It also provided a way to draw simple graphics. Unlike more modern displays, the console was a vector drawing system rather than a raster system. Line were drawn by specifying a start and end point. The consoles had a single font where each glyph was a series of vectors.]]

The '''CDC 6600''' was a [[mainframe computer]] from [[Control Data Corporation]], first delivered in 1964 to the Lawrence Radiation Laboratory, part of the University of California at Berkeley. It was used primarily for high-energy nuclear physics research, particularly for the analysis of nuclear events photographed inside the Alvarez bubble chamber. The very first CDC 6600 was delivered about one year earlier to Conseil Européen pour la Recherche Nucléaire ([[CERN]]) near Geneva, Switzerland, also for use in high-energy nuclear physics research.
It is generally considered to be the first successful [[supercomputer]], outperforming its fastest predecessor, [[IBM 7030 Stretch]], by about three times. With performance of about 1 [[FLOPS|megaFLOPS]],<ref>[http://www.princeton.edu/~achaney/tmve/wiki100k/docs/CDC_6600.html]</ref> it remained the world's fastest computer from 1964 to 1969, when it relinquished that status to its successor, the [[CDC 7600]].

The system organization of the CDC 6600 was used for the simpler (and slower) [[CDC 6400]], and later a version containing two 6400 processors known as the CDC 6500. These machines were instruction-compatible with the 6600, but ran slower due to a much simpler and more sequential processor design. The entire family is now referred to as the [[CDC 6000 series]]. The [[CDC 7600]] was originally to be compatible as well, starting its life as the CDC 6800, but during the design, compatibility was dropped in favor of outright performance. While the 7600 CPU remained compatible with the 6600, allowing portable user code, the peripheral processor units (PPUs) were different, requiring a different operating system.

A CDC 6600 is on display at the [[Computer History Museum]] in [[Mountain View, California]].

==History and impact==
{{main|Control Data Corporation}}

CDC's first products were based on the machines designed at [[Engineering Research Associates|ERA]], which [[Seymour Cray]] had been asked to update after moving to CDC. After an experimental machine known as the ''Little Character'', they delivered the [[CDC 1604]], one of the first commercial [[Transistor computer|transistor-based computers]], and one of the fastest machines on the market. Management was delighted, and made plans for a new series of machines that were more tailored to business use; they would include instructions for character handling and record keeping for instance. Cray was not interested in such a project, and set himself the goal of producing a new machine that would be 50 times faster than the 1604. When asked to complete a detailed report on plans at one and five years into the future, he wrote back that his five year goal was "to produce the largest computer in the world", "largest" at that time being synonymous with "fastest", and that his one year plan was "to be one-fifth of the way".

Taking his core team to new offices nearby the original CDC headquarters, they started to experiment with higher quality versions of the "cheap" [[transistor]]s Cray had used in the 1604. After much experimentation, they found that there was simply no way the [[germanium]]-based transistors could be run much faster than those used in the 1604. The "business machine" that management had originally wanted, now forming as the [[CDC 3000]] series, pushed them about as far as they could go. Cray then decided the solution was to work with the then-new [[silicon]]-based transistors from [[Fairchild Semiconductor]], which were just coming onto the market and offered dramatically improved switching performance.

During this period, CDC grew from a startup to a large company and Cray became increasingly frustrated with what he saw as ridiculous management requirements. Things became considerably more tense in 1962 when the new [[CDC 3600]] started to near production quality, and appeared to be exactly what management wanted, when they wanted it. Cray eventually told CDC's CEO, [[William Norris (CEO)|William Norris]] that something had to change, or he would leave the company. Norris felt he was too important to lose, and gave Cray the green light to set up a new lab wherever he wanted.

After a short search, Cray decided to return to his home town of [[Chippewa Falls, WI]], where he purchased a block of land and started up a new lab. Although this process introduced a fairly lengthy delay in the design of his new machine, once in the new lab, without management interference, things started to progress quickly. By this time, the new transistors were becoming quite reliable, and modules built with them tended to work properly on the first try. Working with Jim Thornton, who was the system architect and the 'hidden genius' behind the 6600, the machine soon took form.

More than 100 CDC 6600s were sold over the machine's lifetime. Many of these went to various [[nuclear bomb]]-related labs, and quite a few found their way into university computing labs. Cray immediately turned his attention to its replacement, this time setting a goal of 10 times the performance of the 6600, delivered as the [[CDC 7600]]. The later [[CDC Cyber]] 70 and 170 computers were very similar to the CDC 6600 in overall design and were nearly completely backwards compatible.

==Description==
Typical machines of the era used a single [[Central processing unit|CPU]] to drive the entire system. A typical program would first load data into memory (often using pre-rolled library code), process it, and then write it back out. This required the CPUs to be fairly complex in order to handle the complete set of instructions they would be called on to perform. A complex CPU implied a large CPU, introducing signalling delays while information flowed between the individual modules making it up. These delays set a maximum upper limit on performance, the machine could only operate at a cycle speed that allowed the signals time to arrive at the next module.

Cray took another approach. At the time, CPUs generally ran slower than the [[main memory]] they were attached to. For instance, a processor might take 15 cycles to multiply two numbers, while each memory access took only one or two. This meant there was a significant time where the main memory was idle. It was this idle time that the 6600 exploited.

Instead of trying to make the CPU handle all the tasks, the 6600 CPUs handled arithmetic and logic only. This resulted in a much smaller CPU which could operate at a higher clock speed. Combined with the faster switching speeds of the silicon transistors, the new CPU design easily outperformed everything then available. The new design ran at 10 MHz (100 ns cycle), about ten times faster than other machines on the market. In addition to the clock being faster, the simple processor executed instructions in fewer clock cycles; for instance, the CPU could complete a multiplication in ten cycles.

However, the CPU could only execute a limited number of simple instructions. A typical CPU of the era had a [[Complex instruction set computer|complex instruction set]], which included instructions to handle all the normal "housekeeping" tasks such as memory access and [[input/output]]. Cray instead implemented these instructions in separate, simpler processors dedicated solely to these tasks, leaving the CPU with a much smaller instruction set. (This was the first of what later came to be called [[reduced instruction set computer]] (RISC) design.) By allowing the CPU, peripheral processors (PPs) and I/O to operate in parallel, the design considerably improved the performance of the machine. Under normal conditions a machine with several processors would also cost a great deal more. Key to the 6600's design was to make the I/O processors, known as ''peripheral processors'' (PPs), as simple as possible. The PPs were based on the simple 12-bit [[CDC 160A]], which ran much slower than the CPU, gathering up data and "squirting" it into main memory at high speed via dedicated hardware.

The machine as a whole operated in a fashion known as ''barrel and slot'', the "barrel" referring to the ten PPs, and the "slot" the main CPU. For any given slice of time, one PP was given control of the CPU, asking it to complete some task (if required). Control was then handed off to the next PP in the barrel. Programs were written, with some difficulty, to take advantage of the exact timing of the machine to avoid any "dead time" on the CPU. With the CPU running much faster than on other computers, each memory access took ten CPU clock cycles to complete, so by using ten PPs, each PP was guaranteed one memory access per machine cycle.

The 10 PPs were implemented virtually; there was CPU hardware only for a single PP. This CPU hardware was shared and operated on 10 PP register sets which represented each of the 10 PP ''states'' (similar to modern [[temporal multithreading|multithreading]] processors). The PP ''[[barrel processor|register barrel]]'' would "rotate", with each PP register set presented to the "slot" which the actual PP CPU occupied. The shared CPU would execute all or some portion of a PP's instruction whereupon the barrel would "rotate" again, presenting the next PP's register set (state). Multiple "rotations" of the barrel were needed to complete an instruction. A complete barrel "rotation" occurred in 1000 nanoseconds (100 nanoseconds per PP), and an instruction could take from 1 to 5 "rotations" of the barrel to be completed, or more if it was a data transfer instruction.

The basis for the 6600 CPU is what would today be referred to as a [[RISC]] system, one in which the processor is tuned to do instructions which are comparatively simple and have limited and well-defined access to memory. The philosophy of many other machines was toward using instructions which were complicated — for example, a single instruction which would fetch an operand from memory and add it to a value in a register. In the 6600, loading the value from memory would require one instruction, and adding it would require a second one. While slower in theory due to the additional memory accesses, the fact that in well-scheduled code, multiple instructions could be processing in parallel offloaded this expense. This simplification also forced programmers to be very aware of their memory accesses, and therefore code deliberately to reduce them as much as possible.

===Central Processor (CP)===
{| class="infobox" style="font-size:88%"
|-
|style="text-align:center" |''CDC 6x00 registers''
|-
|
{| style="font-size:88%;"
|-
| style="width:10px; text-align:center;"| 59
| style="width:160px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 17
| style="width:70px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 00
| style="width:auto;" | ''(bit position)''
|-
|colspan="6" | '''General purpose registers''' ''(60 bits)''
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X0
| style="width:auto; background:white; color:black;"| Register 0
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X1
| style="width:auto; background:white; color:black;"| Register 1
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X2
| style="width:auto; background:white; color:black;"| Register 2
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X3
| style="width:auto; background:white; color:black;"| Register 3
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X4
| style="width:auto; background:white; color:black;"| Register 4
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X5
| style="width:auto; background:white; color:black;"| Register 5
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X6
| style="width:auto; background:white; color:black;"| Register 6
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X7
| style="width:auto; background:white; color:black;"| Register 7
|-
|colspan="6" | '''Address registers''' ''(18 bits)''
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A0
| style="width:auto; background:white; color:black;"| Address 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A1
| style="width:auto; background:white; color:black;"| Address 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A2
| style="width:auto; background:white; color:black;"| Address 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A3
| style="width:auto; background:white; color:black;"| Address 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A4
| style="width:auto; background:white; color:black;"| Address 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A5
| style="width:auto; background:white; color:black;"| Address 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A6
| style="width:auto; background:white; color:black;"| Address 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A7
| style="width:auto; background:white; color:black;"| Address 7
|-
|colspan="6" | '''Scratchpad registers''' ''(18 bits)''
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B0 ''(all bits zero)''
| style="width:auto; background:white; color:black;"| Scratchpad 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B1
| style="width:auto; background:white; color:black;"| Scratchpad 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B2
| style="width:auto; background:white; color:black;"| Scratchpad 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B3
| style="width:auto; background:white; color:black;"| Scratchpad 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B4
| style="width:auto; background:white; color:black;"| Scratchpad 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B5
| style="width:auto; background:white; color:black;"| Scratchpad 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B6
| style="width:auto; background:white; color:black;"| Scratchpad 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B7
| style="width:auto; background:white; color:black;"| Scratchpad 7
|-
|colspan="6" | '''Program counter''' ''(18 bits)''
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center;" colspan="3"| PC
| style="background:white; color:black;"| '''P'''rogram '''C'''ounter
|}

|}

The Central Processor (CP) and main memory of the 6400, 6500, and 6600 machines had a 60-bit word length. The Central Processor had eight general purpose [[60-bit]] [[processor register|registers]] X0 through X7, eight [[18-bit]] address registers A0 through A7, and eight 18-bit scratchpad registers B0 through B7. B0 was held permanently at zero by the hardware; many programmers found it useful to set B1 to 1 and then treat it as similarly inviolate.

The CP had no instructions for input and output, which are accomplished through Peripheral Processors (below). No opcodes were specifically dedicated to loading or storing memory; this occurred as a side effect of assignment to certain A registers. Setting A1 through A5 loaded the word at that address into X1 through X5 respectively; setting A6 or A7 stored a word from X6 or X7. No side effects were associated with A0.
A separate hardware load/store unit handled the actual data movement independently of the operation of the instruction stream, allowing other operations to complete while memory was being accessed, which required (best case) eight cycles.

The 6600 CP included 10 parallel functional units, allowing multiple instructions to be worked on at the same time. Today, this is known as a [[superscalar]] design, but it was unique for its time. Unlike most modern CPU designs, functional units were not pipelined; the functional unit would become busy when an instruction was "issued" to it and would remain busy for the entire time required to execute that instruction. (By contrast, the CDC 7600 introduced pipelining into its functional units.) In the best case, an instruction could be issued to a functional unit every 100 ns clock cycle. The system read and decoded instructions from memory as fast as possible, generally faster than they could be completed, and fed them off to the units for processing. The units were:
*floating point multiply (2 copies)
*floating point divide
*floating point add
*"long" integer add
*incrementers (2 copies; performed memory load/store)
*shift
*boolean logic
*branch

Floating-point operations were given pride of place in this [[Computer architecture|architecture]]: the CDC 6600 (and kin) stand virtually alone in being able to execute a 60-bit [[floating point]] multiplication in time comparable to that for a program branch.

Previously executed instructions were saved in an eight-word [[CPU cache|cache]], called the "stack". In-stack jumps were quicker than out-of-stack jumps because no memory fetch was required. The stack was flushed by an unconditional jump instruction, so unconditional jumps at the ends of loops were conventionally written as conditional jumps that would always succeed.

The system used a 10-megahertz clock, but used a [[clock signal#4-phase clock|four-phase signal]], so the system could at times effectively operate at 40 MHz. A floating-point multiplication took ten cycles, a division took 29, and the overall performance, taking into account memory delays and other issues, was about 3 [[MFLOPS]]. Using the best available compilers, late in the machine's history, [[FORTRAN]] programs could expect to maintain about 0.5 MFLOPS.

===Memory organization===

User programs are restricted to use only a contiguous area of main memory. The portion of memory to which an executing program has access is controlled by the ''RA'' (Relative Address) and ''FL'' (Field Length) registers which are not accessible to the user program. When a user program tries to read or write a word in central memory at address ''a'', the processor will first verify that a is between 0 and FL-1. If it is, the processor accesses the word in central memory at address RA+a. This process is known as base-bound relocation; each user program sees core memory as a contiguous block words with length FL, starting with address 0; in fact the program may be anywhere in the physical memory. Using this technique, each user program can be moved ("relocated") in main memory by the operating system, as long as the RA register reflects its position in memory. A user program which attempts to access memory outside the allowed range (that is, with an address which is not less than FL) will trigger an interrupt, and will be terminated by the operating system. When this happens, the operating system may create a [[core dump]] which records the contents of the program's memory and registers in a file, allowing the developer of the program a means to know what happened. Note the distinction with [[virtual memory]] systems; in this case, the entirety of a process's addressable space must be in core memory, must be contiguous, and its size cannot be larger than the real memory capacity.

All but the first seven [[CDC 6000 series]] machines could be configured with an optional Extended Core Storage (ECS) system. ECS was built from a different variety of core memory than was used in the central memory. This made it economical for it to be both larger and slower. The primary reason was that ECS memory was wired with only two wires per core (contrast with 5 for central memory), Because it performed very wide transfers, its sequential transfer rate was the same as that of the small core memory. A 6000 CPU could directly perform block memory transfers between a user's program (or operating system) and the ECS unit. Wide data paths were used, so this was a very fast operation. Memory bounds were maintained in a similar manner as central memory — with an RA/FL mechanism maintained by the operating system. ECS could be used for a variety of purposes, including containing user data arrays that were too large for central memory, holding often-used files, swapping, and even as a communication path in a multi-mainframe complex.

===Peripheral Processors (PPs)===
To handle the 'household' tasks which other designs put in the CPU, Cray included ten other processors, based partly on his earlier computer, the [[CDC 160A]]. These machines, called Peripheral Processors, or PPs, were full computers in their own right, but were tuned to performing [[Input/output|I/O]] tasks and running the operating system. One of the PPs was in overall control of the machine, including control of the program running on the main CPU, while the others would be dedicated to various I/O tasks — quite similarly to [[I/O channel]]s in [[IBM mainframe]]s of the time. When the program needed to perform some sort of I/O, it instead loaded a small program into one of these other machines and let it do the work. The PP would then inform the CPU when the task was complete with an interrupt.

Each PP included its own memory of 4096 [[12-bit]] words. This memory served for both for I/O buffering and program storage, but the execution units were shared by 10 PPs, in a configuration called the [[Barrel processor|Barrel and slot]]. This meant that the execution units (the "slot") would execute one instruction cycle from the first PP, then one instruction cycle from the second PP, etc. in a round robin fashion. This was done both to reduce costs, and because access to CP memory required 10 PP clock cycles: when a PP accesses CP memory, the data is available next time the PP receives its slot time.

===Wordlengths, characters===
The central processor had [[60-bit]] words, whilst the peripheral processors had [[12-bit]] words. CDC used the term "byte" to refer to 12-bit entities used by peripheral processors; characters were 6-bit, and central processor instructions were either 15 bits, or 30 bits with a signed 18-bit address field, the latter allowing for a directly addressable memory space of 128K words of central memory (converted to modern terms, with 8-bit bytes, this is 0.94 MB). The signed nature of the address registers limited an individual program to 128K words. (Later CDC 6000-compatible machines could have 256K or more words of central memory, budget permitting, but individual user programs were still limited to 128K words of CM.) Central processor instructions started on a word boundary when they were the target of a jump statement or subroutine return jump instruction, so no-operations were sometimes required to fill out the last 15, 30 or 45 bits of a word.

The 6-bit characters, in an encoding called [[display code]], could be used to store up to 10 characters in a word. They permitted a character set of 64 characters, which is enough for all upper case letters, digits, and some punctuation. Certainly, enough to write [[FORTRAN]], or print financial or scientific reports. There were actually two variations of the [[display code]] character sets in use, 64-character and 63-character. The 64-character set had the disadvantage that two consecutive ':' (colon) characters might be interpreted as the end of a line if they fell at the end of a 10-byte word. A later variant, called [[CDC display code#6/12 display code|6/12 display code]], was also used in the [[CDC Kronos|Kronos]] and [[NOS (software)|NOS]] timesharing systems to allow full use of the [[ASCII]] character set in a manner somewhat compatible with older software.

With no byte addressing instructions at all, code had to be written to pack and shift characters into words. The very large words, and comparatively small amount of memory, meant that programmers would frequently economize on memory by packing data into words at the bit level.

It is interesting to note that due to the large word size, and with 10 characters per word, it was often faster to process words full of characters at a time — rather than unpacking/processing/repacking them. For example, the CDC [[COBOL]] compiler was actually quite good at processing decimal fields using this technique. These sorts of techniques are now commonly used in the 'multi-media' instructions of current processors.

===Physical design===
[[Image:CDCcordwood1.jpg|thumb|right|300px|A CDC 6600 cordwood logic module containing 64 silicon transistors. The coaxial connectors are test points. The module is cooled conductively via the front panel. The 6600 model contained nearly 6,000 such modules.<ref>Understanding Computers: Speed and Power 1990, p. 17.</ref>]]

The machine was built in a plus-sign-shaped cabinet with a pump and heat exchanger in the outermost {{convert|18|in|cm|abbr=on}} of each of the four arms. Cooling was done with [[Freon]] circulating within the machine and exchanging heat to an external chilled water supply. Each arm could hold four chassis, each about {{convert|8|in|cm|abbr=on}} thick, hinged near the center, and opening a bit like a book. The intersection of the "plus" was filled with cables which interconnected the chassis. The chassis were numbered from 1 (containing all 10 PPUs and their memories, as well as the 12 rather minimal I/O channels) to 16. The main memory for the CPU was spread over many of the chassis. In a system with only 64K words of main memory, one of the arms of the "plus" was omitted.

The logic of the machine was packaged into modules about {{convert|2.5|in|mm|abbr=on}} square and about {{convert|1|in|cm|abbr=on}} thick. Each module had a connector (30 pins, two vertical rows of 15) on one edge, and six test points on the opposite edge. The module was placed between two aluminum cold plates to remove heat. The module itself consisted of two parallel printed circuit boards, with components mounted either on one of the boards or between the two boards. This provided a very dense package; somewhat difficult to repair, but with good heat removal. It was known as [[Printed circuit board#.22Cordwood.22 construction|cordwood construction]].

==Operating system and programming==
There was a sore point with the 6600 [[operating system]] support — slipping timelines. The machines originally ran a very simple [[batch processing|job-control]] system known as ''COS'' ([[Chippewa Operating System]]), which was quickly "thrown together" based on the earlier [[CDC 3000]] operating system in order to have something running to test the systems for delivery. However the machines were intended to be delivered with a much more powerful system known as ''SIPROS'' (for Simultaneous Processing Operating System), which was being developed at the company's System Sciences Division in [[Los Angeles]]. Customers were impressed with SIPROS's feature list, and many had SIPROS written into their delivery contracts.

SIPROS turned out to be a major fiasco. Development timelines continued to slip, costing CDC major amounts of profit in the form of delivery delay penalties. After several months of waiting with the machines ready to be shipped, the project was eventually cancelled. The programmers who had worked on COS had little faith in SIPROS and had continued working on improving COS.

[[Operating system development]] then split into two camps. The CDC-sanctioned evolution of COS was undertaken at the [[Sunnyvale, California]] software development lab. Many customers eventually took delivery of their systems with this software, then known as ''[[SCOPE (software)|SCOPE]]'' (Supervisory Control Of Program Execution). (Some Control Data Field Engineers used to refer to SCOPE as ''Sunnyvale's Collection Of Programming Errors''). SCOPE version 1 was, essentially, dis-assembled COS; SCOPE version 2 included new device and file system support; SCOPE version 3 included permanent file support, EI/200 remote batch support, and INTERCOM [[time sharing]] support. SCOPE always had significant reliability and maintainability issues.

The underground evolution of COS took place at the [[Arden Hills, Minnesota]] assembly plant. ''MACE'' ([Greg] Mansfield And [Dave] Cahlander Executive) was written largely by a single programmer in the off-hours when machines were available. Its feature set was essentially the same as COS and SCOPE 1. It retained the earlier COS file system, but made significant advances in code modularity to improve system reliability and adaptiveness to new storage devices. MACE was never an official product, although many customers were able to wrangle a copy from CDC.

MACE was later used as the basis of ''[[CDC Kronos|Kronos]]'', named after the [[Chronos|Greek god of time]]. The main marketing reason for its adoption was the development of its TELEX [[time sharing]] feature and its BATCHIO remote batch feature. Kronos continued to use the COS/SCOPE 1 file system with the addition of a permanent file feature.

An attempt to unify the SCOPE and Kronos operating system products produced ''[[NOS (software)|NOS]]'', (Network Operating System). NOS was intended to be the sole operating system for all CDC machines, a fact CDC promoted heavily. Many SCOPE customers remained software-dependent on the SCOPE architecture, so CDC simply renamed it ''[[NOS/BE]]'' (Batch Environment), and were able to claim that everyone was thus running NOS. In practice, it was far easier to modify the Kronos code base to add SCOPE features than the reverse.

The assembly plant environment also produced other operating systems which were never intended for customer use. These included the engineering tools SMM for hardware testing, and KALEIDOSCOPE, for software [[smoke testing]]. Another commonly used tool for CDC Field Engineers during testing was MALET (Maintenance Application Language for Equipment Testing), which was used to stress test components and devices after repairs and/or servicing by engineers. Testing conditions often used hard disk packs and magnetic tapes which were deliberately marked with errors to determine if the errors would be detected by MALET and the engineer.

==See also==
* [[History of supercomputing]]

== Notes ==
{{Reflist}}

==References==
*Grishman, Ralph (1974). ''Assembly Language Programming for the Control Data 6000 Series and the Cyber 70 Series''. New York, NY: Algorithmics Press. [http://www.bitsavers.org/pdf/cdc/cyber/books/Grishman_CDC6000AsmLangPgmg.pdf]
*[http://ed-thelen.org/comp-hist/CDC-6600-R-M.html#TOC/ CONTROL DATA 6400/6500/6600 COMPUTER SYSTEMS Reference Manual]
*Thornton, J. (1963). ''Considerations in Computer Design - Leading up to the Control Data 6600'' [http://www.bitsavers.org/pdf/cdc/cyber/cyber_70/thornton_6600_paper.pdf]
*Thornton, J. (1970). ''Design of a Computer—The Control Data 6600''. Glenview, IL: Scott, Foresman and Co. [http://www.bitsavers.org/pdf/cdc/cyber/books/DesignOfAComputer_CDC6600.pdf]
*(1990) ''Understanding Computers: Speed and Power, a TIME LIFE series'' ISBN 0809475863

==External links==
*[http://purl.umn.edu/104327 Neil R. Lincoln with 18 Control Data Corporation (CDC) engineers on computer architecture and design], [[Charles Babbage Institute]], University of Minnesota. Engineers include Robert Moe, Wayne Specker, Dennis Grinna, Tom Rowan, Maurice Hutson, Curt Alexander, Don Pagelkopf, Maris Bergmanis, Dolan Toth, Chuck Hawley, Larry Krueger, Mike Pavlov, Dave Resnick, Howard Krohn, Bill Bhend, Kent Steiner, Raymon Kort, and Neil R. Lincoln. Discussion topics include [[CDC 1604]], CDC 6600, [[CDC 7600]], [[CDC 8600]], [[CDC STAR-100]] and [[Seymour Cray]].
*[http://research.microsoft.com/~gbell/Computer_Structures__Readings_and_Examples/00000509.htm Parallel operation in the Control Data 6600, James Thornton]
*[http://research.microsoft.com/users/gbell/craytalk/sld001.htm Presentation of the CDC 6600 and other machines designed by Seymour Cray] – by C. [[Gordon Bell]] of Microsoft Research (formerly of DEC)

{{DEFAULTSORT:Cdc 6600}}
[[Category:1964 introductions]]
[[Category:CDC hardware|6600]]
[[Category:Supercomputers]]
[[Category:Transistorized computers]]
[[Category:Superscalar microprocessors]]

CDC 6600

2013-07-09T01:48:32Z

Loadmaster: /* Central Processor (CP) */ Added CPU register table/infobox

{{no footnotes|date=March 2013}}
[[File:CDC 6600.jc.jpg|thumb|300px|right| The CDC 6600. Behind the system console are two of the "arms" of the plus-sign shaped cabinet with the covers opened. Individual modules can be seen inside. The racks holding the modules are hinged to give access to the racks behind them. Each arm of the machine had up to four such racks. On the right is the cooling system.]]
[[Image:Control Data 6600 mainframe.jpg|thumb|right|300px|A CDC 6600 system console. The displays were driven through software, primarily to provide text display (in a choice of three sizes). It also provided a way to draw simple graphics. Unlike more modern displays, the console was a vector drawing system rather than a raster system. Line were drawn by specifying a start and end point. The consoles had a single font where each glyph was a series of vectors.]]

The '''CDC 6600''' was a [[mainframe computer]] from [[Control Data Corporation]], first delivered in 1964 to the Lawrence Radiation Laboratory, part of the University of California at Berkeley. It was used primarily for high-energy nuclear physics research, particularly for the analysis of nuclear events photographed inside the Alvarez bubble chamber. The very first CDC 6600 was delivered about one year earlier to Conseil Européen pour la Recherche Nucléaire ([[CERN]]) near Geneva, Switzerland, also for use in high-energy nuclear physics research.
It is generally considered to be the first successful [[supercomputer]], outperforming its fastest predecessor, [[IBM 7030 Stretch]], by about three times. With performance of about 1 [[FLOPS|megaFLOPS]],<ref>[http://www.princeton.edu/~achaney/tmve/wiki100k/docs/CDC_6600.html]</ref> it remained the world's fastest computer from 1964 to 1969, when it relinquished that status to its successor, the [[CDC 7600]].

The system organization of the CDC 6600 was used for the simpler (and slower) [[CDC 6400]], and later a version containing two 6400 processors known as the CDC 6500. These machines were instruction-compatible with the 6600, but ran slower due to a much simpler and more sequential processor design. The entire family is now referred to as the [[CDC 6000 series]]. The [[CDC 7600]] was originally to be compatible as well, starting its life as the CDC 6800, but during the design, compatibility was dropped in favor of outright performance. While the 7600 CPU remained compatible with the 6600, allowing portable user code, the peripheral processor units (PPUs) were different, requiring a different operating system.

A CDC 6600 is on display at the [[Computer History Museum]] in [[Mountain View, California]].

==History and impact==
{{main|Control Data Corporation}}

CDC's first products were based on the machines designed at [[Engineering Research Associates|ERA]], which [[Seymour Cray]] had been asked to update after moving to CDC. After an experimental machine known as the ''Little Character'', they delivered the [[CDC 1604]], one of the first commercial [[Transistor computer|transistor-based computers]], and one of the fastest machines on the market. Management was delighted, and made plans for a new series of machines that were more tailored to business use; they would include instructions for character handling and record keeping for instance. Cray was not interested in such a project, and set himself the goal of producing a new machine that would be 50 times faster than the 1604. When asked to complete a detailed report on plans at one and five years into the future, he wrote back that his five year goal was "to produce the largest computer in the world", "largest" at that time being synonymous with "fastest", and that his one year plan was "to be one-fifth of the way".

Taking his core team to new offices nearby the original CDC headquarters, they started to experiment with higher quality versions of the "cheap" [[transistor]]s Cray had used in the 1604. After much experimentation, they found that there was simply no way the [[germanium]]-based transistors could be run much faster than those used in the 1604. The "business machine" that management had originally wanted, now forming as the [[CDC 3000]] series, pushed them about as far as they could go. Cray then decided the solution was to work with the then-new [[silicon]]-based transistors from [[Fairchild Semiconductor]], which were just coming onto the market and offered dramatically improved switching performance.

During this period, CDC grew from a startup to a large company and Cray became increasingly frustrated with what he saw as ridiculous management requirements. Things became considerably more tense in 1962 when the new [[CDC 3600]] started to near production quality, and appeared to be exactly what management wanted, when they wanted it. Cray eventually told CDC's CEO, [[William Norris (CEO)|William Norris]] that something had to change, or he would leave the company. Norris felt he was too important to lose, and gave Cray the green light to set up a new lab wherever he wanted.

After a short search, Cray decided to return to his home town of [[Chippewa Falls, WI]], where he purchased a block of land and started up a new lab. Although this process introduced a fairly lengthy delay in the design of his new machine, once in the new lab, without management interference, things started to progress quickly. By this time, the new transistors were becoming quite reliable, and modules built with them tended to work properly on the first try. Working with Jim Thornton, who was the system architect and the 'hidden genius' behind the 6600, the machine soon took form.

More than 100 CDC 6600s were sold over the machine's lifetime. Many of these went to various [[nuclear bomb]]-related labs, and quite a few found their way into university computing labs. Cray immediately turned his attention to its replacement, this time setting a goal of 10 times the performance of the 6600, delivered as the [[CDC 7600]]. The later [[CDC Cyber]] 70 and 170 computers were very similar to the CDC 6600 in overall design and were nearly completely backwards compatible.

==Description==
Typical machines of the era used a single [[Central processing unit|CPU]] to drive the entire system. A typical program would first load data into memory (often using pre-rolled library code), process it, and then write it back out. This required the CPUs to be fairly complex in order to handle the complete set of instructions they would be called on to perform. A complex CPU implied a large CPU, introducing signalling delays while information flowed between the individual modules making it up. These delays set a maximum upper limit on performance, the machine could only operate at a cycle speed that allowed the signals time to arrive at the next module.

Cray took another approach. At the time, CPUs generally ran slower than the [[main memory]] they were attached to. For instance, a processor might take 15 cycles to multiply two numbers, while each memory access took only one or two. This meant there was a significant time where the main memory was idle. It was this idle time that the 6600 exploited.

Instead of trying to make the CPU handle all the tasks, the 6600 CPUs handled arithmetic and logic only. This resulted in a much smaller CPU which could operate at a higher clock speed. Combined with the faster switching speeds of the silicon transistors, the new CPU design easily outperformed everything then available. The new design ran at 10 MHz (100 ns cycle), about ten times faster than other machines on the market. In addition to the clock being faster, the simple processor executed instructions in fewer clock cycles; for instance, the CPU could complete a multiplication in ten cycles.

However, the CPU could only execute a limited number of simple instructions. A typical CPU of the era had a [[Complex instruction set computer|complex instruction set]], which included instructions to handle all the normal "housekeeping" tasks such as memory access and [[input/output]]. Cray instead implemented these instructions in separate, simpler processors dedicated solely to these tasks, leaving the CPU with a much smaller instruction set. (This was the first of what later came to be called [[reduced instruction set computer]] (RISC) design.) By allowing the CPU, peripheral processors (PPs) and I/O to operate in parallel, the design considerably improved the performance of the machine. Under normal conditions a machine with several processors would also cost a great deal more. Key to the 6600's design was to make the I/O processors, known as ''peripheral processors'' (PPs), as simple as possible. The PPs were based on the simple 12-bit [[CDC 160A]], which ran much slower than the CPU, gathering up data and "squirting" it into main memory at high speed via dedicated hardware.

The machine as a whole operated in a fashion known as ''barrel and slot'', the "barrel" referring to the ten PPs, and the "slot" the main CPU. For any given slice of time, one PP was given control of the CPU, asking it to complete some task (if required). Control was then handed off to the next PP in the barrel. Programs were written, with some difficulty, to take advantage of the exact timing of the machine to avoid any "dead time" on the CPU. With the CPU running much faster than on other computers, each memory access took ten CPU clock cycles to complete, so by using ten PPs, each PP was guaranteed one memory access per machine cycle.

The 10 PPs were implemented virtually; there was CPU hardware only for a single PP. This CPU hardware was shared and operated on 10 PP register sets which represented each of the 10 PP ''states'' (similar to modern [[temporal multithreading|multithreading]] processors). The PP ''[[barrel processor|register barrel]]'' would "rotate", with each PP register set presented to the "slot" which the actual PP CPU occupied. The shared CPU would execute all or some portion of a PP's instruction whereupon the barrel would "rotate" again, presenting the next PP's register set (state). Multiple "rotations" of the barrel were needed to complete an instruction. A complete barrel "rotation" occurred in 1000 nanoseconds (100 nanoseconds per PP), and an instruction could take from 1 to 5 "rotations" of the barrel to be completed, or more if it was a data transfer instruction.

The basis for the 6600 CPU is what would today be referred to as a [[RISC]] system, one in which the processor is tuned to do instructions which are comparatively simple and have limited and well-defined access to memory. The philosophy of many other machines was toward using instructions which were complicated — for example, a single instruction which would fetch an operand from memory and add it to a value in a register. In the 6600, loading the value from memory would require one instruction, and adding it would require a second one. While slower in theory due to the additional memory accesses, the fact that in well-scheduled code, multiple instructions could be processing in parallel offloaded this expense. This simplification also forced programmers to be very aware of their memory accesses, and therefore code deliberately to reduce them as much as possible.

===Central Processor (CP)===
{| class="infobox" style="font-size:88%"
|-
|style="text-align:center" |''CDC 6x00 registers''
|-
|
{| style="font-size:88%;"
|-
| style="width:10px; text-align:center;"| 59
| style="width:160px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 17
| style="width:70px; text-align:center;"| . . .
| style="width:10px; text-align:center;"| 00
| style="width:auto;" | ''(bit position)''
|-
|colspan="6" | '''General purpose registers''' ''(60 bits)'' 
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X0
| style="width:auto; background:white; color:black;"| Register 0
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X1
| style="width:auto; background:white; color:black;"| Register 1
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X2
| style="width:auto; background:white; color:black;"| Register 2
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X3
| style="width:auto; background:white; color:black;"| Register 3
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X4
| style="width:auto; background:white; color:black;"| Register 4
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X5
| style="width:auto; background:white; color:black;"| Register 5
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X6
| style="width:auto; background:white; color:black;"| Register 6
|- style="background:silver;color:black"
| style="text-align:center" colspan="5"| X7
| style="width:auto; background:white; color:black;"| Register 7
|-
|colspan="6" | '''Address registers''' ''(18 bits)'' 
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A0
| style="width:auto; background:white; color:black;"| Address 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A1
| style="width:auto; background:white; color:black;"| Address 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A2
| style="width:auto; background:white; color:black;"| Address 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A3
| style="width:auto; background:white; color:black;"| Address 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A4
| style="width:auto; background:white; color:black;"| Address 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A5
| style="width:auto; background:white; color:black;"| Address 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A6
| style="width:auto; background:white; color:black;"| Address 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| A7
| style="width:auto; background:white; color:black;"| Address 7
|-
|colspan="6" | '''Scratchpad registers''' ''(18 bits)'' 
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B0 ''(all bits zero)''
| style="width:auto; background:white; color:black;"| Scratchpad 0
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B1
| style="width:auto; background:white; color:black;"| Scratchpad 1
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B2
| style="width:auto; background:white; color:black;"| Scratchpad 2
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B3
| style="width:auto; background:white; color:black;"| Scratchpad 3
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B4
| style="width:auto; background:white; color:black;"| Scratchpad 4
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B5
| style="width:auto; background:white; color:black;"| Scratchpad 5
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B6
| style="width:auto; background:white; color:black;"| Scratchpad 6
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center" colspan="3"| B7
| style="width:auto; background:white; color:black;"| Scratchpad 7
|-
|colspan="6" | '''Instruction pointer''' ''(18 bits)'' 
|- style="background:silver;color:black"
| style="text-align:center;background:#FFF" colspan="2"|  
| style="text-align:center;" colspan="3"| PC
| style="background:white; color:black;"| '''P'''rogram '''C'''ounter
|}

|}

The Central Processor (CP) and main memory of the 6400, 6500, and 6600 machines had a 60-bit word length. The Central Processor had eight general purpose [[60-bit]] [[processor register|registers]] X0 through X7, eight [[18-bit]] address registers A0 through A7, and eight 18-bit scratchpad registers B0 through B7. B0 was held permanently at zero by the hardware; many programmers found it useful to set B1 to 1 and then treat it as similarly inviolate.

The CP had no instructions for input and output, which are accomplished through Peripheral Processors (below). No opcodes were specifically dedicated to loading or storing memory; this occurred as a side effect of assignment to certain A registers. Setting A1 through A5 loaded the word at that address into X1 through X5 respectively; setting A6 or A7 stored a word from X6 or X7. No side effects were associated with A0.
A separate hardware load/store unit handled the actual data movement independently of the operation of the instruction stream, allowing other operations to complete while memory was being accessed, which required (best case) eight cycles.

The 6600 CP included 10 parallel functional units, allowing multiple instructions to be worked on at the same time. Today, this is known as a [[superscalar]] design, but it was unique for its time. Unlike most modern CPU designs, functional units were not pipelined; the functional unit would become busy when an instruction was "issued" to it and would remain busy for the entire time required to execute that instruction. (By contrast, the CDC 7600 introduced pipelining into its functional units.) In the best case, an instruction could be issued to a functional unit every 100 ns clock cycle. The system read and decoded instructions from memory as fast as possible, generally faster than they could be completed, and fed them off to the units for processing. The units were:
*floating point multiply (2 copies)
*floating point divide
*floating point add
*"long" integer add
*incrementers (2 copies; performed memory load/store)
*shift
*boolean logic
*branch

Floating-point operations were given pride of place in this [[Computer architecture|architecture]]: the CDC 6600 (and kin) stand virtually alone in being able to execute a 60-bit [[floating point]] multiplication in time comparable to that for a program branch.

Previously executed instructions were saved in an eight-word [[CPU cache|cache]], called the "stack". In-stack jumps were quicker than out-of-stack jumps because no memory fetch was required. The stack was flushed by an unconditional jump instruction, so unconditional jumps at the ends of loops were conventionally written as conditional jumps that would always succeed.

The system used a 10-megahertz clock, but used a [[clock signal#4-phase clock|four-phase signal]], so the system could at times effectively operate at 40 MHz. A floating-point multiplication took ten cycles, a division took 29, and the overall performance, taking into account memory delays and other issues, was about 3 [[MFLOPS]]. Using the best available compilers, late in the machine's history, [[FORTRAN]] programs could expect to maintain about 0.5 MFLOPS.

===Memory organization===

User programs are restricted to use only a contiguous area of main memory. The portion of memory to which an executing program has access is controlled by the ''RA'' (Relative Address) and ''FL'' (Field Length) registers which are not accessible to the user program. When a user program tries to read or write a word in central memory at address ''a'', the processor will first verify that a is between 0 and FL-1. If it is, the processor accesses the word in central memory at address RA+a. This process is known as base-bound relocation; each user program sees core memory as a contiguous block words with length FL, starting with address 0; in fact the program may be anywhere in the physical memory. Using this technique, each user program can be moved ("relocated") in main memory by the operating system, as long as the RA register reflects its position in memory. A user program which attempts to access memory outside the allowed range (that is, with an address which is not less than FL) will trigger an interrupt, and will be terminated by the operating system. When this happens, the operating system may create a [[core dump]] which records the contents of the program's memory and registers in a file, allowing the developer of the program a means to know what happened. Note the distinction with [[virtual memory]] systems; in this case, the entirety of a process's addressable space must be in core memory, must be contiguous, and its size cannot be larger than the real memory capacity.

All but the first seven [[CDC 6000 series]] machines could be configured with an optional Extended Core Storage (ECS) system. ECS was built from a different variety of core memory than was used in the central memory. This made it economical for it to be both larger and slower. The primary reason was that ECS memory was wired with only two wires per core (contrast with 5 for central memory), Because it performed very wide transfers, its sequential transfer rate was the same as that of the small core memory. A 6000 CPU could directly perform block memory transfers between a user's program (or operating system) and the ECS unit. Wide data paths were used, so this was a very fast operation. Memory bounds were maintained in a similar manner as central memory — with an RA/FL mechanism maintained by the operating system. ECS could be used for a variety of purposes, including containing user data arrays that were too large for central memory, holding often-used files, swapping, and even as a communication path in a multi-mainframe complex.

===Peripheral Processors (PPs)===
To handle the 'household' tasks which other designs put in the CPU, Cray included ten other processors, based partly on his earlier computer, the [[CDC 160A]]. These machines, called Peripheral Processors, or PPs, were full computers in their own right, but were tuned to performing [[Input/output|I/O]] tasks and running the operating system. One of the PPs was in overall control of the machine, including control of the program running on the main CPU, while the others would be dedicated to various I/O tasks — quite similarly to [[I/O channel]]s in [[IBM mainframe]]s of the time. When the program needed to perform some sort of I/O, it instead loaded a small program into one of these other machines and let it do the work. The PP would then inform the CPU when the task was complete with an interrupt.

Each PP included its own memory of 4096 [[12-bit]] words. This memory served for both for I/O buffering and program storage, but the execution units were shared by 10 PPs, in a configuration called the [[Barrel processor|Barrel and slot]]. This meant that the execution units (the "slot") would execute one instruction cycle from the first PP, then one instruction cycle from the second PP, etc. in a round robin fashion. This was done both to reduce costs, and because access to CP memory required 10 PP clock cycles: when a PP accesses CP memory, the data is available next time the PP receives its slot time.

===Wordlengths, characters===
The central processor had [[60-bit]] words, whilst the peripheral processors had [[12-bit]] words. CDC used the term "byte" to refer to 12-bit entities used by peripheral processors; characters were 6-bit, and central processor instructions were either 15 bits, or 30 bits with a signed 18-bit address field, the latter allowing for a directly addressable memory space of 128K words of central memory (converted to modern terms, with 8-bit bytes, this is 0.94 MB). The signed nature of the address registers limited an individual program to 128K words. (Later CDC 6000-compatible machines could have 256K or more words of central memory, budget permitting, but individual user programs were still limited to 128K words of CM.) Central processor instructions started on a word boundary when they were the target of a jump statement or subroutine return jump instruction, so no-operations were sometimes required to fill out the last 15, 30 or 45 bits of a word.

The 6-bit characters, in an encoding called [[display code]], could be used to store up to 10 characters in a word. They permitted a character set of 64 characters, which is enough for all upper case letters, digits, and some punctuation. Certainly, enough to write [[FORTRAN]], or print financial or scientific reports. There were actually two variations of the [[display code]] character sets in use, 64-character and 63-character. The 64-character set had the disadvantage that two consecutive ':' (colon) characters might be interpreted as the end of a line if they fell at the end of a 10-byte word. A later variant, called [[CDC display code#6/12 display code|6/12 display code]], was also used in the [[CDC Kronos|Kronos]] and [[NOS (software)|NOS]] timesharing systems to allow full use of the [[ASCII]] character set in a manner somewhat compatible with older software.

With no byte addressing instructions at all, code had to be written to pack and shift characters into words. The very large words, and comparatively small amount of memory, meant that programmers would frequently economize on memory by packing data into words at the bit level.

It is interesting to note that due to the large word size, and with 10 characters per word, it was often faster to process words full of characters at a time — rather than unpacking/processing/repacking them. For example, the CDC [[COBOL]] compiler was actually quite good at processing decimal fields using this technique. These sorts of techniques are now commonly used in the 'multi-media' instructions of current processors.

===Physical design===
[[Image:CDCcordwood1.jpg|thumb|right|300px|A CDC 6600 cordwood logic module containing 64 silicon transistors. The coaxial connectors are test points. The module is cooled conductively via the front panel. The 6600 model contained nearly 6,000 such modules.<ref>Understanding Computers: Speed and Power 1990, p. 17.</ref>]]

The machine was built in a plus-sign-shaped cabinet with a pump and heat exchanger in the outermost {{convert|18|in|cm|abbr=on}} of each of the four arms. Cooling was done with [[Freon]] circulating within the machine and exchanging heat to an external chilled water supply. Each arm could hold four chassis, each about {{convert|8|in|cm|abbr=on}} thick, hinged near the center, and opening a bit like a book. The intersection of the "plus" was filled with cables which interconnected the chassis. The chassis were numbered from 1 (containing all 10 PPUs and their memories, as well as the 12 rather minimal I/O channels) to 16. The main memory for the CPU was spread over many of the chassis. In a system with only 64K words of main memory, one of the arms of the "plus" was omitted.

The logic of the machine was packaged into modules about {{convert|2.5|in|mm|abbr=on}} square and about {{convert|1|in|cm|abbr=on}} thick. Each module had a connector (30 pins, two vertical rows of 15) on one edge, and six test points on the opposite edge. The module was placed between two aluminum cold plates to remove heat. The module itself consisted of two parallel printed circuit boards, with components mounted either on one of the boards or between the two boards. This provided a very dense package; somewhat difficult to repair, but with good heat removal. It was known as [[Printed circuit board#.22Cordwood.22 construction|cordwood construction]].

==Operating system and programming==
There was a sore point with the 6600 [[operating system]] support — slipping timelines. The machines originally ran a very simple [[batch processing|job-control]] system known as ''COS'' ([[Chippewa Operating System]]), which was quickly "thrown together" based on the earlier [[CDC 3000]] operating system in order to have something running to test the systems for delivery. However the machines were intended to be delivered with a much more powerful system known as ''SIPROS'' (for Simultaneous Processing Operating System), which was being developed at the company's System Sciences Division in [[Los Angeles]]. Customers were impressed with SIPROS's feature list, and many had SIPROS written into their delivery contracts.

SIPROS turned out to be a major fiasco. Development timelines continued to slip, costing CDC major amounts of profit in the form of delivery delay penalties. After several months of waiting with the machines ready to be shipped, the project was eventually cancelled. The programmers who had worked on COS had little faith in SIPROS and had continued working on improving COS.

[[Operating system development]] then split into two camps. The CDC-sanctioned evolution of COS was undertaken at the [[Sunnyvale, California]] software development lab. Many customers eventually took delivery of their systems with this software, then known as ''[[SCOPE (software)|SCOPE]]'' (Supervisory Control Of Program Execution). (Some Control Data Field Engineers used to refer to SCOPE as ''Sunnyvale's Collection Of Programming Errors''). SCOPE version 1 was, essentially, dis-assembled COS; SCOPE version 2 included new device and file system support; SCOPE version 3 included permanent file support, EI/200 remote batch support, and INTERCOM [[time sharing]] support. SCOPE always had significant reliability and maintainability issues.

The underground evolution of COS took place at the [[Arden Hills, Minnesota]] assembly plant. ''MACE'' ([Greg] Mansfield And [Dave] Cahlander Executive) was written largely by a single programmer in the off-hours when machines were available. Its feature set was essentially the same as COS and SCOPE 1. It retained the earlier COS file system, but made significant advances in code modularity to improve system reliability and adaptiveness to new storage devices. MACE was never an official product, although many customers were able to wrangle a copy from CDC.

MACE was later used as the basis of ''[[CDC Kronos|Kronos]]'', named after the [[Chronos|Greek god of time]]. The main marketing reason for its adoption was the development of its TELEX [[time sharing]] feature and its BATCHIO remote batch feature. Kronos continued to use the COS/SCOPE 1 file system with the addition of a permanent file feature.

An attempt to unify the SCOPE and Kronos operating system products produced ''[[NOS (software)|NOS]]'', (Network Operating System). NOS was intended to be the sole operating system for all CDC machines, a fact CDC promoted heavily. Many SCOPE customers remained software-dependent on the SCOPE architecture, so CDC simply renamed it ''[[NOS/BE]]'' (Batch Environment), and were able to claim that everyone was thus running NOS. In practice, it was far easier to modify the Kronos code base to add SCOPE features than the reverse.

The assembly plant environment also produced other operating systems which were never intended for customer use. These included the engineering tools SMM for hardware testing, and KALEIDOSCOPE, for software [[smoke testing]]. Another commonly used tool for CDC Field Engineers during testing was MALET (Maintenance Application Language for Equipment Testing), which was used to stress test components and devices after repairs and/or servicing by engineers. Testing conditions often used hard disk packs and magnetic tapes which were deliberately marked with errors to determine if the errors would be detected by MALET and the engineer.

==See also==
* [[History of supercomputing]]

== Notes ==
{{Reflist}}

==References==
*Grishman, Ralph (1974). ''Assembly Language Programming for the Control Data 6000 Series and the Cyber 70 Series''. New York, NY: Algorithmics Press. [http://www.bitsavers.org/pdf/cdc/cyber/books/Grishman_CDC6000AsmLangPgmg.pdf]
*[http://ed-thelen.org/comp-hist/CDC-6600-R-M.html#TOC/ CONTROL DATA 6400/6500/6600 COMPUTER SYSTEMS Reference Manual]
*Thornton, J. (1963). ''Considerations in Computer Design - Leading up to the Control Data 6600'' [http://www.bitsavers.org/pdf/cdc/cyber/cyber_70/thornton_6600_paper.pdf]
*Thornton, J. (1970). ''Design of a Computer—The Control Data 6600''. Glenview, IL: Scott, Foresman and Co. [http://www.bitsavers.org/pdf/cdc/cyber/books/DesignOfAComputer_CDC6600.pdf]
*(1990) ''Understanding Computers: Speed and Power, a TIME LIFE series'' ISBN 0809475863

==External links==
*[http://purl.umn.edu/104327 Neil R. Lincoln with 18 Control Data Corporation (CDC) engineers on computer architecture and design], [[Charles Babbage Institute]], University of Minnesota. Engineers include Robert Moe, Wayne Specker, Dennis Grinna, Tom Rowan, Maurice Hutson, Curt Alexander, Don Pagelkopf, Maris Bergmanis, Dolan Toth, Chuck Hawley, Larry Krueger, Mike Pavlov, Dave Resnick, Howard Krohn, Bill Bhend, Kent Steiner, Raymon Kort, and Neil R. Lincoln. Discussion topics include [[CDC 1604]], CDC 6600, [[CDC 7600]], [[CDC 8600]], [[CDC STAR-100]] and [[Seymour Cray]].
*[http://research.microsoft.com/~gbell/Computer_Structures__Readings_and_Examples/00000509.htm Parallel operation in the Control Data 6600, James Thornton]
*[http://research.microsoft.com/users/gbell/craytalk/sld001.htm Presentation of the CDC 6600 and other machines designed by Seymour Cray] – by C. [[Gordon Bell]] of Microsoft Research (formerly of DEC)

{{DEFAULTSORT:Cdc 6600}}
[[Category:1964 introductions]]
[[Category:CDC hardware|6600]]
[[Category:Supercomputers]]
[[Category:Transistorized computers]]
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Star Wars: Das Erwachen der Macht

2012-11-02T22:16:42Z

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{{AfDM|page=Star Wars Episode VII (2nd nomination)|year=2012|month=October|day=30|substed=yes|origtag=afdx}}

'''''Star Wars Episode VII''''' is an upcoming sequel to ''[[Star Wars Episode VI: Return of the Jedi]]'' with an anticipated release date of 2015. It will be the first ''[[Star Wars]]'' film produced following [[George Lucas]]' retirement from [[Lucasfilm]] and the first ''Star Wars'' film distributed by [[Walt Disney Pictures]]. It is intended to be the first film in a planned [[Star Wars sequel trilogy|sequel trilogy]] with subsequent installments released two to three years apart.<ref name="RinzlerWinding">{{cite web | url=http://starwarsblog.starwars.com/index.php/2012/10/30/the-long-winding-and-shapeshifting-trail-to-episodes-vii-viii-ix/ | title=The Long, Winding, and Shapeshifting Trail to Episodes VII, VIII & IX | publisher=StarWars.com | accessdate=October 30, 2012 | author=Rinzler, J.W.}}</ref>

==Production==
===Background===
{{See also|Star Wars sequel trilogy}}
Plans for ''Episode VII'' date back nearly 40 years.<ref name="RinzlerWinding" /> Following the success of [[Star Wars Episode IV: A New Hope|the first ''Star Wars'' film]] in 1977, George Lucas expanded his plans for the ''Star Wars'' saga to include three trilogies rather than two.<ref name="RinzlerWinding" /> The third "sequel" trilogy would comprise ''Episodes VII'', ''VIII'', and ''IX''. However, after completing the [[Star Wars#Original trilogy|original trilogy]] and [[Star Wars#Prequel trilogy|prequel trilogy]], Lucas revealed that he had no detailed plans for a sequel trilogy and lacked enthusiasm for completing it himself. Negative fan reactions to these latter projects contributed to this lack of interest.<ref name="NYTRollCredits">{{cite web |url=http://www.nytimes.com/2012/01/22/magazine/george-lucas-red-tails.html?pagewanted=3&_r=1|title=George Lucas Is Ready to Roll the Credits |last=Curtis|first=Bryan|date=2012-01-17|publisher=''[[nytimes.com]]'' |accessdate=2012-09-09}}</ref> As Lucas took steps to retire from Lucasfilm in 2012, he experienced a change of heart, noting, "It's now time for me to pass ''Star Wars'' on to a new generation of filmmakers."<ref name="NYTimesDisneyBuy" /> Lucas revealed plans to film ''Episodes VII'', ''VIII'', and IX to series stars [[Mark Hamill]] and [[Carrie Fisher]] in August 2012 in confidence. At that time, he indicated that he would not direct the films and that Lucasfilm president [[Kathleen Kennedy (film producer)|Kathleen Kennedy]] would produce. He did not discuss Fisher and Hamill's involvement in the production.<ref>{{cite web |publisher=Entertainment Weekly |last=Rottenberg |first=Josh |title=Mark Hamill weighs in on the future of 'Star Wars' -- EXCLUSIVE |date=October 31, 2012 |accessdate=November 1, 2012 |url=http://insidemovies.ew.com/2012/10/31/mark-hamill-star-wars-episode-vii-disney/}}</ref> The production of ''Episode VII'' was announced by [[The Walt Disney Company]] on October 30, 2012.<ref name="ForbesNewSWMovie">{{cite news | url=http://www.forbes.com/sites/dorothypomerantz/2012/10/30/disney-planning-new-star-wars-movie-with-lucasfilms-purchase/ | title=Disney Planning New 'Star Wars' Movie With Lucasfilm Purchase | work=[[Forbes]] | date=October 30, 2012 | accessdate=October 30, 2012 | author=Pomerantz, Dorothy}}</ref><ref>{{cite web|title=New Star Wars Movies Announced as Disney Enters Agreement to Acquire Lucasfilm Ltd.|url=http://www.starwars.com/news/new-star-wars-movies-announced-as-disney-enters-agreement-to-acquire-lucasfilm-ltd/index.html|work=StarWars.com|publisher=StarWars.com|accessdate=31 October 2012}}</ref> The announcement was made in a press release along with news of the acquisition of [[Lucasfilm]] for US$4.05 billion in stock and cash.<ref name="NYTimesDisneyBuy">{{cite news | url=http://mediadecoder.blogs.nytimes.com/2012/10/30/disney-buying-lucas-films-for-4-billion/?hp | title=Disney Buying Lucasfilm for $4 Billion | work=[[New York Times]] | date=October 30, 2012 | accessdate=October 30, 2012 | author=Ciepley, Michael}}</ref>

===Development===
{{As of|2012}}, the film is in "early-stage development" with an anticipated release date of 2015.<ref name="FoxNews">{{cite web | url=http://www.foxnews.com/entertainment/2012/10/30/disney-buying-tar-wars-maker-lucasfilm-for-405b/ | title=Disney buying 'Star Wars' maker Lucasfilm for $4.05B| work=[[FOX News]] | accessdate=October 30, 2012 | author=[[Associated Press]]}}</ref><ref name="CNNDisneyBuy">{{cite news | url=http://money.cnn.com/2012/10/30/technology/disney-buys-lucasfilm/index.html?hpt=hp_t2 | title=Disney to buy Lucasfilm for $4 billion | work=[[CNN]] | date=October 30, 2012 | accessdate=October 30, 2012 | author=Cowley, Stacey}}</ref> Lucasfilm president Kathleen Kennedy will serve as executive producer of the film, and Lucas will serve as creative consultant.<ref name="ForbesNewSWMovie" />

==Plot==
''Episode VII'' is expected to be an entirely original story, and not based on the various novels, graphic novels, and other materials that have previously been considered [[Star Wars canon|''Star Wars'' canon]].<ref name="E! Online">{{cite news | url=http://www.eonline.com/news/358685/star-wars-7-plot-will-be-an-original-story-says-lucasfilm-source | title=Star Wars 7 Plot Will Be "an Original Story," Says Lucasfilm Source | work=[[E! Online]] | date=October 30, 2012 | accessdate=October 30, 2012 | author=Gornstein, Leslie}}</ref> As part of the purchase, Disney received an "extensive story treatment" by Lucas. In a video interview that was part of the purchase announcement, Kennedy stated that she and Lucas had already met with writers to discuss the film script.<ref>{{cite web |publisher=Collider.com |url=http://collider.com/star-wars-episode-7/207393/ |title=George Lucas and Kathleen Kennedy Talk the Future of STAR WARS; Reveal They’ve Already Met with Writers for EPISODE VII |last=Chitwood |first=Adam |date=October 30, 2012 |accessdate=November 1, 2012}}</ref>

Lucas biographer Dale Pollock, who had read the original twelve stories written by Lucas, noted that the new episodes would involve character [[Luke Skywalker]] in his 30s or 40s. In an interview, Pollock stated that "the three most exciting stories were 7, 8 and 9. They had propulsive action, really interesting new worlds, new characters. I remember thinking, 'I want to see these 3 movies.'"<ref name="thewrap-2012-10-30">{{cite web |publisher=The Wrap |title='Star Wars' 7, 8 and 9 Are 'The Most Exciting,' Says George Lucas Biographer (Exclusive) |url=http://www.thewrap.com/movies/column-post/star-wars-7-8-and-9-are-most-exciting-says-george-lucas-biographer-exclusive-63006 |date=October 30, 2012 |accessdate=November 1, 2012 |last=Waxman |first=Sharon}}</ref> Pollock expects the screenplays to follow the treatments written by Lucas.<ref name="thewrap-2012-10-30"/>

==References==
{{Reflist}}

==External links==
{{Portal|Star Wars|Film}}
* {{IMDb title|id=2488496}}

{{Star Wars}}

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[[Category:Star Wars episodes|7]]
[[Category:Upcoming films]]
[[Category:Films distributed by Disney]]
[[Category:Lucasfilm films]]
[[Category:Sequel films]]

[[es:Star Wars: Episode VII]]
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0,999…

2011-12-27T21:21:14Z

Loadmaster: slight improvements

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[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' (which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math>, '''0.(9)''', or as "0.9" followed by any number of 9s in the repeating decimal) denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|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.

In fact, every nonzero, terminating decimal has an equal twin representation with trailing 9s, such as 8.32 and 8.31999... The terminating decimal representation is almost always preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other [[radix|bases]].

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. Nonetheless, some students question or reject it. Some can be persuaded by an [[argument from authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning, to accept that the two are equal. However, some remain unconvinced, and seek further justification.

The equality of 0.999... and 1 is closely related to the absence of nonzero [[infinitesimal]]s in the real number system. Some alternative number systems, such as the [[hyperreal]]s, do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be non-zero numbers whose difference from 1 is less than any positive rational number. Although the real numbers are the most common object of study in the field of [[mathematical analysis]], the hyperreals and other mathematical structures also have applications.

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111…, in which the digits repeat without end. This decimal yields a quick proof for {{nowrap|1=0.999… = 1}}. Multiplication of 9 times 1 produces 9 in each digit, so {{nowrap|9 × 0.111…}} equals 0.999… and {{nowrap|9 × {{frac|1|9}}}} equals 1, so {{nowrap|1=0.999… = 1}}:

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

Another form of this proof multiplies {{nowrap|1=⅓ = 0.333…}} by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It should be noted that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the [[Archimedean property]] of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (exponentiation)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by the smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of —rather than independent alternatives to— the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the real numbers: that there are no nonzero [[infinitesimal]]s. 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 therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to the real numbers, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note however that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> [[A. H. Lightstone]] developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

The standard definition of the number 0.999... is the [[limit of a sequence|limit of the sequence]] 0.9, 0.99, 0.999, ... A different definition considers the equivalence class [(0.9, 0.99, 0.999, ...)] of this sequence in the [[ultrapower construction]], which corresponds to a number that is infinitesimally smaller than 1. More generally, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative interpretation of "0.999...":
:<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}.</math><ref>Katz & Katz 2010</ref>
All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>
[[Jose Benardete]] in his book ''Infinity: An essay in metaphysics'' argues that some natural pre-mathematical intuitions cannot be expressed if one is limited to an overly restrictive number system:
:The intelligibility of the continuum has been found--many times over--to require that the domain of real numbers be enlarged to include infinitesimals. This enlarged domain may be styled the domain of continuum numbers. It will now be evident that .9999... does not equal 1 but falls infinitesimally short of it. I think that .9999... should indeed be admitted as a ''number'' ... though not as a ''real'' number.<ref>{{cite book|first=José Amado |last=Benardete |title=Infinity: An essay in metaphysics |publisher=Clarendon Press |year=1964 |page=279 |url=http://books.google.com/books?id=wMgtAAAAMAAJ&&hl=en&ei=3lTSTqSPGMrE4gTNwI1g&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDIQ6AEwAA}}</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the [[sign and magnitude]] or [[ones' complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: 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)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |jstor=2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0 |issue=2}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |jstor=2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |jstor=2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |jstor=2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |jstor=2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: 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)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |jstor=2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{Cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |jstor=493261 |doi=10.2307/493261 |postscript={{inconsistent citations}}}}
*{{Cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |jstor=40248503 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936 |postscript={{inconsistent citations}}}}
*{{Cite book |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student’s understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146 |postscript={{inconsistent citations}}}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
*{{Cite journal |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |chapter=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |title=Research issues in undergraduate mathematics learning |postscript={{inconsistent citations}}}}
* {{cite arxiv | eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 | class=math.NT }}
*{{Cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501 |issue=3 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999…=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf |postscript={{inconsistent citations}}}}
*{{Cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103 |postscript={{inconsistent citations}}}}
*{{Cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354 |postscript={{inconsistent citations}}}}
*{{Cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |jstor=749807 |doi=10.2307/749807 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6 |postscript={{inconsistent citations}}}}
*{{Cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33 |postscript={{inconsistent citations}}}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real numbers]]
[[Category:Articles containing proofs]]

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0,999…

2011-07-05T17:40:08Z

Loadmaster: restored removed image, please discuss this change on the talk page

{{Portal:Mathematics/Featured article template}}
[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''', which may also be written as '''0.9''', <math alt="0.9 with dot over the 9" style="position:relative;top:-.3em">\scriptstyle\mathbf{0}.\mathbf{\dot{9}}</math> or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks to students. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument from authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification.

The equality of 0.999... and 1 is closely related to the fact that there are no nonzero [[infinitesimal]] real numbers. Some alternative number systems, such as the [[hyperreals]], do contain nonzero infinitesimals. In these systems, unlike in the reals, there can be numbers whose difference from 1 is less than any rational number. Other systems, known as the [[p-adic numbers|''p''-adic numbers]], have a different form of "decimal expansions" which behave quite differently than expansions of real numbers. Although the real numbers are the most common object of study in the field of [[mathematical analysis]], the hyperreals and ''p''-adics both have applications in that area.

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
Algebraic proofs showing that 0.999... represents the number 1 use concepts such as [[Fraction (mathematics)|fractions]], [[long division]], and digit manipulation to build transformations preserving equality from 0.999... to 1.

===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===
When a number in decimal notation is multiplied by 10, the digits do not change but each digit moves one place to the left. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator cancels, i.e. the result is 9 − 9 = 0 for each such digit. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number; it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by the smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest newspaper column ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of – rather than independent alternatives to – the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the real numbers: that there are no nonzero [[infinitesimal]]s. 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 therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to the real numbers, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> [[A. H. Lightstone]] developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. As a consequence the number 0.999...;...999... = 1. With this type of decimal representation, not every expansion represents a number. In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>\underset{H}{0.\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{H}}</math>,<ref>Katz & Katz 2010</ref>
All such interpretations of "0.999..." are infinitely close to 1. [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some computing binary number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: 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)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |jstor=2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268 |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0-387-96014-7}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |jstor=2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |jstor=2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |jstor=2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |jstor=2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |jstor=2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |jstor=2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: 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)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |jstor=2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf |accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi= |accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |jstor=2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |jstor=2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0-665-54808-7}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf |accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf |accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{Citation |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
*{{Citation |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |jstor=3618786 |doi=10.2307/3618786}}
*{{Citation |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |jstor=493261 |doi=10.2307/493261}}
*{{Citation |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |jstor=40248503}}
*{{Citation |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |jstor=2004936 |doi=10.2307/2004936}}
*{{Citation |last=Edwards |first=B. |year=1997 |chapter=An undergraduate student’s understanding and use of mathematical definitions in real analysis |editor=Dossey, J., Swafford, J.O., Parmentier, M., Dossey, A.E. |title=Proceedings of the 19th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education |volume=1 |publisher=ERIC Clearinghouse for Science, Mathematics and Environmental Education |location=Columbus, OH |pages=17–22}}
*{{Citation |last=Eisenmann |first=Petr |year=2008 |title=Why is it not true that 0.999... < 1? |journal=The Teaching of Mathematics |volume=11 |issue=1 |pages=35–40 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/20/tm1114.pdf}}
*{{Citation |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
*{{Citation |last=Ferrini-Mundy |first=J. |last2=Graham |first2=K. |year=1994 |chapter=Research in calculus learning: Understanding of limits, derivatives and integrals |journal=MAA Notes |volume=33 |pages=31–45 |editor1-first=J. |editor1-last=Kaput |editor2-first=E. |editor2-last=Dubinsky |title=Research issues in undergraduate mathematics learning}}
* {{cite arxiv | eprint=math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |year=2006 }}
*{{Citation |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4 |arxiv=1003.1501}}
*{{Citation |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |jstor=3616921 |doi=10.2307/3616921}}
*{{Citation |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |jstor=3619940 |doi=10.2307/3619940}}
*{{Citation |first=Maria Angeles |last=Navarro |first2=Pedro Pérez |last2=Carreras |year=2010 |title=A Socratic methodological proposal for the study of the equality 0.999…=1 |journal=The Teaching of Mathematics |volume=13 |issue=1 |pages=17–34 |url=http://elib.mi.sanu.ac.rs/files/journals/tm/24/tm1312.pdf}}
*{{Citation |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
*{{Citation |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |jstor=2975103 |doi=10.2307/2975103}}
*{{Citation |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |doi=10.1007/BF00240986 |jstor=3482354}}
*{{Citation |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |jstor=749807 |doi=10.2307/749807}}
*{{Citation |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
*{{Citation |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-10-18T15:27:37Z

Loadmaster: /* In alternative number systems{{anchor|Alternative number systems}} */ many rationals have equal values but different binary tree paths

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', '''0.9̇''' or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infintesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as ''nonstandard'' intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

This is in fact true of the binary expansions of many rational numbers, where the values of the numbers are equal but the corresponding binary tree paths are different. For example, 0.10111...2 = 0.11000...2, which are both equal to {{frac|3|4}}, but the first representation corresponds to the binary tree path LRLRRR... while the second corresponds to the different path LRRLLL....

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
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* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0665548087}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-10-18T15:14:19Z

Loadmaster: /* Infinitesimals */ added missing trailing "..."

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', '''0.9̇''' or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infintesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: 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)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |url=http://jstor.org/stable/2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |url=http://jstor.org/stable/2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |url=http://jstor.org/stable/2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |url=http://jstor.org/stable/2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |url=http://jstor.org/stable/2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |url=http://jstor.org/stable/2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |url=http://jstor.org/stable/2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: 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)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0665548087}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-10-18T15:10:25Z

Loadmaster: "non-zero" → "nonzero" for consistency

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', '''0.9̇''' or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infintesimal; but since the reals do not contain nonzero infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
* {{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*: This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p. ix)
* {{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: 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)
* {{cite book |author=Bartle, R. G. and D. R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*: This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp. vii–viii)
* {{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
* {{cite book |author=[[Elwyn Berlekamp|Berlekamp, E. R.]]; [[John Horton Conway|J. H. Conway]]; and [[Richard K. Guy|R. K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
* {{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.3019}}
* {{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*: This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp. ix-xi, 119)
* {{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
* {{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
* {{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*: This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p. vii)
* {{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
* {{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |url=http://jstor.org/stable/2309468 |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
* {{cite journal |last1=Dubinsky |first1=Ed |last2=Weller |first2=Kirk |last3=McDonald |first3=Michael |last4=Brown |first4=Anne |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
* {{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5 |jstor=4145268}}
* {{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*: An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp. xi–xii)
* {{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
* {{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |url=http://jstor.org/stable/2687285 |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
* {{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
* {{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
* {{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
* {{cite book |last1=Griffiths |first1=H. B. |last2=Hilton |first2=P. J. | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6 | id={{LCC|QA37.2|G75}}}}
*: This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp. vii, xiv)
* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
* {{cite journal |last=Kempner |first=A. J. |title=Anormal Systems of Numeration |url=http://jstor.org/stable/2300532 |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
* {{cite journal |last1=Komornik |first1=Vilmos |last2=Loreti |first2=Paola |title=Unique Developments in Non-Integer Bases |url=http://jstor.org/stable/2589246 |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
* {{cite journal |last=Leavitt |first=W. G. |title=A Theorem on Repeating Decimals |url=http://jstor.org/stable/2314251 |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
* {{cite journal |last=Leavitt |first=W. G. |title=Repeating Decimals |url=http://jstor.org/stable/2686394 |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
* {{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
* {{cite journal |last=Lightstone |first=A. H. |title=Infinitesimals |url=http://jstor.org/stable/2316619 |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
* {{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: 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)
* {{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*: A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp. x-xiii)
* {{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
* {{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*: Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p. xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p. 30)
* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
* {{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
* {{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
* {{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |url=http://jstor.org/stable/2324393 |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0665548087}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-10-17T17:34:24Z

Loadmaster: /* Infinitesimals */ reals do not contain infinitesimals

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as '''0.9''', '''0.9̇''' or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an [[Argument_from_authority|appeal to authority]] from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However, some are often uneasy enough that they seek further justification. The students' reasoning for denying or affirming the equality is typically based on their intuition that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|right|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. Specifically, the difference 1 − 0.999... must be smaller than any positive real number, so it must be an infintesimal; but since the reals do not contain infinitesimals, the difference is therefore zero, and therefore the two values are the same.

However, there are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
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* {{cite conference |last1=Pinto |first1=Márcia |last2=Tall |first2=David |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
* {{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*: A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p. ix)
* {{cite journal |doi=10.2307/2690144 |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |url=http://jstor.org/stable/2690144 |journal=Mathematics Magazine |volume=51 |issue=2 |month=March |year=1978 |pages=90–98 }}
* {{cite book |last1=Smith |first1=Charles |last2=Harrington |first2=Charles |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115 |isbn=0665548087}}
* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

{{Link FA|hu}}
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Pseudosphäre

2010-10-02T03:32:57Z

Loadmaster: added link for Surface of revolution

In [[geometry]], the term '''pseudosphere''' is used to describe various surfaces with constant negative [[gaussian curvature]]. Depending on context, it can refer to either a theoretical surface of constant negative curvature, to a tractricoid, or to a hyperboloid.
__TOC__

== Theoretical Pseudosphere ==
In its general interpretation, a '''pseudosphere''' of radius ''R'' is any surface of [[Gaussian curvature|curvature]] −1/''R''2 (precisely, a [[complete metric space|complete]], [[simply connected]] surface of that curvature), by analogy with the sphere of radius ''R'', which is a surface of curvature 1/''R''2. The term was introduced by [[Eugenio Beltrami]] in his 1868 paper on models of [[hyperbolic geometry]]<ref>{{Citation
| first=Eugenio
| last=Beltrami
| title=Saggio sulla interpretazione della geometria non euclidea
| journal=Gior. Mat.
| volume=6
| pages=248–312
| language=Italian
| year=1868
}} 
(Also {{Citation
| first=Eugenio
| last=Beltrami
| title=Opere Matematiche
| volume=1
| pages=374–405
| language=Italian
| isbn=1418184349
}}; 
{{Citation
| first=Eugenio
| last=Beltrami
| title=Essai d'interprétation de la géométrie noneuclidéenne
| journal=Ann. École Norm. Sup. 6
| year=1869
| pages=251–288
| language=French
| url=http://smf4.emath.fr/Publications/AnnalesENS/1_6/html/
}})</ref>.

== Tractricoid ==
[[Image:Pseudosphere.png|right|frame|Tractricoid]]
The term is also used to refer to a certain surface called the '''tractricoid''': the result of [[surface of revolution|revolving]] a [[tractrix]] about its [[asymptote]].

It is a [[Mathematical singularity|singular space]] (the equator is a singularity), but away from the singularities, it has constant negative [[Gaussian curvature]] and therefore is locally [[isometry|isometric]] to a [[hyperbolic plane]].

The name "pseudosphere" comes about because it is a [[dimension|two-dimensional]] [[surface]] of constant negative curvature just like a sphere with positive Gauss curvature.
Just as the [[sphere]] has at every point a [[negative and non-negative numbers|positively]] curved geometry of a [[dome]] the whole pseudosphere has at every point the [[negative and non-negative numbers|negatively]] curved geometry of a [[saddle surface|saddle]].

As early as 1639 [[Christian Huygens]] found that the volume and the surface area of the pseudosphere are finite,<ref>{{cite book
|title=Computational optimization: a tribute to Olvi Mangasarian, Volume 1
|first1=Olvi L.
|last1=Mangasarian
|first2=Jong-Shi
|last2=Pang
|publisher=Springer
|year=1999
|isbn=0-792-38480-6
|page=324
|url=http://books.google.com/books?id=kJa15IMxAoIC}}, [http://books.google.com/books?id=kJa15IMxAoIC&pg=PA324 Chapter 17, page 324]
</ref> despite the infinite extent of the shape along the axis of rotation. For a given edge [[radius]] ''R'', the [[area]] is 4π''R''2 just as it is for the sphere, while the [[volume]] is 2/3 π''R''3 and therefore half that of a sphere of that radius.<ref>{{cite book
|title=Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences
|edition=2
|first1=F.
|last1=Le Lionnais
|publisher=Courier Dover Publications
|year=2004
|isbn=0-486-49579-5
|page=154
|url=http://books.google.com/books?id=pCYDhbhu1O0C}}, [http://books.google.com/books?id=pCYDhbhu1O0C&pg=PA154 Chapter 40, page 154]
</ref><ref>{{MathWorld|title=Pseudosphere|urlname=Pseudosphere}}</ref>

== Hyperboloid ==
In some sources that use the [[Hyperboloid model]] of the hyperbolic plane, the [[hyperboloid]] is referred to as a '''pseudosphere'''<ref>{{Citation
| first=Elman
| last=Hasanov
| year=2004
| title=A new theory of complex rays
| journal=IMA J Appl Math
| volume=69
| pages=521–537
| issn=1464-3634
| url=http://imamat.oxfordjournals.org/cgi/reprint/69/6/521
}}</ref>.
This usage of the word is because the hyperboloid can be thought of as a sphere of imaginary radius, embedded in a [[Minkowski space]].

==See also==
*[[Dini's surface]]
*[[Hyperboloid structure]]
*[[Sine–Gordon equation]]
*[[Sphere]]
*[[Surface of revolution]]

==References==
<div class="references-small">
<references/>
* {{cite book|author=Henderson, D. W. and Taimina, D.|title=Aesthetics and Mathematics|publisher=Springer-Verlag|year=2006|chapter=[http://dspace.library.cornell.edu/bitstream/1813/2714/1/2003-4.pdf Experiencing Geometry: Euclidean and Non-Euclidean with History]}}
</div>

==External links==
*[http://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html Non Euclid]
*[http://www.cabinetmagazine.org/issues/16/crocheting.php Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina ]
*[http://www.maths.manchester.ac.uk/~kd/ Prof. C.T.J. Dodson's web site at University of Manchester]
*[http://www.maths.manchester.ac.uk/~kd/geomview/dini.html Interactive demonstration of the pseudosphere] (at the [[University of Manchester]])

[[Category:Differential geometry]]
[[Category:Surfaces]]

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0,999…

2010-08-25T16:33:50Z

Loadmaster: replaced representation that can be rendered as regular text, which looks better in some browsers

[[File:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999...''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9} \,\!</math>, <math style="position:relative;top:-.35em">0.\dot{9} \,\!</math> or '''0.(9)''', denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the symbols ''0.999...'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[Positional notation|positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this apparently paradoxical equality among students. Some reject it due to their intuitions that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999... eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999..., but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111..., in which the digits repeat without end. This decimal yields a quick proof for 0.999... = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111... equals 0.999... and 9 × {{frac|1|9}} equals 1, so 0.999... = 1:

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

Another form of this proof multiplies {{frac|1|3}} = 0.333... by 3.

===Digit manipulation===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999... equals 9.999..., which is 9 greater than the original number. To see this, consider that in subtracting 0.999... from 9.999..., each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999... = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999... = 1 but 0.333... < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p. 186</ref> William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp. 39–41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking".<ref>Richman p. 396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999... and 1.000... both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5 \dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999... one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p. 61, Theorem 3.26; J. Stewart p. 706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[File:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, ...) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999...<ref>Grattan-Guinness p. 69; Bonnycastle p. 177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p. 706, Rudin p. 61, Protter and Morrey p. 213, Pugh p. 180, J.B. Conway p. 31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, ...) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step, that {{frac|1|10''n''}} → 0 as ''n'' → ∞, is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small".<ref>Davies p. 175; Smith and Harrington p. 115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999... itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[File:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000... = 0.222...]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], ..., [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, ..., and one writes

:<math>x = b_0.b_1b_2b_3 \dots</math>

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.<ref>Beals p. 22; I. Stewart p. 34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3... is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp. 60–62; Pedrick p. 29; Sohrab p. 46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3... to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, ...}.<ref>Apostol pp. 9, 11–12; Beals p. 22; Rosenlicht p. 27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999... = 1 again. Tom Apostol concludes,

<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p. 12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s – 0, 1, 2, 3, and so on – begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p. 10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p. 17 or Rudin p. 17. For viewpoints on logic, Pugh p. 10, Rudin p.ix, or Munkres p. 30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typically the definitions allow
{ x : x < 1 } to be a cut but not { x : x ≤ 1 } (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp. 398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, ...), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p. 386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,..., it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton p. 395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999... = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999... = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p. 408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111... equals 1, and in base 3 (the [[ternary numeral system]]) 0.222... equals 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these generalizations from the start.<ref>Protter and Morrey p. 503; Bartle and Sherbert p. 61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000... and 0.101010..., and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000.... This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650.... In this base, 1 = 0.11010011001011010010110011010011...; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p. 636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p. 611; Petkovšek p. 409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111... = 1.111....
*In the reverse [[factorial number system]] (using bases 2,3,4,... for positions ''after'' the decimal point), 1 = 1.000... = 0.1234....

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999.../1.000... pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow "9" in each position while forbidding an infinite succession of "9"s). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp. 410–411</ref>

==Applications==
One application of 0.999... as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857... and 142 + 857 = 999.
*1/73 = 0.0136986301369863... and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p. 301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp. 1–3; Leavitt 1967 pp. 669, 673; Shrader-Frechette pp. 96–98</ref>

[[File:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000..., since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222..., since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p. 97; Alligood, Sauer, and Yorke pp. 150–152. Protter and Morrey (p. 507) and Pedrick (p. 29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999... A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p. 60) and Mankiewicz (p. 151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p. 50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p. 50, Pugh p. 98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p. 119; Tall and Schwarzenberger p. 6. The last suggestion is due to Burrell (p. 28): "Perhaps the most reassuring of all numbers is 1 ... So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp. 6–7; Tall 2000 p. 221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p. 6; Tall 2000 p. 221</ref>

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 [[counterexample]]s to better understand 0.999...

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999... as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p. 221</ref>

Of the elementary proofs, multiplying 0.333... = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999... = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp. 10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p. 5, Edwards and Ward pp. 416–417</ref> Others still are able to prove that 1⁄3 = 0.333..., but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99... = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp. 137–141</ref>

As part of Ed Dubinsky's [[APOS theory]] of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999... as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999... may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999... and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999... have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>sci.math</tt>, arguing over 0.999... is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999... via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999... features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p. 27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999....
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999... = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p. 60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999... and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999... = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999... behaves (if, indeed, a number expressed as "0.999..." is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999... = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999... depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999... = 1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A. H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp. 245–247</ref> Lightstone shows how to associate to each number a sequence of digits,

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

indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999..., he shows the real number 1/3 is represented by 0.333...;...333... which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333...;...000..." and "0.999...;...000..." do not correspond to any number.

At the same time, the hyperreal number {{nowrap|1 = ''u''''H''=0.999...;...999000...,}} with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1 = 0.9,}} {{nowrap|1=''u''2 = 0.99,}} {{nowrap|1=''u''3 = 0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999...":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, ...)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999....<ref>Stewart 2009, p. 175; the full discussion of 0.999... is spread through pp. 172–175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999... < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL... is 0.0101012... = 1/3. However, the value of LRLLL... (corresponding to 0.111...2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR... or 0.000...2.<ref>Berlekamp, Conway, and Guy (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.</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999... simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999... < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999... + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp. 397–399</ref>

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

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

When asked about 0.999..., novices often believe there should be a "final 9," believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999....<ref>Gardiner p. 98; Gowers p. 60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[File:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, ...) converging to −1. The 10-adic analogue is ...999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion ...999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + ...999 = ...000 = 0, and so ...999 = −1.<ref name="Fjelstad11">Fjelstad p. 11</ref> Another derivation uses a geometric series. The infinite series implied by "...999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp. 14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999... = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = ...999 then 10''x'' =  ...990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999... = 1{{nowrap end}} (in the reals) and {{nowrap begin}}...999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p. 901</ref> one may add the two equations and arrive at {{nowrap begin}}...999.999... = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp. 902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999... and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999..., resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p. 51, Maor p. 17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999..., and it also stirs up contention. While most authors choose to define 0.999..., almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp. 47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p. 54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p. 34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Finitism]]
* [[Infinity]]
{{Col-2-of-3}}
* [[Geometric series]]
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
{{Col-3-of-3}}
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
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* {{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999... less than 1? |journal=The Montana Mathematics Enthusiast |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
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*: 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)
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* {{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
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*: 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, nodecreasing 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)
* {{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*: While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p. 10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
* {{cite journal |last1=Renteln |first1=Paul |last2=Dundes |first2=Allan |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |issue=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
* {{cite journal |doi=10.2307/2690798 |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999... = 1? |url=http://jstor.org/stable/2690798 |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999... = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
* {{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
* {{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (pp. 27–31) as infinite decimals with 0.999... = 1 as part of the definition.
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*: 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)
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* {{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
* {{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
* {{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
* {{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*: This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p. v) It omits proofs of the foundations of calculus.
* {{cite journal |last1=Tall |first1=D. O. |last2=Schwarzenberger |first2=R. L. E.|title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |issue=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |issue=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
* {{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
* {{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
* {{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
* {{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
* {{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
* {{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999... and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
* {{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
* {{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*: This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999...}} falling short of 1 by an infinitesimal {{nowrap|0.000...1.}}
* {{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |pages=259 |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |volume=74 |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
* {{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
* {{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
* {{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1–3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
* {{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
* {{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
* {{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
* {{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
* {{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999... = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2010-05-08T02:46:38Z

Loadmaster: Corrected example added by LokiClock (talk)

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math style="position:relative;top:-.35em"> 0.\bar{9}</math>, <math style="position:relative;top:-.35em">0.\dot{9}</math> or <math style="position:relative;top:-.2em"> 0.(9)\,\!</math>, denotes a [[real number]] that can be shown to be [[1 (number)|the number one]]. In other words, the notations ''0.999…'' and ''1'' represent the same number. [[mathematical proof|Proofs]] of this [[equality (mathematics)|equality]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s (e.g., 0.222… = 1 in base 3), and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999…. The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. The non-terminating form is more convenient for understanding the decimal expansions of certain [[fraction (mathematics)|fraction]]s and, in base three, for the structure of the ternary [[Cantor set]], a simple [[fractal]]. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any [[positional numeral system]] for the real numbers contains infinitely many numbers with multiple representations.

The equality 0.999…=1 has long been accepted by mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students. Some reject it due to their intuitions that each number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] numbers should exist, or that the expansion of 0.999… eventually terminates. These intuitions fail in the real numbers, but alternate number systems can be constructed bearing some of them out. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Algebraic proofs{{anchor|Proofs}}{{anchor|Algebraic}}==
===Fractions and long division{{anchor|Fractions}}===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like {{frac|1|9}} becomes a recurring decimal, 0.111…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 9 times 1 produces 9 in each digit, so 9 × 0.111… equals 0.999… And 9 × {{frac|1|9}} equals 1, so 0.999… = 1.

Another form of this proof multiplies {{frac|1|3}} = 0.3… by 3. Written out in equations this time:

:<math>
\begin{align}
\frac{1}{3} & =0.333\dots \\
3 \times \frac{1}{3} & = 3 \times 0.333\dots \\
1 & = 0.999\dots
\end{align}
</math>

===Digit manipulation{{anchor|Digit manipulation}}===

When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 greater than the original number. To see this, consider that in subtracting 0.999… from 9.999…, each of the digits after the decimal separator the result is 9 − 9, which is 0. The final step uses algebra:

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1
\end{align}
</math>

===Discussion===
Although these proofs demonstrate that 0.999… = 1, the extent to which they ''explain'' the equation depends on the audience. In introductory arithmetic, such proofs help explain why we have 0.999… = 1 but 0.333… < 0.4. And in introductory algebra, the proofs help explain why the general method of converting between fractions and repeating decimals works. But the proofs shed little light on the fundamental relationship between decimals and the numbers they represent, which underlies the question of how two different decimals can be said to be equal at all.<ref>This argument is found in Peressini and Peressini p.186</ref> William Byers argues that a student who agrees that 0.999… = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation.<ref>Byers pp.39-41</ref> Fred Richman argues that the first argument "gets its force from the fact that most people have been conditioned to accept the first line without thinking."<ref>Richman pp.396</ref>

Once a representation scheme is defined, it can be used to justify the rules of decimal arithmetic used in the above proofs. Moreover, one can directly demonstrate that the decimals 0.999… and 1.000… both represent the same real number, it is built into the definition. This is done below.

==Analytic proofs{{anchor|Analytic}}==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1\left({\tfrac{1}{10}}\right) + b_2\left({\tfrac{1}{10}}\right)^2 + b_3\left({\tfrac{1}{10}}\right)^3 + b_4\left({\tfrac{1}{10}}\right)^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1 \,\!</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

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

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the [[quaternary numeral system|base-4]] fraction sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Digit manipulation|algebraic proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:<ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

===Nested intervals and least upper bounds===
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

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.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

==Proofs from the construction of the real numbers{{anchor|Based on the construction of the real numbers}}==
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found to be less than, greater than, or equal to another number.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

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

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

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>Richman</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate. He also notes that typical the definitions allow <math>\{\ x:x<1\}</math> to be a cut but not <math>\{x: x \le 1\}</math> (or vice versa) "Why do that? Precisely to rule out the existence of distinct numbers 0.9* and 1. [...] So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure where the two are not equal. Although it is consistent, many of the common rules of decimal arithmetic no longer hold, for example the fraction 1/3 has no representation; see "[[#Alternative number systems|Alternative number systems]]" below.

===Cauchy sequences===
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==

The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik–Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue–Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the reverse [[factorial number system]] (using bases 2,3,4,… for positions ''after'' the decimal point), 1 = 1.000… = 0.1234….

===Impossibility of unique representation===

That all these different number systems suffer from multiple representations for some real numbers can be attributed to a fundamental difference between the real numbers as an ordered set and collections of infinite strings of symbols, ordered [[lexicographic ordering|lexicographically]]. Indeed the following two properties account for the difficulty:

* If an [[interval (mathematics)|interval]] of the [[real number]]s is [[partition of a set|partitioned]] into two non-empty parts ''L'', ''R'', such that every element of ''L'' is (strictly) less than every element of ''R'', then either ''L'' contains a largest element or ''R'' contains a smallest element, but not both.
* The collection of infinite [[string (computer science)|string]]s of symbols taken from any finite "alphabet", lexicographically ordered, can be partitioned into two non-empty parts ''L'', ''R'', such that every element of ''L'' is less than every element of ''R'', while ''L'' contains a largest element ''and'' ''R'' contains a smallest element. Indeed it suffices to take two finite [[prefix (computer science)|prefix]]es (initial substrings) ''p''1, ''p''2 of elements from the collection such that they differ only in their final symbol, for which symbol they have successive values, and take for ''L'' the set of all strings in the collection whose corresponding prefix is at most ''p''1, and for ''R'' the remainder, the strings in the collection whose corresponding prefix is at least ''p''2. Then ''L'' has a largest element, starting with ''p''1 and choosing the largest available symbol in all following positions, while ''R'' has a smallest element obtained by following ''p''2 by smallest symbol in all positions.

The first point follows from basic properties of the real numbers: ''L'' has a [[supremum]] and ''R'' has an [[infimum]], which are easily seen to be equal; being a real number it either lies in ''R'' or in ''L'', but not both since ''L'' and ''R'' are supposed to be [[disjoint sets|disjoint]]. The second point generalizes the 0.999…/1.000… pair obtained for ''p''1 = "0", ''p''2 = "1". In fact one need not use the same alphabet for all positions (so that for instance [[mixed radix]] systems can be included) or consider the full collection of possible strings; the only important points are that at each position a [[finite set]] of symbols (which may even depend on the previous symbols) can be chosen from (this is needed to ensure maximal and minimal choices), and that making a valid choice for any position should result in a valid infinite string (so one should not allow '9' in each position while forbidding an infinite succession of '9's). Under these assumptions, the above argument shows that an [[monotonic|order preserving]] map from the collection of strings to an interval of the real numbers cannot be a [[bijection]]: either some numbers do not correspond to any string, or some of them correspond to more than one string.

Marko Petkovšek has proved that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1….So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>

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 [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://us.blizzard.com/en-us/company/press/pressreleases.html?040401 |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2009-11-16}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==In alternative number systems{{anchor|Alternative number systems}}==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: that there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for [[hyperreal number]]s in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each number a sequence of digits,
:0.d1d2d3…;…d∞−1d∞d∞+1…,
indexed by the [[hypernatural]] numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s. Lightstone shows that in this system, the expressions "0.333…;…000…" and "0.999…;…000…" do not correspond to any number.

At the same time, the hyperreal number ''u''''H''=0.999…;…999000…, with last digit 9 at infinite [[hypernatural]] rank ''H'', satisfies a strict inequality {{nowrap|''u''''H'' < 1.}} Indeed, the sequence {{nowrap|1=''u''1=0.9,}} {{nowrap|1=''u''2=0.99,}} {{nowrap|1=''u''3=0.999,}} etc. satisfies {{nowrap|1=''u''''n'' = 1 − 10−''n'',}} hence by the transfer principle {{nowrap|1=u''H'' = 1 − 10−''H'' < 1.}} Accordingly, Karin Katz and [[Mikhail Katz]] have proposed an alternative evaluation of "0.999…":
:<math>.\underset{[\mathbb{N}]}{\underbrace{999\ldots}}\; = 1\;-\;\frac{1}{10^{[\mathbb{N}]}}</math>,
where <math>[\mathbb{N}]</math> is an infinite hypernatural given by the sequence {{nowrap|(1, 2, 3, …)}} modulo some [[ultrafilter]].<ref>Katz & Katz 2010</ref> [[Ian Stewart (mathematician)|Ian Stewart]] characterizes this interpretation as an "entirely reasonable" way to rigorously justify the intuition that "there's a little bit missing" from 1 in 0.999….<ref>Stewart 2009, p.175; the full discussion of 0.999… is spread through pp.172-175.</ref> Along with Katz & Katz, Robert Ely also questions the assumption that students' ideas about {{nowrap|0.999… < 1}} are erroneous intuitions about the real numbers, interpreting them rather as nonstandard intuitions that could be valuable in the learning of calculus.<ref>Katz & Katz (2010b)</ref><ref>R. Ely (2010)</ref>

===Hackenbush===
[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (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, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

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

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

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Geometric series]]
{{Col-2-of-3}}
* [[Limit (mathematics)]]
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
* [[Finitism]]

{{col-end}}

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

==References==
{{refbegin|colwidth=30em}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton UP |isbn=0-691-12738-7}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468 |issue=9}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF |issue=5}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170 |isbn=0387960147}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285 |issue=1}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy |authorlink=William Timothy Gowers |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last1=Katz |first1=K. |last2=Katz |first2=M. |author2-link=Mikhail Katz |year=2010a |title=When is .999… less than 1? |journal=[[The Montana Mathematics Enthusiast]] |volume=7 |issue=1 |pages=3–30 |url=http://www.math.umt.edu/TMME/vol7no1/}}
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 |issue=10}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 |issue=7 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 |issue=6 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 |issue=4 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 |issue=3 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite book |first1=Anthony |last1=Peressini |first2=Dominic |last2=Peressini |editor=Bart van Kerkhove, Jean Paul van Bendegem |chapter=Philosophy of Mathematics and Mathematics Education |title=Perspectives on Mathematical Practices |publisher=Springer |isbn=978-1-4020-5033-6 |year=2007 |series=Logic, Epistemology, and the Unity of Science |volume=5}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 |issue=5 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and [[Charles B. Morrey, Jr.|C.B. Morrey]] |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}} This book gives a "careful rigorous" introduction to real analysis. It gives the axioms of the real numbers and then constructs them (p 27-31) as infinite decimals with 0.999…=1 as part of the definition.
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=Ian |title=Professor Stewart's Hoard of Mathematical Treasures |year=2009 |publisher=Profile Books |isbn=978-1-84668-292-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{cite journal |journal=Journal of Statistical Physics |volume=47 |issue=3/4 |year=1987 |title=One-dimensional model of the quasicrystalline alloy |first=S. E. |last=Burkov |doi=10.1007/BF01007518 |pages=409}}
*{{cite journal |title=81.15 A Case of Conflict |first=Bob |last=Burn |journal=The Mathematical Gazette |volume=81 |issue=490 |month=March |year=1997 |pages=109–112 |url=http://www.jstor.org/stable/3618786 |doi=10.2307/3618786}}
*{{cite journal |title=The Age of Newton: An Intensive Interdisciplinary Course |author=J. B. Calvert, E. R. Tuttle, Michael S. Martin, Peter Warren |journal=The History Teacher |volume=14 |issue=2 |month=February |year=1981 |pages=167–190 |url=http://www.jstor.org/stable/493261 |doi=10.2307/493261}}
*{{cite journal |first1=Younggi |last1=Choi |first2=Jonghoon |last2=Do |title=Equality Involved in 0.999… and (-8)⅓ |journal=For the Learning of Mathematics |volume=25 |issue=3 |month=November |year=2005 |pages=13–15, 36 |url=http://www.jstor.org/stable/40248503}}
*{{cite journal |title=Rational Approximations to π |author=K. Y. Choong, D. E. Daykin, C. R. Rathbone |journal=Mathematics of Computation |volume=25 |issue=114 |month=April |year=1971 |pages=387–392 |url=http://www.jstor.org/stable/2004936 |doi=10.2307/2004936}}
*{{cite journal |last=Ely |first=Robert |year=2010 |title=Nonstandard student conceptions about infinitesimals |journal=Journal for Research in Mathematics Education |volume=41 |issue=2 |pages=117–146}}
*:This article is a field study involving a student who developed a Leibnizian-style theory of infinitesimals to help her understand calculus, and in particular to account for {{nowrap|0.999…}} falling short of 1 by an infinitesimal {{nowrap|0.000…1.}}
*{{cite journal |first1=Karin Usadi |last1=Katz |first2=Mikhail G. |last2=Katz |year=2010b |title=Zooming in on infinitesimal 1 − .9.. in a post-triumvirate era |journal=[[Educational Studies in Mathematics]] |doi=10.1007/s10649-010-9239-4}} See also arXiv:1003.1501.
*{{cite journal |title=Infinite processes in elementary mathematics: How much should we tell the children? |first=Tony |last=Gardiner |journal=The Mathematical Gazette |volume=69 |issue=448 |month=June |year=1985 |pages=77–87 |url=http://www.jstor.org/stable/3616921 |doi=10.2307/3616921}}
*{{cite journal |title=Real Mathematics: One Aspect of the Future of A-Level |first=John |last=Monaghan |journal=The Mathematical Gazette |volume=72 |issue=462 |month=December |year=1988 |pages=276–281 |url=http://www.jstor.org/stable/3619940 |doi=10.2307/3619940}}
*{{cite journal |first=Malgorzata |last=Przenioslo |title=Images of the limit of function formed in the course of mathematical studies at the university |journal=Educational Studies in Mathematics |volume=55 |issue=1-3 |month=March |year=2004 |doi=10.1023/B:EDUC.0000017667.70982.05 |pages=103–132}}
*{{cite journal |title=Using Self-Similarity to Find Length, Area, and Dimension |first=James T. |last=Sandefur |journal=The American Mathematical Monthly |volume=103 |issue=2 |month=February |year=1996 |pages=107–120 |url=http://www.jstor.org/stable/2975103 |doi=10.2307/2975103}}
*{{cite journal |last=Sierpińska |first=Anna |title=Humanities students and epistemological obstacles related to limits |journal=Educational Studies in Mathematics |volume=18 |issue=4 |year=1987 |month=November |pages=371–396 |url=http://links.jstor.org/sici?sici=0013-1954%28198711%2918%3A4%3C371%3AHSAEOR%3E2.0.CO%3B2-%23 |doi=10.1007/BF00240986}}
*{{cite journal |title=Mathematical Beliefs and Conceptual Understanding of the Limit of a Function |first=Jennifer Earles |last=Szydlik |journal=Journal for Research in Mathematics Education |volume=31 |issue=3 |month=May |year=2000 |pages=258–276 |url=http://www.jstor.org/stable/749807 |doi=10.2307/749807}}
*{{cite journal |first=David O. |last=Tall |title=Dynamic mathematics and the blending of knowledge structures in the calculus |journal=ZDM Mathematics Education |year=2009 |volume=41 |issue=4 |pages=481–492 |doi=10.1007/s11858-009-0192-6}}
*{{cite journal |first=David O. |last=Tall |title=Intuitions of infinity |journal=Mathematics in School |month=May |year=1981 |pages=30–33}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999…] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999… ?= 1 FAQ]
* [http://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1]

{{featured article}}

[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Nigger in the Woodpile

2010-04-26T02:31:13Z

Loadmaster: /* Scientific Literature */ said → wrote

[[Image:The Nigger in the Woodpile.jpg|thumb|A racist parody, titled "The Nigger in the Woodpile", of Republican efforts to play down the antislavery plank in their 1860 platform.]]
A '''nigger in the woodpile''' (or '''fence''') is an English [[figure of speech]] formerly commonly used in the [[United States]] and elsewhere.{{Fact|date=February 2009}} It means ''"some fact of considerable importance that is not disclosed - something suspicious or wrong"''.

Less commonly it may refer to an ''"undisclosed black ancestry"''.

==Origin==
Both the 'fence' and 'woodpile' variants developed about the same time in the period of 1840-50 when the [[Underground Railroad]] was flourishing successfully, and although the evidence is slight it is presumed that they derived from actual instances of the concealment of [[fugitive slaves]] in their flight north under piles of firewood or within hiding places in stone fences.<ref>Heavens to Betsy" (1955, Harper & Row) by Charles Earle Funk </ref>

==Film==
An American film comedy entitled ''A Nigger in the woodpile'' was released in 1904.<ref>[http://www.imdb.com/title/tt0232190/ IMDb entry]</ref>


In ''[[My Little Chickadee]]'' (1940) [[W. C. Fields]] remarks, "Hmm. There's an Ethiopian in the fuel supply."

In ''[[Not So Dumb]]'' (1930), actress [[Marion Davies]]' dim-bulb character says, "I knew there was a woodpile in the nigger."

In ''Porky's Railroad'' (1937), a train quickly passes a pile of wood, thus blowing away all the wood and revealing a black man with large white eyes.

==Literature==
In [[Agatha Christie]]'s ''[[They Do It with Mirrors]]'', Inspector Curry asks the phrase of [[Miss Marple]] in relation to Gina's white GI husband, Wally. The phrase is uttered by Judge Wargrave in ''[[And Then There Were None]]''. In Christie's ''[[Dumb Witness]]'' (copyright 1937), the phrase is the title of Chapter 18 and it is uttered by [[Hercule Poirot]] who also asks if it is a "saying."

Can be found in [[W. Somerset Maugham]]'s ''[[The Razor's Edge]]'' on p. 305 when Gray is talking to author about a new business deal.
"As soon as I get to New York I'll fly down to Texas to give it the once over, and you bet I'll keep my eyes peeled for a nigger in the woodpile before I cough up any of Isabel's dough."

In the original version of [[The Hardy Boys]] ''[[The House on the Cliff]]'', Frank Hardy uses the phrase in chapter 9 in regards to a suspicious circumstance.

== Scientific Literature ==
In the first footnote of a 1956 article in the Journal of Symbolic Logic, [[Robin Gandy]] notes that [[Alan Turing]] ''always spoke of the [[axiom of extensionality]] as being 'the nigger in the woodpile'.'' <ref name="gandyjsl56">http://projecteuclid.org/euclid.jsl/1183732302 On the Axiom of Extensionality--Part II</ref>

The computer scientist [[Edsger Wybe Dijkstra|E. W. Dijkstra]] also used the phrase in a paper, where he wrote, "A main nigger in the woodpile is the invention —in Europe— and the subsequent proliferation —primarily in the USA— of the term 'software engineering'."<ref>http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD690.html E.W.Dijkstra Archive: The pragmatic engineer versus the scientific designer</ref>

== Recent usage==
===United Kingdom===
In the UK in recent years, the occasional use of this phrase by public figures has normally been followed by an apology.<ref>[http://www.telegraph.co.uk/news/main.jhtml?xml=/news/2007/01/25/ustandard125.xml ''Insurance boss apologises for racist remark'' Daily Telegraph, Jan 25, 2007] - phrase used by an executive of [[Standard Life]]</ref><ref>[http://news.independent.co.uk/media/article335041.ece ''BBC apologises for general's 'racist remark' in radio interview'', The Independent, Dec 24, 2005] - phrase used by General [[Patrick Cordingley]]</ref><ref>[http://www.guardian.co.uk/comment/story/0,3604,628596,00.html Gary Younge, ''Not while racism exists'', The Guardian, Jan 7, 2002] - use attributed to [[Germaine Greer]]</ref>
* 2007, Bedfordshire County counciller Rhys Goodwin, stepped down as chairman of the environment and economic development committee: "...During a debate on heavy goods vehicle traffic in the county, he wanted to argue that a particular problem in Bedfordshire is the amount of trucks on the roads connected with quarrying. But he used the unfortunate figure of speech before sheepishly rephrasing his point.'<ref>[http://www.bedfordtoday.co.uk/bed-news/Councillor-quits-over-race-clanger.3517770.jp "Councillor quits over race clanger", www.bedfordtoday.co.uk, 23 November 2007]</ref> Goodwin, who was 74 at the time, said: ''"There was no racist intent at all. For 50 years of my life that was common parlance, with no more a derogatory connotation than the symbol on a jar of marmalade."''<ref>[http://www.mk-news.co.uk/bedsonsunday-news/DisplayArticle.asp?ID=237123 "'Racist' remark gets support", Bedfordshire on Sunday]</ref>
* 2008, Lord Dixon Smith, Conservative frontbencher, used the phrase in a debate on the Housing and Regeneration Bill: "Of course, the nigger in the woodpile, as the noble Baroness, Lady Hamwee, has already pointed out, is that it still incorporates what I call the hangover of the new towns legislation." He immediately apologised to the House. His Lordship, also in his seventies, later commented to journalists that the phrase had been "in common parlance when I was younger".<ref>[http://news.bbc.co.uk/1/hi/uk_politics/7497097.stm"Peer's apology over racist phrase"]</ref>
* 2009, Dick Denby, of Dick Denby Transport uttered this phrase on the BBC Radio 2 Jeremy Vine show (Tuesday, 1 December) during a discussion on the merits of 83 foot long HGV's. To his credit he did say that perhaps he should not have used said phrase. Jeremy Vine agreed he should not have used it and later apologised to Radio 2 listeners who might have been offended.

===Ireland===
In November, 2007, in relation to a debate on the Gaelic Players Association, [[Fine Gael]] Senator Paul Coughlan asked "Can the leader kick it into play and give members an update? Who is the nigger in the woodpile?". There was no call for an apology.

===Australia===
David Lord, an ABC News Radio presenter was forced to apologise after using the expression. On 22 February 2007, [[Alan Jones (radio broadcaster)|Alan Jones]], another radio presenter, was heard to use the same phrase.<ref>[http://www.abc.net.au/mediawatch/transcripts/s1898662.htm Media Watch: Alan Moans (16/04/2007)]</ref> There was no call for an apology.

===United States===

On September 18, 2009, the Town Board of Bethlehem, New York, confirmed that Police Chief Lou Corsi used the phrase during a phone conversation with Under Sheriff John Mahan, and that the conversation had been recorded.<ref> [http://wnyt.com/article/stories/S1145858.shtml?cat=300 "Police chief caught on tape using slur"] </ref>

On December 6, 2009, Joel Barbee made a reference to it in his cartoon "The Woods Pile".<ref> [http://www.nationalfreepress.org/The-Woods-Pile "The National Free Press"] </ref>

==References==
{{Reflist|2}}
*
* "I Hear America Talking" by Stuart Berg Flexner (Von Nostrand Reinhold Co., New York, 1976).
==External links==
*[http://loc.harpweek.com/LCPoliticalCartoons/DisplayCartoonMedium.asp?MaxID=&UniqueID=4&Year=1860&YearMark= Racist parody of Republican platform] from 1860 Presidential campaign, in ''[[Harper's Weekly]]''
*[http://www.utpjournals.com/product/utq/714/714_cariou.htm Epistemology of the Woodpile], ''[[University of Toronto]] Quarterly''
*[http://hnn.us/articles/1828.html#11130301 History News Network blog post] about origins of term
*[http://hnn.us/articles/1368.html#11180302 History News Network blog post] about a recent controversy

[[Category:English phrases]]

Hamilton (Texas)

2010-04-05T18:06:24Z

Loadmaster: /* Siehe auch */

{{Infobox Ort in den Vereinigten Staaten
| Name = Hamilton
| Stadtspitzname =
| Bundesstaat = Texas
| County = Hamilton County
| Bild1 = Hamilton-TX-CourtHouse.jpg
| Bildgröße1 =
| Bildbeschreibung1 = Hamilton County Courthouse
| Siegel =
| Flagge =
| Karte = [[Bild:Hamilton County Hamilton.svg|300px]]
| Beschriftung Karte = Lage von Hamilton in Texas
| Beschriftung Positionskarte =
| Breitengrad = 31/42/12/N
| Längengrad = 98/07/13/W
| Gründung = 1858
| Zeitzone = UTC-6
| Einwohner = 2977
| Stand = 2000
| Metropolregion =
| Stand Metropolregion =
| Fläche = 7.5
| Landfläche = 7.3
| Höhe = 356
| Gliederung =
| Postleitzahl = 76531
| Vorwahl = 254
| Typ = City
| Fips = 48-31952
| Gnis = 1337274
| Website = hamiltontexas.com
| Bürgermeister =
| Bild2 =
| Bildgröße2 =
| Bildbeschreibung2 =
}}
'''Hamilton''' ist eine Stadt und der [[County Seat|Sitz der Countyverwaltung]] des [[Hamilton County (Texas)|Hamilton County]]s im [[Bundesstaat der Vereinigten Staaten|US-Bundesstaat]] [[Texas]] in den [[Vereinigte Staaten|Vereinigten Staaten]].

== Geografie ==
Die Stadt liegt etwas östlich des geografischen Zentrums von Texas an der Zusammenführung des [[U.S. Highway 281]] mit den Highways 22 und 36 sowie den Farm Roads 218 und 932 im Zentrum des Countys. Der Ort hat eine Gesamtfläche von 7,5 km², wovon 0,2 km² Wasserfläche ist.

== Geschichte ==
Der Ort wurde 1858 gegründet und benannt nach [[James Hamilton junior|James Hamilton]], einem Gouverneur von [[South Carolina]]. Drei Jahre später wurde das erste Postbüro eingerichtet und 1873 war die Einwohnerzahl auf 200 gestiegen.

== Demografische Daten ==
Nach der [[United States Census 2000|Volkszählung im Jahr 2000]] lebten hier 2.977 Menschen in 1.227 Haushalten und 779 Familien. Die Bevölkerungsdichte betrug 406,2 Einwohner pro km². Ethnisch betrachtet setzte sich die Bevölkerung zusammen aus 95,13% weißer Bevölkerung, 0,07% Afroamerikanern, 0,17% amerikanischen Ureinwohnern, 0,34% Asiaten, 0,00% Bewohnern aus dem pazifischen Inselraum und 3,29% aus anderen ethnischen Gruppen. Etwa 1,01% waren gemischter Abstammung und 6,75% der Bevölkerung waren spanischer oder lateinamerikanischer Abstammung.

Von den 1.227 Haushalten hatten 28,6% Kinder unter 18 Jahre, die im Haushalt lebten. 49,7% davon waren verheiratete, zusammenlebende Paare. 10,4% waren allein erziehende Mütter und 36,5% waren keine Familien. 34,2% aller Haushalte waren Singlehaushalte und in 21,2% lebten Menschen, die 65 Jahre oder älter waren. Die Durchschnittshaushaltsgröße betrug 2,33 und die durchschnittliche Größe einer Familie belief sich auf 3,00 Personen.

25,8% der Bevölkerung waren unter 18 Jahre alt, 6,3% von 18 bis 24, 23,5% von 25 bis 44, 20,8% von 45 bis 64, und 23,6% die 65 Jahre oder älter waren. Das Durchschnittsalter war 41 Jahre. Auf 100 weibliche Personen aller Altersgruppen kamen 85,3 männliche Personen. Auf 100 Frauen im Alter von 18 Jahren und darüber kamen 77,7 Männer.

Das jährliche Durchschnittseinkommen eines Haushalts betrug 26.585 [[United States Dollar|USD]], das Durchschnittseinkommen einer Familie 38.702 USD. Männer hatten ein Durchschnittseinkommen von 27.074 USD gegenüber den Frauen mit 17.500 USD. Das Prokopfeinkommen betrug 15.012 USD. 15,9% der Bevölkerung und 12,2% der Familien lebten unterhalb der Armutsgrenze. Davon waren 23,0% Kinder und Jugendliche unter 18 Jahren und 16,1% waren 65 oder älter.

== Siehe auch ==
*[[Liste der Städte in Texas]]

== Weblinks ==
*[http://www.tshaonline.org/handbook/online/articles/HH/hgh2.html Handbook of Texas], engl.

[[Kategorie:County Seat in Texas]]

[[en:Hamilton, Texas]]
[[ht:Hamilton, Texas]]
[[nl:Hamilton (Texas)]]
[[pt:Hamilton (Texas)]]
[[vo:Hamilton (Texas)]]

Pepsodent

2009-12-08T00:51:49Z

Loadmaster: added :File:Pepsodent-0179c.jpg

[[File:Pepsodent-0179c.jpg|thumb|right|Pepsodent toothpaste]]
[[Image:Irium.jpg|thumb|right|US Pepsodent TV advertisement]]
'''Pepsodent''' is a brand of [[toothpaste]] with a [[wintergreen]] flavor. It was formerly owned by [[Unilever]] (but, since 2003, by [[Church and Dwight]] in USA).

It was advertised for its purported properties fighting tooth decay, attributed in advertisements to the supposed ingredient ''Irium''. Irium is another word for [[sodium lauryl sulfate]], an inexpensive [[ion]]ic [[surfactant]]<ref name="Susan_Budavari">Susan Budavari, Maryadele J. O'Neil, Ann Smith, Patricia E. Heckelman, Joanne F. Kinneary. 1996. The Merck Index, twelfth edition. Merk & Co., Inc.: White house Station, NJ. Page 1478</ref>. However, in a 1994 speech, then-[[Federal Communications Commission|FCC]] chairman [[Reed Hundt]] claimed that the "Irium" mentioned in Pepsodent advertisements "didn't exist".<ref>{{cite paper |author=Reed E. Hundt |authorlink=Reed Hundt |title=Address Before the NAB Radio Show |date=[[October 13]] [[1994]] |url=http://www.fcc.gov/Speeches/Hundt/spreh432.txt |accessdate=2007-08-21 }}</ref>

Another ingredient, "I.M.P." was purported to whiten teeth. Its best-known slogan was “You'll wonder where the yellow went / when you brush your teeth with Pepsodent!” British comedian [[Jasper Carrott]] referred to the slogan in one of his [[Stand-up comedy|stand-up]] routines, saying “On your tongue - that's where the yellow went!”

Pepsodent was a very popular brand before the mid [[1950s|'50s]], but its makers were slow to add [[fluoride]] to its formula to counter the rise of other highly promoted brands such as [[Crest (brand)|Crest]] and [[Gleem toothpaste]] by [[Procter & Gamble]], and Colgate's [[eponym]]ous product; sales of Pepsodent plummeted. Today Pepsodent is a “value brand” marketed primarily in discount stores and retails for roughly half the price of similarly-sized tubes of Crest or of Colgate.

In the 1930s a massive animated neon advertising sign, featuring a young girl on a swing, hung on a building in [[Times Square]] in [[New York City]]. This ad was re-created for the climax of the 2005 film ''[[King Kong (2005 film)|King Kong]]''.

The product was discontinued in South Africa in 1974 but was revived in 1976 with a new ad slogan "Gets Your Teeth Their Whitest" featuring celebrity endorsers [[Rita Moreno]], [[Steve Lawrence]], and others. The popular slogan was also changed in South Africa to "You'll wonder where the dullness went / when you polish your teeth with Pepsodent".

Pepsodent is still sold as a Unilever property in India[http://www.hll.com/brands/pepsodent.asp], Indonesia [http://www.pepsodent.co.id], Chile[http://www.pepsodent.cl/sitio.html], Finland[http://www.pepsodent.fi/], and several other countries.

==In popular culture==
Pepsodent was so popular that [[Rodgers and Hammerstein]] included a reference to it in their long-running 1949 hit musical ''[[South Pacific (musical)|South Pacific]]'', when the Seabees sing about the native woman Bloody Mary:

''Bloody Mary's chewing betel nuts'' 
''And she don't use Pepsodent''.

It's also referenced in some versions of [[Cole Porter]]'s song ''[[You're The Top]]'':

''You're the baby grand of a lady and a gent,'' 
''You're an old Dutch master, you're Mrs. Astor,'' 
''You're Pepsodent''

==References==
{{reflist}}

==External links==
* [http://adclassix.com/classictvcommercials/1948pepsodenttoothpaste.htm Advertisement video]
* [http://www.springbokradio.com/ADSPEPSODENT.html 1972 South African radio commercial]
* [http://www.churchdwight.com/Conprods/oralcare/ Church and Dwight Oral Care website]

[[Category:Brands of toothpaste]]
[[Category:Church & Dwight brands]]

[[es:Pepsodent]]
[[fi:Pepsodent]]
[[sv:Pepsodent]]

Kurdische Alphabete

2009-10-29T16:36:12Z

Loadmaster: /* Yekgirtú (IS) */ bold

The '''Kurdish alphabet''' is a [[writing system]] for the [[Kurdish language]]. Three systems currently exist. The form used in [[Turkey]] was derived from the [[Latin alphabet]] by [[Jaladat Ali Badirkhan|Jeladet Ali Bedirkhan]] in 1932, and thus is also called the '''Bedirxan script''' or more properly [[Hawar (magazine)|Hawar]]. It is used by [[Kurds]] in [[Turkey]] and [[Syria]]. The [[Sorani]] alphabet is used by Kurds in Iraq and Iran, and there is also a recent alphabet called [[Yekgirtú]] which attempts to unify these.

==Hawar alphabet==
The alphabet of the Kurmanji Kurdish dialect contains 31 letters:

A, B, C, Ç, D, E, Ê, F, G, H, I, Î, J, K, L, M, N, O, P, Q, R, S, Ş, T, U, Û, V, W, X, Y, Z

a, b, c, ç, d, e, ê, f, g, h, i, î, j, k, l, m, n, o, p, q, r, s, ş, t, u, û, v, w, x, y, z

'''A''': The "o" in Ab'''o'''ve Or the "a" in B'''a'''r

'''B''': The "B" in '''B'''all or '''B'''ook

'''C''': The "J" in '''J'''ob

'''Ç''': The "Ch" in '''Ch'''eck

'''D''': The "D" '''D'''oor or '''D'''esk

'''E''': The "A" in '''A'''ccept and "e" in T'''e'''st

'''Ê''': the "e" in H'''e'''ll or "Ai" in '''Ai'''r

'''F''': The "F" in '''F'''ar

'''G''': The "G" in '''G'''rass

'''H''': The "H" in '''H'''ot

'''I''': The "e" in Op'''e'''n

'''Î''': The "'''ee'''" in Fl'''ee''' and S'''ee'''k

'''J''': The "s" in Plea'''s'''ure or Per'''s'''ian

'''K''': The "C" in '''C'''op or "K" in '''K'''angaroo

'''L''': The "L" in '''L'''ove

'''M''': The "M" in '''M'''en

'''N''': The "N" in '''N'''ever

'''O''': The "O" in '''O'''ld or F'''o'''rt

'''P''': The "P" in '''P'''olice

'''Q''': The "Q" in Ira'''q'''

'''R''': The "R" in '''R'''apid

'''S''': The "S" in '''S'''tar

'''Ş''': The "Sh" in '''Sh'''oes

'''T''': The "T" in '''T'''ower

'''U''': The "u" in K'''u'''rd

'''Û''': The "oo" in B'''oo'''t or Br'''oo'''m

'''V''': The "V" in '''V'''ehicle or '''V'''al'''v'''e

'''W''': the "W" in '''W'''ater or '''W'''ood

'''X''': No English equivalent. German "'''Ch'''"

'''Y''': The "Y" in '''Y'''es

'''Z''': The "Z" in '''Z'''ero

There are eight vowels in this alphabet—three short and five long. For the vowels that can be both short and long, the long vowels are represented using a [[circumflex]]. The short vowels are (E, I, U) and the long ones are (A, Ê, Î, O, Û).

When presenting this alphabet in his magazine "Hawar", Jeladet Ali Bedirkhan proposed using ḧ, ẍ and ' for غ ,ح, and ع, sounds which he judged to be "non-Kurdish" (see [http://www.nefel.com/epirtuk/pdf/celadet_ali_bedir_xan_elfabeugramer_02.pdf?NR:122] page 12,13). These three glyphs do not have the status of letter and serve to represent these sounds when they are indispensable to comprehension.

The [[Turkey|Turkish]] state does not recognise the alphabet, and use of the letters ''Q'', ''W'', ''X'' which do not exist in the [[Turkish alphabet]] has led to persecution in 2000 and 2003 (see [http://www.ihf-hr.org/viewbinary/viewdocument.php?download=1&doc_id=6391], p. 8, and [http://www.rsf.org/rsf/uk/html/mo/cplp/cp/000300.html]). Since September 2003, many Kurds have applied to the courts seeking to change their names to Kurdish ones written with the letters ''Q'', ''W'', and ''X'' but eventually failed.<ref name=cla3a>{{cite web | url = http://www.unhchr.ch/minorities/statements10/CLA3a.doc | title = Submission to the Sub-Commission on Promotion and Protection of Human Rights: Working Group of Minorities; Tenth Session, Agenda Item 3 (a) | first = Saniye | last = Karakaş | coauthors = [[Diyarbakır]] Branch of the Contemporary Lawyers Association | publisher = [[United Nations Commission on Human Rights]] | year=2004 | month= March | accessdate = 2006-11-07 | format = [[Microsoft Word|MS Word]]|quote = Kurds have been officially allowed since September 2003 to take Kurdish names, but cannot use the letters x, w, or q, which are common in Kurdish but do not exist in Turkey's version of the Latin alphabet. [...] Those letters, however, are used in Turkey in the names of companies, TV and radio channels, and trademarks. For example [[Turkish Army]] has company under the name of [[AXA]] [[OYAK]] and there is [[Show TV|SHOW TV]] television channel in Turkey.}}</ref>

==Soraní alphabet==
The [[Soraní]] Kurdish dialect is mainly written using a modified Arabic-based alphabet with 33 letters. Unlike the standard [[Arabic alphabet]], which is an [[abjad]], Soraní is a true [[alphabet]] in which vowels are mandatory, making the script easy to read. Yet it is not a complete representation of Kurdish sounds, as it is missing short /i/ (as in English ''bit''), and is also unable to differentiate between /w/ and short /u/ as well between /y/ and /î/. However it does show the two [[pharyngeal consonant]]s, as well as a [[voiced velar fricative]] used in Kurdish.


ى ,ێ ,ە ,ﮪ ,ﻭﻭ ,ﻭ ,ﯙ ,ﻥ ,ﻡ ,ڵ ,ﻝ ,ﮒ ,ﮎ ,ﻕ ,ڤ ,ﻑ ,ﻍ ,ﻉ ,ﺵ ,ﺱ ,ﮊ ,ﺯ ,ڕ ,ﺭ ,ﺩ ,ﺥ ,ﺡ ,ﭺ ,ﺝ ,ﺕ ,ﭖ ,ﺏ ,ﺍ

Kurds in [[Iraq]] and [[Iran]] mainly use this alphabet, though the Kurdish Latin alphabet is also in use.

==Cyrillic alphabet==
A third system, used for the few (Kurmanji-speaking) Kurds in the former [[Soviet Union]], uses a modified [[Cyrillic alphabet]], consisting of 32 letters:

А, Б, В, Г, Г', Д, Е, Ә, Ә', Ж, З, И, Й, К, К', Л, М, Н, О, Ö, П, П', Р, Р', С, Т, Т', У, Ф, Х, Һ, Һ', Ч, Ч', Ш, Щ, Ь, Э, Q, W

==Armenian alphabet==
From 1921 to 1929 the [[Armenian alphabet]] was used for Kurdish languages in the [[Soviet Armenia]].<ref>{{ru icon}} [http://www.krugosvet.ru/articles/81/1008155/1008155a1.htm Курдский язык (''Kurdish language'')], Кругосвет (''Krugosvet'')</ref>

Then it was replaced to [[Janalif]]-like Latin alphabet during [[Latinisation (USSR)|Latinisation campaign]].

==Uniform Turkic Alphabet adaptation for Kurdish==
In 1928 Kurdish language in all [[USSR]], including [[Armenian SSR]], was switched to Latin alphabet, containing some additional Cyrillic characters: '''a, b, c, ç, d, e, ә, f, g, г, h, i, ь, j, k, {{Unicode|ʀ}}, l, m, {{Unicode|ɴ}}, o, ө, w, p, n, q, ч, s, ш, ц, t, u, y, v, x, z, {{Unicode|ƶ}}'''. In 1929 it was reformed and was replaced by
<ref>{{ru icon}} Культура и письменность Востока (Eastern Culture and Literature). 1928, №2.</ref>:

{| style="font-family:Arial Unicode MS; font-size:1.4em; border-color:#000000; border-width:1px; border-style:solid; border-collapse:collapse; background-color:#F8F8EF"
| style="width:3em; text-align:center; padding: 3px;" | A a
| style="width:3em; text-align:center; padding: 3px;" | B b
| style="width:3em; text-align:center; padding: 3px;" | C c
| style="width:3em; text-align:center; padding: 3px;" | Є є
| style="width:3em; text-align:center; padding: 3px;" | Ç ç
| style="width:3em; text-align:center; padding: 3px;" | D d
| style="width:3em; text-align:center; padding: 3px;" | E e
| style="width:3em; text-align:center; padding: 3px;" | Ә ә
|-
| style="width:3em; text-align:center; padding: 3px;" | Ә́ ә́
| style="width:3em; text-align:center; padding: 3px;" | F f
| style="width:3em; text-align:center; padding: 3px;" | G g
| style="width:3em; text-align:center; padding: 3px;" | Ƣ ƣ
| style="width:3em; text-align:center; padding: 3px;" | H h
| style="width:3em; text-align:center; padding: 3px;" | Ħ ħ
| style="width:3em; text-align:center; padding: 3px;" | I i
| style="width:3em; text-align:center; padding: 3px;" | J j
|-
| style="width:3em; text-align:center; padding: 3px;" | K k
| style="width:3em; text-align:center; padding: 3px;" | {{Unicode|K̡ k̡}}
| style="width:3em; text-align:center; padding: 3px;" | L l
| style="width:3em; text-align:center; padding: 3px;" | M m
| style="width:3em; text-align:center; padding: 3px;" | N n
| style="width:3em; text-align:center; padding: 3px;" | O o
| style="width:3em; text-align:center; padding: 3px;" | Ö ö
| style="width:3em; text-align:center; padding: 3px;" | P p
|-
| style="width:3em; text-align:center; padding: 3px;" | {{Unicode|Ṕ ṕ}}
| style="width:3em; text-align:center; padding: 3px;" | Q q
| style="width:3em; text-align:center; padding: 3px;" | R r
| style="width:3em; text-align:center; padding: 3px;" | S s
| style="width:3em; text-align:center; padding: 3px;" | Ş ş
| style="width:3em; text-align:center; padding: 3px;" | T t
| style="width:3em; text-align:center; padding: 3px;" | {{Unicode|T̡ t̡}}
| style="width:3em; text-align:center; padding: 3px;" | U u
|-
| style="width:3em; text-align:center; padding: 3px;" | Û û
| style="width:3em; text-align:center; padding: 3px;" | V v
| style="width:3em; text-align:center; padding: 3px;" | W w
| style="width:3em; text-align:center; padding: 3px;" | X x
| style="width:3em; text-align:center; padding: 3px;" | Y y
| style="width:3em; text-align:center; padding: 3px;" | Z z
| style="width:3em; text-align:center; padding: 3px;" | Ƶ ƶ
| style="width:3em; text-align:center; padding: 3px;" | Ь ь
|}

==''Yekgirtú (IS)''==
The ''Yekgirtú'' (Yekgirtí, yekgirig) alphabet is a recent devised writing system by Kurdish Academy of Language. It has many advantages compared to the Kurmanji and Sorani alphabets. It is adapted for all Kurdish dialects and not exclusive to just one, and is therefore called ''Yekgirtú'', which means "unified." It is also better adapted to the vowel-rich Kurdish language than is the Arabic script.

The Kurdish Academy of Language [KAL] realises that there are too many shortcomings with current Kurdish writing systems. These include workability, cross dialectal usage, and a lack of International IT-based Standards and representation for Kurdish. To avoid the communication obstacles presented by the existence of various Kurdish writing systems, KAL has introduced a standard '''Kurdish Unified Alphabet''' (Yekgirtú) based on International [[ISO-8859-1]] Standards. This modern Kurdish (IS) alphabet contained some minor changes in the existing Latin based alphabet and adopting new sings. The new signs were introduced to improve the flexibility of the writing system in Kurdish. This effort was undertaken as part of KAL's broad endeavour to revive and promote the use of the Kurdish language for the benefit of young Kurds. The system devised and presented here by KAL is simple and adequate for the purpose of communicating via the Internet and any electronic media.

The development of the Unified Kurdish Alphabet has proceeded along three lines. First one letter has been designated for each sound (with the exception of digraph characters such as velar [ll], trill [rr], "jh" and "sh"). Second, no diacritical marks have been allowed that are difficult to convey via the Internet without the use of specialised programs. Specifically, all characters in the unified alphabet have been chosen carefully from the ISO-8859-1 "[[Latin 1]]" system for West European languages in order to ensure that the Kurdish characters follow one single global standard only. Loanwords need to naturalise and comply with common global Kurdish spelling rules whilst local exceptional pronunciations are also justified. The Kurdish Unified Alphabet contains 34 characters including 4 [[digraph]] cases (jh, ll, rr, sh) and 4 characters with [[diacritics]] (é, í, ú, ù). It represents 9 vowels (a, e, é, i, í, o, u, ú, ù) and 25 consonants.

A, B, C, D, E, É, F, G, H, I, Í, J, Jh, K, L, ll, M, N, O, P, Q, R, rr, S, Sh, T, U, Ú, Ù, V, W, X, Y, Z

Recently it has been used more than the Arabic script on Kurdish TV.

==Comparison of Kurmanjí, Yekgirtú and Sorani alphabets==
{| class="wikitable"
|-
! Latin Kurmanjî
! Yekgirtú
! Cyrillic Kurmanjí
! Sorani - Stand-alone
! Sorani - Initial
! Sorani - Medial
! Sorani - Final
! [[IPA]]
|- align="center"
| A,a 
| A,a 
| A,a 
| ا
| ئا
| colspan="1" | —
|ـا
| {{IPA|[aː]}}

|- align="center"
| B,b 
| B,b 
| Б,б 
| ﺏ
| ﺑ
| ـبـ
| ـب
| {{IPA|[b]}}
|- align="center"
| C,c 
| J,j 
| Щ,щ 
| {{lang|ar|ﺝ}}
| {{lang|ar|ﺟ}}
| {{lang|ar| ـجـ}}
| {{lang|ar| ـج}}
| {{IPA|[d͡ʒ]}}
|- align="center"
| Ç,ç 
| C,c 
| Ч,ч 
| {{lang|ar|چ }}
| {{lang|ar|ﭼ}}
| {{lang|ar| ـچـ }}
| {{lang|ar| ـچ}}
| {{IPA|[t͡ʃ]}}
|- align="center"
| D,d 
| D,d 
| Д,д 
| {{lang|ar|ﺩ}}
| colspan="2" | —
| {{lang|ar| ــد}}
| {{IPA|[d]}}
|- align="center" ﮫﺋﮫ ﮦ
| E,e 
| E,e 
| Ә,ә 
| {{lang|ar|ﮦ}}
| {{lang|ar|ﺋﮫ}}
| colspan="1" | —
| {{lang|ar|ﮫ}}
| {{IPA|[ɛː]}}
|- align="center"
| Ê,ê 
| É,é 
| E,e(Э э) 
| {{lang|ar|ێ}}
| {{lang|ar|ئێـ}}
| {{lang|ar|ـێـ}}
| {{lang|ar|ـێ}}
| {{IPA|[e]}}
|- align="center"
| F,f 
| F,f 
| Ф,ф 
| {{lang|ar|ﻑ}}
| {{lang|ar|ﻓ}}
| {{lang|ar| ـفـ }}
| {{lang|ar|ﻒ}}
| {{IPA|[f]}}
|- align="center"
| G,g 
| G,g 
| Г,г 
| {{lang|ar|ﮒ}}
| {{lang|ar|ﮔ}}
| {{lang|ar|ـگـ}}
| {{lang|ar| ـگ }}
| {{IPA|[ɡ]}}
|- align="center"
| H,h 
| H,h 
| h,h 
| {{lang|ar|ﻫ}}
| {{lang|ar|ﻫ}}
| {{lang|ar| ـهـ}}
| {{lang|ar|ـهـ}}
| {{IPA|[h]}}
|- align="center"
| Ḧ,ḧ <ref name="Bedirxan 2002">{{cite web | url = http://www.nefel.com/epirtuk/pdf/celadet_ali_bedir_xan_elfabeugramer_02.pdf?NR:122 | title = Elfabeya kurdî & Bingehên gramera kurdmancî | first = Celadet Ali | last = Bedirxan | coauthors = [[Stockholm]] Arif Zêrevanî | publisher = [http://www.nefel.com NEFEL] | year=2002 | format = [[pdf]] }}</ref>
| H',h' 
| h’,h’ 
| {{lang|ar|ح}}
| {{lang|ar|حـ}}
| {{lang|ar| ـحـ}}
| {{lang|ar|ـح}}
| [[ħ|{{IPA|[ħ]}}]]
|- align="center"
| — 
| ' 
| — 
| {{lang|ar|ع}}
| {{lang|ar|عـ}}
| {{lang|ar| ـعـ}}
| {{lang|ar|ـع}}
| [[ʕ|{{IPA|[ʕ]}}]]
|- align="center"
| I,i 
| I,i 
| Ь,ь 
| colspan="4" | —
| [[ɯ|{{IPA|[ɯ]}}]]
|- align="center"
| Î,î 
| Í,í 
| И,и 
| {{lang|ar|ﯼ}}
| {{lang|ar|ﺋﯾ}}
| {{lang|ar|ـيـ}}
| {{lang|ar|ﯽ}}
| {{IPA|[iː]}}
|- align="center"
| J,j 
| Jh,jh 
| Ж,ж 
| {{lang|ar|ﮊ}}
| colspan="2" | —
| {{lang|ar| ـژ }}
| [[ʒ|{{IPA|[ʒ]}}]]
|- align="center"
| K,k 
| K,k 
| K,k 
| {{lang|ar|ﮎ}}
| {{lang|ar|ﮐ}}
| {{lang|ar|ـکـ}}
| {{lang|ar|ﮏ}}
| {{IPA|[k]}}
|- align="center"
| L,l 
| L,l 
| Л,л 
| {{lang|ar|ﻝ}}
| {{lang|ar|ﻟ}}
| {{lang|ar|ـلـ}}
| {{lang|ar| ـل }}
| {{IPA|[l]}}
|- align="center"
| colspan="1" | —
| ll 
| Л’,л’ 
| {{lang|ar|ڵ}}, {{lang|ar|ڶ}}
| {{lang|ar|ڵــ}}, {{lang|ar|ڶــ}}
| {{lang|ar|ـڵـ}}, {{lang|ar|ـڶـ}}
| {{lang|ar| ـڵ}}, {{lang|ar| ـڶ}}
| {{IPA|[lˁ]}}
|- align="center"
| M,m 
| M,m 
| M,m 
| {{lang|ar|ﻡ}}
| {{lang|ar|ﻣ}}
| {{lang|ar| ـمـ }}
| {{lang|ar|ـم}}
|{{IPA|[m]}}
|- align="center"
| N,n 
| N,n 
| Н,н 
| {{lang|ar|ﻥ}}
| {{lang|ar|ﻧ}}
| {{lang|ar|ـنـ}}
| {{lang|ar| ـن}}
|{{IPA|[n]}}
|- align="center"
| O,o 
| O,o 
| O,o 
| {{lang|ar|ۆ}}
| {{lang|ar|ئۆ}}
| colspan="1" | -
| {{lang|ar|ـۆ}}
| {{IPA|[o]}}

|- align="center"
| P,p 
| P,p 
| П,п 
| {{lang|ar|پ}}
| {{lang|ar|پــ}}
| {{lang|ar| ـپـ }}
| {{lang|ar| ـپ}}
| {{IPA|[p]}}
|- align="center"
| Q,q 
| Q,q 
| Q,q 
| {{lang|ar|ﻕ}}
| {{lang|ar|ﻗ}}
| {{lang|ar|ـقـ }}
| {{lang|ar| ـق}}
| {{IPA|[q]}}
|- align="center"
| R,r 
| R,r 
| P,p 
| {{lang|ar|ﺭ}}
| colspan="2" | —
| {{lang|ar|ـر}}
| {{IPA|[r]}}
|- align="center"
| colspan="1" | —
| rr 
| Р’,р’ 
| {{lang|ar|ڕ}}, {{lang|ar|ڒ}}, {{lang|ar|ڔ}}
| colspan="2" | —
| {{lang|ar| ـڕ}}, {{lang|ar| ـڒ}}, {{lang|ar| ـڔ}}
| {{IPA|[r]}}
|- align="center"
| S,s 
| S,s 
| C,c 
| {{lang|ar|ﺱ}}
| {{lang|ar|ﺳ}}
| {{lang|ar|ـسـ}}
| {{lang|ar| ـس}}
| {{IPA|[s]}}
|- align="center"
| Ş,ş 
| Sh,sh 
| Ш,ш 
| {{lang|ar|ﺵ}}
| {{lang|ar|ﺷ}}
| {{lang|ar| ـشـ }}
| {{lang|ar|ـش }}
| {{IPA|[ʃ]}}
|- align="center"
| T,t 
| T,t 
| T,т 
| {{lang|ar|ﺕ}}
| {{lang|ar|ﺗ}}
| {{lang|ar| ـتـ }}
| {{lang|ar|ـت}}
| {{IPA|[t]}}
|- align="center"
| U,u 
| U,u 
| Ö,ö 
| {{lang|ar|ﻭ}}
| colspan="2" | —
| {{lang|ar|ـو}}
| {{IPA|[œ]}}
|- align="center"
| Û,û 
| Ú,ú 
| У,у 
| {{lang|ar|ﻭﻭ, ۇ}}
| colspan="2" | —
| {{lang|ar|ـوﻭ, ـۇ}}
| {{IPA|[uː]}}
|- align="center"
| colspan="1" | —
| colspan="1" | Ù,ù 
| colspan="1" | —
| {{lang|ar|ۈ}}
| colspan="2" | —
| {{lang|ar|ـۈ}}
| [[ʉ|{{IPA|[ʉː]}}]]
|- align="center"
| V,v 
| V,v 
| B,в 
| {{lang|ar|ڤ, ۋ}}
| {{lang|ar| ڤـ}}
| {{lang|ar|ـڤـ }}
| {{lang|ar|ـڤ ,ـۋ }}
| {{IPA|[v]}}
|- align="center"
| W,w 
| W,w 
| W,w 
| {{lang|ar|ﻭ}}
| colspan="2" | —
| {{lang|ar|ـو}}
| {{IPA|[w]}}
|- align="center"
| X,x 
| X,x 
| X,x 
| {{lang|ar|ﺥ}}
| {{lang|ar|ﺧ}}
| {{lang|ar|ـخـ}}
| {{lang|ar|ـخ}}
| {{IPA|[x]}}
|- align="center"
| Ẍ,ẍ <ref name="Bedirxan 2002"/>
| X',x'
| Ѓ,ѓ 
| {{lang|ar|ﻍ}}
| {{lang|ar|ﻏ}}
| {{lang|ar|ـغـ}}
| {{lang|ar|ـغ}}
| [[ʁ|{{IPA|[ʁ]}}]]
|- align="center"
| Y,y 
| Y,y 
| Й,й 
| {{lang|ar|ﯼ}}
| {{lang|ar|يـ}}
| colspan="2" | —
| {{IPA|[j]}}
|- align="center"
| Z,z 
| Z,z 
| З,з 
| {{lang|ar|ﺯ}}
| colspan="2" | —
| {{lang|ar| ـز}}
| {{IPA|[z]}}
|}

== References ==
{{reflist}}

== External links ==
* [http://kurdishacademy.org/?q=node/145 KAL - A table of the various Kurdish alphabets]
* [http://www.omniglot.com/writing/kurdish.htm Kurdish language, alphabet and pronunciation]
* [http://www.kurditgroup.org/downloads.php?cid=2 Kurdish Unicode Fonts]

{{Kurdish language}}

{{DEFAULTSORT:Kurdish Alphabet}}
[[Category:Latin-derived alphabets]]
[[Category:Arabic-derived alphabets]]
[[Category:Kurdish language]]

[[ar:أبجدية كردية]]
[[da:De kurdiske alfabeter]]
[[de:Kurmandschi-Alphabet]]
[[ku:Alfabeyên kurdî]]
[[arz:كوردى]]
[[pl:Alfabet kurdyjski]]
[[ru:Курдская письменность]]
[[ckb:ئەلفوبێکانی کوردی]]
[[tg:Алифбои курдӣ]]
[[tr:Kürt alfabesi]]
[[zh:库尔德语字母]]

Holozän-Kalender

2009-10-02T15:54:04Z

Loadmaster: /* Top */ example, added "AD" for clarity

{{redirect3|H.E.|For other uses of H.E., see [[He (disambiguation)]]}}{{redirect3|Holocene era|For the geological epoch, see [[Holocene Epoch]]}}
The '''Holocene calendar''', popular term for the '''''Holocene Era''''' or '''''Human Era''''', is a year numbering system similar to [[astronomical year numbering]] but adds 10,000, placing its first year at the start of the ''Human Era'' (HE, the beginning of human [[civilization]]) the approximation of the [[Holocene|Holocene Epoch]] (HE, post [[Ice Age]]) for easier [[geological]], [[archaeological]], [[dendrochronological]] and [[historical]] dating. The current [[Gregorian Calendar|Gregorian]] year can be transformed by simply placing a 1 before it (e.g., [[Anno Domini|AD]] {{ #expr: {{{year|<noinclude>{{CURRENTYEAR}}</noinclude><includeonly>{{PAGENAME}}</includeonly>}}} }} becomes {{ #expr: {{{year|<noinclude>{{CURRENTYEAR}}</noinclude><includeonly>{{PAGENAME}}</includeonly>}}}+10000 }} HE). The ''Human Era'' was first proposed by [[Cesare Emiliani]] in 1993 {{nowrap|(11993 HE)}}.<ref>Cesare Emiliani, "[http://adsabs.harvard.edu/abs/1993Natur.366..716E Calendar Reform]", ''Nature'' '''366''' (1993) 716.</ref><ref>[http://www.readersadvice.com/mmeade/house/holocene.html The Holocene Calendar] at Meerkat Meade</ref><ref>[http://weseman.com/page.php?form=human_era_calendar The Human Era Calendar] by Harry and Svetlana Weseman</ref>

==Motivation==
Cesare Emiliani's proposal for a [[calendar reform]] sought to solve a number of problems with the current [[Anno Domini]] and [[Common Era]]s, which number the years of the commonly accepted world calendar. The issues include:
* The ''Anno Domini'' and ''Common Eras'' begin at the presumed year of the birth of [[Jesus Christ]]. This Christian aspect (especially the use of ''Before Christ'' and ''Anno Domini'') can be offensive to non-Christians.<ref>[http://www.religioustolerance.org/ce.htm Controversy over the use of "CE/BCE" or "AD/BC" dating notation] at Religious Tolerance.org</ref>
* Biblical scholarship is virtually unanimous that the birth of Jesus Christ would actually have been a few years prior to AD 1. This makes the calendar inaccurate insofar as Christian dates are concerned.
* There is no [[year zero]] as 1 BC is followed immediately by AD 1.
* BC years are counted down when moving from past to future, thus 44 BC is after 250 BC. This makes calculating date ranges in the Holocene era across the BC/AD boundary more complicated than in the HE.

Instead, HE places its [[Epoch (reference date)|epoch]] or year one of the current [[era]] to 10,000 BC. This is a rough approximation of the start of the current [[Epoch (geology)|geologic epoch]], the [[Holocene]] (the name means ''entirely recent''). The motivation for this is that human civilization (e.g., the first settlements, agriculture, etc.) is believed to have arisen entirely within this time. All key dates in human history can then be listed using a simple increasing date scale with smaller dates always occurring before larger dates.

===Conversion===
Conversion to the Holocene Era from Julian or Gregorian AD years can be achieved by adding 10,000. BC years are converted by subtracting the BC year from 10,001.

A useful validity check is that the last digit of BC and HE equivalents must add up to 1 or 11.

{| {{fintabell}}
|-
| '''Events'''
| '''Julian or Gregorian years'''
| ''' Holocene Era''' '''Human Era'''
|-
| End of the [[Paleolithic]] Period, All continents (except [[Antarctica]]) inhabited, [[Neolithic Revolution|Agriculture]] and the domestication of animals begins.
| c. 10000 BC
| c. 1 HE
|-
| Earliest [[PPNA Wall of Jericho|walled city]] ([[Jericho]])
| c. 9000 BC
| c. 1000 HE
|-
| First copper found in [[Middle East]] - beginning of [[Copper Age]]
| c. 6000 BC
| c. 4000 HE
|-
| Beginning of [[Indus Valley Civilization]]
| c. 3000 BC
| c. 7000 HE
|-
| Probable date of the completion of the [[Pyramid of Djoser|first Egyptian pyramid]]
| 2611 BC
| 7390 HE
|-
| Beginning of [[Xia Dynasty]] in [[China]]
| c. 2100 BC
| c. 7900 HE
|-
| Foundation of [[Rome]]
| 753 BC
| 9248 HE
|-
| First [[Maya civilization#Writing system|Central American writing systems]]
| c. 400 BC
| c. 9600 HE
|-
| Empire of [[Ashoka the Great|Asoka]]
| 273 BC
| 9728 HE
|-
| Imperial China, [[Qin dynasty]]
| 221 BC
| 9780 HE
|-
| Last year of [[Before Christ|BC]] era
| 1 BC
| 10000 HE
|-
| First year of [[Anno Domini]] era
| AD 1
| 10001 HE
|-
| [[Migration Period]] begins, leading to the [[Fall of Rome]]
| AD 300/476
| 10300/10476 HE
|-
| [[Turkic migration]]s begin
| c. AD 500
| c. 10500 HE
|-
| [[Muslim conquests]] begin
| AD 632
| 10632 HE
|-
| [[Great Zimbabwe]] built
| c. AD 1000
| c. 11000 HE
|-
| [[Hindu-Arabic numerals]] introduced to [[Europe]]
| AD 1202
| 11202 HE
|-
| [[Black Death]] decimates Asia and Europe
| AD 1340s
| 11340s HE
|-
| [[Age of Discovery|European expansion and colonization]] begins
| AD 1419
| 11419 HE
|-
| [[Voyages of Christopher Columbus|European discovery of the New World]]
| AD 1492
| 11492 HE
|-
| Fall of the [[Inca Empire]]
| AD 1572
| 11572 HE
|-
| [[United States Declaration of Independence|America declared independence from Britain]]
| AD 1776
| 11776 HE
|-
| [[Second Industrial Revolution]]
| c. AD 1850
| c. 11850 HE
|-
| [[World War II|Second World War]] and [[nuclear technology|nuclear fission]]
| AD 1939-1945
| 11939-11945 HE
|-
| [[Sputnik 1|First artificial satellite (Sputnik I)]]
| AD 1957
| 11957 HE
|-
| [[Human spaceflight|First human in space]]
| AD 1961
| 11961 HE
|-
| [[Apollo 11|First human landing on the Moon]]
| AD 1969
| 11969 HE
|-
| Current year
| AD {{CURRENTYEAR}}
| {{#expr:{{{year|<noinclude>{{CURRENTYEAR}}</noinclude><includeonly>{{PAGENAME}}</includeonly>}}}+10000 }} HE
|}

== See also ==
* [[Before Present]]
* [[Calendar Era]]
* [[Common Era]]
* [[Julian Period]]

== References ==
{{reflist}}
* {{cite book|author=David Ewing Duncan|title=The Calendar|year=1999|publisher=|pages=331–332|isbn=1-85702-979-8}}
* {{cite book|author=Duncan Steel|title=Marking Time: The Epic Quest to Invent the Perfect Calendar|year=2000|publisher=|pages=149–151|url=http://books.google.com/books?id=fsni_qV-FJoC&printsec=frontcover&pg=PA149}}
* {{cite book|author=Günther A. Wagner|title=''Age Determination of Young Rocks and Artifacts: Physical and Chemical Clocks in Quaternary Geology and Archeology''|year=1998|publisher=Springer|url=http://books.google.com/books?id=ADuZDCa08kwC&pg=PA48|page=48}}
* [http://timelines.ws/0A1MILL_3300BC.HTML Timeline of World History]
* "[http://www.blackwell-synergy.com/doi/abs/10.1111/j.1365-2451.2004.00457.x News and comment]", ''Geology Today'', '''20'''/3 (2004) 89–96.

{{Time Topics}}
{{Time measurement and standards}}
{{Chronology}}

[[Category:Archaeology]]
[[Category:Calendar eras]]
[[Category:Chronology]]
[[Category:Geochronology]]
[[Category:Holocene]]
[[Category:Proposed calendars]]
[[Category:Jōmon period]]

[[cs:Holocénový letopočet]]
[[es:Calendario holoceno]]
[[fa:تقویم هولوسین]]
[[fr:Calendrier holocène]]
[[it:Calendario olocenico]]
[[nl:Holocene kalender]]
[[sh:Holocenski kalendar]]
[[fi:Holoseenin ajanlasku]]
[[sv:Holocen era]]
[[zh-yue:全新世紀年]]
[[zh:全新世紀年]]

Compiler: Prinzipien, Techniken und Werkzeuge

2009-06-07T20:52:54Z

Loadmaster: undo edit by 209.208.18.97, rm apparently irrelevant link to Hackers_(film)

{{Infobox Book
| name = Compilers: Principles, Techniques, and Tools
| title_orig =
| translator =
| image = [[Image:purple_dragon_book_b.jpg|thumbnail|right|200px]]
| image_caption = The cover of the second edition, showing a knight and dragon
| author = [[Alfred V. Aho]], [[Monica S. Lam]], [[Ravi Sethi]], ''and'' [[Jeffrey D. Ullman]]
| illustrator =
| cover_artist =
| country =
| language = [[English language|English]]
| series =
| subject =
| genre =
| publisher = [[Pearson Education, Inc]]
| pub_date = 1986, ''2006''
| english_pub_date =
| media_type =
| pages =
| isbn = ISBN 0-201-10088-6, ISBN 0-321-48681-1
| oclc =
| preceded_by =
| followed_by =
}}

'''Compilers: Principles, Techniques, and Tools''' <ref>Aho, Sethi, Ullman, ''Compilers: Principles, Techniques, and Tools'', Addison-Wesley, 1986. ISBN 0-201-10088-6</ref> is a famous [[computer science]] textbook by [[Alfred V. Aho]], [[Ravi Sethi]], and [[Jeffrey D. Ullman]] about [[compiler]] construction. Although two decades have passed since the publication of the first edition, it is widely regarded as the classic definitive [[compiler]] technology text.

It is known as the '''Dragon Book''' because its covers depict a [[knight]] and a [[western dragon|dragon]] in battle, a metaphor for conquering complexity. The first edition is informally called the “red dragon book” to distinguish it from the second edition and from Aho & Ullman’s ''[[Principles of Compiler Design]]'' (1977, sometimes known as the “green dragon book” because the dragon on its cover is green).

A new edition of the book was published in August 2006.

Topics covered in the first edition include:
*[[Compiler]] structure
*[[Lexical analysis]] (including [[regular expression]]s and [[finite state automaton|finite automata]])
*[[Syntax analysis]] (including [[context-free grammar]]s, [[LL parser]]s, [[Bottom-up parsing|bottom-up parser]]s, and [[LR parser]]s)
*[[Syntax-directed translation]]
*[[Type checking]] (including [[type conversions]] and [[polymorphism (computer science)|polymorphism]])
*[[Run-time environment]] (including [[parameter passing]], [[symbol tables]], and [[storage allocation]])
*[[Code generation]] (including [[intermediate code generation]])
*[[Compiler optimization|Code optimization]]

==Second edition==
Following in the tradition of its two predecessors, the second edition features a dragon and a knight on its cover, designed by Strange Tonic Productions; for this reason, the series of books is commonly known as the ''[[Dragon Book]]''s. Different editions in the series are further distinguished by the color of the dragon. This edition is informally known as the '''purple dragon'''. [[Monica S. Lam]] of [[Stanford University]] became a co-author with this edition.

The second edition includes several additional topics that are not covered in the first edition. New topics include
* directed translation
* new data flow analyses
* [[parallel machine]]s
* [[Just-in-time compilation|JIT]] compiling
* [[Garbage collection (computer science)|garbage collection]]
* new case studies.

==See also==
*[[Programming language]]
*[http://members.optusnet.com.au/clausen/photos/minimalist/dragon.jpg Compilers in Fashion]

==References==
<div class="references-small">
<references/></div>

==External links==
* [http://dragonbook.stanford.edu/ Book Website at Stanford with link to Errata]
* [http://wps.aw.com/aw_aho_compilers_2/0,11227,2663889-,00.html Sample chapters from the second edition.]
* The 2006 edition: ISBN 0-321-48681-1
* [http://www.pearson.ch/1471/9780321491695/Compilers_Principles_Techniques_and.aspx Pearson Education] (International edition, cover has no dragon)

[[Category:1986 books]]
[[Category:2006 books]]
[[Category:Computer books]]
[[Category:Compiler theory]]

[[ar:المترجمات : مبادئ وتقنيات وأدوات]]
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0,999…

2009-05-26T17:33:35Z

Loadmaster: rv grammatically incorrect edit by 90.213.53.53

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math> denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|one]]. In other words: the notations ''0.999…'' and ''1 '' actually represent the same real number. This [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. [[mathematical proof|Proofs]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

The fact that certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
====Fractions and long division====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

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

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, 0.999… must equal 1 (by the [[transitive property]]). Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

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

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

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

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.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

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

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
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 [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

0.999… features also in mathematical folklore, specifically in the following joke:<ref>Renteln and Dundes, p.27</ref>
<blockquote>
Q: How many mathematicians does it take to screw in a lightbulb?
</blockquote>
<blockquote>
A: 0.999999….
</blockquote>

==Alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref> Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (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, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

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

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

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a [[field (algebra)|field]] for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Digit manipulation|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
{{refbegin|2}}
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |author=Renteln, Paul and Allan Dundes |year=2005 |month=January |title=Foolproof: A Sampling of Mathematical Folk Humor |journal=[[Notices of the AMS]] |volume=52 |number=1 |pages=24–34 |url=http://www.ams.org/notices/200501/fea-dundes.pdf|doi=|format=PDF|accessdate=2009-05-03}}
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF|accessdate=2009-05-03}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
{{refend}}

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2009-03-30T17:10:18Z

Loadmaster: rv incorrect edits by 85.23.75.227

[[Image:999 Perspective.svg|300px|right]]

In [[mathematics]], the [[repeating decimal]] '''0.999…''' which may also be written as <math>0.\bar{9} , 0.\dot{9}</math> or <math> 0.(9)\,\!</math> denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|one]]. In other words: the notations ''0.999…'' and ''1 '' actually represent the same real number. This [[Equality (mathematics)|equality]] has long been accepted by professional mathematicians and taught in textbooks. [[mathematical proof|Proofs]] have been formulated with varying degrees of [[mathematical rigour]], taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

The fact that certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all [[integer]] [[radix|base]]s, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s, such as 28.3287 and 28.3286999…. For simplicity, the terminating decimal is almost always the preferred representation, contributing to a misconception that it is the ''only'' representation. Even more generally, any [[positional numeral system]] contains infinitely many numbers with multiple representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this equality among students, many of whom initially question or reject this equality. Many are persuaded by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common [[erroneous intuitions]] about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] real numbers should exist, or that the expansion of 0.999… eventually terminates. Number systems that bear out some of these intuitions can be constructed, but only outside the standard [[real number]] system used in elementary, and most higher, mathematics. Indeed, some settings contain numbers that are "just shy" of 1; these are generally unrelated to 0.999…, but they are of considerable interest in [[mathematical analysis]].

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic—[[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]]—uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Misinterpreting the meaning of the use of the "…" ([[ellipsis]]) in 0.999… accounts for some of the misunderstanding about its equality to 1. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated, which can only be interpreted as a number by using the mathematical concept of [[limit (mathematics)|limit]]s. As a result, in conventional mathematical usage, the value assigned to the notation "0.999…" is defined to be the [[real number]] which is the [[limit of a sequence|limit]] of the [[convergent sequence]] (0.9, 0.99, 0.999, 0.9999, …).

Unlike the case with integers and finite decimals, other notations can also express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

There are many proofs that 0.999… = 1, of varying degrees of [[mathematical rigour]]. A short sketch of one rigorous proof can be simply stated as follows. Consider that two [[real number]]s are identical [[if and only if]] their difference is equal to zero. Most people would agree that the difference between 0.999… and 1, if it exists at all, must be very small. By considering the convergence of the sequence above, we can show that the magnitude of this difference must be smaller than any positive quantity, and it can be shown (see [[Archimedean property]] for details) that the only real number with this property is 0. Since the difference is 0 it follows that the numbers 1 and 0.999… are identical. The same argument also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

==Proofs==
===Algebraic===
====Fractions====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

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

:{| style="wikitable"
|<math>
\begin{align}
0.333\dots &{} = \frac{1}{3} \\
3 \times 0.333\dots &{} = 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &{} = 1
\end{align}
</math>
|width="25px"|
|width="25px" style="border-left:1px solid silver;"|
|<math>
\begin{align}
0.111\dots & {} = \frac{1}{9} \\
9 \times 0.111\dots & {} = 9 \times \frac{1}{9} = \frac{9 \times 1}{9} \\
0.999\dots & {} = 1
\end{align}
</math>
|}

A more compact version of the same proof is given by the following equations:

:<math>
1 = \frac{9}{9} = 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
</math>

Since both equations are valid, by the [[transitive property]], 0.999… must equal 1. Similarly, 3/3 = 1, and 3/3 = 0.999…. So, 0.999… must equal 1.

====Digit manipulation====

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

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

:<math>
\begin{align}
x &= 0.999\ldots \\
10 x &= 9.999\ldots \\
10 x - x &= 9.999\ldots - 0.999\ldots \\
9 x &= 9 \\
x &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

The validity of the digit manipulations in the above two proofs does not have to be taken on faith or as an axiom; it follows from the fundamental relationship between decimals and the numbers they represent. This relationship, which can be developed in several equivalent manners, already establishes that the decimals 0.999… and 1.000... both represent the same number.

===Analytic===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the [[convergent series|convergence]] theorem concerning [[geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A [[sequence]] (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that <math>\lim_{n\to\infty} \frac{1}{10^n} = 0</math> — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

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.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

===Based on the construction of the real numbers===
{{further|[[Construction of the real numbers]]}}

Some approaches explicitly define real numbers to be certain [[construction of the real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: [[Dedekind cut]]s and [[Cauchy sequence]]s. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

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

In the [[Dedekind cut]] approach, each real number ''x'' is defined as the [[infinite set]] of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |month=October | year=2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Alternative number systems|Alternative number systems]]" below.

====Cauchy sequences====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that have the [[Cauchy sequence]] property using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Generalizations==
The result that 0.999… = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are [[Dense set|dense]].<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, the [[Komornik-Loreti constant]] ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in elementary [[number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the [[Cantor set]]]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

==Skepticism in education==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an [[infinitesimal]] but non-zero amount.
*Some students believe that the value of a [[convergent series]] is at best an approximation, that <math>0.\bar{9} \approx 1</math>.
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 [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor [[David O. Tall]], who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fractions|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==In popular culture==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1⁄3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1⁄3 and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fools' Day]] 2004 that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

==Alternative number systems==
Although the real numbers form an extremely useful [[number system]], the decision to interpret the notation "0.999…" as naming a real number is ultimately a convention, and [[Timothy Gowers]] argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

[[Non-standard analysis]] provides a number system with a full array of infinitesimals (and their inverses).<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone developed a decimal expansion for the non-standard real numbers in (0, 1)∗.<ref>Lightstone pp.245–247</ref>. Lightstone shows how to associate to each extended real number a sequence of digits
:0.d1d2d3…;…d∞−1d∞d∞+1…
indexed by the extended natural numbers. While he does not directly discuss 0.999…, he shows the real number 1/3 is represented by 0.333…;…333… which is a consequence of the [[transfer principle]]. Multiplying by 3, one obtains analogous facts for expansions with repeating 9s.
The hyperreal number ''u''''H''=0.999…;…999000… with ''H''-infinitely many 9s, for some infinite [[hyperinteger]] ''H'', satisfies a strict inequality ''u''''H'' < 1.{{Fact|date=January 2009}} Indeed, the sequence u1=0.9, u2=0.99, u3=0.999, etc. satisfies un = 1 - 1/n, hence by the transfer principle uH = 1 - 1/H < 1. Lightstone shows that in this system 0.333...;...000... and 0.999...;...000... are not numbers.

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between [[Hackenbush strings]] and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.0101012… = 1/3. However, the value of LRLLL… (corresponding to 0.111…2) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000…2.<ref>Berlekamp, Conway, and Guy (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, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

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

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

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which they write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> However, there is a system that contains an infinite string of 9s including a last 9.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1. The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

As a final extension, since {{nowrap begin}}0.999… = 1{{nowrap end}} (in the reals) and {{nowrap begin}}…999 = −1{{nowrap end}} (in the 10-adics), then by "blind faith and unabashed juggling of symbols"<ref>DeSua p.901</ref> one may add the two equations and arrive at {{nowrap begin}}…999.999… = 0.{{nowrap end}} This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has a "[[point at infinity]]". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |isbn=0-7167-1088-9 |page=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{Col-begin}}
{{Col-1-of-3}}
* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
{{Col-2-of-3}}
* [[Informal mathematics|Naive mathematics]]
* [[Non-standard analysis]]
{{Col-3-of-3}}
* [[Real analysis]]
* [[Series (mathematics)]]
{{col-end}}

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |isbn=0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |isbn=0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |isbn=0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |isbn=0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |isbn=0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |isbn=0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |isbn=0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format= |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903 |doi=10.2307/2309468}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |doi=10.1007/s10649-005-0473-0}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf |doi=10.2307/4145268|format=PDF}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |isbn=0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format= |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |doi=10.2307/2687285}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |isbn=0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |isbn=0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |isbn=0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | isbn=0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format= |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 |doi=10.2307/2300532 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format= |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 |doi=10.2307/2589246 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format= |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 |doi=10.2307/2314251 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format= |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 |doi=10.2307/2686394 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format= |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 |doi=10.2307/2316619 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |isbn=0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |isbn=3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |isbn=0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |isbn=0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite conference |last=Núñez |first=Rafael |title=Do Real Numbers Really Move? Language, Thought, and Gesture: The Embodied Cognitive Foundations of Mathematics |year=2006 |booktitle=18 Unconventional Essays on the Nature of Mathematics |publisher=Springer |pages=160–181 |url=http://www.cogsci.ucsd.edu/~nunez/web/publications.html | id=ISBN 978-0-387-25717-4}}
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |isbn=0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format= |journal=[[American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 |doi=10.2307/2324393 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf|format=PDF}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |isbn=0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |isbn=0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces [[Dedekind cut]]s as soon as possible, saying of the axiomatic treatment, "This is something of a fraud, considering that the entire structure of analysis is built on the real number system." (p.10) After proving the least upper bound property and some allied facts, cuts are not used in the rest of the book.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format= |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|isbn=0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |isbn=0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |isbn=0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format= |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |isbn=0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |isbn=0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |isbn=0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |authorlink=David O. Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf|format=PDF}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf|format=PDF}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |isbn=0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]
* [http://www.dpmms.cam.ac.uk/~wtg10/decimals.html What is so wrong with thinking of real numbers as infinite decimals?]
* [http://us.metamath.org/mpegif/0.999....html Theorem 0.999...] on [[Metamath]]
* [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ]
{{featured article}}



[[Category:One]]
[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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Creation Evidence Museum

2009-02-03T22:44:59Z

Loadmaster: link creationist museum

{{coord|32.237125|-97.806301|type:landmark_region:US-TX|display=title}}
[[Image:Creation Evidence Museum Original.jpg|right|thumb|Temporary museum building]]
[[Image:Creation Evidence Museum.jpg|right|thumb|New museum building]]
[[Image:Hyperbaric Biosphere.jpg|right|thumb|The first hyperbaric biosphere]]
The '''Creation Evidence Museum''', originally '''Creation Evidences Museum''',<ref name="Observer">[http://www.dallasobserver.com/1996-12-12/news/footprints-of-fantasy/ "Footprints of Fantasy"], by Kaylois Henry, ''[[Dallas Observer]]'', [[December 12]], [[1996]]. Retrieved [[August 15]], [[2008]].</ref> is a [[creationist museum|museum]] in [[Glen Rose, Texas]], [[United States|USA]] founded in 1984 by [[Carl Baugh]] for the purpose of researching and displaying purported evidence for [[creationism]], specifically that the Earth is only 6000 years old, and that humans and [[dinosaurs]] coexisted.<ref name="Chronicle"/> These views [[Level of support for evolution#Scientific support|contradict]] the scientific consensus in relevant fields.<ref name=nihrecord>[http://nihrecord.od.nih.gov/newsletters/2006/07_28_2006/story03.htm ''Finding the Evolution in Medicine''], Cynthia Delgado, [[National Institutes of Health|NIH]] Record, July 28, 2006.</ref><ref name="Newsweek_1987_Martz_McDaniel">As reported by [[Newsweek]]: "By one count there are some 700 [[scientist]]s (out of a total of 480,000 U.S. earth and life scientists) who give credence to [[creation science|creation-science]], the general theory that complex life forms did not evolve but appeared 'abruptly'," in "Keeping [[God]] out of the Classroom ([[Washington DC|Washington]] and bureau reports)",
Larry Martz & Ann McDaniel, [[Newsweek]] CIX(26): 23-24, June 29, 1987, ISSN 0028-9604</ref> Also the museum exhibits dispute the scientific conclusions that the [[age of the Earth]] is approximately 4.5 billion years, and that the dinosaurs [[Cretaceous–Tertiary extinction event|became extinct]] 65.5 million years before [[human evolution|human beings arose]].<ref name=iapstatement> {{cite web|url=http://www.interacademies.net/Object.File/Master/6/150/Evolution%20statement.pdf |title=IAP STATEMENT ON THE TEACHING OF EVOLUTION |accessdate=2007-10-02 |format=PDF }}</ref>

==History and projects==

The museum was founded by Carl Baugh, a [[young earth creationist]], after coming to Glen Rose in 1982 to research claims of alleged [[fossilized]] human footprints and [[dinosaur]] footprints in the [[limestone]] banks of the [[Paluxy River]], near [[Dinosaur Valley State Park]]. He claims to have excavated 475 dinosaur footprints and 86 human footprints<ref name="Roadside America">[http://www.roadsideamerica.com/story/8196 "Creation Evidence Museum, Glen Rose, Texas"], [[Roadside America]]. Retrieved [[August 15]], [[2008]].</ref>, which form the basis of the museum as well as other exhibits.<ref name="Democrat"/><ref name="Chronicle">[http://www.austinchronicle.com/gyrobase/Issue/story?oid=oid%3A283058 "Creationism Alive and Kicking in Glen Rose"], by Greg Beets, [[August 5]], [[2005]], ''[[Austin Chronicle]]''.</ref> The prints have been examined by scientists who concluded they are a combination of admitted hoaxes and misidentified tracks.<ref>[[Massimo Pigliucci]] ''[[Denying Evolution: Creationism, Scientism, and the Nature of Science]]'' (Sinauer, 2002, p. 246): ISBN 0878936599 </ref><ref name="tracks">{{cite news | url=http://www.star-telegram.com/news/story/820344.html | title=Human footprints along with dinosaur tracks? | publisher=[[Star-Telegram]] |date=August 10, 2008 | first= | last= | accessdate = 2008-08-10}}</ref>

Since 1993 the museum has been housed in a doublewide [[trailer (vehicle)|trailer]], while a new [[greenhouse]]-like building is being constructed with donations.<ref name="Observer"/> Baugh remains the director and main speaker for CEM.

The museum sponsors continuing [[paleontological]] and [[archaeological]] excavations among other research projects, including a hunt for living [[pterodactyls]] in [[Papua New Guinea]],<ref name="Observer"/><ref name="LATimes">[http://articles.latimes.com/2005/aug/27/local/me-dinosaurs27 "Adam, Eve and T. Rex"], by Ashley Powers, [[August 27]], [[2005]], ''[[Los Angeles Times]]''. Retrieved [[August 14]], [[2008]].</ref> and expeditions to [[Israel]].<ref name="Cleburne Times Review">[http://www.cleburnetimesreview.com/archivesearch/local_story_036154557.html "Bronstein, head of Israeli Department of Antiquities, lectures in Glen Rose"], John Watson, ''[[Cleburne Times-Review]]'', [[February 05]], [[2006]].</ref> He does not have any [[accredited]] degrees.<ref>{{cite news | url=http://www.toarchive.org/faqs/credentials.html | title=Some Questionable Creationist Credentials | publisher=[[talk.origins]]|date= May 31, 2002 | first=Brett | last=Vickers | accessdate = 2007-02-19}}</ref> Materials from the museum have been recommended by the [[National Council on Bible Curriculum in Public Schools]],<ref name="Time">[http://www.time.com//time/printout/0,8816,1601845,00.html "The Case for Teaching The Bible"], by David Van Biema, ''[[Time]]'' Magazine, [[March 22]], [[2007]].</ref> but the NCBCPS curriculum has been deemed "unfit for use in public school classrooms."<ref>Amanda Colleen Brown, "[http://www.bibleliteracy.org/site/News/bibl_newsBaylorLawReview071010.htm Losing My Religion: The Controversy over Bible Classes in Public Schools]," Baylor Law Review 59 (2007): 193-240.</ref> A Creation Evidence Museum poster is displayed in the [[Tehran]] museum of [[natural history]].<ref name="Science">[http://www.sciencemag.org/cgi/search?src=hw&site_area=sci&fulltext=%22Creation+Evidence+Museum%22&search_submit.x=16&search_submit.y=8&search_submit=go "SCIENCE IN IRAN: Picking a Path Among the Fatwas"], John Bohannon, ''[[Science (journal)|Science]]'', [[21 July]] [[2006]], 313: 292–293 [DOI: 10.1126/science.313.5785.292] (in News Focus).</ref>

One of the museum's research projects is a "hyperbaric [[biosphere]]", a sealed chamber designed to reproduce the ostensible atmosphere of the Earth before [[Noah's Ark|the Flood]], which will allegedly allow extended lifespans, and larger physical sizes. It is claimed to have tripled the lifespan of fruit-flies, and detoxified copperhead snakes. A much larger version is under construction in the new building.<ref name="Chronicle"/><ref name="Roadside America"/>

In 2001 Baugh and Creation Evidence Museum were featured on [[The Daily Show]] where Baugh likened human history to [[The Flintstones]] and the show poked fun at his claims about the hyperbaric biosphere, pterodactyl expeditions, and dinosaurs.<ref name="DailyShow">{{cite news | url=http://www.thedailyshow.com/video/index.jhtml?videoId=105921&title=Tyrannosaurus-Redux | title=Tyrannosaurus Redux | publisher=[[The Daily Show]] |date=November 14, 2001 | first= | last= | accessdate = 2007-02-19}}</ref> On March 11 2006, [[KDFW]], a local affiliate of [[Fox TV]], in [[Dallas Fort Worth]] aired a news report on Baugh's museum and claims.<ref>[http://www.myfoxdfw.com/myfox/pages/Home/Detail;jsessionid=BCB443D95A9CBDBC8C565B17983235DA?contentId=2659180&version=2&locale=EN-US&layoutCode=VSTY&pageId=1.1.1&sflg=1 "Lone Star Adventure: Creation Evidence Museum"], [[KDFW]], a local affiliate of [[Fox TV]], in [[Dallas Fort Worth]] aired on [[March 11]], [[2006]]</ref>

== Exhibits ==
Displays in the museum include:
* The "London Artifact", an alleged [[Out-of-place artifact|out of place artifact]] of a "18th century miner's hammer" found in million-year-old [[Ordovician]] rock (he has also claimed it is in [[Cretaceous]] rock) found in 1934 in [[London, Texas]].<ref name="NCSE-Hammer">{{cite news | url=http://www.ncseweb.org/resources/articles/3868_issue_15_volume_5_number_1__4_23_2003.asp#If%20I%20Had%20a%20Hammer| title=If I Had a Hammer | publisher=[[National Center for Science Education]] |date=Issue 15 (Volume 5, Number 1 - Winter 1985) | first= | last= | accessdate = 2007-02-19}}</ref> It was examined by scientists who concluded: "The stone is real, and it looks impressive to someone unfamiliar with geological processes. ... Minerals in solution can harden around an intrusive object dropped in a crack or simply left on the ground if the source rock (in this case, reportedly Ordovician) is chemically soluble."<ref name="Hammer-London">{{cite news | url=http://paleo.cc/paluxy/hammer.htm | title=The London Hammer: An Alleged Out-of-Place Artifact | publisher=The Paluxy Dinosaur/"Man Track" Controversy |date=14 July 2006 | first= | last= | accessdate = 2007-02-19}}</ref>
* The "Burdick Track", allegedly a human footprint in Cretaceous rock. Glen J. Kuban and Geologist Gregg Wilkerson wrote that anatomic errors on it indicate that is was [[carve]]d from limestone, similar to other tracks that were carved in Glen Rose.<ref>{{cite news | url=http://paleo.cc/paluxy/wilker6.htm | title=The "Burdick Print" | publisher=The Paluxy Dinosaur/"Man Track" Controversy |date= 2008 | first=Glen | last=Kuban | accessdate = 2008-08-19}}</ref>
* The "Fossilized Human Finger", allegedly a finger with tissues [[Fossilized#Replacement_and_recrystallization|replaced]] by Cretaceous stone. There is doubt about its authenticity since it was not found [[in situ]], and cannot be conclusively associated with Cretaceous formations.<ref name="FingerFake">{{cite news | url=http://www.toarchive.org/indexcc/CC/CC120.html | title=Claim CC120: Baugh's Cretaceous fossil finger | publisher=[[talk.origins]]|date= 2001-2-18| first= | last= | accessdate = 2008-02-19}}</ref> Even if it were real, it does not provide evidence for creation or against evolution.<ref name="FingerFake"/>
* The "Meister Print", allegedly a human sandal print crushing two [[trilobites]] in [[slate]]. The print is "questionable on several accounts" such as the shallowness of the print, spall patterns, striding sequence, and similarities to the Wheeler formation.<ref>{{cite news | url=http://www.toarchive.org/faqs/paluxy/meister.html | title=The "Meister Print" | publisher=[[talk.origins]]|date= 1998 | first= | last= | accessdate = 2008-02-19}}</ref>
* The "Hand Print in Stone", allegedly a hand print in Cretaceous rock.<ref name="CEM Displays">[http://75.125.60.6/~creatio1/index.php?option=com_content&task=view&id=9&Itemid=11 "Creation Evidence Museum Online — Museum Displays"], official site. Retrieved [[August 15]], [[2008]].</ref> Baugh has provided no evidence it was [[in situ]] in any Cretaceous bed, nor allowed experts to inspect it.<ref>{{cite news | url=http://paleo.cc/paluxy/hand.htm | title=Alleged Human Hand Print in Cretaceous Rock | publisher=[[talk.origins]]|date= 2006 | first= | last= | accessdate = 2008-02-19}}</ref> Creationists have been critical of it too.<ref name="AIC">[http://www.answersincreation.org/rebuttal/cem/cem.htm "Creation Science Rebuttals — Creation Evidence Museum Lacks Evidence!"], by Greg Neyman, [[Answers in Creation]]. Retrieved [[August 15]], [[2008]].</ref>
* The "Alvis Delk Cretaceous Footprint", allegedly a human footprint partially overlapped by an [[Acrocanthosaurus]] dinosaur footprint in Glen Rose limestone.<ref name="CEM Delk">[http://75.125.60.6/~creatio1/index.php?option=com_content&task=view&id=48&Itemid=24 "Creation Evidence Museum Online – Alvis Delk Cretaceous Footprint"], official site. Retrieved [[August 15]], [[2008]].</ref><ref name="Democrat">[http://www.weatherforddemocrat.com/archivesearch/local_story_210103057.html "Rock solid proof?"], by David May, ''[[Weatherford Democrat]]'', [[July 28]], [[2008]]. Also printed as [http://www.mineralwellsindex.com/homepage/local_story_210093256.html "Rock-solid proof?"], in ''[[Mineral Wells Index]]'', [[July 28]], [[2008]].</ref><ref name="MWI Aug 11 2008">[http://www.mineralwellsindex.com/local/local_story_224120721.html "One step at a time"], by David May, [[August 11]], [[2008]], ''[[Mineral Wells Index]]''. Retrieved [[August 15]], [[2008]].</ref><ref name="MWI Aug 12 2008">[http://www.mineralwellsindex.com/local/local_story_225091209.html "Rock's finders discovering celebrity not always pleasant"], by David May, [[August 12]], [[2008]], ''[[Mineral Wells Index]]''. Retrieved [[August 15]], [[2008]].</ref> This was deemed "not a convincing human footprint in ancient rock" by biologist Glen J. Kuban and called a "blatant fake" by biologist [[PZ Myers]].<ref>{{cite news | url=http://scienceblogs.com/pharyngula/2008/07/transparent_fakery.php | title=Transparent fakery | publisher=[[Pharyngula (blog)]] |date=July 28, 2008 | first= | last= | accessdate = 2007-02-19}}</ref>

The "Burdick track" and "fossilized finger" were featured on the controversial [[NBC]] program "[[The Mysterious Origins of Man]]", aired in [[1996]] and hosted by [[Charlton Heston]].<ref name="Kuban">[http://paleo.cc/paluxy/nbc.htm "A Review of NBC's 'The Mysterious Origins of Man'"], 1996, Glen J. Kuban. Retrieved [[August 15]], [[2008]].</ref> Creationist [[Ken Ham]] criticized the production in the February 1996 [[Answers in Genesis]] newsletter in a review titled "Hollywood's 'Moses' Undermines [[Book of Genesis|Genesis]]."<ref>{{cite news | url=http://www.csicop.org/sb/9603/origins.html | title=NBC's Origins Show | publisher=[[Committee for Skeptical Inquiry]] |date= March 1996 | first=Dave | last=Thomas | accessdate = 2007-02-19}}</ref> Ham attacked fellow creationist Baugh's claims, saying, "According to leading creationist researchers, this evidence is open to much debate and needs much more intensive research. One wonders how much of the information in the program can really be trusted!"<ref name="SkepticalOrigins">{{cite news | url=http://www.csicop.org/sb/9603/origins.html | title=NBC's Origins Show | publisher=[[Committee for Skeptical Inquiry]] |date= March 1996 | first=Dave | last=Thomas | accessdate = 2007-02-19}}</ref>

== Criticism ==

All of the museum exhibits have been strongly criticized as incorrectly identified dinosaur prints, other fossils, or outright forgeries.<ref name="AIC"/><ref name="Kuban"/><ref name="Armstrong">[http://www.asa3.org/ASA/PSCF/1989/PSCF3-89Armstrong.html "Seeking Ancient Paths"], John R. Armstrong, PSCF 41 (March 1989): 33–35, [[American Scientific Affiliation]]. Retrieved [[August 15]], [[2008]].</ref><ref name="Hastings">[http://www.asa3.org/ASA/PSCF/1988/PSCF9-88Hastings.html "The Rise and Fall of the Paluxy Mantracks"], by Ronnie J. Hastings, PSCF 40 (September 1988): 144–154. [[American Scientific Affiliation]]. Retrieved [[August 15]], [[2008]].</ref>
In 2008, a descendant of a family that found many original Paluxy River dinosaur tracks in the 1930s claimed that her grandfather had faked many of them, including the "Alvis Delk Cretaceous Footprint".<ref name="Star-Telegram">[http://www.star-telegram.com/news/columnists/bud_kennedy//story/820344.html "Human footprints along with dinosaur tracks?"], by Bud Kennedy, [[August 10]], [[2008]], ''[[Fort Worth Star-Telegram]]''.</ref>

Both scientists and creationists have criticized Baugh's claims. In 1982–1984, several scientists, including J.R. Cole, L.R. Godfrey, R.J. Hastings, and S.D. Schafersman, examined Baugh's purported "mantracks" as well as others provided by creationists in Glen Rose. In the course of the examination "Baugh contradicted his own earlier reports of the locations of key discoveries" and many of the supposed prints "lacked characteristics of human footprints." After a three-year investigation of the tracks and Baugh's specimens, the scientists concluded there was no evidence of any of Baugh's claims or any "dinosaur-man tracks".<ref name="NCSE-Examine">{{cite news | url=http://www.ncseweb.org/resources/articles/3868_issue_15_volume_5_number_1__4_23_2003.asp | title=Creation/Evolution | publisher=[[National Center for Science Education]] |date=Issue 15 (Volume 5, Number 1 — Winter 1985) | first= | last= | accessdate = 2007-02-19}}</ref>

Creationist organizations such as [[Answers in Genesis]] have criticized Baugh's claims saying he "muddied the water for many Christians...People are being misled."<ref name="Observer"/> Don Batten, of [[Creation Ministries International]] wrote: "Some Christians will try to use Baugh's 'evidences' in witnessing and get 'shot down' by someone who is scientifically literate. The ones witnessed to will thereafter be wary of all creation evidences and even more inclined to dismiss Christians as [[nut case]]s not worth listening to."<ref>{{cite news | url=http://paleo.cc/paluxy/whatbau.htm | title=What About Carl Baugh?| publisher=[[Creation Ministries International]] |date= 1998 | first= | last= | accessdate =2007-05-17}}</ref> Answers in Genesis (AiG) lists the "Paluxy tracks" as arguments "we think creationists should NOT use" [emphasis in original].<ref name="AIGCriticism">{{cite news | url=http://www.answersingenesis.org/home/area/faq/dont_use.asp | title=Arguments we think creationists should NOT use | publisher=[[Answers In Genesis]] |date= 2008 | first= | last= | accessdate =2007-05-17}}</ref> Also [[Answers In Creation]] reviewed Baugh's museum and concluded "the main artifacts they claim show a young earth reveal that they are deceptions, and in many cases, not even clever ones."<ref name="AICReview">{{cite news | url=http://www.answersincreation.org/rebuttal/cem/cem.htm | title=Review of Carl Baugh's Museum | publisher=[[Answers In Creation]] |date= 2008 | first= | last= | accessdate =2007-05-17}}</ref>

== See also ==
* [[Creation Museum]] — Similar, larger, museum in Northern [[Kentucky]]
* [[Glen_Rose_Formation#Human_footprint_hoax|Glen Rose dinosaur-human hoax]]
* [[List of museums in Texas]]

== References ==
{{reflist|2}}

== External links ==
* [http://www.creationevidence.org/ Creation Evidence Museum Online] — official site.

[[Category:Creationist museums]]
[[Category:Museums established in 1984]]
[[Category:Museums in Texas]]

Od (Unix)

2009-01-05T18:49:49Z

Loadmaster: links

{{lowercase}}

'''od''' is an [[octal|''o''ctal]] ''d''umping program for [[Unix]] and Unix-like systems. It can also dump [[hexadecimal]] or [[decimal]] data.

od is one of the earliest Unix programs, having appeared in version 1 AT&T Unix. It is also specified in the [[POSIX]] standards (see external link below). The implementation for od used on [[Linux]] systems is usually provided by [[GNU Core Utilities]].

=== Example session ===
Normally a dump of an executable file is very long. The [[head (Unix)|head]] program prints out the first few lines of the output. Here is an example of a dump of the [[Hello world program|"Hello world" program]], [[pipe (computing)|pipe]]d through head.

% od hello|head
0000000 042577 043114 000401 000001 000000 000000 000000 000000
0000020 000002 000003 000001 000000 101400 004004 000064 000000
0000040 003610 000000 000000 000000 000064 000040 000006 000050
0000060 000033 000030 000006 000000 000064 000000 100064 004004
0000100 100064 004004 000300 000000 000300 000000 000005 000000
0000120 000004 000000 000003 000000 000364 000000 100364 004004
0000140 100364 004004 000023 000000 000023 000000 000004 000000
0000160 000001 000000 000001 000000 000000 000000 100000 004004
0000200 100000 004004 002121 000000 002121 000000 000005 000000
0000220 010000 000000 000001 000000 002124 000000 112124 004004

==See also==
*[[Hex editor]]

==External links==
*[http://www.linuxmanpages.com/man1/od.1.php The program's [[manpage]]]
*[http://linux-documentation.com/en/man/man1p/od.html POSIX standard for od]

[[Category:Unix software]]

Delegate (CLI)

2008-12-12T17:14:13Z

Loadmaster: /* See also */ added Delegate method

{{Cleanup-jargon|date=December 2007}}
A '''delegate''' is a form of [[Type safety|type-safe]] [[function pointer]] used by the [[.NET Framework]]. Delegates specify a [[Method (computer science)|method]] to call and optionally an [[Object (computer science)|object]] to call the method on. They are used, among other things, to implement [[Callback (computer science)|callbacks]] and [[event listener]]s.

== Implementation ==
Although internal [[implementation]]s may vary, delegate [[Object (computer science)|instances]] can be thought of as a [[tuple]] of an [[Object (computer science)|object]] and a [[Method (computer science)|method]] [[pointer]] and a [[Reference (computer science)|reference]] (possibly null) to another delegate. Hence a reference to one delegate is possibly a reference to multiple delegates. When the first delegate has finished, if its chain reference is not null, the next will be invoked, and so on until the list is complete. This pattern allows an [[Event-driven programming| event]] to have overhead scaling easily from that of a single reference up to dispatch to a list of delegates, and is widely used in the [[.NET Framework]].

Performance of delegates used to be much slower than a [[Virtual function|virtual]] or [[Interface (computer science)|interface]] method call (6 to 8 times slower in Microsoft's 2003 benchmarks),<ref>{{cite web
| url=http://msdn2.microsoft.com/en-us/library/ms973852
| title=Writing Faster Managed Code: Know What Things Cost
| publisher=Microsoft
| last=Gray|first=Jan
| date=[[June 2003]]
| accessdate=2007-09-09}}</ref> but it is now about the same as interface calls.<ref>{{cite web
| url=http://www.sturmnet.org/blog/archives/2005/09/01/
| title=Delegate calls vastly sped up in .NET 2
|last=Sturm|first=Oliver
| date=[[2005-09-01]]
| accessdate=2007-09-09}}</ref> This means there is a small added overhead compared to direct method invocations.

Delegates are a variation of [[Closure (computer science)|closures]].

== See also ==

* [[Delegate method]]
* [[Delegation (programming)]]

==References==
{{reflist|2}}

== External links ==
*[http://msdn2.microsoft.com/en-us/library/system.delegate.aspx MSDN documentation for Delegates]
*[http://java.sun.com/docs/white/delegates.html Sun's critic of Delegates]
*[http://msdn.microsoft.com/en-us/vjsharp/bb188664.aspx Microsoft answer to Sun]
*[http://blog.monstuff.com/archives/000037.html Inner workings of Delegates]
*[http://perfectjpattern.sourceforge.net/dp-delegates.html PerfectJPattern Open Source Project], Provides componentized i.e. context-free and type-safe implementation of the Delegates Pattern in Java

[[Category:.NET framework]]
[[Category:Programming constructs]]

[[ru:Делегат (программирование)]]

Joe the Plumber

2008-10-17T16:11:00Z

Loadmaster: /* External links */ link to article with full text of conversation

{{pp-semi|small=yes}}
{{Current|date=October 2008}}

{{Infobox Person
|name = Joe Wurzelbacher
|image =
|image_size = 200px
|caption =
|birth_name =
|birth_date =
|birth_place =
|residence = [[Holland, Ohio|Holland]], [[Ohio]]
|nationality = American
|ethnicity =
|citizenship = United States of America
|other_names = Joe the Plumber
|known_for =
|education =
|alma_mater =
|employer = Newell Plumbing & Heating (A. W. Newell Inc.)
|occupation = assistant to plumbing contractor
|years_active =
|home_town =
|salary =
|networth =
|boards =
|religion =
|spouse =
|partner =
|children =
|parents =
|relations =
|signature =
|website =
|footnotes =
}}
'''Samuel Joseph Wurzelbacher''' (from [[Holland, Ohio|Holland]], [[Ohio]]) is an employee of Newell Plumbing & Heating, a [[plumbing]] firm.<ref name = TBlicense>{{cite news|url=http://toledoblade.com/apps/pbcs.dll/article?AID=/20081016/NEWS09/810160418|title='Joe the plumber' isn’t licensed|first =Larry| last = Vellequette |coauthors = Troy, Tom|publisher=[[Toledo Blade]]|date=October 16, 2008}}</ref> Wurzelbacher was mentioned by [[Republican Party (United States)|Republican]] [[United States]] [[United States Senator|Senator]] [[John McCain]] and [[Democratic Party (United States)|Democratic]] Senator [[Barack Obama]] 23 times<ref name = VoA>{{cite news|url = http://www.voanews.com/english/2008-10-16-voa59.cfm|title = 'Joe the Plumber' - Unexpected Star of US Presidential Debate|first = Cindy |last = Saine|date = 16 October 2008|publisher = Voice of America: VoA News}} </ref> (usually as "'''Joe the Plumber'''") during the third and final [[United States presidential election debates, 2008#October 15: Third presidential debate (Hofstra University – Hempstead, New York)|presidential debate]] of 2008.<ref name ="ABC-News">{{cite web
|url= http://abcnews.go.com/GMA/Vote2008/story?id=6047360&page=1
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "America's Overnight Sensation Joe the Plumber Owes $1,200 in Taxes"
|publisher = [[ABC News]]
}}</ref>

==Encounter with Obama==
A few days before the October 15, 2008 presidential debate, Obama was meeting residents in Wurzelbacher's neighborhood.<ref name=TBlicense/> Wurzelbacher, who had been playing football with his son in his front yard at the time, asked Obama about his tax plan.<ref name=TBlicense/> As a [[Fox News Channel]] camera maintained tight focus on the interaction, Wurzelbacher said he was upset about his interpretation of Obama's tax plan and suggested that such a plan would be at odds with "the American dream".<ref name = NYT>{{cite news|url = http://www.nytimes.com/2008/10/16/us/politics/16plumber.html|title = Plumber From Ohio Is Thrust Into Spotlight |last = Rohter|first = Larry|publisher = New York Times|date = October 15, 2008 |accessdate = 2008-10-17}}</ref> Wurzelbacher said, "I’m getting ready to buy a company that makes $250,000 to $280,000 a year. Your new tax plan is going to tax me more, isn’t it?"<ref name=realdeal>{{cite news |url=http://www.nytimes.com/2008/10/17/us/politics/17joe.html |title=Real Deal on ‘Joe the Plumber’ Reveals New Slant |work=The New York Times |date=2008-10-16 |author=Rohter, Larry. |accessdate=2008-10-16 }}</ref> Wurzelbacher suggested that making more money should not result in paying a higher tax rate and asked Obama whether he would support a [[flat tax]] plan.

Obama responded with an explanation of how his tax plan would affect a small business in this bracket. He stated,
{{cquote|It's not that I want to punish your success. I just want to make sure that everybody who is behind you, that they've got a chance at success, too. And I think that when we spread the wealth around, it's good for everybody.<ref>{{cite news|url = http://ap.google.com/article/ALeqM5ha9qKZFKzFvH74G3c5PxRG_lHRIAD93RC8FG0|title =McCain, Obama get tough, personal in final debate|last = Fouhy|first = Beth|date = 15 October 2008|accessdate = 2008-10-17|publisher = The Associated Press}} </ref>}}
Obama's choice of words were suggested to have evoked the [[populism|populist]] "[[Share Our Wealth]]" movement of [[Huey Long]].<ref name="Pethokoukis">{{cite news|url=http://www.usnews.com/blogs/capital-commerce/2008/10/16/did-barack-spread-the-wealth-obama-just-blow-the-election.html|title=Did Barack "Spread the Wealth" Obama Just Blow the Election?|last=Pethokoukis|first=James|date=2008-10-16|publisher=[[U.S. News & World Report]]|accessdate=2008-10-17}}</ref> [[Steve Schmidt]], McCain's chief campaign strategist, commented that this statement would be a focus of their campaign in its final weeks.<ref name="Nagourney">{{cite news|url=http://www.nytimes.com/2008/10/17/us/politics/17campaign.html?_r=1&oref=slogin|title=Polls Cause Campaigns to Change Their Itineraries |last=Nagourney|first=Adam|coauthors=Jim Rutenberg|date=2008-10-16|publisher=[[The New York Times]]|accessdate=2008-10-17}}</ref>

==Press coverage==
Asked by [[Katie Couric]] of [[CBS Evening News]] on October 15 whether Obama's proposed $250,000 tax threshold would affect him, Wurzelbacher replied: "Not right now at presently, but, you know, question, so he's going to do that now for people who make $250,000 a year. When's he going to decide that $100,000 is too much, you know? I mean, you're on a slippery slope here. You vote on somebody who decides that $250,000 and you're rich? And $100,000 and you're rich? I mean, where does it end?"<ref name="CBS-News"/> He also said, “I asked the question but I still got a tap dance ... almost as good as [[Sammy Davis, Jr.]]” <ref name ="CBS-News">{{cite web
|url= http://www.cbsnews.com/blogs/2008/10/16/politics/horserace/entry4525242.shtml
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Joe The Plumber's Chat With Couric"
|publisher = [[CBS News]]
}}</ref>

On October 16, Wurzelbacher appeared on ''[[Your World with Neil Cavuto]]'' on Fox News. [[Neil Cavuto|Cavuto]] asked if Wurzelbacher was persuaded by Obama's plan. Wurzelbacher said that he was not and that he was more frightened upon hearing it. Wurzelbacher suggested that Obama's plan was [[Socialism|socialist]] in nature.<ref name = NYT/>

Wurzelbacher also appeared on ''[[Good Morning America]]'' the day following the debate (October 16). [[Diane Sawyer]] asked him if he was taking home $250,000 now, Wurzelbacher said with a laugh "No, not even close."<ref name ="ABC-vid">{{cite web
|url= http://abcnews.go.com/Video/playerIndex?id=6047458
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Meet Joe the Plumber"
|publisher = [[ABC News]]
}}</ref> Sawyer asked Wurzelbacher if he had been contacted by the McCain campaign before his encounter with Obama. Wurzelbacher said "I have been contacted by them and asked to show up at a rally." <ref name ="ABC-vid"/>

Wurzelbacher held a press conference at his home the morning following the debates, where he refused to express support for either candidate. "I'm not telling anybody anything" about which candidate he prefers, he said, adding, "It's a private booth. I want the American people to vote for who they want to vote for."<ref name = LATimes>{{cite web |url=http://www.latimes.com/news/politics/la-na-campaign17-2008oct17,0,373757.story |title='Joe the Plumber' still a topic for McCain, Obama |accessdate=2008-10-16 |date= 16 October 2008|work= |publisher=[[Los Angeles Times]]|first = Seema |last = Mehta |coauthors = Michael Muskal}}</ref> He reportedly had been registered in 1992 under the name "Samuel Joseph Worzelbacher".<ref name="toledoblade">{{cite web
|url=http://www.toledoblade.com/apps/pbcs.dll/article?AID=/20081016/NEWS09/810160418
|title="'Joe the Plumber' is focus of presidential debate's first few minutes"
|author=Bridget Tharp and Mark Zaborney
|publisher=Toledo Blade
|date=2008-10-16
|accessdate=2008-10-16}}</ref> He told an [[Associated Press]] reporter that in the most recent Republican primary, he backed McCain.<ref name = APlicense/>

==Plumbing career and licensing issues==
An [[Associated Press]] article revealed that Wurzelbacher does not have a plumber's license or [[apprentice]]ship.<ref name ="Yahoo-news">{{cite web
|url= http://news.yahoo.com/s/ap/20081017/ap_on_re_us/joe_the_plumber;_ylt=Avwwyr5JRUHYIXm2bhAYz5RH2ocA
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Is 'Joe the Plumber' a plumber? That's debatable"
|publisher = [[Yahoo]]
}}</ref>
He claimed he does not need a license because he works for someone else. [[Lucas County, Ohio|Lucas County]], where Wurzelbacher and his employer reside, requires plumbers who perform work including sanitary drainage, water supply, storm drainage, and natural gas piping, to have licenses.<ref>{{cite web|url = http://www.contractor-licensing.com/ohio/plumbing-license.html|title = Ohio Plumbing License|publisher = National Contractors Pre-Licensing Services, Inc.|accessdate = 2008-10-17}}</ref> Neither Wurzelbacher nor his employer are licensed there, said Cheryl Schimming of Lucas County Building Regulations, which handles plumber licenses in parts of the county outside Toledo.<ref name = TBlicense/><ref name = APlicense/>
Local 50 of the [[United Association|United Association of Plumbers, Steamfitters and Service Mechanics]], whose membership endorsed Obama, indicated that Wurzelbacher applied for an apprentice program in 2003 but never completed the work.<ref>{{cite news|url=http://www.washingtonpost.com/wp-dyn/content/article/2008/10/16/AR2008101603614_2.html|title=After Debate, Glare Of Media Hits Joe|author=Barnes, Robert|publisher=[[Washington Post]]|accessdate=October 16, 2008}}</ref>

Wurzelbacher is one of two employees of a small plumbing firm, Newell Plumbing and Heating Co. of Toledo.<ref name=realdeal/> This is the company he described to Obama as making more than $250,000 per year;<ref name = APlicense>{{cite news|url = http://ap.google.com/article/ALeqM5gJsPHiQlgYvAsrHz9mvHJlezQJLwD93RONUO0|title = 'Joe the Plumber' says he has no plumbing license|last = Seewer|first = John|publisher = The Associated press|date = 16 October 2008|accessdate = 2008-10-17}}</ref> however, according to Bloomberg News, the company's net profit in reality is between $150,000 and $200,000, and would not see a tax hike under the Obama plan.<ref>{{cite news |url= http://www.bloomberg.com/apps/news?pid=20601087&sid=aC4j3T5.s_eQ&refer=home |title= `Joe the Plumber,' Obama Tax-Plan Critic, Owes Taxes |author= [[Ryan J. Donmoyer]] |publisher= [[Bloomberg.com]] |date= 2008-10-16 |accessdate= 2008-10-17}}</ref> MSNBC reported an even lower estimate: "Ohio business records show the company’s estimated total annual revenue as only $100,000. Actual ''taxable income'' would be even less than that."<ref>{{cite news |url= http://www.msnbc.msn.com/id/27221645/ |title= ‘Joe the plumber’ and Obama’s tax plan |publisher= [[msnbc.com]] |date= 2008-10-16 |accessdate= 2008-10-17}}</ref> He reported that the idea of buying the company was discussed during his job interview six years prior.<ref name= TBlicense/>

In his interview on ABC's ''[[Good Morning America]]'' on October 16, Wurzelbacher stated that his two current plumbing jobs were for a gas station and for a shopping center, Levis Commons.<ref name = "ABC-vid"/>

==Views on taxation==
In an October interview, Wurzelbacher said, "You know, a lot of the stuff that our government is doing right now is all about taxation without representation, and, you know, the last time that happened a couple guys got together and threw the Brits out."<ref>[http://www.kare11.com/news/news_article.aspx?storyid=527094 'Who is "Joe the Plumber"?', KARE11, October 16, 2008]</ref>

==References==
{{reflist|3}}

==External links==
* [http://www.liveleak.com/view?i=1a4_1224166209 Full, unedited video of the conversation between Joe the Plumber and Obama]
* [http://blogs.abcnews.com/politicalpunch/2008/10/spread-the-weal.html Full text] of the conversation, Jake Tapper, [[ABC News]] online, 2008-10-14

{{United States presidential election, 2008}}

{{DEFAULTSORT:Wurzelbacher, Joe}}
[[Category:United States presidential election, 2008]]
[[Category:John McCain]]
[[Category:Ohio Republicans]]
[[Category:Plumbers]]
[[Category:People from Lucas County, Ohio]]
[[Category:Living people]]

Joe the Plumber

2008-10-17T16:05:08Z

Loadmaster: /* Views on taxation */ punctuation

{{pp-semi|small=yes}}
{{Current|date=October 2008}}

{{Infobox Person
|name = Joe Wurzelbacher
|image =
|image_size = 200px
|caption =
|birth_name =
|birth_date =
|birth_place =
|residence = [[Holland, Ohio|Holland]], [[Ohio]]
|nationality = American
|ethnicity =
|citizenship = United States of America
|other_names = Joe the Plumber
|known_for =
|education =
|alma_mater =
|employer = Newell Plumbing & Heating (A. W. Newell Inc.)
|occupation = assistant to plumbing contractor
|years_active =
|home_town =
|salary =
|networth =
|boards =
|religion =
|spouse =
|partner =
|children =
|parents =
|relations =
|signature =
|website =
|footnotes =
}}
'''Samuel Joseph Wurzelbacher''' (from [[Holland, Ohio|Holland]], [[Ohio]]) is an employee of Newell Plumbing & Heating, a [[plumbing]] firm.<ref name = TBlicense>{{cite news|url=http://toledoblade.com/apps/pbcs.dll/article?AID=/20081016/NEWS09/810160418|title='Joe the plumber' isn’t licensed|first =Larry| last = Vellequette |coauthors = Troy, Tom|publisher=[[Toledo Blade]]|date=October 16, 2008}}</ref> Wurzelbacher was mentioned by [[Republican Party (United States)|Republican]] [[United States]] [[United States Senator|Senator]] [[John McCain]] and [[Democratic Party (United States)|Democratic]] Senator [[Barack Obama]] 23 times<ref name = VoA>{{cite news|url = http://www.voanews.com/english/2008-10-16-voa59.cfm|title = 'Joe the Plumber' - Unexpected Star of US Presidential Debate|first = Cindy |last = Saine|date = 16 October 2008|publisher = Voice of America: VoA News}} </ref> (usually as "'''Joe the Plumber'''") during the third and final [[United States presidential election debates, 2008#October 15: Third presidential debate (Hofstra University – Hempstead, New York)|presidential debate]] of 2008.<ref name ="ABC-News">{{cite web
|url= http://abcnews.go.com/GMA/Vote2008/story?id=6047360&page=1
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "America's Overnight Sensation Joe the Plumber Owes $1,200 in Taxes"
|publisher = [[ABC News]]
}}</ref>

==Encounter with Obama==
A few days before the October 15, 2008 presidential debate, Obama was meeting residents in Wurzelbacher's neighborhood.<ref name=TBlicense/> Wurzelbacher, who had been playing football with his son in his front yard at the time, asked Obama about his tax plan.<ref name=TBlicense/> As a [[Fox News Channel]] camera maintained tight focus on the interaction, Wurzelbacher said he was upset about his interpretation of Obama's tax plan and suggested that such a plan would be at odds with "the American dream".<ref name = NYT>{{cite news|url = http://www.nytimes.com/2008/10/16/us/politics/16plumber.html|title = Plumber From Ohio Is Thrust Into Spotlight |last = Rohter|first = Larry|publisher = New York Times|date = October 15, 2008 |accessdate = 2008-10-17}}</ref> Wurzelbacher said, "I’m getting ready to buy a company that makes $250,000 to $280,000 a year. Your new tax plan is going to tax me more, isn’t it?"<ref name=realdeal>{{cite news |url=http://www.nytimes.com/2008/10/17/us/politics/17joe.html |title=Real Deal on ‘Joe the Plumber’ Reveals New Slant |work=The New York Times |date=2008-10-16 |author=Rohter, Larry. |accessdate=2008-10-16 }}</ref> Wurzelbacher suggested that making more money should not result in paying a higher tax rate and asked Obama whether he would support a [[flat tax]] plan.

Obama responded with an explanation of how his tax plan would affect a small business in this bracket. He stated,
{{cquote|It's not that I want to punish your success. I just want to make sure that everybody who is behind you, that they've got a chance at success, too. And I think that when we spread the wealth around, it's good for everybody.<ref>{{cite news|url = http://ap.google.com/article/ALeqM5ha9qKZFKzFvH74G3c5PxRG_lHRIAD93RC8FG0|title =McCain, Obama get tough, personal in final debate|last = Fouhy|first = Beth|date = 15 October 2008|accessdate = 2008-10-17|publisher = The Associated Press}} </ref>}}
Obama's choice of words were suggested to have evoked the [[populism|populist]] "[[Share Our Wealth]]" movement of [[Huey Long]].<ref name="Pethokoukis">{{cite news|url=http://www.usnews.com/blogs/capital-commerce/2008/10/16/did-barack-spread-the-wealth-obama-just-blow-the-election.html|title=Did Barack "Spread the Wealth" Obama Just Blow the Election?|last=Pethokoukis|first=James|date=2008-10-16|publisher=[[U.S. News & World Report]]|accessdate=2008-10-17}}</ref> [[Steve Schmidt]], McCain's chief campaign strategist, commented that this statement would be a focus of their campaign in its final weeks.<ref name="Nagourney">{{cite news|url=http://www.nytimes.com/2008/10/17/us/politics/17campaign.html?_r=1&oref=slogin|title=Polls Cause Campaigns to Change Their Itineraries |last=Nagourney|first=Adam|coauthors=Jim Rutenberg|date=2008-10-16|publisher=[[The New York Times]]|accessdate=2008-10-17}}</ref>

==Press coverage==
Asked by [[Katie Couric]] of [[CBS Evening News]] on October 15 whether Obama's proposed $250,000 tax threshold would affect him, Wurzelbacher replied: "Not right now at presently, but, you know, question, so he's going to do that now for people who make $250,000 a year. When's he going to decide that $100,000 is too much, you know? I mean, you're on a slippery slope here. You vote on somebody who decides that $250,000 and you're rich? And $100,000 and you're rich? I mean, where does it end?"<ref name="CBS-News"/> He also said, “I asked the question but I still got a tap dance ... almost as good as [[Sammy Davis, Jr.]]” <ref name ="CBS-News">{{cite web
|url= http://www.cbsnews.com/blogs/2008/10/16/politics/horserace/entry4525242.shtml
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Joe The Plumber's Chat With Couric"
|publisher = [[CBS News]]
}}</ref>

On October 16, Wurzelbacher appeared on ''[[Your World with Neil Cavuto]]'' on Fox News. [[Neil Cavuto|Cavuto]] asked if Wurzelbacher was persuaded by Obama's plan. Wurzelbacher said that he was not and that he was more frightened upon hearing it. Wurzelbacher suggested that Obama's plan was [[Socialism|socialist]] in nature.<ref name = NYT/>

Wurzelbacher also appeared on ''[[Good Morning America]]'' the day following the debate (October 16). [[Diane Sawyer]] asked him if he was taking home $250,000 now, Wurzelbacher said with a laugh "No, not even close."<ref name ="ABC-vid">{{cite web
|url= http://abcnews.go.com/Video/playerIndex?id=6047458
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Meet Joe the Plumber"
|publisher = [[ABC News]]
}}</ref> Sawyer asked Wurzelbacher if he had been contacted by the McCain campaign before his encounter with Obama. Wurzelbacher said "I have been contacted by them and asked to show up at a rally." <ref name ="ABC-vid"/>

Wurzelbacher held a press conference at his home the morning following the debates, where he refused to express support for either candidate. "I'm not telling anybody anything" about which candidate he prefers, he said, adding, "It's a private booth. I want the American people to vote for who they want to vote for."<ref name = LATimes>{{cite web |url=http://www.latimes.com/news/politics/la-na-campaign17-2008oct17,0,373757.story |title='Joe the Plumber' still a topic for McCain, Obama |accessdate=2008-10-16 |date= 16 October 2008|work= |publisher=[[Los Angeles Times]]|first = Seema |last = Mehta |coauthors = Michael Muskal}}</ref> He reportedly had been registered in 1992 under the name "Samuel Joseph Worzelbacher".<ref name="toledoblade">{{cite web
|url=http://www.toledoblade.com/apps/pbcs.dll/article?AID=/20081016/NEWS09/810160418
|title="'Joe the Plumber' is focus of presidential debate's first few minutes"
|author=Bridget Tharp and Mark Zaborney
|publisher=Toledo Blade
|date=2008-10-16
|accessdate=2008-10-16}}</ref> He told an [[Associated Press]] reporter that in the most recent Republican primary, he backed McCain.<ref name = APlicense/>

==Plumbing career and licensing issues==
An [[Associated Press]] article revealed that Wurzelbacher does not have a plumber's license or [[apprentice]]ship.<ref name ="Yahoo-news">{{cite web
|url= http://news.yahoo.com/s/ap/20081017/ap_on_re_us/joe_the_plumber;_ylt=Avwwyr5JRUHYIXm2bhAYz5RH2ocA
|date=[[2008-10-16]]
|accessdate = 2008-10-16
|title = "Is 'Joe the Plumber' a plumber? That's debatable"
|publisher = [[Yahoo]]
}}</ref>
He claimed he does not need a license because he works for someone else. [[Lucas County, Ohio|Lucas County]], where Wurzelbacher and his employer reside, requires plumbers who perform work including sanitary drainage, water supply, storm drainage, and natural gas piping, to have licenses.<ref>{{cite web|url = http://www.contractor-licensing.com/ohio/plumbing-license.html|title = Ohio Plumbing License|publisher = National Contractors Pre-Licensing Services, Inc.|accessdate = 2008-10-17}}</ref> Neither Wurzelbacher nor his employer are licensed there, said Cheryl Schimming of Lucas County Building Regulations, which handles plumber licenses in parts of the county outside Toledo.<ref name = TBlicense/><ref name = APlicense/>
Local 50 of the [[United Association|United Association of Plumbers, Steamfitters and Service Mechanics]], whose membership endorsed Obama, indicated that Wurzelbacher applied for an apprentice program in 2003 but never completed the work.<ref>{{cite news|url=http://www.washingtonpost.com/wp-dyn/content/article/2008/10/16/AR2008101603614_2.html|title=After Debate, Glare Of Media Hits Joe|author=Barnes, Robert|publisher=[[Washington Post]]|accessdate=October 16, 2008}}</ref>

Wurzelbacher is one of two employees of a small plumbing firm, Newell Plumbing and Heating Co. of Toledo.<ref name=realdeal/> This is the company he described to Obama as making more than $250,000 per year;<ref name = APlicense>{{cite news|url = http://ap.google.com/article/ALeqM5gJsPHiQlgYvAsrHz9mvHJlezQJLwD93RONUO0|title = 'Joe the Plumber' says he has no plumbing license|last = Seewer|first = John|publisher = The Associated press|date = 16 October 2008|accessdate = 2008-10-17}}</ref> however, according to Bloomberg News, the company's net profit in reality is between $150,000 and $200,000, and would not see a tax hike under the Obama plan.<ref>{{cite news |url= http://www.bloomberg.com/apps/news?pid=20601087&sid=aC4j3T5.s_eQ&refer=home |title= `Joe the Plumber,' Obama Tax-Plan Critic, Owes Taxes |author= [[Ryan J. Donmoyer]] |publisher= [[Bloomberg.com]] |date= 2008-10-16 |accessdate= 2008-10-17}}</ref> MSNBC reported an even lower estimate: "Ohio business records show the company’s estimated total annual revenue as only $100,000. Actual ''taxable income'' would be even less than that."<ref>{{cite news |url= http://www.msnbc.msn.com/id/27221645/ |title= ‘Joe the plumber’ and Obama’s tax plan |publisher= [[msnbc.com]] |date= 2008-10-16 |accessdate= 2008-10-17}}</ref> He reported that the idea of buying the company was discussed during his job interview six years prior.<ref name= TBlicense/>

In his interview on ABC's ''[[Good Morning America]]'' on October 16, Wurzelbacher stated that his two current plumbing jobs were for a gas station and for a shopping center, Levis Commons.<ref name = "ABC-vid"/>

==Views on taxation==
In an October interview, Wurzelbacher said, "You know, a lot of the stuff that our government is doing right now is all about taxation without representation, and, you know, the last time that happened a couple guys got together and threw the Brits out."<ref>[http://www.kare11.com/news/news_article.aspx?storyid=527094 'Who is "Joe the Plumber"?', KARE11, October 16, 2008]</ref>

==References==
{{reflist|3}}

==External links==
* [http://www.liveleak.com/view?i=1a4_1224166209 Full, unedited video of the conversation between Joe the Plumber and Obama]

{{United States presidential election, 2008}}

{{DEFAULTSORT:Wurzelbacher, Joe}}
[[Category:United States presidential election, 2008]]
[[Category:John McCain]]
[[Category:Ohio Republicans]]
[[Category:Plumbers]]
[[Category:People from Lucas County, Ohio]]
[[Category:Living people]]

Frank Buckles

2008-03-07T18:39:43Z

Loadmaster: /* Biography */ minor link fix

{{Cleanup|date=March 2008}}
{{Infobox Military Person
|name= Frank Buckles
|lived= {{birth date|1901|02|1}} –
|image= [[Image:Frank Buckles WW1 at 16 edited.jpg|200px]]
|caption= Frank Buckles at age 16
|nickname=
|placeofbirth= [[Harrison County, Missouri]], [[United States]]
|placeofdeath=
|allegiance= [[United States|United States of America]]
|branch= [[United States Army]]
|serviceyears= [[1917]] – [[1920]]
|rank= [[Corporal]]
|unit=
|commands=
|battles= [[World War I]] [[World War II]]
|awards= [[Légion d'honneur]]
|relations=
|laterwork=
}}
'''Frank Woodruff Buckles''' (born [[February 1]] [[1901]]) is, at age {{age|1901|2|1}}, the last known surviving [[United States|American]]-born veteran of [[World War I|the First World War]] <ref>[http://www.orlandosentinel.com/news/orl-mww1vets2807may28,0,1647526.story?coll=orl-news-headlines]</ref>.
[[Image:Frank Buckles at 106.jpg|thumb|left|Buckles pictured at 103, was awarded the French Legion of Honor military decoration]]

==Biography==
Buckles is the last living WWI U.S. veteran to finish basic training and be stationed overseas prior to the end of the war. The US Library of Congress included him in its Veterans History Project that has audio, video and pictorial information on Buckles' experiences in both World War I and [[World War II|the Second World War]], and which includes a full 148-minute video interview. <ref>[http://lcweb2.loc.gov/diglib/vhp-stories/loc.natlib.afc2001001.01070/#vhp:clip,
[[May 29]], [[2007]], Library of Congress, Veterans History Project].</ref>

He was born in [[Harrison County, Missouri]], and enlisted at the beginning of the United States' involvement in World War I in April 1917. Only fifteen at the time of his enlistment, Buckles lied and said he was 21. Before being accepted into the army, he was turned down by the marines due to his weight. During his time in service for the [[United States Army]], Frank was stationed in the United States, [[United Kingdom]], Germany, and [[France]]. Buckles was sent to France in 1917 at age 16, where he was a driver; after the [[Armistice with Germany (Compiègne)|Armistice]] was signed in 1918, he escorted prisoners of war back to Germany. In 1919, after the war had ended, Frank Buckles was stationed in Germany, and he was discharged from service in 1920 having achieved the rank of [[corporal]]. In [[World War II|the Second World War]], in the [[1940s]], Buckles was a civilian working for an American shipping line. He was captured by the [[Japan]]ese, however, and spent three years in a Japanese prison camp during most of that war.<ref>[http://www.usatoday.com/news/nation/2007-03-27-cover-ww1-vet_N.htm?POE=NEWISVA "'One of the last': WWI vet recalls Great War", USAToday.com, [[March 27]], [[2007]], Andrea Stone.]</ref>

Buckles has at least one interview on a daily basis. He has stated in many interviews that he doesn't understand why people in the twenty-first century are in such a rush. He commented "What's the hurry?". Also, he does not own a television and has stated that people today watch too much television. He has said the worst president in his opinion was [[William McKinley|McKinley]]. Once asked about Nixon, he replied "He said a few bad things here and there." When asked on how he could live so long, he replied "Hope". On a daily basis he lifts 2-pound weights and does stretches in the morning. He does, according to his care taker, do around 50 sit-ups before he gets up in the morning.

Buckles was awarded the [[légion d'honneur]] by then French president [[Jacques Chirac]], and he currently lives in [[Charles Town, West Virginia]]. His story was featured on the [[Memorial Day]] 2007 episode of [[NBC Nightly News]]. He was also at the 2007 Memorial Day parade in [[Washington, D.C.]], riding in a buggy. Buckles stated in an interview with ''[[The Washington Post]]'' that he feels that the United States should only go to war when "it's an emergency." <ref>[http://www.washingtonpost.com/wp-dyn/content/article/2007/11/11/AR2007111101576.htmlWorld War I Veteran Reflects on Lessons]</ref> <ref>[http://thinkprogress.org/2007/11/12/106-year-old-wwi-veteran-speaks-on-the-iraq-war 106-year old WWI veteran speaks on the Iraq war]</ref> On [[March 6]], 2008, he met with President Bush at the White House.<ref>[http://www.cnn.com/2008/US/03/06/oldest.american.vet/index.html Bush thanks WWI veteran for 'love for America']</ref> The same day, he attended the opening of a [[Pentagon]] exhibit featuring photos of nine World War I veterans. Of the group, only Buckles and Canadian veteran [[John Babcock]] survive. Babcock was unable to attend.<ref name="ABCNews">{{cite news|title = Last doughboy gets Presidential 'Thank You'|publisher = abcnews.com| date = [[2008-03-06]]|url = http://abcnews.go.com/WN/Story?id=4404661|accessdate = 2008-03-07}}</ref>

==See also==
* [[Surviving veterans of World War I]]

==References==
{{reflist}}

==External links==
* [http://www.nytimes.com/2007/11/12/opinion/12rubin.html Over There — and Gone Forever] (New York Times Op-Ed about Frank Buckles, written by Richard Rubin and published on November 12, 2007)

{{DEFAULTSORT:Buckles, Frank}}
[[Category:1901 births]]
[[Category:American centenarians]]
[[Category:American military personnel of World War I]]
[[Category:American military personnel of World War II]]
[[Category:Légion d'honneur recipients]]
[[Category:Living people]]
[[Category:People from Harrison County, Missouri]]
[[Category:People from Jefferson County, West Virginia]]
[[Category:World War II prisoners of war]]
[[Category:United States Army soldiers]]

[[fi:Frank Buckles]]

Pseudosphäre

2007-12-30T05:00:50Z

Loadmaster: moved interactive webpage to External Links

[[Image:Pseudosphere.png|right|frame|Partial Pseudosphere]]
In [[geometry]], a '''pseudosphere''' of radius ''R'' is a surface of curvature −1/''R''2 (precisely, a [[complete metric space|complete]], [[simply connected]] surface of that curvature), by analogy with the sphere of radius ''R'', which is a surface of curvature 1/''R''2. The term was introduced by [[Eugenio Beltrami]] in his 1868 paper on models of [[hyperbolic geometry]]<ref>E. Beltrami, Saggio sulla interpretazione della geometria non euclidea, Gior. Mat. 6, 248–312 (Also Op. Mat. 1, 374-405; Ann. École Norm. Sup. 6 (1869), 251-288).</ref>.

The term is also used to refer to what is traditionally called a '''tractricoid''': the result of revolving a [[tractrix]] about its [[asymptote]], which is the subject of this article.

It is a [[Singularity|singular space]] (the equator is a singularity), but away from the singularities, it has constant negative [[Gaussian curvature]] and therefore is locally [[isometry|isometric]] to a [[hyperbolic plane]].

It also denotes the entire set of points of an infinite [[hyperbolic space]] which is one of the three models of [[Riemannian geometry]]. This can be viewed as the assemblage of continuous [[Saddle point|saddle]] shapes to [[infinity]]. The further outward from the symmetry axis, the more increasingly ruffled the [[manifold]] becomes. This makes it very hard to represent a pseudosphere in the [[Euclidean space]] of drawings. A trick mathematicians have come up with to represent it is called the '''[[Poincaré half-plane model|Poincaré model of hyperbolic geometry]]'''. By increasingly shrinking the pseudosphere as it goes further out towards the cuspidal edge, it will fit into a circle, called the '''[[Poincaré disk]]'''; with the "edge" representing infinity. This is usually [[tessellate]]d with [[equilateral]] [[triangle]]s, or other [[polygons]] which become increasingly distorted towards the edges, such that some vertices are shared by more polygons than is normal under [[Euclidean geometry]]. (In normal [[curvature|flat]] space only six equilateral triangles, for instance, can share a [[Vertex (geometry)|vertex]] but on the Poincaré disk, some points can share eight triangles as the total of the [[angle]]s in a narrow triangle of geodesic arcs is now less than 180°). Reverting the triangles back to their normal shape yields various bent sections of the pseudosphere.At any point the product of two principal radii of curvature is constant. Along lines of zero normal curvature geodesic torsion is constant by virtue of Beltrami-Enneper theorem.

The name "pseudosphere" comes about because it is a [[dimension|two-dimensional]] [[surface]] of constant negative curvature just like a sphere with positive Gauss curvature. It has same formulas for area and volume (''R'' = edge radius) 4π''R''2 and 4π''R''3/3 of the full surface in spite of the opposite Gauss curvature sign. Just as the [[sphere]] has at every point a [[negative and non-negative numbers|positively]] curved geometry of a [[dome]] the whole pseudosphere has at every point the [[negative and non-negative numbers|negatively]] curved geometry of a [[saddle surface|saddle]].

==References==
<div class="references-small">
* {{cite book|author=Henderson, D. W. and Taimina, D.|title=Aesthetics and Mathematics|publisher=Springer-Verlag|year=2006|chapter=[http://dspace.library.cornell.edu/bitstream/1813/2714/1/2003-4.pdf Experiencing Geometry: Euclidean and Non-Euclidean with History]}}
</div>

==See also==
*[[Sphere]]
*[[Hyperboloid structure]]

==References==
<references/>

==External links==
*[http://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html Non Euclid]
*[http://www.cabinetmagazine.org/issues/16/crocheting.php Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina ]
*[http://www.maths.manchester.ac.uk/~kd/ Prof. C.T.J. Dodson's web site at University of Manchester]
*[http://www.maths.manchester.ac.uk/~kd/geomview/dini.html Interactive demonstration of the pseudosphere] (at the [[University of Manchester]])

[[Category:Differential geometry]]
[[Category:Surfaces]]

[[it:Pseudosfera]]
[[ru:Псевдосфера]]
[[sv:Pseudosfär]]

0,999…

2007-11-13T18:45:50Z

Loadmaster: /* Introduction */ fraction consistency

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[recurring decimal]] '''0.999…''' , which is also written as <math>0.\bar{9} , 0.\dot{9}</math> or <math>\ 0.(9)</math>, denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|1]]. In other words, "0.999…" represents the same number as the symbol "1". The equality has long been accepted by professional mathematicians and taught in textbooks. Various [[mathematical proof|proof]]s of this identity have been formulated with varying [[Rigour#Mathematical rigour|rigour]], preferred development of the real numbers, background assumptions, historical context, and target audience.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this [[Equality (mathematics)|equality]] among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] quantities should exist, or that the expansion of 0.999… eventually terminates.

The non-uniqueness of such expansions is not limited to the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. For reasons of simplicity, the terminating decimal is almost always the preferred representation, further contributing to the misconception that it is the ''only'' representation. In fact, once infinite expansions are allowed, all [[positional numeral system]]s contain an infinity of ambiguous numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

Number systems in which 0.999… is strictly [[less than]] 1 can be constructed, but only outside the standard [[real number]] system which is used in elementary mathematics.

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

The meaning of "…" ([[ellipsis]]) in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated. In particular, 0.999… indicates the [[limit (mathematics)|limit]] of the [[sequence]] (0.9,0.99,0.999,0.9999,…) (or, equivalently, <math>\sum^{\infty}_{k=1} 9\times 10^{-k}</math>). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1.

There are many proofs that 0.999… = 1. Before demonstrating this using algebraic methods, consider that two [[real number]]s are identical [[if and only if]] their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a [[Interval (mathematics)|closed interval]] defined by the above sequence and the [[triangle inequality]]). Thus the difference is 0 and the numbers are identical. This also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

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

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount. (''i.e. 1 - 0.999… = 10-∞'')
:Note: It is important to know that 10-∞ has no mathematical meaning, a common error made by students. You may however determine the limit as k→-∞ of 10k, which is 0.
*Some students believe that the value of a [[convergent series]] is an approximation, not the actual value.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fraction proof|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==Proofs==
===Algebra===
==== Fractions ====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so <math>0.999\dots = 1</math>.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

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

:{| style="wikitable"
|

<math>
\begin{align}
0.333\dots &= \frac{1}{3} \\
3 \times 0.333\dots &= 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &= 1
\end{align}
</math>

|width="50px"|

||

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

|}

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

:<math>
\begin{align}
\frac{9}{9} &= 1 \\
\frac{9}{9} &= 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
\end{align}
</math>
Since both equations are valid, by the [[transitive property]], 0.999… must equal 1. Similarly, ³/3 = 1, and ³/3 = 0.999…. So, 0.999… must equal 1.

==== Digit manipulation ====

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

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

:<math>
\begin{align}
c &= 0.999\ldots \\
10 c &= 9.999\ldots \\
10 c - c &= 9.999\ldots - 0.999\ldots \\
9 c &= 9 \\
c &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

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

=== Real analysis ===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebraic proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

=== Real numbers ===
{{main|Construction of real numbers}}
All proofs given above have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.
* The proof on fractions assumes that <math>1/3 = 0.333...</math>, how do we know it's true? Why can't there be an infinitely small number <math>\epsilon</math> such that <math>1/3 = 0.333... + \epsilon/3</math>?
* The proof on digit manipulation says "To prove that the manipulations also work for infinite decimals, one needs the methods of real analysis." So the proof depends on something unproved.
* The proof on infinite series says: "The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
* The nested interval proof uses nested intervals theorem, which is just another form of Archimedean property.
In fact it is impossible to prove rigorously that 0.999... equals 1 with a naive intuitive approach to numbers. This gives a motivation to more serious mathematical constructions. If we wish to prove or disprove the statement, we must have a precise definition of real numbers.

Some approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

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

In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Different answers from alternative number systems|Different answers from alternative number systems]]" below.

==== Cauchy sequences ====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

===Generalizations===
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers ⅓, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via ⅓ and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fool's Day]] [[2004]] that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

== Different answers from alternative number systems ==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

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

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

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has point "infinity". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{commons|0.999...}}

* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
* [[informal mathematics|naive mathematics]]
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}

* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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0,999…

2007-11-13T18:44:42Z

Loadmaster: /* Introduction */ 1/3, 1/9, etc.

[[Image:999 Perspective.png|300px|right]]

In [[mathematics]], the [[recurring decimal]] '''0.999…''' , which is also written as <math>0.\bar{9} , 0.\dot{9}</math> or <math>\ 0.(9)</math>, denotes a [[real number]] [[equality (mathematics)|equal]] to [[1 (number)|1]]. In other words, "0.999…" represents the same number as the symbol "1". The equality has long been accepted by professional mathematicians and taught in textbooks. Various [[mathematical proof|proof]]s of this identity have been formulated with varying [[Rigour#Mathematical rigour|rigour]], preferred development of the real numbers, background assumptions, historical context, and target audience.

In the last few decades, researchers of [[mathematics education]] have studied the reception of this [[Equality (mathematics)|equality]] among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example that each real number has a unique [[decimal expansion]], that nonzero [[infinitesimal]] quantities should exist, or that the expansion of 0.999… eventually terminates.

The non-uniqueness of such expansions is not limited to the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. For reasons of simplicity, the terminating decimal is almost always the preferred representation, further contributing to the misconception that it is the ''only'' representation. In fact, once infinite expansions are allowed, all [[positional numeral system]]s contain an infinity of ambiguous numbers. For example, 28.3287 is the same number as 28.3286999…, 28.3287000, or many other representations. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple [[fractal]], the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

Number systems in which 0.999… is strictly [[less than]] 1 can be constructed, but only outside the standard [[real number]] system which is used in elementary mathematics.

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

The meaning of "…" ([[ellipsis]]) in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated. In particular, 0.999… indicates the [[limit (mathematics)|limit]] of the [[sequence]] (0.9,0.99,0.999,0.9999,…) (or, equivalently, <math>\sum^{\infty}_{k=1} 9\times 10^{-k}</math>). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1.

There are many proofs that 0.999… = 1. Before demonstrating this using algebraic methods, consider that two [[real number]]s are identical [[if and only if]] their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a [[Interval (mathematics)|closed interval]] defined by the above sequence and the [[triangle inequality]]). Thus the difference is 0 and the numbers are identical. This also explains why 0.333… = 1⁄3, 0.111… = 1⁄9, etc.

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

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1.…So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6–7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999… as having a fixed value which is less than 1 by an infinitely small amount. (''i.e. 1 - 0.999… = 10-∞'')
:Note: It is important to know that 10-∞ has no mathematical meaning, a common error made by students. You may however determine the limit as k→-∞ of 10k, which is 0.
*Some students believe that the value of a [[convergent series]] is an approximation, not the actual value.
These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10–14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1⁄3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416–417</ref> Others still are able to prove that 1⁄3 = 0.333…, but, upon being confronted by the [[#Fraction proof|fractional proof]], insist that "logic" supersedes the mathematical calculations.

[[Joseph Mazur]] tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137–141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1⁄3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261–262</ref>

==Proofs==
===Algebra===
==== Fractions ====

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so <math>0.999\dots = 1</math>.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

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

:{| style="wikitable"
|

<math>
\begin{align}
0.333\dots &= \frac{1}{3} \\
3 \times 0.333\dots &= 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &= 1
\end{align}
</math>

|width="50px"|

||

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

|}

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

:<math>
\begin{align}
\frac{9}{9} &= 1 \\
\frac{9}{9} &= 9 \times \frac{1}{9} = 9 \times 0.111\dots = 0.999\dots
\end{align}
</math>
Since both equations are valid, by the [[transitive property]], 0.999… must equal 1. Similarly, ³/3 = 1, and ³/3 = 0.999…. So, 0.999… must equal 1.

==== Digit manipulation ====

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

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

:<math>
\begin{align}
c &= 0.999\ldots \\
10 c &= 9.999\ldots \\
10 c - c &= 9.999\ldots - 0.999\ldots \\
9 c &= 9 \\
c &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

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

=== Real analysis ===
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

====Infinite series and sequences====
{{further|[[Decimal representation]]}}

Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999…) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebraic proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is ''defined'' to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "…when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such [[heuristic]]s are often interpreted by students as implying that 0.999… itself is less than 1.

====Nested intervals and least upper bounds====
{{further|[[Nested intervals]]}}

[[Image:999 Intervals C.svg|right|thumb|250px|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60–62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11–12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
<blockquote>
The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum.<ref>Apostol p.12</ref>
</blockquote>

=== Real numbers ===
{{main|Construction of real numbers}}
All proofs given above have certain problems and aren't really rigorous mathematical proofs. Let's take a closer look.
* The proof on fractions assumes that <math>1/3 = 0.333...</math>, how do we know it's true? Why can't there be an infinitely small number <math>\epsilon</math> such that <math>1/3 = 0.333... + \epsilon/3</math>?
* The proof on digit manipulation says "To prove that the manipulations also work for infinite decimals, one needs the methods of real analysis." So the proof depends on something unproved.
* The proof on infinite series says: "The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the Archimedean property." Here we have some axiom, that magically solves the infinitesimal problem.
* The nested interval proof uses nested intervals theorem, which is just another form of Archimedean property.
In fact it is impossible to prove rigorously that 0.999... equals 1 with a naive intuitive approach to numbers. This gives a motivation to more serious mathematical constructions. If we wish to prove or disprove the statement, we must have a precise definition of real numbers.

Some approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

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

In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17–20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form <math>\begin{align}1-(\tfrac{1}{10})^n\end{align}</math>.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number
<math>\begin{align}\tfrac{a}{b}<1\end{align}</math>, which implies <math>\begin{align}\tfrac{a}{b}<1-(\tfrac{1}{10})^b\end{align}</math>. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398–399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Different answers from alternative number systems|Different answers from alternative number systems]]" below.

==== Cauchy sequences ====
{{further|[[Cauchy sequence]]}}

Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes smaller than any positive rational.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

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

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that
:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

===Generalizations===
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any such system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410–411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1–3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96–98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point ²⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150–152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers ⅓, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via ⅓ and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company issued a "press release" on [[April Fool's Day]] [[2004]] that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Two proofs are then offered, based on limits and multiplication by 10.

== Different answers from alternative number systems ==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
<blockquote>
However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.<ref>Gowers p.60</ref>
</blockquote>
One can define other number systems using different rules or new objects; in some such number systems, the above proofs would need to be reinterpreted and one might find that, in a given number system, 0.999… and 1 might not be identical. However, many number systems are extensions of — rather than independent alternatives to — the real number system, so 0.999… = 1 continues to hold. Even in such number systems, though, it is worthwhile to examine alternative number systems, not only for how 0.999… behaves (if, indeed, a number expressed as "0.999…" is both meaningful and unambiguous), but also for the behavior of related phenomena. If such phenomena differ from those in the real number system, then at least one of the assumptions built into the system must break down.

===Infinitesimals===
{{main|Infinitesimal}}

Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε² = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439–442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245–247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79–80, 307–311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397–399</ref>

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

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

When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number which many write as "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]

The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14–15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902–903</ref>

==Related questions==

* [[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
* [[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the [[extended complex plane]], i.e. the [[Riemann sphere]], has point "infinity". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47–57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
* [[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref>

==See also==
{{commons|0.999...}}

* [[Decimal representation]]
* [[Infinity]]
* [[Limit (mathematics)]]
* [[informal mathematics|naive mathematics]]
* [[Non-standard analysis]]
* [[Real analysis]]
* [[Series (mathematics)]]

==Notes==
{{reflist|2}}

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439–450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999… is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900–903}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253–266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411–425 |url=http://www.wou.edu/~wardm/FromMonthlyMay2004.pdf}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11–15 |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610–617 }}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636–639 }}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669–673 }}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299–308 }}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242–251 }}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408–411 }}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57–64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396–400 }} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90–98 }}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44–49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2–18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210–230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

==External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}

* [http://www.cut-the-knot.org/arithmetic/999999.shtml .999999… = 1?] from [[cut-the-knot]]
* [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]
* [http://mathcentral.uregina.ca/QQ/database/QQ.09.00/joan2.html Proof of the equality based on arithmetic]
* [http://descmath.com/diag/nines.html Repeating Nines]
* [http://qntm.org/pointnine Point nine recurring equals one]
* [http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Articles containing proofs]]

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National Football League Cheerleading

2007-06-14T23:14:22Z

Loadmaster: link Blast! (musical)

For many [[National Football League|NFL]] teams, their [[franchise]] also includes a cheerleading squad. [[Cheerleaders]] are a popular addition that can give a team more coverage/airtime, popular local support and increased media image.
According to most NFL cheerleading sites, cheerleading is classified as a [[part-time]] job. However, this "part-time" job is a substantial commitment of time for practice, camp, games, appearances, photo shoots, and charity events.

Most often, cheerleaders have completed or are attending a [[university]], and continue on to other careers after cheering for an average of 1-4 [[seasons]].

Apart from their main duties of cheering during the football games, the cheerleaders have many other responsibilities. Nearly every team member is available for appearances at schools, events, conferences, etc., for a set fee. An anticipated annual event is the release of each squad's calendar, featuring members for each month in swimsuits, lingerie, or uniforms. As well as being a mainstay of American football culture, the cheerleaders are one of the biggest entertainment groups to regularly perform for the [[U.S. Military]] overseas. All performances and tours are enlisted by the [[United Service Organizations|USO]]. Teams send their [[variety show]], an elite group of their best members, to perform combination shows of [[dance]], music, [[baton (twirling)|baton twirling]], [[acrobatics]], [[gymnastics]], and more. In February 2007, the Buffalo Jills even sent a squad of 8 along with their choreographer into the war zone of Iraq.

As of 2006, a competition strictly for NFL cheerleaders was introduced on [[The NFL Network]], called ''[[NFL Cheerleader Playoffs]]''. Two girls from each cheerleading team compete against other mini-teams in various athletic events. This includes kayaking, 100 yd. dash, obstacle courses, and more.

==NFL Cheerleading==
===Misconceptions===
Cheerleaders are strictly prohibited from dating NFL players. However, most often the cheerleaders don't even see or come into contact with the players. It was recommended, in the FAQ section of the [[Miami Dolphins]] audition page, that if a woman's intention in joining a cheer squad was to date a NFL player, she should simply become a [[groupie]], not a cheerleader.

===Magazine Coverage===
Of all the American [[men's magazines]], [[Maxim (magazine)|Maxim]] most regularly features NFL cheerleaders. While mostly famous for their pin-up style photographs of said cheerleaders, the articles comprise the bulk of modern cheerleader biographical material. Cheerleaders featured in Maxim include:
*[[Meghan Vasconcellos]], [[Maxim (magazine)|Maxim]], Sept. 2006
*[[Bonnie-Jill Laflin]], [[Maxim (magazine)|Maxim]] Hot 100,<ref>Maxim Hot 100 (2005). [http://www.freejose.com/lists/maxim/2005/] Retrieved February 9, 2007.</ref> 2005

==Teams==
Listed by name, with corresponding NFL football team.

{| class="wikitable"
|-
! Name
! Established
! NFL Team
|-
| Arizona Cardinals Cheerleaders
| 1977
| [[Arizona Cardinals]]
|-
| Atlanta Falcons Cheerleaders
| 1976
| [[Atlanta Falcons]]
|-
| Baltimore Ravens Cheerleaders*
| 1998
| [[Baltimore Ravens]]
|-
| [[Buffalo Jills]]
| 1967 They existed as the Buffalo Bills Cheerleaders from 1960-1965
| [[Buffalo Bills]]
|-
| Carolina Topcats
| 1996
| [[Carolina Panthers]]
|-
| Cincinnati '''Ben-Gals'''<ref>[http://www.bengals.com/cheerleaders/index.asp]</ref>
| 1976
| [[Cincinnati Bengals]]
|-
| [[Dallas Cowboys Cheerleaders]]
| 1972<ref>Dallas Cheerleaders History (2007). [http://www.dallascowboys.com/cheerleaders/history.cfm] Retrieved February 8, 2007.</ref>
| [[Dallas Cowboys]]
|-
| Denver Broncos Cheerleaders
| 1977
| [[Denver Broncos]]
|-
| Houston Texans Cheerleaders
| 2002
| [[Houston Texans]]
|-
| Indianapolis Colts Cheerleaders
| 1977
| [[Indianapolis Colts]]
|-
| Jacksonville ROAR
| 1995
| [[Jacksonville Jaguars]]
|-
| Kansas City Chiefs Cheerleaders
| 1960s<ref>Kansas City Chiefs Cheerleaders History (2007). [http://www.kcchiefs.com/cheerleaders/history/] Retrieved February 8, 2007.</ref>
| [[Kansas City Chiefs]]
|-
| Miami Dolphins Cheerleaders
| 1966<ref>Miami Cheerleaders History (2007). [http://www.miamidolphins.com/newsite/cheerleaders/cheerleaderhistory/cheerleaderhistory.asp] Retrieved February 8, 2007.</ref>
| [[Miami Dolphins]]
|-
| Minnesota Vikings Cheerleaders
| 1984 professional / official
| [[Minnesota Vikings]]
|-
| New England Patriots Cheerleaders
| 1977
| [[New England Patriots]]
|-
| New Orleans '''Saintsations'''
| 1977
| [[New Orleans Saints]]
|-
| Oakland '''Raiderettes'''
| 1961<ref>Oakland Raiderettes History (2007). [http://www.raiderdrive.com/reunion_rekindles_reminiscences.htm] Retrieved February 8, 2007.</ref>
| [[Oakland Raiders]]
|-
| [[Pittsburgh Steelerettes]]
| 1960-1969<ref>Steelerettes History (2007). [http://www.steelerettes.com/1969.htm] Retrieved February 8, 2007.</ref>
| [[Pittsburgh Steelers]]
|-
| Philadelphia Eagles Cheerleaders
| 1948<ref>Philidelphia Eagles Cheerleaders History (2007). [http://www.fundinguniverse.com/company-histories/Philadelphia-Eagles-Company-History.html] Retrieved February 8, 2007.</ref>
| [[Philadelphia Eagles]]
|-
| Rams Cheerleaders
| 1974
| [[St. Louis Rams]]
|-
| San Diego '''Charger Girls'''
| 1976<ref>Charger Girls History (2007). [http://www.chargers.com/charger_girls/general_info.htm] Retrieved February 8, 2007.</ref>
| [[San Diego Chargers]]
|-
| [[San Francisco Gold Rush]]
| 1979 (as a coed quad before becoming an all-girl squad in 1983<ref>Gold Rush History (2007). [http://www.49ers.com/cheerleaders/history.php?section=CH%20History] Retrieved February 8, 2007.</ref>
| [[San Fransisco 49ers]]
|-
| Seattle '''[[Sea Gals]]'''
| 1976<ref>Sea Gals History (2007). [http://www.seahawks.com/SeaGals/History.aspx] Retrieved February 8, 2007.</ref>
| [[Seattle Seahawks]]
|-
| Tampa Bay Buccaneers Cheerleaders ''(formerly the '''SwashBuclers''' from 1976-1999)''
| 1976<ref>[http://www.buccaneers.com/cheerleaders/cheermain.aspx] Retrieved February 15, 2007.</ref>
| [[Tampa Bay Buccaneers]]
|-
| Tennessee Titans Cheerleaders
| 1975
| [[Tennessee Titans]]
|-
|Washington Redskins Cheerleaders
|1962<ref>Redskin Cheerleader History (2007). [http://www.redskins.com/cheerleaders/cheerleaderhistory.jsp] Retrieved February 8, 2007.</ref>
|[[Washington Redskins]]
|}
<nowiki>*</nowiki> Ravens Cheerleading Squad is technically a Co-ed Stunt and All-Female Dance squad.

==Notable Cheerleaders==
===Arizona Cardinals===
*[[Danielle Demski]], [[Miss Arizona USA]], 2004

===Atlanta Falcons===
*Nicole Duncan, [[Georgia State University]] Cheerleading Coach<ref>Georgia State Cheerleading (2007). [http://www.georgiastatesports.com/ViewArticle.dbml?SPSID=60906&SPID=5636&DB_OEM_ID=12700&ATCLID=620456] Retrieved February 9, 2007.</ref>

*Whitney Frink, Hollywood TV Producer

===Baltimore Ravens===
*[[Stacy Keibler]], Former professional [[World Wrestling Entertainment|WWE]] wrestler

===Dallas Cowboys Cheerleaders===
[[Image:Dallas-001216-N-1110A-513.jpg|thumb|right|200px|The [[Dallas Cowboys Cheerleaders]] onboard the [[USS Harry S. Truman (CVN-75)|USS Harry S. Truman]] on [[December 16]], [[2000]]]]
*[[Lezlie Deane]], actress, founder of techno group [[Fem2fem]]
*[[Bonnie-Jill Laflin]], actress/model
*[[Michelle Parma]], actress, MTV's ''[[Road Rules: Europe]]''
*[[Sarah Shahi]], (1999-2000),<ref>Sarah Shahi (2007). [http://www.cheern.com/news/11_2006/cheerleading_one_step_away_stardom/index.php] Retrieved February 9, 2007.</ref> actress, plays Carmen on ''[[The L Word]]'', second season
*[[Jill Marie Jones]], actress, plays Toni On [[Girlfriends]]
*[[ Kristin Holt]], television personality, entertainment news correspondent.

===Denver Broncos===
*[[Katee Doland]], [[Miss Colorado USA]] 2001
*Tatiana Anderson, Host of [[ESPN]]'S TV show ''Kiana's Flex Appeal''<ref>Tatiana Anderson (2007). [http://www.shapefit.com/fitness-models-tatiana-anderson.html] Retrieved February 9, 2007.</ref>

===Miami Dolphins===
*[[Shannon Ford]], [[Miss Florida USA]] 2002<ref>Miss Florida USA (2007). [http://www.missfloridausa.com/home.htm] Retrieved February 8, 2007.</ref>
*[[Suzy Tavarez]], On-Air Personality, LA radio station [[KIIS-FM]]

===New England Patriots===
*[[Kristin Gauvin]], [[Miss Massachusetts]] 2005<ref>Pageant History (2007). [http://missmass.org/] Retrieved February 8, 2007.</ref>

===Oakland Raiderettes===
*[[Danielle Gamba]], [[Playboy]] Cyber Girl of the Month, October 2004.
*Jennifer (Jenn) Grijalva, actress, [[MTV]]'s [[The_Real_World:_Denver|The Real World]] (season 18).

===Saint Louis Rams===
*[[Lisa Guerrero]], (1980s) American sports broadcaster, actress, model
*[[Jenilee Harrison]],<ref>Jenilee Harrison (2007). [http://www.tv.com/jenilee-harrison/person/6380/biography.html] Retrieved February 9, 2007.</ref> (1978-1980), actress, ''[[Three's Company]]''

===San Diego Chargers===
*[[Charisma Carpenter]], actress, played [[Cordelia Chase]] on ''[[Buffy The Vampire Slayer]]''

===San Francisco Gold Rush===
*[[Angela King-Twitero]], author of ''The Business of Professional Cheerleading'',<ref>Angela King (2007). [http://www.angelakingdesigns.com/aboutakd.html] Retrieved February 8, 2007.</ref> and dance costume designer (eight NFL Cheerleading teams wear her designs)<ref>Angela King Designs (2007). [http://www.angelakingdesigns.com/nfl.html] Retrieved February 8, 2007.</ref>Angela transitioned from cheerleader to director of the Gold Rush Cheerleaders, and lead the team from 1992-1997. She also was the founding co-director of the NFL Pro Bowl Cheerleaders from 1992-1997.

===Tennessee Titans===
*Dr. Monica Williams, [[Vanderbilt University]] cancer research fellow<ref>Dr. Williams (2007). [http://www.findarticles.com/p/articles/mi_m1077/is_1_58/ai_97997758] Retrieved February 8, 2007.</ref>

===Washington Redskins Cheerleaders===
[[Image:Washington-Redskins-041129-N-7469S-002.jpg|400px|right|thumb|'''Washington Redskins Cheerleaders''' perform for U.S. servicemen onboard Naval Support Activity (NSA) Bahrain]]
*Debbie Barrigan (1994, 1995, 1999, 2000, 2001), ''[[Blast! (musical)|Blast!]]'' dance troupe member<ref>Debbie Barrigan (2007). [http://www.redskins.com/cheerleaders/wherearetheynow.jsp;jsessionid=NIOMBOHLBCLG] Retrieved February 9, 2007.</ref>
*[[Michaé Holloman]], (2000- ), [[Miss Maryland USA]] 2007<ref>Miss Maryland USA (2007). [http://www.missmarylandusa.com/] Retrieved February 9, 2007.</ref>
*Kristianna Nichols, Mrs. America 1992<ref>Mrs. America USA (2007). [http://www.mrsamerica.com/] Retrieved February 8, 2007.</ref>

[[Image:Pro Bowl 2006 cheerleaders.jpg|right|thumb|[[2006 Pro Bowl]] cheerleaders]]

==Pro Bowl==

Each year, one squad member from every NFL team is chosen to participate in the collective [[Pro Bowl]] cheerleading squad. Traditionally, this is the highest honor of talent and popularity an NFL cheerleader can receive.

==2006 Pro Bowl AFC Cheerleading Squad<ref>Pro Bowl Roster (2007). [http://www.nfl.com/probowl/cheerleaders/afc] Retrieved February 8, 2007.</ref>==
*Dorothy Lee, Baltimore Ravens
*Eileen Stachowiak, Buffalo Bills
*Melissa Scalia, Cincinnati Bengals
*Renee Herlocker, Denver Broncos
*Rolanda Johnson, Houston Texans
*Brandi Jones, Indianapolis Colts
*Aubrey Moore, Jacksonville Jaguars
*Amy Day, Kansas City Chiefs
*Danielle O'Connell, Miami Dolphins
*Nicole Schell, New England Patriots
*Nikki Rogers, Oakland Raiders
*Lacy Harrison, San Diego Chargers
*Kerry Arrington, Tennesee Titans

==2006 Pro Bowl NFC Cheerleading Squad<ref>Pro Bowl Roster (2007). [http://www.nfl.com/probowl/cheerleaders/nfc] Retrieved February 8, 2007.</ref>==
*Kristi Gauble, Arizona Cardinals
*Melanie Snare (Sutton), Atlanta Falcons
*Amy Woodard, Carolina Panthers
*Lynlee Allen, Dallas Cowboys
*Theresa Baugus, Minnesota Vikings
*Lindsey Williams, New Orleans Saints
*Tara Keating, Philadelphia Eagles
*Kristin Beisel (Wolff), San Francisco 49ers
*Trina Mills, Seattle Seahawks
*Lacey Baldwin, St. Louis Rams
*Jennifer Abbott, Tampa Bay Buccaneers
*Courtney DeYoung, Washington Redskins

==2005 Pro Bowl Cheerleading Squad==
*Heather Joy, Arizona Cardinals
*Kim Kennedy, Atlanta Falcons
*Jamie Ringer, Buffalo Bills
*Shannon McClattie, Carolina Panthers
*Tara Wilson, Cincinnati Bengals
*Brandi Redmond, Dallas Cowboys
*Sarah Silva, Denver Broncos
*Julie Rainbolt, Houston Texans
*Jennifer Trock, Indianapolis Colts
*Jill Cottingham, Jacksonville Jaguars
*Kendrea White, Kansas City Chiefs
*Jackie Villarino, Miami Dolphins
*Erin Frey, Minnesota Vikings
*Allison Preston, New England Patriots
*Deryn Derbigny, New Orleans Saints
*Kristin Medwick, Oakland Raiders
*Monica Devlin, Philadelphia Eagles
*Lisa Simmons, San Diego Chargers
*Jany Collaco, San Francisco 49ers
*Kiara Bright, Seattle Seahawks
*Sommer Harris, St. Louis Rams
*Leigh Vollmer, Tampa Bay Buccaneers
*Jenita Smith, Tennessee Titans
*Jamilla Keene, Washington Redskins
*Brandi Redmond, Dallas Cowboys

==2000 Pro Bowl Cheerleading Squad==
*Katie Currier, Arizona Cardinals
*Jillian Edwards, Atlanta Falcons
*Meah Pace, Baltimore Ravens
*Julie Fanutti, Buffalo Bills
*Nicole Price, Carolina Panthers
*Nikki Lanzetta, Cincinnati Bengals
*Megan Willsey, Dallas Cowboys
*Marie Nesbitt, Denver Broncos
*Carrie Vogel, Indianapolis Colts
*Stephanie Archibald, Jacksonville Jaguars
*Rosie Hannan, Kansas City Chiefs
*Suzanne Bierwith, Miami Dolphins
*Angela Parkos, Minnesota Vikings
*Kalen Mace, New England Patriots
*Lani Quagliano, New Orleans Saints
*Patty Herrera, Oakland Raiders
*Cheryl Williams, Philidelphia Eagles
*Michelle Steptoe, St. Louis Rams
*Susan Macy, San Diego Chargers
*Antoinette Bertolani, San Fransisco 49ers
*Angela Adto, Seattle Seahawks
*Kristin Turner, Tampa Bay Buccaneers

==References==
<div class="references-small">
<references />
</div>

==See also==
*[[List of cheerleaders]]

==External links==
*[http://buffalojills.net/ Buffalo Jills Official Site]
*[http://www.bengals.com/cheerleaders/index.html Cincinnati Bengals Cheerleaders Official Site]
*[http://dallascowboys.com/cheerleaders/ Dallas Cowboys Cheerleaders Official Site]
*[http://www.kcchiefs.com/cheerleaders/ Kansas City Chiefs Cheerleaders Official Site]
*[http://www.patriots.com/cheerleaders/ Patriots Cheerleaders Official Site]
*[http://www.philadelphiaeagles.com/cheerleaders/home.jsp?id=30770 Philidelphia Eagles Cheerleaders Official Site]
*[http://www.raiders.com/raiderettes/line1.jsp Raiderettes Official Site]
*[http://cheerleaders.stlouisrams.com/cheerleaders/ Saint Louis Rams Cheerleaders Site]
*[http://www.49ers.com/cheerleaders/2006_squad.php?section=Cheerleaders San Fransisco Gold Rush Official Site]
*[http://www.seahawks.com/SeaGals/SeaGals.aspx/ Sea Gals Official Site]
*[http://www.redskins.com/cheerleaders/ Washington Redskins Cheerleaders Official Site]
*[http://www.nfl.com/cheerleaderplayoffs NFL Cheerleader Playoffs]
*[http://www.cheerleader.com/Teams.html]
*[http://nflcheerleader.blogspot.com/ Pro Cheerleader Blog]
*http://cheerrate.007ihost.com/ Rate the Cheerleader (The web's first cheerleader rating site)

[[Category:National Football League cheerleaders| ]]
[[Category:National Football League]]
[[Category:Cheerleaders]]
[[Category:American cheerleaders]]

Böser Clown

2007-04-16T15:02:28Z

Loadmaster: /* The evil clown in popular culture */ Killjoy (movie) link

{{unreferenced|article|date=November 2006}}
[[Image:Pennywiseclown.JPG|thumb|250px|[[It (monster)|Pennywise]] in [[Stephen King]]'s ''[[It (novel)|It]]'']]
The image of the '''evil clown''' is a recent development in [[United States|American]] [[popular culture]] in which the playful [[trope (literature)|trope]] of the [[clown]] is rendered as disturbing through the use of horror elements and dark humor.

==Background==
Many people find [[clown]]s disturbing rather than amusing. Many children are afraid of disguised, exaggerated, or costumed figures — even [[Santa Claus]]. Clown costumes tend to exaggerate the facial features and some body parts, such as hands and feet. This can be read as monstrous or deformed as easily as it can be read as comical. At the same time, the clown act is often represented as drunken, reckless, or simply [[insanity|insane]] — that of the giggling [[mania]]c. This includes the notorious Canio who murdered Nedda and Silvio (recorded in Leoncavallo's opera ''[[Pagliacci]]''.)

An extreme fear of clowns is known as [[coulrophobia]].

It can also be said one's response to a clown might depend on where it's seen. At a [[circus]] or a party, a clown is normal and one may find a clown funny. The same clown knocking on one's front door late one evening is more likely to generate fear or distress than laughter or amusement. This effect is summed up in a quote often attributed to actor [[Lon Chaney, Sr.]]: "There is nothing funny about a clown in the moonlight."

The idea of an evil clown can also be used in comedy. Since a clown is supposed to be funny, it is considered ironic if a clown has nothing whatsoever funny about him. Thus, a clown that is not funny can be hilarious.

== John Wayne Gacy==
[[John Wayne Gacy]], ([[March 17]], [[1942]] – [[May 10]], [[1994]]) was an American [[serial killer]]. He was convicted and later executed for the [[rape]] and [[murder]] of 33 boys and men, 27 of whom he buried in a crawl space under the floor of his house, while others were found in nearby rivers, between [[1972]] and his arrest in [[1978]]. He became notorious as the "Killer Clown" because of the many block parties he attended, entertaining children in a clown suit and makeup, under the name of "Pogo the Clown."

The wide publicity of Gacy's crimes is often presumed to have a strong influence on the idea of an evil clown.

== The evil clown in popular culture ==
The image of the evil clown appears to have gained notoriety to the extent of becoming a [[cliché]].

Major examples of "evil clown" imagery are:
*In the popular anime [[Yu-Gi-Oh!]], the monster "Saggi the Dark Clown" was a hideous clown that had a habit of cackling wickedly upon being summoned.
*In the TV show [[Ace Lightning]], one of the villain Lord Fear's assistants is a clown.
*In the [[Beatles]]' [[animated film]] ''[[Yellow Submarine (film)|Yellow Submarine]]'', large clowns serve as [[artillery]] for the [[Blue Meanies]].
*[[It (monster)|Pennywise]] in [[Stephen King]]'s [[1986]] [[novel]] ''[[It (novel)|It]]'', and the [[It (1990 film)|made-for-TV movie]] based on the book.
*The Clowns are a sadistic rival motorcycle gang in [[Katsuhiro Ôtomo]]'s 1988 anime ''[[Akira (film)|Akira]],'' and Ôtomo's original manga that he based it on.
*Musical groups such as the [[Insane Clown Posse]], [[Twiztid]], [[Dangerous Toys]], and [[Shawn "Clown" Crahan]] of [[Slipknot (band)|Slipknot]] who impersonate clowns in a "creepy" manner.
*[[Papa Lazarou]] of [[BBC]] sitcom, ''[[The League of Gentlemen (comedy)|The League of Gentlemen]]'', the wife-stealing Circus Master of the Pandemonium Carnival. Papa Lazarou's specific make-up, however, has greater resemblance to a [[minstrel]]'s.
*[[The Wack Pack|Yucko the Clown]], a frequent guest on the [[Howard Stern]] show, is a racist clown who has also appeared on Jimmy Kimmel Live, as well as his own series on MTV2 called Stankervision. In addition to his racist jokes, he prides himself on never washing his clown costume and his rank odor.
*[[Koko the Killer Clown]], a featured attraction at a [[sideshow]] on Coney Island. [http://abcnews.go.com/Entertainment/WolfFiles/story?id=91615&page=3]
*[[In Living Color]] featured [[Homey the Clown|Homey D. Clown]], a short-tempered, ex-con drug addict with a violent brand of comedy.
*The [[science fiction|sci fi]] [[film|movie]] ''[[Killer Klowns from Outer Space]]'' (1988).
[[Image:Jokerkillingjoke.png|275px|thumb|right|The Joker, the archetypal clown supervillain]]
*[[Joker (comics)|The Joker]] who is the greatest enemy of [[Batman]], is a murderously insane [[supervillain]] with a disturbing clown-like appearance. Appeared in ''Batman #1'' ([[1940]]), three decades before [[John Wayne Gacy]].
*"Kinko The Clown", a song about a "kid loving" clown inspired by Gacy.
*The clown doll in the first of the ''[[Poltergeist movies|Poltergeist]]'' movies, which becomes possessed by a [[ghost]] and attempts to strangle a young boy.
*In "[[Treehouse of Horror III|Treehouse of Horror]]", a [[parody]] of ''[[Trilogy of Terror]]'', [[Homer Simpson]] purchased a talking [[Krusty the Klown]] doll which attempted to kill him because a switch in the toy's back was set to "evil."
*Zombie clowns who drowned in a circus train accident return to eat the living in ''Dead Clowns,'' a low-budget horror movie made in the US and released in the UK in 2003.
*The Chief Clown (Ian Reddington) in the [[1988]] ''[[Doctor Who]]'' serial ''[[The Greatest Show in the Galaxy]]'' is a high-voiced maniac [[assassination|assassin]] with an army of [[robot]] clowns. He drove a hearse. Sinister clown imagery had previously appeared in the programme in ''[[The Celestial Toymaker]]'' and ''[[The Deadly Assassin]]''.
*The ''[[Star Trek: Voyager]]'' episode "[[The Thaw (Voyager episode)|The Thaw]]" features a mocking clown ([[Michael McKean]]) who presides over a [[Edgar Allan Poe|Poe]]-influenced [[virtual reality]] kingdom. He feeds on the [[fear]] of those wired into the VR, and is determined to keep them under his power.
* [[Doink the Clown]], the [[stage name]] of a [[professional wrestling|professional wrestler]] of the [[1990s]]. He wrestled in clown costume and makeup, but for the most part portrayed a [[heel (professional wrestling)|heel]], one who wrestles in an unsportsmanlike manner. He also had an evil twin who came from underneath the ring at [[WrestleMania IX]].
*[[Buggy the Clown]], a villainous clown [[pirate]] captain with supernatural powers, from the ''[[One Piece]]'' [[manga]] and [[anime]].
*The serial killer clown named [[Sweet Tooth (Twisted Metal)|Sweet Tooth]] from [[Sony Computer Entertainment|Sony's]] video game series, ''[[Twisted Metal (series)|Twisted Metal]]''.
*The clown doll in the video game ''[[Alone in the Dark 2]]'', which gets animated and kills by strangling. It is very probably based on [[Poltergeist]].
*Khan, the assassin in the [[adventure game]] ''[[Broken Sword: The Shadow of the Templars]]''; he is dressed as a clown playing [[accordion]] who bombs a [[France|French]] [[café]] in the beginning.
*The [[Evil Clown of Middletown]], a giant metal sign in [[Middletown Township, New Jersey]] from an aborted grocery store.
*In the ''[[Spawn (comics)|Spawn]]'' comics, the demon, [[Violator (comics)|Violator]] uses the human form of a fat clown called Clown.
*[[Superman]] has faced a villainous fat clown named Tringle in some comic issues.
*The character [[Piedmon|Piemon]] (Piedmon in the [[English language|English]] version) from the first TV series of the ''[[Digimon]]'' animated TV series, [[Digimon Adventure]].
*In the [[PlayStation 2|PS2]] [[computer role-playing game|RPG]] ''[[Dark Cloud 2]]'', an evil clown known as Flotsam pursues the hero with the aid of a giant clown robot called [[Halloween]] and an army of knife-wielding clowns.
*[[Mega Man (character)|Mega Man]] ([[Rockman]]) has fought a [[robot master]] called Clown Man and an unnamed fat robot clown whose head was badly attached. Later, in the [[Mega Man Battle Network]] series, Mega Man fights against two clown-themed Navis: ColorMan (known as WackoMan in the American dub of the [[Megaman NT Warrior|game-based anime]]), and CircusMan.
*In the arcade game ''[[Tumblepop]]'', the first boss (in the [[Russia]] levels) is a giant clown held by balloons who juggles with [[bomb]]s.
*In the arcade game ''[[CarnEvil]]'', the player must fight many varieties of undead clowns and is frequently taunted by a disembodied clown's head.
*Coco Demento, one of the students in ''[[¡Mucha Lucha!]]'', has the appearance of an evil clown, and was even shown as a bad guy in his first appearance. Afterwards, though, he is not considered a villain by the rest of the cast, in spite of his practical jokes. It was later revealed that he once hung out with some really evil clowns whom he is now afraid of.
*[[Mr. Giggles]], Mischief and Stumpy were evil clowns in the ''[[TimeSplitters]]'' series.
*In [[Rockstar Games|Rockstar]]'s Western-themed shooter ''[[Red Dead Revolver]]'' the protagonist faces off against a troupe of shotgun-wielding midget clowns early in the game.
*In [[Tim Burton]]'s ''[[The Nightmare Before Christmas]],'' the residents of Halloween Town include the so-called "Clown with the tear-away face," a hideous, seemingly undead clown. As his name suggests, he is able to remove his face, showing hs hollow head.
*In the cartoon ''[[The Powerpuff Girls]]'', the girls meet a jolly clown in multicolored attire nicknamed Rainbow the Clown. He later gets splashed by [[bleach]], which turns his appearance [[Black and white (colours)|black and white]], makes him insanely evil, and gives him the power to turn anything he touches into black and white, becoming Mr. Mime. Although he turns back to normal at the end of the episode, the girls still beat him up, an act which offended many viewers. Apparently he was set free, as he is seen in latter episodes in crew shots.
[[Image:BirthdayBandit.jpg|thumb|160px|The Birthday Bandit from ''Teamo Supremo''.]]
*A recurring villain on ''[[Teamo Supremo]]'' is [[List of recurring villains from Teamo Supremo#The Birthday Bandit|The Birthday Bandit]], a clown outraged with having gone through poor birthdays himself and intent on ruining other people's birthdays.
*In the movie ''[[Scary Movie 2]]'', an evil clown attacked Ray. This is another parody of both the evil toy clown scene in [[Poltergeist movies|Poltergeist]] and [[It (monster)|Pennywise]] from ''[[It (novel)|It]]''.
*A clown doll in the first [[Ghoulies]] movie is possessed by Ghoulie in the end.
*In the ''[[Aqua Teen Hunger Force]]'' episode "The Clowning", the ATHF's neighbor Carl gets a wig containing alien clown DNA which gradually deforms Carl, givng him giant feet, red hair and white skin with purple blothches, giving him the appearance of a ghastly clown. The episode also featured the keeper of the evil clown wig, a robotic [[List of Aqua Teen Hunger Force villains#Bingo|disembodied clown head]].
*In an early episode of ''[[Mighty Morphin Power Rangers]]'', [[Rita Repulsa]]'s Pineoctopus Monster takes the form of Pineapple the Clown. He would appear again during a [[Public Service Announcement|PSA]].
*In the movie [[Saw (2004 movie)|Saw]] and its sequels, [[Jigsaw Killer|the antagonist]] sends a [[Billy the Jigsaw Puppet|clown-like doll]] on a tricycle to deliver macabre messages to his victims.
*The character of [[Captain Spaulding (Rob Zombie)|Captain Spaulding]] in the [[Rob Zombie]] directed films [[House of 1000 Corpses]] and [[The Devil's Rejects]].
*[[Rudy the Clown]], a grotesque-looking clown is the main enemy in the [[Game Boy Color]] game [[Wario Land 3]]
[[Image:clownhouse.jpg|275px|thumb|right|Movie poster for the 1989 film ''[[Clownhouse]]''.]]
*In the [[Victor Salva]] film ''[[Clownhouse]]'', lunatics murder three circus clowns: Cheezo, Bippo, and Dippo, and assume their identities while terrorizing children.
*In the [[manga]] series [[Angel Sanctuary]], the [[demon]] [[Belial]] appears in makeup and attire similar to that of a [[clown]] or mime.
*[[Frenchy the Clown]], star of ''Evil Clown Comics'', a recurring feature in [[National Lampoon]] magazine in the late 1980s and early 1990s, first appearing in the June 1988 issue.
*[[Kefka Palazzo#Kefka Palazzo|Kefka Palazzo]], the main villain of the [[SNES]] game [[Final Fantasy VI]], wears face paint and clothing reminiscent of a clown.
* Phillipe, (a circus clown) is the boss of the England area in the [[SNES]] game [[Final Fight 2]].
*The movies ''[[Xtro]]'', ''[[Killjoy (movie)|Killjoy]]'', ''S.I.C.K. Serial Insane Clown Killer'', ''Fear of Clowns'' and ''Mr. Jingles'' all feature evil clowns.
* Freako the Clown is a nemesis in the ''[[Mighty Max (TV series)|Mighty Max]]'' episode 'Clown without Pity'
* [[The Anubis Gates]], a book by Tim Powers, features Horrabin as one of its villains. Horrabin's a sewer gang leader with clown make-up and he also walks on stilts.
*[[Beautiful Stories for Ugly Children]] Vol. 1, issue 1, a comic book put out in the late 1980s by Piranha Press, tells the tale of several clowns who are suspected of burning down the bigtop. They go on a weekend drinking binge with a two-headed woman and some poodles in a stolen car, and get into fights with mimes and bikers before the story is resolved.
*Doctor Whiteface, the head of the [[Guilds of Ankh-Morpork|Ankh-Morpork Guild of Fools]] in [[Terry Pratchett]]'s ''[[Discworld]]'' novels is a sinister, and possibly evil, clown, described as "the one who never gets in the way of the custard", and terrifying even to other clowns.
*In the movie [[Batman Returns]], the [[Penguin (comics)|Penguin's]] henchmen are a street gang known as the Red Triangle, formerly a traveling circus act. The gang has a circus theme, most members being clowns.
*In [[Dexter's Laboratory]], an episode parodies [[werewolf]] lore, but instead of a wolf, Dexter is bitten by and transforms into a clown, and causes mayhem throughout the city.
*The villains in Dean Koontz's novel ''[[Life Expectancy (novel)|Life Expectancy]]'' are the homicidal clown Konrad Beezo and his son, Punchinello.
*In the game [[Dragon Quest VIII: Journey of the Cursed King]], the main villain is a psychopathic clown going by the name of Dhoulmagus.
*An ''[[Extreme Ghostbusters]]'' episode called ''Killjoys'' features paranormal clowns that feed on laughter. Whenever a person laughs, a long green tentacle comes out of the clown's mouth and sucks the victim up leaving only their clothes behind. The faces of the victims appear on the clowns' hands.
* In Nintendo's "Super Punch Out!" one of the opponents on the world circus is named Mad Clown. He wears a clown costume in the ring and often attacks by throwing juggled balls.
* Ouchy the Clown, purveyor of "Adult Clown Services." [http://www.ouchytheclown.com/]
* In [[Capcom]]'s "[[Dead Rising]]", one of the Psychopath bosses you face is an insane, dual chainsaw-wielding clown named Adam.
[[Image:PhanPhanHittei.PNG|right|thumb|180px|Whippy from Kirby: Right Back at Ya!]]
*An episode of [[Kirby: Right Back at Ya!]] that didn't make it to the United States until Season 4 of the English dub features a psycho clown named [[Hittei (Kirby)|Whippy]] as the monster of that episode, and the 3rd-to-last monster the [[Cloaked Nightmare]] sent on [[Kirby (Nintendo)|Kirby]]. He is the head of the [[Majuu]] school. After being downloaded by [[King Dedede]] to help tame a rampaging and cowardly Phan Phan, he goes over to hypnotize Phan Phan and rides on his head, chasing after Kirby with his whip as well as endless supplies of spiked balls from Phan Phan's trunk. He is defeated later by Throw Kirby. Also, the four leaders of the Monster Training School are evil clowns.
* Canio, the main character in Leoncavallo's opera [[Pagliacci]], a Commedia Dell'Arte performer who, after discovering his wife Nedda's infidelity, murders both her and her lover Silvio while on stage as Pagliaccio (Italian for clown)
*In 2000, [[Universal Orlando|Universal Orlando's]] [[Halloween Horror Nights (Orlando)|Halloween Horror Nights]] event created their first advertising icon named Jack Schmidt, an evil clown and mass murderer who spent years on the circuit of sideshows, luring small children and women to their deaths through his circus acts. He was brought back in 2001 and 2006.
*One of the human villains in [[Danny Phantom]] is an evil clown-like ringmaster named [[Freakshow (Danny Phantom)|Freakshow]] who used a magic staff to make Danny evil in [[Control Freaks (Danny Phantom)|Control Freaks]] and returned in [[Reality Trip]] with a magic gauntlet that enabled him to warp reality to be a big-top nightmare.
*In the [[television program|TV series]] ''[[Are You Afraid of the Dark?]]'', an episode features Zeebo the clown, a cigar-smoking clown-ghost haunting a fun-house. Zeebo is mentioned again in several other episodes. Also, one episode features the "Crimson Clown", a doll that haunts children who lie or steal.
*The cover of the [[Reel Big Fish]] album "[[Cheer Up!]]" depicts an evil, cigar-smoking clown grabbing a child by the shirt and instructing him to cheer up.
*In the Simpson's episode, [[Lisa's First Word]], Homer makes Bart a clown-themed bed, but due to the shoddy work, looks evil. In Bart's [[nightmare]], it says "If you should die before you wake..." and chuckles insanely. The next day, baby Bart is curled up on the couch, chanting "Can't sleep. Clown will eat me."
*Shock rocker [[Alice Cooper]] released a song about cannabalistic clowns titled "Can't Sleep, the Clowns Will Eat Me," on certain versions of his heavy metal albums [[Brutal Planet]] and [[Dragontown]].
* The logo of Graffix, a manufacturer of water pipes and related merchandise, is the skull-like head of a menacing, fanged clown.
* An episode of [[Ben 10]] features as its villain [[List of secondary characters in Ben 10#Zombozo|Zombozo]], a clown who feeds on happiness, draining all joy from his victims and leaving them shambling, hopless drifters.

==See also==
*[[Mump and Smoot]]
*[[Bouffon]]

[[Category:Clowns]]
[[Category:Stock characters]]

[[fr:Clown maléfique]]
[[it:Clown malvagio]]
[[lt:Velniškas klounas]]

J-Live

2007-03-22T19:36:34Z

Loadmaster: link LP album

{{Unreferenced|date=December 2006}}




'''J-Live''' (also know as '''Justice Allah''', legal name '''Jean-Jacques Cadet''') is an [[Rapper|MC]], [[Disc Jockey|DJ]], and [[Record producer|producer]] from [[New York]], and also the founder of [[Triple Threat Productions]].

==Early Career and ''The Best Part''==
J-Live first appeared on the [[underground hip-hop]] scene in [[1995]], when he was a [[freshman]] [[English studies|English]] major at the [[University at Albany, The State University of New York]]. His first single "Longevity" appeared on [[Mark Farina]]'s ''[[Mushroom Jazz]]'' CD, and follow-up singles "Braggin' Writes" and "Hush the Crowd" garnered widespread attention, getting him printed in the "Unsigned Hype" section of ''[[The Source (magazine)|The Source]]''. This led to a record contract with Raw Shack Records. His first album, ''[[The Best Part]]'', recorded from [[1995]] to [[1999]], featured production by [[Prince Paul]], [[DJ Premier]], and [[Pete Rock]]. Due to problems with his record label, J-Live left the label and the album was shelved. He moved to [[Payday]] Records, but when Payday's parent company [[London Records]] was bought from [[Universal Music Group]] by [[Warner Music Group|WEA]], the album was again shelved. In 2001, copies surfaced as bootlegs and several were of such high quality it was rumored that J-Live himself was behind them. By the fall of 2001, The Best Part was finally officially released after five years of label problems. It was released on Triple Threat Productions, J-Live's own record label.

==''All of the Above'' to ''The Hear After''==
In [[2002]] his second album, ''All of the Above'' was released on the label, Coup D'etat. It featured production by [[DJ Spinna]], [[Usef Dinero]], and [[DJ Jazzy Jeff]]. In [[2003]] J-Live simultaneously released a two [[EP]]s, ''Always Has Been'' and ''Always Will Be''. ''Always Has Been'' was a collection of J-Live's six early singles which did not appear on any of his albums. ''Always Will Be'' consisted of 8 new songs. Both EPs were released on the label, Triple Threat Productions. Most recently J-Live released his third [[LP album|LP]], ''The Hear After'' in [[2005]] on the label, Penalty (Ryko).

==Personal Life==
After completion of his English degree from SUNY Albany, J-Live became a [[middle school]] [[English studies|English]] teacher in [[Brownsville and Bushwick, Brooklyn]]. He later left teaching for the full time pursuit of his hip-hop career. J-Live is currently living in [[Philadelphia]], PA. He is married and has three children. J-Live is a member of the [[The Nation of Gods and Earths]].

==Albums==
* ''[[The Best Part]]'' (2001)
* ''All of the Above'' (2002)
* ''Always Has Been'' (2003)
* ''Always Will Be'' (2003)
* ''The Hear After'' (2005)

== Appearances ==
* ''The Truth'' by [[Handsome Boy Modeling School]]
* ''Trackrunners'' by [[Asheru & Blue Black of Unspoken Heard]]
* ''Great Live Caper'' by [[J. Rawls]]
* ''Azucar (Remix)'' by [[Soulive]]
* ''Sub-Level'' by [[Rob Swift]]
* ''Give It Up'' by [[Think Differently Music]]
* ''Brand Nu Live'' by [[DJ Nu-Mark]]
* ''31st February'' by [[The Nextmen]]

== External links ==
*[http://www.j-livemusic.com/ J-Live Official Website]

{{DEFAULTSORT:Jlive}}

[[Category:African American musicians]]
[[Category:American rappers]]
[[Category:Five Percenters]]
[[Category:Living people]]

0,999…

2007-03-12T21:02:16Z

Loadmaster: /* Dedekind cuts */ fraction formatting

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], the [[recurring decimal]] '''0.999…''' , which is also written as <math>0.\bar{9} , 0.\dot{9}</math> or <math>\ 0.(9)</math>, denotes a [[real number]]. Notably, this number is [[equality (mathematics)|equal]] to [[1 (number)|1]]. In other words, "0.999…" represents the same number as the symbol "1". Various [[mathematical proof|proof]]s of this identity have been formulated with varying [[Rigour#Mathematical rigour|rigour]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality has long been accepted by professional mathematicians and taught in textbooks. In the last few decades, researchers of [[mathematics education]] have studied the reception of this [[equation]] among students. A great many question or reject the equality, at least initially. Many are swayed by textbooks, teachers and arithmetic reasoning as below to accept that the two are equal. However, they are often uneasy enough that they offer further justification. The students' reasoning for denying or affirming the equality is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique [[decimal expansion]] must correspond to a unique number, an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, an inability to understand [[limit (mathematics)|limits]] or simply the belief that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of these students' intuitive expectations can occur in other number systems. There has even been described a system in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two decimal expansions is not a peculiarity of the decimal system alone. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in [[Non-integer representation|non-integer bases]]. Nor is this phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. For reasons of simplicity, the terminating decimal is almost always the preferred representation, further contributing to the misconception that it is the ''only'' representation. In fact, once infinite expansions are allowed, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. As with integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

The meaning of "…" ([[ellipsis]]) in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated. In particular, 0.999… indicates the [[limit (mathematics)|limit]] of the [[sequence]] (0.9,0.99,0.999,0.9999,…) (or, equivalently, the sum of all terms of the form 9 × 0.1''k'' for integers k=1 to infinity). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1.

There are many proofs that 0.999…=1. Before demonstrating this using algebraic methods, consider that two [[real number]]s are identical if and only if their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a [[Interval (mathematics)|closed interval]] defined by the above sequence and the [[triangle inequality]]). Thus the difference is 0 and the numbers are identical. This also explains why 0.333... = 1⁄3, etc.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using [[Fraction (mathematics)|fraction]]s, 1⁄3 = 2⁄6. Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).
==Digit manipulation==
=== Fraction proof ===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. And 3 × 1⁄3 equals 1, so <math>0.999\dots = 1</math>.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

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

<math>
\begin{align}
0.333\dots &= \frac{1}{3} \\
3 \times 0.333\dots &= 3 \times \frac{1}{3} = \frac{3 \times 1}{3} \\
0.999\dots &= 1
\end{align}
</math>

=== Algebraic proof ===

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

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

<math>
\begin{align}
c &= 0.999\ldots \\
10 c &= 9.999\ldots \\
10 c - c &= 9.999\ldots - 0.999\ldots \\
9 c &= 9 \\
c &= 1 \\
0.999\ldots &= 1
\end{align}
</math>

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

== Real analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of [[real analysis]]. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form

:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\tfrac{1}{10}}) + b_2({\tfrac{1}{10}})^2 + b_3({\tfrac{1}{10}})^3 + b_4({\tfrac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9(\tfrac{1}{10}) + 9({\tfrac{1}{10}})^2 + 9({\tfrac{1}{10}})^3 + \cdots = \frac{9({\tfrac{1}{10}})}{1-{\tfrac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebraic proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\sum_{k = 1}^n\frac{9}{10^k} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".<ref>Tall and Schwarzenberger pp.6-7; Tall 2000 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2000 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount.
*Some students believe that the value of a [[convergent series]] is an approximation, not the actual value.
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2000 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1⁄3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref> Others still are able to prove that 1/3 = 0.333..., but, upon being confronted by the [[#Fraction proof|fractional proof]], insist that "logic" supersedes the mathematical calculations.

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "[[APOS theory]]" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing ⅓ as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== Real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

== Analogues in other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

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

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. The meaning of 0.999… depends on which structure we use. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Note, however, that, as an extension of the real numbers, the dual numbers still have 0.999…=1. On a related note, while ε exists in dual numbers, so does ε/2, so ε is not "the smallest positive dual number," and, indeed, as in the reals, no such number exists.

Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked about 0.999…, novices often believe there should be a "final 9," believing 1 − 0.999… to be a positive number many term "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers ⅓, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via ⅓ and limits, saying of misconceptions,
<blockquote>
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.<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>
</blockquote>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board … mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]] [[2004]] that it is 1:
<blockquote>
We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers.<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
</blockquote>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. However, division by zero is defined in some other systems, such as [[complex analysis]], where the extended complex plane, i.e. the [[Riemann sphere]], has point "infinity". Here, it makes sense to define 1/0 to be infinity;<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="references-2column">

<references />
</div>

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamical systems is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0-87779-621-1}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== See also ==
* [[Real analysis]]
* [[Non-standard analysis]]
* [[Naive mathematics]]

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

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Sleep (Kommandozeilenbefehl)

2007-03-05T22:54:39Z

Loadmaster: /* See also */ Sleep (operating system)

{{lowercase}}

'''sleep''' is a [[Unix]] [[command line]] program that [[sleep (operating system)|suspends]] program execution for a specified period of time.

The sleep instruction suspends the calling process for at least the specified number of seconds (the default), minutes, hours or days.

==Usage==
sleep '''number'''[suffix]...
or:
sleep option

Where '''number''' is a required floating point number, and suffix is an optional suffix to indicate the time period.

===Suffix===
'''s''' (seconds)
'''m''' (minutes)
'''h''' (hours)
'''d''' (days)

===Options===
'''--help''' display this help and exit
'''--version''' output version information and exit

==Examples==
sleep 5
Causes the current terminal session to wait 5 seconds. The default unit is seconds.

sleep 5h
Causes the current terminal session to wait 5 hours

Note that '''<tt>sleep 5h30m</tt>''' and '''<tt>sleep 5h 30m</tt>''' are illegal since sleep takes only one value and unit as argument. However, '''<tt>sleep 5.5h</tt>''' is allowed.

Possible uses for <tt>sleep</tt> include scheduling tasks and delaying execution to allow a process to start.

==See also==
* [[Sleep (operating system)]]

{{unix commands}}

[[Category:Unix software]]

[[pl:Sleep (Unix)]]

Sleep (Kommandozeilenbefehl)

2007-03-05T22:33:29Z

Loadmaster: Sleep (operating system) link

{{lowercase}}

'''sleep''' is a [[Unix]] [[command line]] program that [[sleep (operating system)|suspends]] program execution for a specified period of time.

The sleep instruction suspends the calling process for at least the specified number of seconds (the default), minutes, hours or days.

==Usage==
sleep '''number'''[suffix]...
or:
sleep option

Where '''number''' is a required floating point number, and suffix is an optional suffix to indicate the time period.

===Suffix===
'''s''' (seconds)
'''m''' (minutes)
'''h''' (hours)
'''d''' (days)

===Options===
'''--help''' display this help and exit
'''--version''' output version information and exit

==Examples==
sleep 5
Causes the current terminal session to wait 5 seconds. The default unit is seconds.

sleep 5h
Causes the current terminal session to wait 5 hours

Note that '''<tt>sleep 5h30m</tt>''' and '''<tt>sleep 5h 30m</tt>''' are illegal since sleep takes only one value and unit as argument. However, '''<tt>sleep 5.5h</tt>''' is allowed.

Possible uses for <tt>sleep</tt> include scheduling tasks and delaying execution to allow a process to start.

==See also==
{{unix commands}}

[[Category:Unix software]]

[[pl:Sleep (Unix)]]

Mattapan (Boston)

2006-10-31T23:32:19Z

Loadmaster: rvv noise by 68.236.84.245

'''Mattapan''' is a neighborhood in [[Boston]], [[Massachusetts]], [[United States]]. Originally part of neighboring [[Dorchester, Massachusetts|Dorchester]], Mattapan was annexed to Boston in [[1870]]. Like other neighborhoods of the time, Mattapan developed, residentially and commercially, as the railroads and streetcars made downtown Boston increasingly more accessible. Predominantly residential, Mattapan is a mix of public housing, small apartment buildings, single homes and two and three family houses. Blue Hill Avenue and Mattapan Square, where Blue Hill Avenue, River Street, and Cummins Highway meet, is the commercial heart of the neighborhood, home to banks, law offices, restaurants, and retail shops.

== Demographic change ==
In the 1960s and '70s Mattapan went through a major change in the makeup of its population. It changed from a predominantly white [[Jewish American|Jewish]] neighborhood to one that is almost entirely black. The years between 1968 and 1970 made up the most dramatic period of ethnic transition. According to Levine and Harmon in their book ''Death of an American Jewish Community'', [[redlining]] the area, blockbusting, and fear in neighborhood residents created by real estate agents brought about panic selling and [[white flight]]. The banking consortium Boston Banks Urban Renewal Group (B-BURG) allegedly drove the Jewish community out of Mattapan and are claimed to bear partial responsibility for the deterioration of the neighborhood, especially along the Blue Hill Avenue corridor.

Again, according to Levine and Harmon, the reason behind this orchestrated attack on the community was to lower market values to buy property, sell the housing with federally guaranteed loans at inflated prices to black families who couldn't afford it, and to get the white community to buy property owned by the banks in the suburbs.

Today Mattapan is seeing another major population shift, albeit a natural turn over of housing, as a large number of [[Haitian]] immigrants continue to move in. Mattapan now has the largest Haitian community in Massachusetts. The neigborhood also faces many problems related to gang violence in modern times. Today community leaders are working with police to control gang violence within the area.

== Transportation ==
The [[Ashmont-Mattapan High Speed Line|Mattapan-Ashmont]] [[Light rail|trolley]] line of the [[Massachusetts Bay Transportation Authority|MBTA]] serves Mattapan as well as several bus routes. The [[Fairmount Line]] of the [[MBTA Commuter Rail]] also serves Mattapan at Morton Street, providing service to downtown Boston and the suburbs. In the near future an additional station at Blue Hill Avenue will be added.{{fact}} Plans for the reconstruction of Mattapan station have been approved and construction should begin in late 2006.{{fact}}

{{Boston neighborhoods}}

==External links==
*[http://ksgaccman.harvard.edu/hotc/displayplace.asp?id=11452 Heart of the City, Mattapan], The Rappaport Institute for Greater Boston at the John F. Kennedy School of Government at Harvard University article on Mattapan
*[http://partners.nytimes.com/books/first/g/gamm-exodus.html Why the Jews Left Boston], New York Times book review of ''Why the Jews Left Boston and the Catholics Stayed''.
*Death of an American Jewish Community (ISBN 0-02-913866-3).
*[http://hcs.harvard.edu/~fup/password/otherplaces.html First Year Urban Project], A Harvard University urban program.

[[Category:Boston neighborhoods]]
[[Category:Streetcar suburbs]]

[[fr:Mattapan (Boston)]]

0,999…

2006-10-31T00:31:15Z

Loadmaster: /* Introduction */ cleanup 9 x 0.1^k

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], the [[real number]] denoted by the [[recurring decimal]] '''0.999…''' (also written <math>0.\bar{9}</math> or <math>0.\dot{9}</math>) is [[equality (mathematics)|exactly equal]] to [[1 (number)|1]]. In other words, "0.999…" represents the same number as the symbol "1". Various [[mathematical proof|proof]]s of this identity have been formulated with varying [[Rigour#Mathematical rigour|rigorousness]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality has long been taught in textbooks, and in the last few decades, researchers of [[mathematics education]] have studied the reception of this [[equation]] among students, who often reject the equality. The students' reasoning is often based on an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, an inability to understand [[limit (mathematics)|limits]] or simply the belief that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There is even a system in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two [[decimal expansion]]s is not a peculiarity of the decimal system alone. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. A similar phenomenon occurs in [[balanced ternary]], where 1/2, instead of 1, has two possible expansions. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Introduction==
0.999… is a number written in the [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. And like integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

The meaning of "[[Ellipsis|…]]" in 0.999… must be precisely specified. The use here is different from the usage in language or in 0.99…9, in which the ellipsis specifies that some ''finite'' portion is left unstated or otherwise omitted. When used to specify a [[recurring decimal]], "…" means that some ''infinite'' portion is left unstated. In particular, 0.999… indicates the [[limit (mathematics)|limit]] of the [[sequence]] (0.9,0.99,0.999,0.9999,…) (or more formally, the sum of all terms of the form 9 × 0.1''k'' for integers k=1 to infinity). Misinterpreting the meaning of 0.999… accounts for some of the misunderstanding about its equality to 1.

There are many proofs that 0.999…=1. Before demonstrating this using algebraic methods, consider this: Two [[real number]]s are identical if and only if their (absolute) difference is not equal to a positive (third) real number. Given any positive value, the difference between 1 and 0.999… is less than this value (which can be formally demonstrated using a [[Interval (mathematics)|closed interval]] defined by the above sequence and the [[triangle inequality]]). Thus the difference is 0 and the numbers are identical. This also explains why 0.333... = 1⁄3, etc.

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

==Digit manipulation==

=== Fraction proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| 0.333… || = 1⁄3
|-
|align="right"| 3 × 0.333… || = 3 × 1⁄3
|-
|align="right"| 0.999… || = 1
|}

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a recurring decimal, 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref> Another form of this proof multiplies 1/9 = 0.111… by 9.

=== Algebraic proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| ''c'' || = 0.999…
|-
|align="right"| 10''c'' || = 9.999…
|-
|align="right"| 10''c'' − ''c'' || = 9.999… − 0.999…
|-
|align="right"| 9''c'' || = 9
|-
|align="right"| ''c'' || = 1
|}
Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''c''. Then 10''c'' − ''c'' = 9. This is the same as 9''c'' = 9. Dividing both sides by 9 completes the proof: ''c'' = 1.<ref name="CME"/>

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

== Calculus and analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form

:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals 9.999...) appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.\,</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity."<ref>Tall and Schwarzenberger pp.6-7; Tall 2001 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount.
*Some students believe that the value of a [[convergent series]] is an approximation, not the actual value.
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2001 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref>

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== Real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at [[undergraduate]] mathematicians.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's ''[[reductio ad absurdum]]''."<ref>Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.</ref> If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including all natural numbers greater than 1) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1/3 and limits, saying of misconceptions,
:"The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

:Nonsense."<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]] [[2004]] that it is 1:
:"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the [[Riemann sphere]], it makes sense to define 1/0 to be infinity.<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> In fact, some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example integers stored in the [[sign and magnitude]] or [[one's complement]] formats, or floating point numbers as specified by the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="small">
<references />
</div>

==References==
<div class="references-small" style="-moz-column-count: 2; column-count: 2;">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamics is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0877796211}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

[[el:0,999...]]
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[[fr:Développement décimal de l'unité]]
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[[zh:证明0.999...等于1]]

0,999…

2006-10-26T23:36:30Z

Loadmaster: /* ''p''-adic numbers */ made "10x = x - 9" a little clearer

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], '''0.999…''' (also denoted <math>0.\bar{9}</math> or <math>0.\dot{9}</math>) is a [[recurring decimal]] [[equality (mathematics)|exactly equal]] to [[1 (number)|1]]. In other words, the symbols "0.999…" and "1" represent the same [[real number]]. A number of [[mathematical proof|proof]]s of this identity have been formulated, which vary as to their level of [[Rigour#Mathematical rigour|rigor]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality has long been taught in textbooks, and in the last few decades, researchers of [[mathematics education]] have studied the reception of this equation among students, who often reject the equality. The students' reasoning is often based on an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, an inability to understand [[limit (mathematics)|limits]] or simply that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two [[Decimal representation|decimal expansion]]s is not a peculiarity of the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. A similar phenomenon occurs in [[balanced ternary]], where 1/2, instead of 1, has two possible expansions. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Digit manipulation==
0.999… is a number written in [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. And like integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using [[fraction]]s, <math>\frac{1}{3} = \frac{2}{6}</math>.

Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

{|class=wikitable align=right
|-
| assertion || 3 × 0.333… = 0.999…
|-
| assertion ||        0.333… = 1⁄3
|-
| step 1 || 3 × 0.333… = 3 × 1⁄3
|-
| proof ||        0.999… = 1
|}

=== Fraction proof ===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a [[recurring decimal]], 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref> Another form of this proof multiplies 1/9 = 0.111… by 9.

=== Algebraic proof ===
{|class=wikitable align=right
|-
| assertion || 10 × 0.999… = 9.999…
|-
| assertion ||                       ''c'' = 0.999…
|-
| step 1 ||                   10''c'' = 9.999…
|-
| step 2 ||             10''c'' - ''c'' = 9.999… − 0.999…
|-
| step 3 ||                    9''c'' = 9
|-
| proof ||                       ''c'' = 1
|}

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''c''. Then 10''c'' − ''c'' = 9. This is the same as 9''c'' = 9. Dividing both sides by 9 completes the proof: ''c'' = 1.<ref name="CME"/>

== Calculus and analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before university level.

One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:<math>b_0.b_1b_2b_3b_4b_5\dots</math>

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals "9·9999999, &c.") appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the [[Limit of a sequence|limit]] concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity."<ref>Tall and Schwarzenberger pp.6-7; Tall 2001 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount.
*Some students believe that the value of a [[convergent series]] is an approximation, not the actual value.
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2001 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref>

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== Real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref>

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at [[undergraduate]] mathematicians.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's ''[[reductio ad absurdum]]''."<ref>Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.</ref> If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:

:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \cdots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>

(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' =  …990, so 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including 2 and 10) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[news:sci.math sci.math]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1/3 and limits, saying of misconceptions,
:"The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

:Nonsense."<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]] [[2004]] that it is 1:
:"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the [[Riemann sphere]], it makes sense to define 1/0 to be infinity.<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> In fact, some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="small">
<references />
</div>

==References==
<div class="references-small">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamics is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0877796211}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

[[el:0,999...]]
[[es:0,9 periódico]]
[[fr:Développement décimal de l'unité]]
[[ja:0.999...が1に等しいことの証明]]
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[[zh:证明0.999...等于1]]

0,999…

2006-10-26T00:15:25Z

Loadmaster: fixed fractions

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], '''0.999…''' (also denoted <math>0.\bar{9}</math> or <math>0.\dot{9}</math>) is a [[recurring decimal]] [[equality (mathematics)|exactly equal]] to [[1 (number)|1]]. In other words, the symbols '0.999…' and '1' represent the same [[real number]]. Mathematicians have formulated numerous [[mathematical proof|proof]]s of this identity, which vary as to their level of [[Rigour#Mathematical rigour|rigor]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality 0.999… = 1 has long been taught in textbooks, and in the last few decades, researchers of [[mathematics education]] have studied the reception of this equation among students, who often reject the equality. The students' reasoning is often based on an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, or simply that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two [[Decimal representation|decimal expansion]]s is not a peculiarity of the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. A similar phenomenon occurs in [[balanced ternary]], where 1/2, instead of 1, has two possible expansions. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Digit manipulation==
0.999… is a number written in [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. And like integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using [[fraction]]s, <math>\frac{1}{3} = \frac{2}{6}</math>.

Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).

{|class=wikitable align=right
|-
| assertion || 3 × 0.333… = 0.999…
|-
| assertion ||        0.333… = 1⁄3
|-
| step 1 || 3 × 0.333… = 3 × 1⁄3
|-
| proof ||        0.999… = 1
|}

=== Fraction proof ===

One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a [[recurring decimal]], 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref> Another form of this proof multiplies 1/9 = 0.111… by 9.

=== Algebraic proof ===
{|class=wikitable align=right
|-
| assertion || 10 × 0.999… = 9.999…
|-
| assertion ||                       ''c'' = 0.999…
|-
| step 1 ||                   10''c'' = 9.999…
|-
| step 2 ||             10''c'' - ''c'' = 9.999… − 0.999…
|-
| step 3 ||                    9''c'' = 9
|-
| proof ||                       ''c'' = 1
|}

Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''c''. Then 10''c'' − ''c'' = 9. This is the same as 9''c'' = 9. Dividing both sides by 9 completes the proof: ''c'' = 1.<ref name="CME"/>

== Calculus and analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before the university level.

One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:''b''0.''b''1''b''2''b''3''b''4''b''5….

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \textstyle\frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals "9·9999999, &c.") appears as early as 1770 in [[Leonhard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the '''base-4''' decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity."<ref>Tall and Schwarzenberger pp.6-7; Tall 2001 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount.
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2001 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref>

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== Real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 that directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref> The following two examples come from less usual sources.

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at [[undergraduate]] mathematicians.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's ''[[reductio ad absurdum]]''."<ref>Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.</ref> If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternative number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \ldots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>
(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a [[doppelgänger]] with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternative representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including 2 and 10) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[[sci.math]]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1/3 and limits, saying of misconceptions,
:"The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

:Nonsense."<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]] [[2004]] that it is 1:
:"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the [[Riemann sphere]], it makes sense to define 1/0 to be infinity.<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> In fact, some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>

==References==
<div class="references-small">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamics is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0877796211}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonhard |authorlink=Leonhard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy|authorlink= William Timothy Gowers|title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

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0,999…

2006-10-25T15:37:17Z

Loadmaster: rvv by 194.29.160.251

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], '''0.999…''' (also denoted <math>0.\bar{9}</math> or <math>0.\dot{9}</math>) is a [[recurring decimal]] [[equality (mathematics)|exactly equal]] to [[1 (number)|1]]. In other words, the symbols '0.999…' and '1' represent the same [[real number]]. Mathematicians have formulated numerous [[mathematical proof|proof]]s of this identity, which vary as to their level of [[Rigour#Mathematical rigour|rigor]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality 0.999… = 1 has long been taught in textbooks, and in the last few decades, researchers of [[mathematics education]] have studied the reception of this equation among students, who often vocally reject the equality. The students' reasoning is often based on an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, or simply that 0.999… should have a last 9. These ideas are false with respect to the [[real number]]s, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two [[Decimal representation|decimal expansion]]s is not a peculiarity of the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Digit manipulation==
0.999… is a number written in [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. And like integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using [[fraction]]s,
:1⁄2 = 3⁄6.
Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).
=== Fraction proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| 0.333… || = 1⁄3
|-
|align="right"| 3 × 0.333… || = 3 × 1⁄3
|-
|align="right"| 0.999… || = 1
|}
One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a [[recurring decimal]], 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref> Another form of this proof multiplies 1/9 = 0.111… by 9.

=== Algebra proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| ''c'' || = 0.999…
|-
|align="right"| 10''c'' || = 9.999…
|-
|align="right"| 10''c'' − ''c'' || = 9.999… − 0.999…
|-
|align="right"| 9''c'' || = 9
|-
|align="right"| ''c'' || = 1
|}
Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''c''. Then 10''c'' − ''c'' = 9. This is the same as 9''c'' = 9. Dividing both sides by 9 completes the proof: ''c'' = 1.<ref name="CME"/>

=== Subtraction proof ===
Another proof is to subtract 0.999… from 1, which yields 0.000… As none of the digits is greater than 0, the result is exactly 0.

== Calculus and analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before the university level.

One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:''b''0.''b''1''b''2''b''3''b''4''b''5….

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \textstyle\frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals "9·9999999, &c.") appears as early as 1770 in [[Leonard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the base-4 decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity."<ref>Tall and Schwarzenberger pp.6-7; Tall 2001 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount. (This is a more sophisticated response.)
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2001 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref>

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== The real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 that directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref> The following two examples come from less usual sources.

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at [[undergraduate]] mathematicians.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's ''[[reductio ad absurdum]]''."<ref>Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.</ref> If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternate number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \ldots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>
(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a [[doppelgänger]] with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternate representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including 2 and 10) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[[sci.math]]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1/3 and limits, saying of misconceptions,
:"The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

:Nonsense."<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]], [[2004]] that it is 1:
:"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the [[Riemann sphere]], it makes sense to define 1/0 to be infinity.<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> In fact, some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>

==References==
<div class="references-small">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamics is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0877796211}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonard |authorlink=Leonard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

[[es:0,9 periódico]]
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0,999…

2006-10-25T15:32:03Z

Loadmaster: /* Fraction proof */ 1/9 fraction

[[Image:999 Perspective.png|300px|right]]
In [[mathematics]], '''0.999…''' (also denoted <math>0.\bar{9}</math> or <math>0.\dot{9}</math>) is a [[recurring decimal]] [[equality (mathematics)|exactly equal]] to [[1 (number)|1]]. In other words, the symbols '0.999…' and '1' represent the same [[real number]]. Mathematicians have formulated numerous [[mathematical proof|proof]]s of this identity, which vary as to their level of [[Rigour#Mathematical rigour|rigor]], preferred development of the real numbers, background assumptions, historical context, and target audience.

The equality 0.999… = 1 has long been taught in textbooks, and in the last few decades, researchers of [[mathematics education]] have studied the reception of this equation among students, who often vocally reject the equality. The students' reasoning is often based on an expectation that [[infinitesimal]] quantities should exist, that [[arithmetic]] may be broken, or simply that 0.999… should have a last 9. These ideas are false with respect to the [[real number]]s, which can be proven by explicitly constructing the reals from the [[rational number]]s, and such constructions can also prove that 0.999… = 1 directly. At the same time, some of the intuitive phenomena can occur in other number systems. There are even systems in which an object that can reasonably be called "0.999…" is strictly [[less than]] 1.

That the number 1 has two [[Decimal representation|decimal expansion]]s is not a peculiarity of the decimal system. The same phenomenon occurs in [[integer]] [[radix|base]]s other than 10, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is the phenomenon unique to 1: every non-zero, terminating decimal has a twin with trailing 9s. In fact, all [[positional numeral system]]s contain an infinity of ambiguous numbers. These various identities have been applied to better understand patterns in the decimal expansions of [[fraction (mathematics)|fraction]]s and the structure of a simple fractal, the [[Cantor set]]. They also occur in a classic investigation of the infinitude of the entire set of real numbers.

==Digit manipulation==
0.999… is a number written in [[decimal]] [[numeral system]], and some of the simplest proofs that 0.999… = 1 rely on the convenient [[arithmetic]] properties of this system. Most of decimal arithmetic — [[addition]], [[subtraction]], [[multiplication]], [[division (mathematics)|division]], and [[inequality|comparison]] — uses manipulations at the digit level that are much the same as those for [[integer]]s. And like integers, any two ''finite'' decimals with different digits mean different numbers (ignoring trailing zeros). In particular, any number of the form 0.99…9, where the 9s eventually stop, is strictly less than 1.

Unlike the case with integers and finite decimals, other notations can express a single number in multiple ways. For example, using [[fraction]]s,
:1⁄2 = 3⁄6.
Infinite decimals, however, can express the same number in at most two different ways. If there are two ways, then one of them must end with an infinite series of nines, and the other must terminate (that is, consist of a recurring series of zeros from a certain point on).
=== Fraction proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| 0.333… || = 1⁄3
|-
|align="right"| 3 × 0.333… || = 3 × 1⁄3
|-
|align="right"| 0.999… || = 1
|}
One reason that infinite decimals are a necessary extension of finite decimals is to represent fractions. Using [[long division]], a simple division of integers like 1⁄3 becomes a [[recurring decimal]], 0.333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999… = 1. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.333… equals 0.999…. But 3 × 1⁄3 equals 1, so 0.999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref> Another form of this proof multiplies 1/9 = 0.111… by 9.

=== Algebra proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| ''c'' || = 0.999…
|-
|align="right"| 10''c'' || = 9.999…
|-
|align="right"| 10''c'' − ''c'' || = 9.999… − 0.999…
|-
|align="right"| 9''c'' || = 9
|-
|align="right"| ''c'' || = 1
|}
Another kind of proof more easily adapts to other repeating decimals. When a number in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.999… equals 9.999…, which is 9 more than the original number. To see this, consider that subtracting 0.999… from 9.999… can proceed digit by digit; in each of the digits after the decimal separator, the result is 9 − 9, which is 0. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.999…, be called ''c''. Then 10''c'' − ''c'' = 9. This is the same as 9''c'' = 9. Dividing both sides by 9 completes the proof: ''c'' = 1.<ref name="CME"/>

=== Subtraction proof ===
{|class="infobox" style="padding:.5em; border:1px solid #ccc" align="right" cellpadding="0" cellspacing="0"
|-
|align="right"| 1 - 0.999… || = 0.000…
|-
|align="right"| 0.000… || = 0
|}
A very simple proof is to subtract 0.999… from 1, which yields 0.000… As none of the decimals is ever greater than 7, the result is exactly 0.

== Calculus and analysis ==
Since the question of 0.999… does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. Rigorous proofs are generally not studied before the university level.

One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of any number of digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999…, the integer part can be summarized as ''b''0 and one can neglect negatives, so a decimal expansion has the form
:''b''0.''b''1''b''2''b''3''b''4''b''5….

It is vital that the fraction part, unlike the integer part, is not limited to a finite number of digits. This is a [[positional notation]], so for example the 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5.

===Infinite series and sequences===
Perhaps the most common development of decimal expansions is to define them as sums of [[infinite series]]. In general:
:<math>b_0 . b_1 b_2 b_3 b_4 \ldots = b_0 + b_1({\textstyle\frac{1}{10}}) + b_2({\textstyle\frac{1}{10}})^2 + b_3({\textstyle\frac{1}{10}})^3 + b_4({\textstyle\frac{1}{10}})^4 + \cdots .</math>

For 0.999… one can apply the powerful [[convergent series|convergence]] theorem concerning [[infinite geometric series]]:<ref>Rudin p.61, Theorem 3.26; J. Stewart p.706</ref>
:If <math>|r| < 1</math> then <math>ar+ar^2+ar^3+\cdots = \textstyle\frac{ar}{1-r}.</math>

Since 0.999… is such a sum with a common ratio <math>r=\textstyle\frac{1}{10}</math>, the theorem makes short work of the question:
:<math>0.999\ldots = 9({\textstyle\frac{1}{10}}) + 9({\textstyle\frac{1}{10}})^2 + 9({\textstyle\frac{1}{10}})^3 + \cdots = \frac{9({\textstyle\frac{1}{10}})}{1-{\textstyle\frac{1}{10}}} = 1.\,</math>
This proof (actually, that 10 equals "9·9999999, &c.") appears as early as 1770 in [[Leonard Euler]]'s ''[[Elements of Algebra]]''.<ref>Euler p.170</ref>

[[Image:base4 333.svg|left|thumb|200px|Limits: The unit interval, including the base-4 decimal sequence (.3, .33, .333, …) converging to 1.]]
The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the [[#Algebra proof|algebra proof]] given above, and as late as 1811, Bonnycastle's textbook ''An Introduction to Algebra'' uses such an argument for geometric series to justify the same maneuver on 0.999….<ref>Grattan-Guinness p.69; Bonnycastle p.177</ref> A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.<ref>For example, J. Stewart p.706, Rudin p.61, Protter and Morrey p.213, Pugh p.180, J.B. Conway p.31</ref>

A sequence (''x''0, ''x''1, ''x''2, …) has a [[limit of a sequence|limit]] ''x'' if the distance |''x'' − ''x''''n''| becomes arbitrarily small as ''n'' increases. The statement that 0.999… = 1 can itself be interpreted and proven as a limit:
:<math>0.999\ldots = \lim_{n\to\infty}0.\underbrace{ 99\ldots9 }_{n} = \lim_{n\to\infty}\left(1-\frac{1}{10^n}\right) = 1-\lim_{n\to\infty}\frac{1}{10^n} = 1.</math><ref>The limit follows, for example, from Rudin p. 57, Theorem 3.20e. For a more direct approach, see also Finney, Weir, Giordano (2001) ''Thomas' Calculus: Early Transcendentals'' 10ed, Addison-Wesley, New York. Section 8.1, example 2(a), example 6(b).</ref>

The last step — that lim 1/10''n'' = 0 — is often justified by the axiom that the real numbers have the [[Archimedean property]]. This limit-based attitude towards 0.999… is often put in more evocative but less precise terms. For example, the 1846 textbook ''The University Arithmetic'' explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 ''Arithmetic for Schools'' says, "...when a large number of 9s is taken, the difference between 1 and .99999… becomes inconceivably small".<ref>Davies p.175; Smith and Harrington p.115</ref> Such heuristics are often interpreted by students as implying that 0.999… itself is less than 1; see [[#Skepticism in education|below]].

===Nested intervals and least upper bounds===
[[Image:999 Intervals C.svg|right|thumb|Nested intervals: in base 3, 1 = 1.000… = 0.222…]]
The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) that are to name it.

If a real number ''x'' is known to lie in the [[closed interval]] [0, 10] (i.e., it is greater than or equal to 0 and less than or equal to 10), one can imagine dividing that interval into ten pieces that overlap only at their endpoints: [0, 1], [1, 2], [2, 3], and so on up to [9, 10]. The number ''x'' must belong to one of these; if it belongs to [2, 3] then one records the digit "2" and subdivides that interval into [2, 2.1], [2.1, 2.2], …, [2.8, 2.9], [2.9, 3]. Continuing this process yields an infinite sequence of [[nested intervals]], labeled by an infinite sequence of digits ''b''0, ''b''1, ''b''2, ''b''3, …, and one writes
:''x'' = ''b''0.''b''1''b''2''b''3…

In this formalism, the fact that 1 = 1.000… and also 1 = 0.999… reflects the fact that 1 lies in both [0, 1] and [1, 2], so one can choose either subinterval when finding its digits. To ensure that this notation does not abuse the "=" sign, one needs a way to reconstruct a unique real number for each decimal. This can be done with limits, but other constructions continue with the ordering theme.<ref>Beals p.22; I. Stewart p.34</ref>

One straightforward choice is the [[nested intervals theorem]], which guarantees that given a sequence of nested, closed intervals whose lengths become arbitrarily small, the intervals contain exactly one real number in their [[intersection (set theory)|intersection]]. So ''b''0.''b''1''b''2''b''3… is defined to be the unique number contained within all the intervals [''b''0, ''b''0 + 1], [''b''0.''b''1, ''b''0.''b''1 + 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.<ref>Bartle and Sherbert pp.60-62; Pedrick p.29; Sohrab p.46</ref>

The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of [[least upper bound]]s or ''suprema''. To directly exploit these objects, one may define ''b''0.''b''1''b''2''b''3… to be the least upper bound of the set of approximants {''b''0, ''b''0.''b''1, ''b''0.''b''1''b''2, …}.<ref>Apostol pp.9, 11-12; Beals p.22; Rosenlicht p.27</ref> One can then show that this definition (or the nested intervals definition) is consistent with the subdivision procedure, implying 0.999… = 1 again. Tom Apostol concludes,
:"The fact that a real number might have two different decimal representations is merely a reflection of the fact that two different sets of real numbers can have the same supremum."<ref>Apostol p.12</ref>

== Skepticism in education ==
Students of mathematics often reject the equality of 0.999… and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of [[infinitesimal]]s. There are many common contributing factors to the confusion:
*Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a [[paradox]], which is amplified by the appearance of the seemingly well-understood number 1.<ref>Bunch p.119; Tall and Schwarzenberger p.6. The last suggestion is due to Burrell (p.28): "Perhaps the most reassuring of all numbers is 1. ...So it is particularly unsettling when someone tries to pass off 0.9~ as 1."</ref>
*Some students interpret "0.999…" (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity."<ref>Tall and Schwarzenberger pp.6-7; Tall 2001 p.221</ref>
*Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since the sequence never reaches its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999…" as meaning the sequence rather than its limit.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
*Some students regard 0.999... as having a fixed value which is less than 1 but by an infinitely small amount. (This is a more sophisticated response.)
These ideas are mistaken in the context of the standard real numbers, although many of them are partially borne out in more sophisticated structures, either invented for their general mathematical utility or as instructive [[counterexample]]s to better understand 0.999….

Many of these explanations were found by professor David Tall, who has studied characteristics of teaching and cognition that lead to some of the misunderstandings he has encountered in his college students. Interviewing his students to determine why the vast majority initially rejected the equality, he found that "students continued to conceive of 0.999… as a sequence of numbers getting closer and closer to 1 and not a fixed value, because 'you haven’t specified how many places there are' or 'it is the nearest possible decimal below 1'".<ref>Tall 2001 p.221</ref>

Of the elementary proofs, multiplying 0.333… = 1/3 by 3 is apparently a successful strategy for convincing reluctant students that 0.999… = 1. Still, when confronted with the conflict between their belief of the first equation and their disbelief of the second, some students either begin to disbelieve the first equation or simply become frustrated.<ref>Tall 1976 pp.10-14</ref> Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999…. For example, one real analysis student was able to prove that 0.333… = 1/3 using a [[supremum]] definition, but then insisted that 0.999… < 1 based on her earlier understanding of long division.<ref>Pinto and Tall p.5, Edwards and Ward pp.416-417</ref>

Joseph Mazur tells the tale of an otherwise brilliant calculus student of his who "challenged almost everything I said in class but never questioned his calculator," and who had come to believe that nine digits are all one needs to do mathematics, including calculating the square root of 23. The student remained uncomfortable with a limiting argument that 9.99… = 10, calling it a "wildly imagined infinite growing process."<ref>Mazur pp.137-141</ref>

As part of Ed Dubinsky's "APOS theory" of mathematical learning, Dubinsky and his collaborators (2005) propose that students who conceive of 0.999… as a finite, indeterminate string with an infinitely small distance from 1 have "not yet constructed a complete process conception of the infinite decimal". Other students who have a complete process conception of 0.999… may not yet be able to "encapsulate" that process into an "object conception", like the object conception they have of 1, and so they view the process 0.999… and the object 1 as incompatible. Dubinsky ''et al.'' also link this mental ability of encapsulation to viewing 1/3 as a number in its own right and to dealing with the set of natural numbers as a whole.<ref>Dubinsky ''et al.'' 261-262</ref>

== The real numbers ==
Other approaches explicitly define real numbers to be certain [[construction of real numbers|structures built upon the rational numbers]], using [[axiomatic set theory]]. The [[natural number]]s — 0, 1, 2, 3, and so on — begin with 0 and continue upwards, so that every number has a successor. One can extend the natural numbers with their negatives to give all the [[integer]]s, and to further extend to ratios, giving the [[rational number]]s. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include [[order theory|ordering]], so that one number can be compared to another and found less than, greater than, or equal.

The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999… = 1 that directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.<ref>The historical synthesis is claimed by Griffiths and Hilton (p.xiv) in 1970 and again by Pugh (p.10) in 2001; both actually prefer Dedekind cuts to axioms. For the use of cuts in textbooks, see Pugh p.17 or Rudin p.17. For viewpoints on logic, Pugh p.10, Rudin p.ix, or Munkres p.30</ref> The following two examples come from less usual sources.

=== Dedekind cuts ===
In the [[Dedekind cut]] approach, each real number ''x'' is the infinite set of all rational numbers that are less than ''x''.<ref>Enderton (p.113) qualifies this description: "The idea behind Dedekind cuts is that a real number ''x'' can be named by giving an infinite set of rationals, namely all the rationals less than ''x''. We will in effect define ''x'' to be the set of rationals smaller than ''x''. To avoid circularity in the definition, we must be able to characterize the sets of rationals obtainable in this way…"</ref> In particular, the real number 1 is the set of all rational numbers that are less than 1.<ref>Rudin pp.17-20, Richman p.399, or Enderton p.119. To be precise, Rudin, Richman, and Enderton call this cut 1*, 1−, and 1''R'', respectively; all three identify it with the traditional real number 1. Note that what Rudin and Enderton call a Dedekind cut, Richman calls a "nonprincipal Dedekind cut".</ref> Every positive decimal expansion easily determines a Dedekind cut: the set of rational numbers which are less than some stage of the expansion. So the real number 0.999… is the set of rational numbers ''r'' such that ''r'' < 0, or ''r'' < 0.9, or ''r'' < 0.99, or ''r'' is less than some other number of the form 1 − (1⁄10)''n''.<ref>Richman p.399</ref> Every element of 0.999… is less than 1, so it is an element of the real number 1. Conversely, an element of 1 is a rational number ''a''/''b'' < 1, which implies ''a''/''b'' < 1 − (1⁄10)''b''. Since 0.999… and 1 contain the same rational numbers, they are the same set: 0.999… = 1.

The definition of real numbers as Dedekind cuts was first published by [[Richard Dedekind]] in 1872.<ref name="MacTutor2">{{cite web |url=http://www-gap.dcs.st-and.ac.uk/~history/PrintHT/Real_numbers_2.html |title=History topic: The real numbers: Stevin to Hilbert |author=J J O'Connor and E F Robertson |work=MacTutor History of Mathematics |date=October 2005 |accessdate=2006-08-30}}</ref>
The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 … = 1?" by Fred Richman in ''[[Mathematics Magazine]]'', which is targeted at [[undergraduate]] mathematicians.<ref>{{cite web |url=http://www.maa.org/pubs/mm-guide.html |title=Mathematics Magazine:Guidelines for Authors |publisher=[[The Mathematical Association of America]] |accessdate=2006-08-23}}</ref> Richman notes that taking Dedekind cuts in any [[dense subset]] of the rational numbers yields the same results; in particular, he uses [[decimal fraction]]s, for which the proof is more immediate: "So we see that in the traditional definition of the real numbers, the equation 0.9* = 1 is built in at the beginning."<ref>Richman pp.398-399</ref> A further modification of the procedure leads to a different structure that Richman is more interested in describing; see "[[#Other number systems|Other number systems]]" below.

=== Cauchy sequences ===
Another approach to constructing the real numbers uses the ordering of rationals less directly. First, the distance between ''x'' and ''y'' is defined as the absolute value |''x'' − ''y''|, where the absolute value |''z''| is defined as the maximum of ''z'' and −''z'', thus never negative. Then the reals are defined to be the sequences of rationals that are [[Cauchy sequence|Cauchy]] using this distance. That is, in the sequence (''x''0, ''x''1, ''x''2, …), a mapping from natural numbers to rationals, for any positive rational δ there is an ''N'' such that |''x''''m'' − ''x''''n''| ≤ δ for all ''m'', ''n'' > ''N''. (The distance between terms becomes arbitrarily small.)<ref>Griffiths & Hilton §24.2 "Sequences" p.386</ref>

If (''x''''n'') and (''y''''n'') are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence (''x''''n'' − ''y''''n'') has the limit 0. Truncations of the decimal number ''b''0.''b''1''b''2''b''3… generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number.<ref>Griffiths & Hilton pp.388, 393</ref> Thus in this formalism the task is to show that the sequence of rational numbers

:<math>\left(1 - 0, 1 - {9 \over 10}, 1 - {99 \over 100}, \dots\right)
= \left(1, {1 \over 10}, {1 \over 100}, \dots \right)</math>

has the limit 0. Considering the ''n''th term of the sequence, for ''n''=0,1,2,…, it must therefore be shown that

:<math>\lim_{n\rightarrow\infty}\frac{1}{10^n} = 0.</math>

This limit is plain;<ref>Griffiths & Hilton pp.395</ref> one possible proof is that for ε = ''a''/''b'' > 0 one can take ''N'' = ''b'' in the definition of the [[limit of a sequence]]. So again 0.9999… = 1.

The definition of real numbers as Cauchy sequences was first published separately by [[Eduard Heine]] and [[Georg Cantor]], also in 1872.<ref name="MacTutor2" /> The above approach to decimal expansions, including the proof that 0.999… = 1, closely follows Griffiths & Hilton's 1970 work ''A comprehensive textbook of classical mathematics: A contemporary interpretation''. The book is written specifically to offer a second look at familiar concepts in a contemporary light.<ref>Griffiths & Hilton pp.viii, 395</ref>

==Other number systems==
Although the real numbers form an extremely useful number system, the decision to interpret the phrase "0.999…" as naming a real number is ultimately a convention, and Timothy Gowers argues in ''Mathematics: A Very Short Introduction'' that the resulting identity 0.999… = 1 is a convention as well:
:"However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic."<ref>Gowers p.60</ref>

One can place constraints on hypothetical number systems where 0.999… ≠ 1, with their new objects or unfamiliar rules, by reinterpreting the above proofs. As Richman puts it, "one man's proof is another man's ''[[reductio ad absurdum]]''."<ref>Richman p.396; emphasis is his. This line appears in a paragraph of the published version that is not present in the earlier preprint.</ref> If 0.999… is to be different from 1, then at least one of the assumptions built into the proofs must break down.

===Infinitesimals===
Some proofs that 0.999… = 1 rely on the [[Archimedean property]] of the standard real numbers: there are no nonzero [[infinitesimal]]s. There are mathematically coherent ordered [[algebraic structure]]s, including various alternatives to standard reals, which are non-Archimedean. For example, the [[dual number]]s include a new infinitesimal element ε, analogous to the imaginary unit ''i'' in the [[complex number]]s except that ε2 = 0. The resulting structure is useful in [[automatic differentiation]]. The dual numbers can be given a [[lexicographic order]], in which case the multiples of ε become non-Archimedean elements.<ref>Berz 439-442</ref> Another way to construct alternatives to standard reals is to use [[topos]] theory and alternative logics rather than [[set theory]] and classical logic (which is a special case). For example, [[smooth infinitesimal analysis]] has infinitesimals with no [[Multiplicative inverse|reciprocal]]s.<ref>{{cite paper|url=http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf|title=An Invitation to Smooth Infinitesimal Analysis|author=John L. Bell |year=2003 |format=PDF |accessdate=2006-06-29}}</ref>

[[Non-standard analysis]] is well-known for including a number system with a full array of infinitesimals (and their inverses) which provide a different, and perhaps more intuitive, approach to [[calculus]].<ref>For a full treatment of non-standard numbers see for example Robinson's ''Non-standard Analysis''.</ref> A.H. Lightstone provided a development of non-standard decimal expansions in 1972 in which every extended real number in (0, 1) has a unique extended decimal expansion: a sequence of digits 0.ddd…;…ddd… indexed by the extended natural numbers. In his formalism, there are two natural extensions of 0.333…, neither of which falls short of 1/3 by an infinitesimal:
:0.333…;…000… does not exist, while
:0.333…;…333… = 1/3 exactly.<ref>Lightstone pp.245-247. He does not explore the possibility repeating 9s in the standard part of an expansion.</ref>

[[Combinatorial game theory]] provides alternative reals as well, with infinite Blue-Red [[Hackenbush]] as one particularly relevant example. In 1974, [[Elwyn Berlekamp]] described a correspondence between Hackenbush strings and binary expansions of real numbers, motivated by the idea of [[data compression]]. For example, the value of the Hackenbush string LRRLRLRL… is 0.010101… = 1/3. However, the value of LRLLL… (corresponding to 0.111…) is infinitesimally less than 1. The difference between the two is the [[surreal number]] 1/ω, where ω is the first [[ordinal number|infinite ordinal]]; the relevant game is LRRRR… or 0.000….<ref>Berlekamp, Conway, and Guy (pp.79-80, 307-311) discuss 1 and 1/3 and touch on 1/ω. The game for 0.111… follows directly from Berlekamp's Rule, and it is discussed by {{cite web |url=http://www.maths.nott.ac.uk/personal/anw/Research/Hack/ |title=Hackenstrings and the 0.999… ≟ 1 FAQ |author=A. N. Walker |year=1999 |accessdate=2006-06-29}}</ref>

===Breaking subtraction===
Another manner in which the proofs might be undermined is if 1 − 0.999… simply does not exist, because subtraction is not always possible. Mathematical structures with an addition operation but not a subtraction operation include [[commutative]] [[semigroup]]s, [[commutative monoid]]s and [[semiring]]s. Richman considers two such systems, designed so that 0.999… < 1.

First, Richman defines a nonnegative ''decimal number'' to be a literal decimal expansion. He defines the [[lexicographical order]] and an addition operation, noting that 0.999… < 1 simply because 0 < 1 in the ones place, but for any nonterminating ''x'', one has 0.999… + ''x'' = 1 + ''x''. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to 1⁄3. After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.<ref>Richman pp.397-399</ref>

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

===''p''-adic numbers===
When asked what 1 − 0.999… might be, students often invent the number "0.000…1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the last 9 in 0.999… would carry all the 9s into 0s and leave a 1 in the ones place. Among other reasons, this idea fails because there is no "last 9" in 0.999….<ref>Gardiner p.98; Gowers p.60</ref> For an infinite string of 9s including a last 9, one must look elsewhere.

[[Image:4adic 333.svg|right|thumb|200px|The 4-adic integers (black points), including the sequence (3, 33, 333, …) converging to −1. The 10-adic analogue is …999 = −1.]]
The [[p-adic number|''p''-adic number]]s are an alternate number system of interest in [[number theory]]. Like the real numbers, the ''p''-adic numbers can be built from the rational numbers via [[Cauchy sequence]]s; the construction uses a different metric in which 0 is closer to ''p'', and much closer to ''pn'', than it is to 1 . The ''p''-adic numbers form a field for prime ''p'' and a [[ring (mathematics)|ring]] for other ''p'', including 10. So arithmetic can be performed in the ''p''-adics, and there are no infinitesimals.

In the 10-adic numbers, the analogues of decimal expansions run to the left. The 10-adic expansion …999 does have a last 9, and it does not have a first 9. One can add 1 to the ones place, and it leaves behind only 0s after carrying through: 1 + …999 = …000 = 0, and so …999 = −1.<ref name="Fjelstad11">Fjelstad p.11</ref> Another derivation uses a geometric series. The infinite series implied by "…999" does not converge in the real numbers, but it converges in the 10-adics, and so one can re-use the familiar formula:
:<math>\ldots999 = 9 + 9(10) + 9(10)^2 + 9(10)^3 + \ldots = \frac{9}{1-10} = -1.</math><ref>Fjelstad pp.14-15</ref>
(Compare with the series [[#Infinite series and sequences|above]].) A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that 0.999… = 1 but was inspired to take the multiply-by-10 proof [[#Algebra proof|above]] in the opposite direction: if ''x'' = …999 then 10''x'' = ''x'' − 9, hence ''x'' = −1 again.<ref name="Fjelstad11" />

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"<ref>DeSua p.901</ref> one may add the two equations and arrive at …999.999… = 0. This equation does not make sense either as a 10-adic expansion or an ordinary decimal expansion, but it turns out to be meaningful and true if one develops a theory of "double-decimals" with eventually-repeating left ends to represent a familiar system: the real numbers.<ref>DeSua pp.902-903</ref>

==Generalizations==
Proofs that 0.999… = 1 immediately generalize in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a [[doppelgänger]] with trailing 9s. For example, 0.24999… equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.<ref>Petkovšek p.408</ref>

Second, a comparable theorem applies in each radix or [[base (mathematics)|base]]. For example, in base 2 (the [[binary numeral system]]) 0.111… equals 1, and in base 3 (the [[ternary numeral system]]) 0.222… equals 1. Textbooks of real analysis are likely to skip the example of 0.999… and present one or both of these generalizations from the start.<ref>Protter and Morrey p.503; Bartle and Sherbert p.61</ref>

Alternate representations of 1 also occur in non-integer bases. For example, in the [[golden ratio base]], the two standard representations are 1.000… and 0.101010…, and there are infinitely many more representations that include adjacent 1s. Generally, for [[almost all]] ''q'' between 1 and 2, there are uncountably many base-''q'' expansions of 1. On the other hand, there are still uncountably many ''q'' (including 2 and 10) for which there is only one base-''q'' expansion of 1, other than the trivial 1.000…. This result was first obtained by [[Paul Erdős]], Miklos Horváth, and István Joó around 1990. In 1998 Vilmos Komornik and Paola Loreti determined the smallest such base, ''q'' = 1.787231650…. In this base, 1 = 0.11010011001011010010110011010011…; the digits are given by the [[Thue-Morse sequence]], which does not repeat.<ref>Komornik and Loreti p.636</ref>

A more far-reaching generalization addresses [[non-standard positional numeral systems|the most general positional numeral systems]]. They too have multiple representations, and in some sense the difficulties are even worse. For example:<ref>Kempner p.611; Petkovšek p.409</ref>
*In the [[balanced ternary]] system, 1/2 = 0.111… = 1.111….
*In the [[factoradic]] system, 1 = 1.000… = 0.1234….
Marko Petkovšek has proved that such ambiguities are necessary consequences of using a positional system: for any system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary [[point-set topology]]"; it involves viewing sets of positional values as [[Stone space]]s and noticing that their real representations are given by [[continuous function (topology)|continuous functions]].<ref>Petkovšek pp.410-411</ref>

==Applications==
One application of 0.999… as a representation of 1 occurs in [[elementary number theory]]. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain [[prime number]]s. Examples include:
*1/7 = 0.142857142857… and 142 + 857 = 999.
*1/73 = 0.0136986301369863… and 0136 + 9863 = 9999.
E. Midy proved a general result about such fractions, now called ''[[Midy's Theorem]]'', in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999…, but at least one modern proof by W. G. Leavitt does. If one can prove that a decimal of the form 0.''b''1''b''2''b''3… is a positive integer, then it must be 0.999…, which is then the source of the 9s in the theorem.<ref>Leavitt 1984 p.301</ref> Investigations in this direction can motivate such concepts as [[greatest common divisor]]s, [[modular arithmetic]], [[Fermat prime]]s, [[order (group theory)|order]] of [[group (mathematics)|group]] elements, and [[quadratic reciprocity]].<ref>Lewittes pp.1-3; Leavitt 1967 pp.669,673; Shrader-Frechette pp.96-98</ref>

[[Image:Cantor base 3.svg|right|thumb|Positions of 1/4, 2/3, and 1 in the Cantor set]]
Returning to real analysis, the base-3 analogue 0.222… = 1 plays a key role in a characterization of one of the simplest [[fractal]]s, the middle-thirds [[Cantor set]]:
*A point in the [[unit interval]] lies in the Cantor set if and only if it can be represented in ternary using only the digits 0 and 2.

The ''n''th digit of the representation reflects the position of the point in the ''n''th stage of the construction. For example, the point 2⁄3 is given the usual representation of 0.2 or 0.2000…, since it lies to the right of the first deletion and to the left of every deletion thereafter. The point 1⁄3 is represented not as 0.1 but as 0.0222…, since it lies to the left of the first deletion and to the right of every deletion thereafter.<ref>Pugh p.97; Alligood, Sauer, and Yorke pp.150-152. Protter and Morrey (p.507) and Pedrick (p.29) assign this description as an exercise.</ref>

Repeating nines also turn up in yet another of Georg Cantor's works. They must be taken into account to construct a valid proof, applying [[Cantor's diagonal argument|his 1891 diagonal argument]] to decimal expansions, of the [[uncountability]] of the unit interval. Such a proof needs to be able to declare certain pairs of real numbers to be different based on their decimal expansions, so one needs to avoid pairs like 0.2 and 0.1999… . A simple method represents all numbers with nonterminating expansions; the opposite method rules out repeating nines.<ref>Maor (p.60) and Mankiewicz (p.151) review the former method; Mankiewicz attributes it to Cantor, but the primary source is unclear. Munkres (p.50) mentions the latter method.</ref> A variant that may be closer to Cantor's original argument actually uses base 2, and by turning base-3 expansions into base-2 expansions, one can prove the uncountability of the Cantor set as well.<ref>Rudin p.50, Pugh p.98</ref>

== In popular culture ==

With the rise of the [[Internet]], debates about 0.999… have escaped the classroom and are commonplace on [[newsgroup]]s and [[message board]]s, including many that nominally have little to do with mathematics. In the newsgroup <tt>[[sci.math]]</tt>, arguing over 0.999… is a "popular sport", and it is one of the questions answered in its [[FAQ]].<ref>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? |accessdate=2006-06-29}}</ref> The FAQ briefly covers 1/3, multiplication by 10, and limits, and it alludes to Cauchy sequences as well.

A 2003 edition of the general-interest [[newspaper column]] ''[[The Straight Dope]]'' discusses 0.999… via 1/3 and limits, saying of misconceptions,
:"The lower primate in us still resists, saying: .999~ doesn't really represent a ''number'', then, but a ''process''. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.

:Nonsense."<ref>{{cite web |url=http://www.straightdope.com/columns/030711.html |title=An infinite question: Why doesn't .999~ = 1? |date=2003-07-11 |author=[[Cecil Adams]] |work=[[The Straight Dope]] |publisher=[[The Chicago Reader]] |accessdate=2006-09-06}}</ref>

''The Straight Dope'' cites a discussion on its own message board that grew out of an unidentified "other message board ... mostly about video games". In the same vein, the question of 0.999… proved such a popular topic in the first seven years of [[Blizzard Entertainment]]'s [[Battle.net]] forums that the company's president, [[Mike Morhaime]], announced at a [[press conference]] on [[April 1]], [[2004]] that it is 1:
:"We are very excited to close the book on this subject once and for all. We've witnessed the heartache and concern over whether .999~ does or does not equal 1, and we're proud that the following proof finally and conclusively addresses the issue for our customers."<ref>{{cite web |url=http://www.blizzard.com/press/040401.shtml |title=Blizzard Entertainment® Announces .999~ (Repeating) = 1 |work=Press Release |publisher=Blizzard Entertainment |date=2004-04-01 |accessdate=2006-09-03}}</ref>
Blizzard's subsequent press release offers two proofs, based on limits and multiplication by 10.

== Related questions ==


*[[Zeno's paradoxes]], particularly the paradox of the runner, are reminiscent of the apparent paradox that 0.999… and 1 are equal. The runner paradox can be mathematically modelled and then, like 0.999…, resolved using a geometric series. However, it is not clear if this mathematical treatment addresses the underlying metaphysical issues Zeno was exploring.<ref>Wallace p.51, Maor p.17</ref>
*[[Division by zero]] occurs in some popular discussions of 0.999…, and it also stirs up contention. While most authors choose to define 0.999…, almost all modern treatments leave division by zero undefined, as it can be given no meaning in the standard real numbers. In other systems, such as the [[Riemann sphere]], it makes sense to define 1/0 to be infinity.<ref>See, for example, J.B. Conway's treatment of Möbius transformations, pp.47-57</ref> In fact, some prominent mathematicians argued for such a definition long before either number system was developed.<ref>Maor p.54</ref>
*[[Negative zero]] is another redundant feature of many ways of writing numbers. In number systems, such as the real numbers, where "0" denotes the additive identity and is neither positive nor negative, the usual interpretation of "−0" is that it should denote the additive inverse of 0, which forces −0 = 0.<ref>Munkres p.34, Exercise 1(c)</ref> Nonetheless, some scientific applications use separate positive and negative zeroes, as do some of the most common computer number systems (for example the [[IEEE floating-point standard]]).<ref>{{cite book |author=Kroemer, Herbert; Kittel, Charles |title=Thermal Physics |edition=2e |publisher=W. H. Freeman |year=1980 |id=ISBN 0-7167-1088-9 |pages=462}}</ref><ref>{{cite web |url=http://msdn.microsoft.com/library/en-us/csspec/html/vclrfcsharpspec_4_1_6.asp |title=Floating point types |work=[[Microsoft Developer Network|MSDN]] C# Language Specification |accessdate=2006-08-29}}</ref> In the case of IEEE floating-point numbers, negative zero represents a value that is too small to represent in the given precision but is, nonetheless, negative. Thus, "negative zero" in IEEE floating-point numbers is not a bona-fide negative zero.

==Notes==
<div class="references-small" style="-moz-column-count:2; column-count:2;">
<references />
</div>

==References==
<div class="references-small">
*{{cite book |author=Alligood, Sauer, and Yorke |year=1996 |title=Chaos: An introduction to dynamical systems |chapter=4.1 Cantor Sets |publisher=Springer |id=ISBN 0-387-94677-2}}
*:This introductory textbook on dynamics is aimed at undergraduate and beginning graduate students. (p.ix)
*{{cite book |last=Apostol |first=Tom M. |year=1974 |title=Mathematical analysis |edition=2e |publisher=Addison-Wesley |id=ISBN 0-201-00288-4}}
*: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)
*{{cite book |author=Bartle, R.G. and D.R. Sherbert |year=1982 |title=Introduction to real analysis |publisher=Wiley |id=ISBN 0-471-05944-7}}
*:This text aims to be "an accessible, reasonably paced textbook that deals with the fundamental concepts and techniques of real analysis." Its development of the real numbers relies on the supremum axiom. (pp.vii-viii)
*{{cite book |last=Beals |first=Richard |title=Analysis |year=2004 |publisher=Cambridge UP |id=ISBN 0-521-60047-2}}
*{{cite book |author=[[Elwyn Berlekamp|Berlekamp, E.R.]]; [[John Horton Conway|J.H. Conway]]; and [[Richard K. Guy|R.K. Guy]] |year=1982 |title=[[Winning Ways for your Mathematical Plays]] |publisher=Academic Press |id=ISBN 0-12-091101-9}}
*{{cite conference |last=Berz |first=Martin |title=Automatic differentiation as nonarchimedean analysis |year=1992 |booktitle=Computer Arithmetic and Enclosure Methods |publisher=Elsevier |pages=439-450 |url=http://citeseer.ist.psu.edu/berz92automatic.html}}
*{{cite book |last=Bunch |first=Bryan H. |title=Mathematical fallacies and paradoxes |year=1982 |publisher=Van Nostrand Reinhold |id=ISBN 0-442-24905-5}}
*:This book presents an analysis of paradoxes and fallacies as a tool for exploring its central topic, "the rather tenuous relationship between mathematical reality and physical reality". It assumes first-year high-school algebra; further mathematics is developed in the book, including geometric series in Chapter 2. Although 0.999... is not one of the paradoxes to be fully treated, it is briefly mentioned during a development of Cantor's diagonal method. (pp.ix-xi, 119)
*{{cite book |last=Burrell |first=Brian |title=Merriam-Webster's Guide to Everyday Math: A Home and Business Reference |year=1998 |publisher=Merriam-Webster |id=ISBN 0877796211}}
*{{cite book |last=Conway |first=John B. |authorlink=John B. Conway |title=Functions of one complex variable I |edition=2e |publisher=Springer-Verlag |origyear=1973 |year=1978 |id=ISBN 0-387-90328-3}}
*:This text assumes "a stiff course in basic calculus" as a prerequisite; its stated principles are to present complex analysis as "An Introduction to Mathematics" and to state the material clearly and precisely. (p.vii)
*{{cite book |last=Davies |first=Charles |year=1846 |title=The University Arithmetic: Embracing the Science of Numbers, and Their Numerous Applications |publisher=A.S. Barnes |url=http://books.google.com/books?vid=LCCN02026287&pg=PA175}}
*{{cite journal |last=DeSua |first=Frank C. |title=A system isomorphic to the reals |format=restricted access |journal=The American Mathematical Monthly |volume=67 |number=9 |month=November |year=1960 |pages=900-903 |url=http://links.jstor.org/sici?sici=0002-9890%28196011%2967%3A9%3C900%3AASITTR%3E2.0.CO%3B2-F}}
*{{cite journal |author=Dubinsky, Ed, Kirk Weller, Michael McDonald, and Anne Brown |title=Some historical issues and paradoxes regarding the concept of infinity: an APOS analysis: part 2 |journal=Educational Studies in Mathematics |year=2005 |volume=60 |pages=253-266 |id={{doi|10.1007/s10649-005-0473-0}}}}
*{{cite journal |author=Edwards, Barbara and Michael Ward |year=2004 |month=May |title=Surprises from mathematics education research: Student (mis)use of mathematical definitions |journal=The American Mathematical Monthly |volume=111 |number=5 |pages=411-425}}
*{{cite book |last=Enderton |first=Herbert B. |year=1977 |title=Elements of set theory |publisher=Elsevier |id=ISBN 0-12-238440-7}}
*:An introductory undergraduate textbook in set theory that "presupposes no specific background". It is written to accommodate a course focusing on axiomatic set theory or on the construction of number systems; the axiomatic material is marked such that it may be de-emphasized. (pp.xi-xii)
*{{cite book |last=Euler |first=Leonard |authorlink=Leonard Euler |origyear=1770 |year=1822 |edition=3rd English edition |title=Elements of Algebra |editor=John Hewlett and Francis Horner, English translators. |publisher=Orme Longman |url=http://books.google.com/books?id=X8yv0sj4_1YC&pg=PA170}}
*{{cite journal |last=Fjelstad |first=Paul |title=The repeating integer paradox |format=restricted access |journal=The College Mathematics Journal |volume=26 |number=1 |month=January |year=1995 |pages=11-15 |url=http://links.jstor.org/sici?sici=0746-8342%28199501%2926%3A1%3C11%3ATRIP%3E2.0.CO%3B2-X |id={{doi|10.2307/2687285}}}}
*{{cite book |last=Gardiner |first=Anthony |title=Understanding Infinity: The Mathematics of Infinite Processes |origyear=1982 |year=2003 |publisher=Dover |id=ISBN 0-486-42538-X}}
*{{cite book |last=Gowers |first=Timothy |title=Mathematics: A Very Short Introduction |year=2002 |publisher=Oxford UP |id=ISBN 0-19-285361-9}}
*{{cite book |last=Grattan-Guinness |first=Ivor |year=1970 |title=The development of the foundations of mathematical analysis from Euler to Riemann |publisher=MIT Press |id=ISBN 0-262-07034-0}}
*{{cite book | last=Griffiths | first=H.B. | coauthors=P.J. Hilton | title=A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation | year=1970 | publisher=Van Nostrand Reinhold | location=London | id=ISBN 0-442-02863-6. {{LCC|QA37.2|G75}}}}
*:This book grew out of a course for [[Birmingham]]-area [[grammar school]] mathematics teachers. The course was intended to convey a university-level perspective on [[mathematics education|school mathematics]], 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 [[ideal theory]], which is not reproduced here. (pp.vii, xiv)
*{{cite journal |last=Kempner |first=A.J. |title=Anormal Systems of Numeration |format=restricted access |journal=The American Mathematical Monthly |volume=43 |number=10 |month=December |year=1936 |pages=610-617 |url=http://links.jstor.org/sici?sici=0002-9890%28193612%2943%3A10%3C610%3AASON%3E2.0.CO%3B2-0}}
*{{cite journal |author=Komornik, Vilmos; and Paola Loreti |title=Unique Developments in Non-Integer Bases |format=restricted access |journal=The American Mathematical Monthly |volume=105 |number=7 |year=1998 |pages=636-639 |url=http://links.jstor.org/sici?sici=0002-9890%28199808%2F09%29105%3A7%3C636%3AUDINB%3E2.0.CO%3B2-G}}
*{{cite journal |last=Leavitt |first=W.G. |title=A Theorem on Repeating Decimals |format=restricted access |journal=The American Mathematical Monthly |volume=74 |number=6 |year=1967 |pages=669-673 |url=http://links.jstor.org/sici?sici=0002-9890%28196706%2F07%2974%3A6%3C669%3AATORD%3E2.0.CO%3B2-0}}
*{{cite journal |last=Leavitt |first=W.G. |title=Repeating Decimals |format=restricted access |journal=The College Mathematics Journal |volume=15 |number=4 |month=September |year=1984 |pages=299-308 |url=http://links.jstor.org/sici?sici=0746-8342%28198409%2915%3A4%3C299%3ARD%3E2.0.CO%3B2-D}}
*{{cite web | url=http://arxiv.org/abs/math.NT/0605182 |title=Midy's Theorem for Periodic Decimals |last=Lewittes |first=Joseph |work=New York Number Theory Workshop on Combinatorial and Additive Number Theory |year=2006 |publisher=[[arXiv]]}}
*{{cite journal |last=Lightstone |first=A.H. |title=Infinitesimals |format=restricted access |journal=The American Mathematical Monthly |year=1972 |volume=79 |number=3 |month=March |pages=242-251 |url=http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C242%3AI%3E2.0.CO%3B2-F}}
*{{cite book |last=Mankiewicz |first=Richard |year=2000 |title=The story of mathematics|publisher=Cassell |id=ISBN 0-304-35473-2}}
*: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)
*{{cite book |last=Maor |first=Eli |title=To infinity and beyond: a cultural history of the infinite |year=1987 |publisher=Birkhäuser |id=ISBN 3-7643-3325-1}}
*:A topical rather than chronological review of infinity, this book is "intended for the general reader" but "told from the point of view of a mathematician". On the dilemma of rigor versus readable language, Maor comments, "I hope I have succeeded in properly addressing this problem." (pp.x-xiii)
*{{cite book |last=Mazur |first=Joseph |title=Euclid in the Rainforest: Discovering Universal Truths in Logic and Math |year=2005 |publisher=Pearson: Pi Press |id=ISBN 0-13-147994-6}}
*{{cite book |last=Munkres |first=James R. |title=Topology |year=2000 |origyear=1975 |edition=2e |publisher=Prentice-Hall |id=ISBN 0-13-181629-2}}
*:Intended as an introduction "at the senior or first-year graduate level" with no formal prerequisites: "I do not even assume the reader knows much set theory." (p.xi) Munkres' treatment of the reals is axiomatic; he claims of bare-hands constructions, "This way of approaching the subject takes a good deal of time and effort and is of greater logical than mathematical interest." (p.30)
*{{cite book |last=Pedrick |first=George |title=A First Course in Analysis |year=1994 |publisher=Springer |id=ISBN 0-387-94108-8}}
*{{cite journal |last=Petkovšek |first=Marko |title=Ambiguous Numbers are Dense |format=restricted access |journal=[[The American Mathematical Monthly|American Mathematical Monthly]] |volume=97 |number=5 |month=May |year=1990 |pages=408-411 |url=http://links.jstor.org/sici?sici=0002-9890%28199005%2997%3A5%3C408%3AANAD%3E2.0.CO%3B2-Q}}
*{{cite conference |author=Pinto, Márcia and David Tall |title=Following students' development in a traditional university analysis course |booktitle=PME25 |pages=v4: 57-64 |year=2001 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001j-pme25-pinto-tall.pdf}}
*{{cite book |author=Protter, M.H. and C.B. Morrey |year=1991 |edition=2e |title=A first course in real analysis |publisher=Springer |id=ISBN 0-387-97437-7}}
*: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, nodecreasing 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)
*{{cite book |last=Pugh |first=Charles Chapman |title=Real mathematical analysis |year=2001 |publisher=Springer-Verlag |id=ISBN 0-387-95297-7}}
*:While assuming familiarity with the rational numbers, Pugh introduces Dedekind cuts 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.
*{{cite journal |first=Fred |last=Richman |year=1999 |month=December |title=Is 0.999… = 1? |format=restricted access |journal=[[Mathematics Magazine]] |volume=72 |issue=5 |pages=396-400 |url=http://links.jstor.org/sici?sici=0025-570X%28199912%2972%3A5%3C396%3AI0.%3D1%3E2.0.CO%3B2-F}} Free HTML preprint: {{cite web |url=http://www.math.fau.edu/Richman/HTML/999.htm |first=Fred|last=Richman|title=Is 0.999… = 1? |date=1999-06-08 |accessdate=2006-08-23}} Note: the journal article contains material and wording not found in the preprint.
*{{cite book |last=Robinson |first=Abraham |authorlink=Abraham Robinson |title=Non-standard analysis |year=1996 |edition=Revised edition |publisher=Princeton University Press|id=ISBN 0-691-04490-2}}
*{{cite book |last=Rosenlicht |first=Maxwell |year=1985 |title=Introduction to Analysis |publisher=Dover |id=ISBN 0-486-65038-3}}
*{{cite book |last=Rudin |first=Walter |authorlink=Walter Rudin |title=Principles of mathematical analysis |edition=3e |year=1976 |origyear=1953 |publisher=McGraw-Hill |id=ISBN 0-07-054235-X}}
*:A textbook for an advanced undergraduate course. "Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones. At the beginning, most students simply fail to appreciate the need for doing this. Accordingly, the real number system is introduced as an ordered field with the least-upper-bound property, and a few interesting applications of this property are quickly made. However, Dedekind's construction is not omitted. It is now in an Appendix to Chapter 1, where it may be studied and enjoyed whenever the time is ripe." (p.ix)
*{{cite journal |last=Shrader-Frechette |first=Maurice |title=Complementary Rational Numbers |format=restricted access |journal=Mathematics Magazine |volume=51 |number=2 |month=March |year=1978 |pages=90-98 |url=http://links.jstor.org/sici?sici=0025-570X%28197803%2951%3A2%3C90%3ACRN%3E2.0.CO%3B2-O}}
*{{cite book |author=Smith, Charles and Charles Harrington |year=1895 |title=Arithmetic for Schools |publisher=Macmillan |url=http://books.google.com/books?vid=LCCN02029670&pg=PA115}}
*{{cite book |last=Sohrab |first=Houshang |title=Basic Real Analysis |year=2003 |publisher=Birkhäuser |id=ISBN 0-8176-4211-0}}
*{{cite book |last=Stewart |first=Ian |title=The Foundations of Mathematics |year=1977 |publisher=Oxford UP |id=ISBN 0-19-853165-6}}
*{{cite book |last=Stewart |first=James |title=Calculus: Early transcendentals |edition=4e |year=1999 |publisher=Brooks/Cole |id=ISBN 0-534-36298-2}}
*:This book aims to "assist students in discovering calculus" and "to foster conceptual understanding". (p.v) It omits proofs of the foundations of calculus.
*{{cite journal |author=D.O. Tall and R.L.E. Schwarzenberger |title=Conflicts in the Learning of Real Numbers and Limits |journal=Mathematics Teaching |year=1978 |volume=82 |pages=44-49 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1978c-with-rolph.pdf}}
*{{cite journal |last=Tall |first=David |authorlink=David Tall |title=Conflicts and Catastrophes in the Learning of Mathematics |journal=Mathematical Education for Teaching |year=1976/7 |volume=2 |number=4 |pages=2-18 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1976a-confl-catastrophy.pdf}}
*{{cite journal |last=Tall |first=David |title=Cognitive Development In Advanced Mathematics Using Technology |journal=Mathematics Education Research Journal |year=2000 |volume=12 |number=3 |pages=210-230 |url=http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot2001b-merj-amt.pdf}}
*{{cite book|last=von Mangoldt|first=Dr. Hans|authorlink =Hans Carl Friedrich von Mangoldt| title=Einführung in die höhere Mathematik|edition=1st ed.|year=1911|publisher=Verlag von S. Hirzel| location=Leipzig|language=German|chapter=Reihenzahlen}}
*{{cite book |last=Wallace |first=David Foster|authorlink =David Foster Wallace |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{Spoken Wikipedia|0.999....ogg|2006-10-19}}
{{commons|0.999...}}
*[http://www.cut-the-knot.org/arithmetic/999999.shtml .999999... = 1?] from [[cut-the-knot]]
*[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]
*[http://descmath.com/diag/nines.html Repeating Nines]
*[http://qntm.org/pointnine Point nine recurring equals one]
*[http://www.warwick.ac.uk/staff/David.Tall/themes/limits-infinity.html David Tall's research on mathematics cognition]

{{featured article}}

[[Category:Mathematics paradoxes]]
[[Category:Real analysis]]
[[Category:Real numbers]]
[[Category:Numeration]]
[[Category:Proofs]]

[[es:0,9 periódico]]
[[fr:Développement décimal de l'unité]]
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[[zh:证明0.999...等于1]]

Flight 93 National Memorial

2006-08-03T18:30:13Z

Loadmaster: /* Controversy */ link "Ed Felt"

{{Infobox_protected_area | name = Flight 93 National Memorial
| iucn_category = V
| image = US_Locator_Blank.svg
| caption =
| locator_x = 232
| locator_y = 69
| location = [[Somerset County, Pennsylvania]], [[United States|USA]]
| nearest_city = [[Somerset, Pennsylvania]]
| lat_degrees = 40
| lat_minutes = 3
| lat_seconds = 3
| lat_direction = N
| long_degrees = 78
| long_minutes = 54
| long_seconds = 13
| long_direction = W
| area = 2200 acres (1000 federal) 8.90 km²
| established = [[September 24]], [[2002]]
| visitation_num = 125,000
| visitation_year = 2005
| governing_body = [[National Park Service]]
}}
'''Flight 93 National Memorial''' protects the site of the crash of [[Aircraft hijacking|hijacked]] [[United Airlines Flight 93]] on [[September 11]], [[2001]] in [[Stonycreek Township, Somerset County, Pennsylvania|Stonycreek Township, Pennsylvania]], approximately 2 miles north of [[Shanksville, Pennsylvania]]. A temporary memorial to the 40 victims of the hijacking was established soon after the crash, with a permanent memorial slated to be constructed and completed by 2011. The current design for the memorial is a modified version of the entry ''Crescent of Embrace'' by Paul and Milena Murdoch.

The crash site is located west of Skyline Road, about 2 1/2 miles south of [[U.S. Route 30]] ([[Lincoln Highway]]), and approximately 2 miles north of Shanksville. The temporary memorial is located on a hill top, 500 yards from the crash site. [[Indian Lake, Pennsylvania|Indian Lake]] is located 1 1/2 miles to the east of the crash site.

==Temporary memorial==
The temporary memorial is within walking distance of the crash site, which is only accessible by the families of the passengers. It consists of a 40 foot (to commemorate the 40 passengers) long chain-link fence on which visitors can leave flags, hats, rosaries, and other items. Next to the fence are several memorials such as marble statues, flags, and a huge cross. There is also a guardrail on which visitors may leave messages.

There is a small building with a guestbook. The building is staffed by Park Service volunteers, called ambassadors, who answer questions. One of the ambassadors is Nevin Lambert, who is one of the two people to see the plane actually crash. He lives about 75 yards from the crash site.

==Permanent memorial==
Of the four aircraft hijacked on September 11, Flight 93 is notable in that it did not reach its intended target, presumed to be in [[Washington, D.C.]], perhaps either the [[United States Capitol]] or the [[White House]]. The passengers had learned about the attacks on the [[World Trade Center]] through cellular telephone calls to family. It is believed that at least [[flight attendant]]s [[Cee Cee Lyles]] and [[Sandra Bradshaw]] and passengers [[Todd Beamer]], [[Mark Bingham]], [[Tom Burnett]], [[Andrew Garcia]], [[Jeremy Glick (September 11 attack victim)|Jeremy Glick]], and [[Richard Guadagno]] (and perhaps others) fought back against the hijackers. The plane crashed into the [[Pennsylvania]] field shortly after 10:00 a.m., killing all on board, but no one on the ground.

[[Image:UAL Flight 93 ceremony.jpg|thumb|left|300px|Wreath-laying ceremony near the site of the crash of Flight 93 on the one-year anniversary of its hijacking.]]
On [[March 7]], [[2002]], Congressman [[John Murtha]] (PA-12) introduced a bill in the [[United States House of Representatives]] to establish a [[National Memorial]] to be developed by a commission, and ultimately administered by the [[National Park Service]]. On April 16, 2002, Senator [[Arlen Specter]] (PA) introduced a version of the "Flight 93 National Memorial Act" in the [[United States Senate|Senate]]. On [[September 10]], [[2002]] the bill passed both houses of [[United States Congress|Congress]]. The final bill specifically excluded the four hijackers from the passengers to be memorialized. When signed by President [[George W. Bush]] on [[September 24]], [[2002]] it became Public Law No. 107-226, and the site was automatically listed on the [[National Register of Historic Places]].

The site of the crash is closed to the general public pending the development of the memorial, but it is accessible to victims' family members. A neighbor has created a temporary memorial on a hilltop, approximately 500 yards north of the crash site. The temporary memorial is on private property, and visitors are asked to remain respectful. Visitors may write their thoughts or simply record their visit in bound books or on comment cards.

Within three years after the act became law, the commission is to submit to the [[United States Secretary of the Interior|Secretary of the Interior]] and Congress a report containing recommendations for the planning, design, construction, and long-term management of a permanent memorial at the crash site. The proposed boundaries of the National Memorial extend from Lambertsville Road to [[U.S. Highway 30]]. It will be approximately 2200 acres, of which about 1200 will be privately held, but protected through partnership agreements.

===Design competition===
====Initial design selection====
The commission decided to select the final design for the memorial through a multi-stage design competition funded by grants from the [[Heinz Foundations]] and the [[John S. and James L. Knight Foundation]]. The competition began on [[September 11]], [[2004]], and over one thousand entries were submitted. In February of 2005, five finalists were selected for further development and consideration. The jury selecting the final design included 15 members and was comprised of family members, design and art professionals, and community and national leaders. Over a three day review period they selected the winning design, and announced it on [[September 7]], [[2005]]. The design entitled ''Crescent of Embrace'' by a design team led by Paul and Milena Murdoch of Los Angeles was chosen.

The design featured a "Tower of Voices," containing 40 wind chimes — one for each passenger and crew member who died. It also consisted of two stands of [[red maple]] trees to line a walkway following the natural bowl shape of the land. Forty separate groves of red and [[sugar maple]]s were to be planted behind the [[crescent]], and eastern [[white oak]] trees for each victim of the September 11 attacks. A black [[slate]] wall would mark the edge of the crash site, where the remains of those who died now rest.

====Controversy====
During their deliberations, the use of the term ''crescent'' did come up. It was raised in the written comments of one person — out of 400 — who viewed the five finalists on display. In addition, Tom Burnett Sr., whose son ([[Tom Burnett]]) died in the crash, said he made an impassioned speech to his fellow jurors about what he felt the crescent represented.
"I explained this goes back centuries as an old-time Islamic symbol," Burnett said. "I told them we'd be a laughing stock if we did this." [http://www.post-gazette.com/pg/05259/572574.stm]

This design choice initially created controversy because the terrorists who hijacked the aircraft were [[Muslim]] and conducted the attacks in the name of [[Islam]]. Although Islam accepts no official icon, the [[crescent]] (which was symbolic of the [[Ottoman Empire]]) is generally recognized as an Islamic symbol and the Red Crescent is used as the Islamic equivalent of the [[International Red Cross and Red Crescent Movement|Red Cross]]. The crescent is also represented on a number of flags of countries with Muslim majorities, including [[Pakistan]].

The winning design's crescent is also oriented toward [[Mecca]]. While the [[Belmont Club]]'s Richard Fernandez noted that this may just be a coincidence he went on to note: "But what a coincidence! Memorials are symbols above all and it may be inappropriate to commemorate Flight 93 with a Red Crescent facing Mecca." [http://fallbackbelmont.blogspot.com/2005/09/flight-93-memorial.html]

The architect asserts that this is coincidental and there is no intent on referencing Muslim symbols. This sentiment has been shared by several victims families as well, such as the family of [[Edward P. Felt|Ed Felt]]. Representative [[Tom Tancredo]] of Colorado has opposed the design's shape "because of the crescent's prominent use as a symbol in Islam." The [[Council on American-Islamic Relations]] has denounced criticism as [[Islamophobia|Islamophobic]]. [http://www.cair-net.org/default.asp?Page=articleView&id=1746&theType=NR]

[[James Lileks]], a journalist and architectural commentator, noted in regard to the winning design: "We don't need giant statues of the guys ramming the drink cart into door. But pedantic though such a monument might be, future generations would infer the plot. All you get from a ''Crescent of Embrace'' is a sorrowful sigh of all-encompassing grief and absolution, as if the lives of all who died on that spot were equal in tragedy. They were not." [http://www.newhousenews.com/archive/lileks091405.html]

Mike Rosen of the [[Rocky Mountain News]] wrote: "On the anniversaries of 9/11, it's not hard to visualize al-Qaeda celebrating the crescent of maple trees, turning red in the fall, "embracing" the Flight 93 crash site. To them, it would be a memorial to their fallen martyrs. Why invite that? Just come up with a different design that eliminates the double meaning and the dispute." [http://www.rockymountainnews.com/drmn/news_columnists/article/0,1299,DRMN_86_4102007,00.html]

====Design modifications====
In response to criticism the designer has agreed to modify the plan. The architect believes that the central elements can be maintained to satisfy criticism. "It's a disappointment there is a misinterpretation and a simplistic distortion of this, but if that is a public concern, then that is something we will look to resolve in a way that keeps the essential qualities," Murdoch, 48, said in a telephone interview to the [[Associated Press]]. [http://www.phillyburbs.com/pb-dyn/news/103-09142005-541451.html]

The redesigned memorial has the plan shape of a circle (as opposed to a crescent) bisected by the flight's trajectory. "The circle enhances the earlier design by putting more emphasis on the crash site, officials said in the newsletter. A break in the trees will symbolize the path the plane took as it crashed." [http://news.yahoo.com/s/ap/20051130/ap_on_re_us/flight93_memorial]

===Construction===
The permanent memorial is planned to be dedicated on either September 11, 2010 or 2011.

==See also==
* [[Pentagon Memorial]] (under construction)
* [[International Freedom Center]], adjacent to [[WTC]] site (abandoned)
* [[Reflecting Absence]], WTC site (proposed)
* [[Tribute in Light]], WTC site (temporary/periodic performance)

==References==
* [http://frwebgate.access.gpo.gov/cgi-bin/getdoc.cgi?dbname=107_cong_public_laws&docid=f:publ226.107 Public Law No. 107-226] ''Flight 93 National Memorial Act''
* [http://www.post-gazette.com/pg/05251/567702.stm Post Gazette article: ''Flight 93 marker design picked'']
* [http://www.usatoday.com/news/nation/2001/09/11/victims-capsules.htm USA Today article: ''United Flight 93 victims at a glance'']
* [http://michellemalkin.com/archives/003513.htm Michelle Malkin: ''Flight 93 Memorial: Seeing is Believing'']
* [http://michellemalkin.com/archives/003543.htm Michelle Malkin: ''Flight 93 Memorial: Design will be Altered'']

==External links==
* Official NPS website: [http://www.nps.gov/flni/ Flight 93 National Memorial]
* [http://www.flight93memorialproject.org/ Flight 93 Memorial Project]
* [http://www.tribune-democrat.com/editorials/local_story_258140301.html?keyword=topstory ''The Johnstown Tribune Democrat'' on changing the name and design]
* [http://news.yahoo.com/s/ap/20051130/ap_on_re_us/flight93_memorial AP Story on Redesigned Flight 93 Memorial, November 30, 2005]

{{Registered Historic Places}}

[[Category:2002 establishments]]
[[Category:National Memorials of the United States]]
[[Category:September 11, 2001 attacks]]
[[Category:Somerset County, Pennsylvania]]
[[Category:Registered Historic Places in Pennsylvania]]