Wikipedia - Benutzerbeiträge [de]

FaceTime

2011-08-21T20:34:41Z

HereToHelp: Undid revision 446040613 by 108.66.65.238 (talk):rm empty ref tags

{{about|the Apple product|the social interaction|face time}}
{{Infobox software
| name = FaceTime
| title = FaceTime
| screenshot = [[File:FaceTime logo.png|120px]]
| caption = FaceTime logo on [[Mac OS X]]
| collapsible =
| author =
| developer = [[Apple Inc.|Apple]]
| released = {{Start date and age|2011|02|24}} (Mac OS X)
| discontinued =
| latest release version = 1.1 (258) (Mac OS X) 
| latest release date = {{Start date and age|2011|07|20}}
| latest preview version =
| latest preview date = 
| programming language =
| operating system = [[iOS (Apple)|iOS]] 4, [[Mac OS X]]
| platform =
| size = 16.8 [[Megabyte|MB]] (Mac OS X)
| language = 16 languages
| status = Active
| genre = [[Video calling]]
| license = [[Proprietary software|Proprietary]]
| website = [http://www.apple.com/iphone/features/facetime.html FaceTime for iPhone] 
[http://www.apple.com/ipodtouch/features/facetime.html FaceTime for iPod Touch] 
[http://www.apple.com/mac/facetime FaceTime for Mac] 
[http://itunes.apple.com/us/app/facetime/id414307850?mt=12 Mac App Store] 
[http://www.apple.com/ipad/built-in-apps/facetime.html FaceTime for iPad]
}}

'''FaceTime''' is a [[video calling]] software application and related protocol developed by [[Apple Inc.|Apple]] for supported mobile devices running the [[IOS (Apple)|iOS]], in addition to Macintosh computers running [[Mac OS X Snow Leopard|Mac OS X]] 10.6.6 and higher. FaceTime is supported on any iOS device with a forward-facing camera (that is, all iOS devices released since the [[iPhone 4]]) and on any Macintosh computer equipped with a '''FaceTime Camera''' (formerly known as an '''[[iSight]] Camera''').

==History==
[[File:FaceTime Logo on iOS 4.3.png|thumb|FaceTime logo on supported [[IOS (Apple)|iOS]] [[IOS_version_history#4.3|4.3]] devices.]]

FaceTime was announced by Apple CEO [[Steve Jobs]] at his [[stevenote|keynote speech]] on June 7, 2010 at the [[WWDC 2010]] in conjunction with the [[iPhone 4]]. Support for the fourth-generation [[iPod Touch]], the first iPod Touch to be equipped with cameras, was announced in conjunction with this device's release on September 8, 2010.

The "FaceTime" name was purchased by Apple from FaceTime Communications, who changed their name to Actiance, Inc.<ref name="FaceTime Communications">{{cite web |url=http://facetime.com/LearnMore.aspx |accessdate=2010-06-07 |title=Our name – FaceTime |author=FaceTime Communications | quote=Apple has announced that it will use 'FaceTime' as the trademark for its new video calling application. Our agreement with Apple to transfer the FaceTime trademark to them comes as we are rebranding our company to better reflect our capabilities. We will be announcing a new name in the coming months.}}</ref>

FaceTime for Mac OS X was announced on October 20, 2010 at the "Back to the Mac" Media event on the Apple Campus.

On February 24, 2011, FaceTime left beta and was listed in the [[Mac App Store]] for [[United States dollar|$]]0.99 ([[Pound sterling|£]]0.59). Apple claims that it intended to provide the application for free, but was prohibited by the [[Sarbanes–Oxley Act]] of 2002 from providing an unadvertised new feature of an already sold product without enduring 'onerous accounting measures'.<ref>http://www.macgasm.net/2011/02/24/apple-charge-facetime/</ref> However, the free beta is still available for download from the Apple servers.<ref>http://mac.softpedia.com/get/Internet-Utilities/FaceTime.shtml - link within last sentence "The page provides access to FaceTime 0.9 Build 92 version".</ref>

On March 2, 2011, FaceTime support was announced for the newly-introduced [[iPad|iPad 2]], which gained forward- and rear-facing cameras.

== Implementation ==
[[File:IPhone call screen.png|thumb|The [[iPhone]]'s screen when a call is in progress, showing the FaceTime button (middle of bottom row).]]

FaceTime works by connecting an [[iPhone 4]], a fourth generation [[iPod touch]], a second generation [[iPad]], or a computer with [[Mac OS X]] to another similar device; the previous generations of [[iPhone]] and [[iPod Touch]] are not currently supported, however on the [[iPhone 3G]] and [[iPhone 3GS]], users can enter facetime:// urls in [[Safari (web browser)|Safari]] to trigger what appears to be a FaceTime call, although it does not truly ever successfully initiate. FaceTime is not compatible with non-Apple devices or any other video calling services. The early 2011 revision of the MacBook Pro introduced high-definition FaceTime which may only be used with devices that have a HD FaceTime camera.

On the iPhone, FaceTime works in the Phone [[application software|application]] instead of being a separate application. It can be activated when in the Phone application by placing a call, and pressing the FaceTime button. The FaceTime button replaced the [[Hold (telephone)|Hold]] button (which is now available by pressing and holding the button). The icon resembles a camcorder. It is also possible to initiate a FaceTime call directly from a contact file.

While the current FaceTime implementation for the iPhone does not officially support FaceTime calls over [[3G]] ([[UMTS]]/[[High Speed Packet Access|HSPA]]), there are 3rd party applications that enable this capability for jailbroken devices.<ref>http://modmyi.com/cydia/com.eini.facebreak</ref>

Before the iOS 5 update, which will become available for end users in fall 2011, FaceTime over 3G can only be achieved by using third party [[iOS jailbreaking|jailbreaking]] commercial software such as the FaceBreak, 3G Unrestrictor or My3G which fools the phone into believing that it is connected via Wi-Fi. These programs are only available on jailbroken iPhones via the Cydia Store. At the moment there are no known free of cost solutions to unrestrict FaceTime. FaceTime uses about three megabytes of data per minute of conversation, so users with a limited data plan who use FaceTime over 3G (only on [[iOS jailbreaking|jailbroken]] devices as of right now) must be careful to not overextend their data limit. However, cellular talk time/minutes aren't used after switching from a voice call to a FaceTime call.

FaceTime on first generation iPad is also possible by porting the FaceTime application for the iPod touch 4th gen to iPad.<ref>{{cite web|url=http://www.9to5mac.com/50959/ipad-facetime-now-possible-with-hack/ |title=iPad FaceTime now possible with hack |publisher= 9 to 5 Mac |date=2011-02-15 |author=Brian Purkiss}}</ref>

iOS 4.1 and FaceTime for Mac, iPhone 4, and iPod Touch (4th Generation) supports the use of an [[Apple ID]] email address to initiate a FaceTime call. The two parties would need two different Apple IDs to initiate a FaceTime call, although multiple IDs can reside under a single iTunes account.

== Standards ==
The FaceTime protocol is partly based on numerous open industry standards.<ref name="AppleInsider">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html |date=2010-06-08 |accessdate=2010-06-09 |title=Inside iPhone 4: FaceTime video calling |publisher=AppleInsider |author=Daniel Eran Dilger}}</ref>

FaceTime is based on numerous technologies:
* [[H.264]] and [[Advanced Audio Coding|AAC]] – video and audio [[codec]]s respectively
* [[Session Initiation Protocol|SIP]] – [[IETF]] signaling protocol for [[VoIP]]
* [[Session Traversal Utilities for NAT|STUN]], [[Traversal Using Relay NAT|TURN]] and [[Interactive Connectivity Establishment|ICE]] – IETF technologies for traversing [[Firewall (computing)|firewalls]] and [[Network address translation|NAT]]
* [[Real-time Transport Protocol|RTP]] and [[Secure Real-time Transport Protocol|SRTP]] – IETF standards for delivering real-time and encrypted media streams for VoIP

Upon the launch of the iPhone 4, Jobs promised that Apple would work in due course with standards bodies to make the FaceTime protocol an "open standard." {{As of|2011|June}}, it is not yet known to have been ratified by any standards body, and the extent of work by Apple with regards to this promise is unclear as Apple has not released technical specifications for the service. FaceTime is not currently supported on any non-Apple devices.

While FaceTime is based on open standards, Apple's FaceTime service requires a client-side certificate.<ref>{{cite web|url=http://www.packetstan.com/2010/07/special-look-face-time-part-3-call.html | date=2010-07-09 | accessdate=2011-03-06 | title=Special Look: Face Time (part 3: Call Connection Initialization)|author=Josh Wright}}</ref> ''I.e.'' while the protocol might become an "open standard", access to Apple's FaceTime service is controlled by Apple.

==Limited availability==
{{As of|2010|October}}, FaceTime is not enabled on devices bought in [[Saudi Arabia]] & [[United Arab Emirates]] possibly due to regulations in these countries which restrict the use of IP-based communications technology. Devices bought elsewhere continue to support FaceTime even in these countries.<ref name="Register 19 October 2010">{{cite news|last=Ray|first=Bill|title=Apple wipes smile off FaceTime in the Middle East|url=http://www.theregister.co.uk/2010/10/19/facetime/|accessdate=19 October 2010|newspaper=[[The Register]]|date=19 October 2010}}</ref>

Although, [[Egypt]], [[Jordan]] & [[Kuwait]] used to disable FaceTime on the iPhone 4. But they are re-enabling the feature back through a carrier update for existing phone owners and pre-enabled on new iPhone 4.

== References ==
{{Reflist}}

{{iOS}}
{{Mac OS X}}
{{IM clients}}

{{DEFAULTSORT:Facetime}}
[[Category:IOS software]]
[[Category:2010 software]]
[[Category:Assistive technology]]
[[Category:Videotelephony]]

[[es:FaceTime]]
[[fr:FaceTime]]
[[it:FaceTime]]
[[ko:페이스타임]]
[[mr:फेसटाइम]]
[[ja:FaceTime]]
[[pt:FaceTime]]
[[ru:FaceTime]]
[[fi:FaceTime]]
[[sv:Facetime]]
[[tr:FaceTime]]

FaceTime

2011-03-19T17:33:52Z

HereToHelp: Undid revision 419646954 by 98.66.255.225 (talk): No.

{{Infobox software
| name = FaceTime
| title = FaceTime
| logo = [[File:FaceTime logo.png|120px]]
| screenshot = 
| caption = [[Mac OS X]] Logo
| collapsible =
| author =
| developer = [[Apple Inc.]]
| released = 
| discontinued =
| latest release version = 1.0.1 (Mac OS X)
| latest release date = {{Start date and age|2011|3|2|df=yes/no}}
| latest preview version =
| latest preview date = 
| programming language =
| operating system = [[Mac OS X]], [[iOS (Apple)|iOS]] 4
| platform = [[Macintosh|Mac]], [[iPhone 4]], and [[iPod Touch]] 4th gen and [[iPad]] 2nd Generation.
| size =
| language =
| status =
| genre = [[Video calling]]
| license = Proprietary
| website = [http://www.apple.com/mac/facetime FaceTime for Mac] [http://www.apple.com/iphone/features/facetime.html FaceTime for iPhone]
}}
{{about|the Apple product|the social interaction|face time}}

'''FaceTime''' is a [[video calling]] software developed by [[Apple Inc.|Apple]] for [[iPhone 4]], the fourth generation [[iPod touch]], the [[iPad 2]] and computers running [[Mac OS X Snow Leopard|Mac OS X]] 10.6.6 and higher. Now, the built-in webcam included on Apple computers is called [[iSight|FaceTime Camera]].

==History==
FaceTime was announced by Apple CEO [[Steve Jobs]] at his [[stevenote|keynote speech]] on June 7, 2010 at the [[WWDC 2010]] in conjunction with the [[iPhone 4]].

The "FaceTime" name was purchased by Apple from FaceTime Communications, who changed their company name.<ref name="FaceTime Communications">{{cite web |url=http://facetime.com/LearnMore.aspx |accessdate=2010-06-07 |title=Our name – FaceTime |author=FaceTime Communications | quote=Apple has announced that it will use 'FaceTime' as the trademark for its new video calling application. Our agreement with Apple to transfer the FaceTime trademark to them comes as we are rebranding our company to better reflect our capabilities. We will be announcing a new name in the coming months.}}</ref>.

FaceTime for Mac OS X was announced on October 20, 2010 at the "Back to the Mac" Media event on the Apple Campus.

On February 24, 2011 FaceTime left beta and was listed in the [[Mac App Store]] for [[United States dollar|$]]0.99 ([[Pound sterling|£]]0.59). Apple had intended to provide the application for free, however, the [[Sarbanes–Oxley Act]], put forth in July 30, 2002, prohibits companies, such as Apple, from providing an unadvertised new feature of an already sold product without enduring 'onerous accounting measures'.<ref>http://www.macgasm.net/2011/02/24/apple-charge-facetime/</ref>

== Implementation ==
[[File:IPhone call screen.png|thumb|The [[iPhone]]'s screen when a call is in progress, showing the FaceTime button (middle of bottom row).]]

FaceTime works by connecting an [[iPhone 4]], fourth Generation [[iPod touch]], a second generation [[iPad]], or a computer with [[Mac OS X]] to another similar device; the previous generations of [[iPhone]] and [[iPod Touch]] are not currently supported, however on the [[iPhone 3G]] and [[iPhone 3GS]], users can enter facetime:// urls in [[Safari (web browser)|Safari]] to trigger what appears to be a FaceTime call, although it does not truly ever successfully initiate. FaceTime is not compatible with non-Apple devices or any other video calling services.

On the iPhone, FaceTime works in the Phone [[application software|application]] instead of being a separate application. It can be activated when in the Phone application by placing a call, and pressing the FaceTime button. The FaceTime button replaced the [[Hold (telephone)|Hold]] button (which is now available by pressing and holding the Mute button). The icon resembles a camcorder. It is also possible to initiate a FaceTime call directly from a contact file.

Apple's FaceTime implementation for the iPhone does not currently support FaceTime calls over [[3G]] ([[UMTS]]/[[High Speed Packet Access|HSPA]]). When announcing FaceTime, Jobs said that "we need to work a little bit with the carriers" before FaceTime could be used over cellular data networks, and said that at least for 2010, it will be Wi-Fi or Ethernet only.<ref>[http://www.computerworld.com/s/article/9177767/Apple_unveils_iPhone_4_touts_FaceTime_video_chat?taxonomyId=11&pageNumber=1 Apple unveils iPhone 4, touts 'FaceTime' video chat], ComputerWorld, Gregg Keizer, June 7, 2010</ref>

FaceTime over 3G can be achieved by using third party [[iOS jailbreaking|jailbreaking]] commercial software such as the FaceBreak, 3G Unrestrictor or My3G which fools the phone into believing that it is connected via Wi-Fi. These programs are only available on jailbroken iPhones via the Cydia Store. At the moment there are no known free of cost solutions to unrestrict FaceTime. FaceTime uses about three megabytes of data per minute of conversation, so users with a limited data plan who use FaceTime over 3G (only on [[iOS jailbreaking|jailbroken]] devices as of right now) must be careful to not overextend their data limit. However, cellular talk time/minutes aren't used after switching from a voice call to a FaceTime call.

FaceTime on iPad original is also possible by porting the FaceTime application for the iPod touch 4th gen to iPad.<ref>{{cite web|url=http://www.9to5mac.com/50959/ipad-facetime-now-possible-with-hack/ |title=iPad FaceTime now possible with hack |publisher= 9 to 5 Mac |date=2011-02-15 |author=Brian Purkiss}}</ref>

iOS 4.1 and FaceTime for Mac, iPhone 4, and iPod Touch (4th Generation) supports the use of an [[Apple ID]] email address to initiate a FaceTime call. The two parties would need two different Apple IDs to initiate a FaceTime call, although multiple IDs can reside under a single iTunes account.

== Standards ==
The FaceTime protocol is based on numerous open industry standards, and Apple has pledged to release it as an open standard allowing other companies to develop around it.<ref name="AppleInsider">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html |date=2010-06-08 |accessdate=2010-06-09 |title=Inside iPhone 4: FaceTime video calling |publisher=AppleInsider |author=Daniel Eran Dilger}}</ref>

FaceTime is based on numerous technologies:
* [[H.264]] and [[Advanced Audio Coding|AAC]] – video and audio [[codec]]s respectively
* [[Session Initiation Protocol|SIP]] – [[IETF]] signaling protocol for [[VoIP]]
* [[Session Traversal Utilities for NAT|STUN]], [[Traversal Using Relay NAT|TURN]] and [[Interactive Connectivity Establishment|ICE]] – IETF technologies for traversing [[Firewall (computing)|firewalls]] and [[Network address translation|NAT]]
* [[Real-time Transport Protocol|RTP]] and [[Secure Real-time Transport Protocol|SRTP]] – IETF standards for delivering real-time and encrypted media streams for VoIP

While FaceTime is based on open standards, Apple's FaceTime services requires a client-side certificate<ref>{{cite web|url=http://www.packetstan.com/2010/07/special-look-face-time-part-3-call.html | date=2010-07-09 | accessdate=2011-03-06 | title=Special Look: Face Time (part 3: Call Connection Initialization)|author=Josh Wright}}</ref>. ''I.e.'' while the protocol might be open, access to Apple's FaceTime service is controlled by Apple.

==Limited availability==
As of October 2010, FaceTime is not enabled on devices bought in [[Kuwait]], [[Saudi Arabia]], [[United Arab Emirates]], [[Qatar]], [[Egypt]] or [[Jordan]], possibly due to regulations in these countries which restrict the use of IP-based communications technology. Devices bought elsewhere continue to support FaceTime even in these countries.<ref name="Register 19 October 2010">{{cite news|last=Ray|first=Bill|title=Apple wipes smile off FaceTime in the Middle East|url=http://www.theregister.co.uk/2010/10/19/facetime/|accessdate=19 October 2010|newspaper=[[The Register]]|date=19 October 2010}}</ref>

== References ==
{{Reflist}}

{{iOS}}
{{Mac OS X}}
{{IM clients}}

{{DEFAULTSORT:Facetime}}
[[Category:IOS software]]
[[Category:2010 software]]
[[Category:Assistive technology]]
[[Category:Videotelephony]]

[[es:FaceTime]]
[[it:FaceTime]]
[[ja:FaceTime]]
[[pt:FaceTime]]
[[ru:FaceTime]]
[[sv:Facetime]]

FaceTime

2010-08-22T16:44:17Z

HereToHelp: Undid revision 380241663 by 96.245.223.106 (talk): Apple's tech specs say nothing of a microphone. Also camera.

'''FaceTime''' is a [[video calling]] software feature for [[iPhone 4]]'s phone application, developed by [[Apple Inc.|Apple]] and announced at [[WWDC 2010]]. It is based on numerous open industry standards and Apple has pledged to release it as an open standard allowing other companies to develop around it.<ref name="AppleInsider">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html |date=2010-06-08 |accessdate=2010-06-09 |title=Inside iPhone 4: FaceTime video calling |publisher=AppleInsider |author=Daniel Eran Dilger}}</ref> Apple has purchased the name "FaceTime" from FaceTime Communications.<ref name="FaceTime Communications">{{cite web |url=http://facetime.com/LearnMore.aspx |accessdate=2010-06-07 |title=Our name – FaceTime |author=FaceTime Communications}}</ref>

== Standards ==

FaceTime is based on numerous technologies:
* [[H.264]] and [[Advanced Audio Coding|AAC]] – video and audio [[codec]]s respectively
* [[Session Initiation Protocol|SIP]] – [[IETF]] signaling protocol for [[VoIP]]
* [[Session Traversal Utilities for NAT|STUN]], [[Traversal Using Relay NAT|TURN]] and [[Interactive Connectivity Establishment|ICE]] – IETF technologies for traversing [[Firewall (computing)|firewalls]] and [[Network address translation|NAT]]
* [[Real-time Transport Protocol|RTP]] and [[Secure Real-time Transport Protocol|SRTP]] – IETF standards for delivering real-time and encrypted media streams for VoIP

== Implementation ==
Currently, FaceTime works only with a [[Wi-Fi]] connection, as there is no official 3G ([[UMTS]]/[[High Speed Packet Access|HSPA]]) data support at the moment. FaceTime can only work by connecting an iPhone 4 to another iPhone 4; the previous generations of [[iPhone]] are not supported. No company other than Apple has made a FaceTime-compatible device. Such, FaceTime communities such as [http://www.ftfun.com/ FTFun], have surfaced, attempting to connect iPhone 4 owners.

FaceTime works in the Phone application, instead of being a separate app. It can be activated when in the Phone application by placing a call, and pressing the FaceTime button. FaceTime is located beneath the Keypad button and is to the right of the Add Call button and replaced the Hold button (which is now available by pressing and holding the Mute button). The icon resembles a camcorder. It is also possible to initiate a FaceTime call directly from a contact file.

Although Apple limits FaceTime to Wi-Fi only, FaceTime over 3G an be achieved by using third party [[iOS jailbreaking|jailbreaking]] software such as 3GUnrestrictor or My3G which fools the phone into believing that it is connected via Wi-Fi. These programs are only available on jailbroken iPhones via the Cydia Store. FaceTime uses about three megabytes of data per minute of conversation, so users with a limited data plan must be careful to not overextend their data limit. However, cellular talk time/minutes aren't used after switching from a voice call to a FaceTime call.

Later releases of iOS 4 have been rumored to have support for a user's Apple ID as opposed to requiring the iPhone 4 phone number to initiate a FaceTime call, however, this has not yet been confirmed by Apple.{{Citation needed|date=August 2010}} Furthermore, there are reports of the FaceTime functionality being incorporated into Apple's [[iPod touch]] and [[iPad]].<ref name="BoyGeniusReport">{{cite web|url=http://www.boygeniusreport.com/2010/07/15/exclusive-apples-facetime-coming-to-ipod-touch-ipad-we-detail-how/ |date=2010-07-15 |accessdate=2010-07-27 |title=Exclusive: Apple’s FaceTime coming to iPod touch, iPad, we detail how |publisher=BoyGeniusReport |author=Boy Genius}}</ref> Current generations of the iPod touch do not have microphones or cameras, meaning only currently unreleased models could support FaceTime.

== References ==

{{Reflist}}

{{iOS}}

{{DEFAULTSORT:Facetime}}
[[Category:IPhone OS software]]
[[Category:2010 software]]
[[Category:Assistive technology]]
[[Category:Videotelephony]]

{{mac-stub}}

[[es:FaceTime]]
[[fr:FaceTime]]
[[it:FaceTime]]
[[pt:FaceTime]]

FaceTime

2010-08-19T10:43:47Z

HereToHelp: /* Implementation */ fix

'''FaceTime''' is a [[video calling]] software feature for [[iPhone 4]]'s phone application, developed by [[Apple Inc.|Apple]] and announced at [[WWDC 2010]]. It is based on numerous open industry standards and Apple has pledged to release it as an open standard allowing other companies to develop around it.<ref name="AppleInsider">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html |date=2010-06-08 |accessdate=2010-06-09 |title=Inside iPhone 4: FaceTime video calling |publisher=AppleInsider |author=Daniel Eran Dilger}}</ref> Apple has purchased the name "FaceTime" from FaceTime Communications.<ref name="FaceTime Communications">{{cite web |url=http://facetime.com/LearnMore.aspx |accessdate=2010-06-07 |title=Our name – FaceTime |author=FaceTime Communications}}</ref>

== Standards ==

FaceTime is based on numerous [[open standard]]s.<ref name=AppleInsider />
* [[H.264]] and [[Advanced Audio Coding|AAC]] – video and audio [[codec]]s respectively
* [[Session Initiation Protocol|SIP]] – [[IETF]] signaling protocol for [[VoIP]]
* [[Session Traversal Utilities for NAT|STUN]], [[Traversal Using Relay NAT|TURN]] and [[Interactive Connectivity Establishment|ICE]] – IETF technologies for traversing [[Firewall (computing)|firewalls]] and [[Network address translation|NAT]]
* [[Real-time Transport Protocol|RTP]] and [[Secure Real-time Transport Protocol|SRTP]] – IETF standards for delivering real-time and encrypted media streams for VoIP

== Implementation ==
Currently, FaceTime works only with a [[Wi-Fi]] connection, as there is no official 3G ([[UTMS]]/[[High Speed Packet Access|HSPA]]) data support at the moment. FaceTime can only work by connecting an iPhone 4 to another iPhone 4; the previous generations of [[iPhone]] are not supported. No company other than Apple has made a FaceTime-compatible device. Such, FaceTime communities such as [http://www.ftfun.com/ FTFun], have surfaced, attempting to connect iPhone 4 owners.

FaceTime works in the Phone application, instead of being a separate app. It can be activated when in the Phone application by placing a call, and pressing the FaceTime button. FaceTime is located beneath the Keypad button and is to the right of the Add Call button and replaced the Hold button (which is now available by pressing and holding the Mute button). The icon resembles a camcorder. It is also possible to initiate a FaceTime call directly from a contact file.

Although Apple limits FaceTime to Wi-Fi only, FaceTime over 3G an be achieved by using third party [[iOS jailbreaking|jailbreaking]] software such as 3GUnrestrictor or My3G which fools the phone into believing that it is connected via Wi-Fi. These programs are only available on jailbroken iPhones via the Cydia Store. FaceTime uses about three megabytes of data per minute of conversation, so users with a limited data plan must be careful to not overextend their data limit. However, cellular talk time/minutes aren't used after switching from a voice call to a FaceTime call.

Later releases of iOS 4 have been rumored to have support for a user's Apple ID as opposed to requiring the iPhone 4 phone number to initiate a FaceTime call, however, this has not yet been confirmed by Apple.{{Citation needed|date=August 2010}} Furthermore, there are reports of the FaceTime functionality being incorporated into Apple's [[iPod touch]] and [[iPad]].<ref name="BoyGeniusReport">{{cite web|url=http://www.boygeniusreport.com/2010/07/15/exclusive-apples-facetime-coming-to-ipod-touch-ipad-we-detail-how/ |date=2010-07-15 |accessdate=2010-07-27 |title=Exclusive: Apple’s FaceTime coming to iPod touch, iPad, we detail how |publisher=BoyGeniusReport |author=Boy Genius}}</ref> Current generations of the iPod touch do not have microphones, meaning only currently unreleased models could support FaceTime.

== References ==

{{Reflist}}

{{iOS}}

{{DEFAULTSORT:Facetime}}
[[Category:IPhone OS software]]
[[Category:2010 software]]
[[Category:Assistive technology]]
[[Category:Videotelephony]]

{{mac-stub}}

[[es:FaceTime]]
[[fr:FaceTime]]
[[it:FaceTime]]
[[pt:FaceTime]]

FaceTime

2010-08-18T11:58:14Z

HereToHelp: /* Implementation */ fring is irrelevant; knowing the number is obvious; cleanup

'''FaceTime''' is a [[video calling]] software feature for [[iPhone 4]]'s phone application, developed by [[Apple Inc.|Apple]] and announced at [[WWDC 2010]]. It is based on numerous open industry standards and Apple has pledged to release it as an open standard allowing other companies to develop around it.<ref name="AppleInsider">{{cite web|url=http://www.appleinsider.com/articles/10/06/08/inside_iphone_4_facetime_video_calling.html |date=2010-06-08 |accessdate=2010-06-09 |title=Inside iPhone 4: FaceTime video calling |publisher=AppleInsider |author=Daniel Eran Dilger}}</ref> Apple has purchased the name "FaceTime" from FaceTime Communications.<ref name="FaceTime Communications">{{cite web |url=http://facetime.com/LearnMore.aspx |accessdate=2010-06-07 |title=Our name – FaceTime |author=FaceTime Communications}}</ref>

== Standards ==

FaceTime is based on numerous [[open standard]]s.<ref name=AppleInsider />
* [[H.264]] and [[Advanced Audio Coding|AAC]] – video and audio [[codec]]s respectively
* [[Session Initiation Protocol|SIP]] – [[IETF]] signaling protocol for [[VoIP]]
* [[Session Traversal Utilities for NAT|STUN]], [[Traversal Using Relay NAT|TURN]] and [[Interactive Connectivity Establishment|ICE]] – IETF technologies for traversing [[Firewall (computing)|firewalls]] and [[Network address translation|NAT]]
* [[Real-time Transport Protocol|RTP]] and [[Secure Real-time Transport Protocol|SRTP]] – IETF standards for delivering real-time and encrypted media streams for VoIP

== Implementation ==
Currently, FaceTime works only with a [[Wi-Fi]] connection, as there is no 3G ([[UTMS]]/[[High Speed Packet Access|HSPA]]) data support at the moment. FaceTime can only work by connecting an iPhone 4 to another iPhone 4; the previous generations of [[iPhone]] are not supported. No company other than Apple has made a FaceTime-compatible device. Such, FaceTime communities such as [http://www.ftfun.com/ FTFun], have surfaced, attempting to connect iPhone 4 owners.

FaceTime works in the Phone application, instead of being a separate app. It can be activated when in the Phone application by placing a call, and pressing the FaceTime button. FaceTime is located beneath the Keypad button and is to the right of the Add Call button and replaced the Hold button (which is now available by pressing and holding the Mute button). The icon resembles a camcorder. It is also possible to initiate a FaceTime call directly from a contact file.

Later releases of iOS 4 have been rumored to have support for a user's Apple ID as opposed to requiring the iPhone 4 phone number to initiate a FaceTime call, however, this has not yet been confirmed by Apple.{{fact}} Furthermore, there are reports of the FaceTime functionality being incorporated into Apple's [[iPod touch]] and [[iPad]].<ref name="BoyGeniusReport">{{cite web|url=http://www.boygeniusreport.com/2010/07/15/exclusive-apples-facetime-coming-to-ipod-touch-ipad-we-detail-how/ |date=2010-07-15 |accessdate=2010-07-27 |title=Exclusive: Apple’s FaceTime coming to iPod touch, iPad, we detail how |publisher=BoyGeniusReport |author=Boy Genius}}</ref> Current generations of the iPod touch do not have microphones, meaning only currently unreleased models could support FaceTime.

Although Apple limits FaceTime to Wi-Fi only, FaceTime over 3G an be achieved by using third party [[iOS jailbreaking|jailbreaking]] software such as 3GUnrestrictor or My3G which fools the phone into believing that it is connected via Wi-Fi. These programs are only available on jailbroken iPhones via the Cydia appstore. FaceTime uses about three megabytes of data per minute of conversation, so users with a limited data plan must be careful to not overextend their data limit.

== References ==

{{reflist}}

{{iOS}}

[[Category:iPhone OS software]]
[[Category:2010 software]]
[[Category:Assistive technology]]
[[Category:Videotelephony]]

[[es:FaceTime]]
[[fr:FaceTime]]
[[it:FaceTime]]
[[pt:FaceTime]]

{{mac-stub}}

IPhone 4

2008-12-20T13:44:39Z

HereToHelp: /* Markteinführung */ Bild:IPhone_3G_Availability.svg Neue Bildshows ursprünglich, 3G, und geplante Verfügbarkeit.

{{Infobox
| Breite = 250px
| Farbe = Silver
| Bild = [[Bild:IPhone bei der Macworld.jpg|250px|iPhone im MP3-Player-Modus]]
| Titel = iPhone
| Bildtext = <div align="center">iPhone der 1. Generation im MP3-Player-Modus</div>
| Stil = 2
| Feldname1 = Hersteller
| Daten1 = [[Apple|Apple Inc.]]
| Feldname2 = Funkverbindungen
| Daten2 = [[Global System for Mobile Communications|GSM]]/[[Enhanced Data Rates for GSM Evolution|EDGE]] (850, 900, 1800, 1900 Mhz); [[Universal Mobile Telecommunications System|UMTS]]/[[High Speed Downlink Packet Access|HSDPA]] (850, 1900, 2100 Mhz)<ref name="g3">ab iPhone 3G</ref>; [[Wireless Local Area Network|WLAN]] [[IEEE 802.11|802.11b/g]]; [[Bluetooth]] 2.0 + EDR
| Feldname3 = Flashspeicher
| Daten3 = 8 [[Byte|GB]] oder 16 GB [[NAND-Flash|NAND]]
| Feldname4 = Prozessor
| Daten4 = [[ARM-Architektur|ARM]] 1176
| Feldname5 = Bildschirm
| Daten5 = 3,5-[[Zoll (Einheit)|Zoll]]-[[Touchscreen]]
320 × 480 [[Pixel]] / 160 [[Dots per inch|DPI]]
| Feldname6 = Digitalkamera
| Daten6 = 2 [[Megapixel]]
| Feldname7 = Gewicht
| Daten7 = 135 g
| Feldname8 = Akkulaufzeit
| Daten8 = [[#Energiespeicher|Laut Hersteller*]]: 6–8 h (Telefonie, Video, Internet) 24 h (nur Audio) 250 h [[Leerlaufverlust|Standby]]
| Feldname9 = Maße (H×B×T)
| Daten9 = 115 × 61 × 11,6 mm
| Feldname10 = Arbeitsspeicher
| Daten10 = 128 [[Byte|MB]] [[Dynamic Random Access Memory|DRAM]]<ref>[http://www.techonline.com/product/underthehood/200001814 „Under the Hood: Inside the Apple iPhone“] (bei techonline.com am 1. Juli 2007, englisch)</ref>
}}
Das '''iPhone''' ist ein von [[Apple]] entwickeltes [[Smartphone]]<ref>Markteinordnung durch Apple, siehe [http://www.apple.com/quicktime/qtv/iphoneroadmap/ ''special presentation'' am 6. März 2008] (englisch)</ref>, das über einen [[Apple iPod|iPod]]-ähnlichen [[Portable Media Player|Medienspieler]] verfügt und weitgehend über den Bildschirm gesteuert wird. Dieser besitzt wie beim [[Apple iPod#iPod touch|iPod touch]] eine [[Multi-Touch]]-Funktionalität, ermöglicht also eine Bedienung mit mehreren Fingern gleichzeitig. Darüber hinaus wird das iPhone nur noch mittels zweier Tasten, einer Wippe und eines Schiebeschalters bedient. Nach Angaben von Apple vereint das iPhone die Funktionen eines [[Breitbildformat|Breitbild]]-Video-iPod-Medienspielers<ref>[http://www.apple.com/de/iphone/features/index.html#ipod Beschreibung auf der offiziellen deutschen Apple-Webseite]</ref> mit denen eines [[Mobiltelefon]]s mit [[Digitalkamera]] und [[Internetzugang]].

Der iPhone-Verkaufsstart in den [[Vereinigte Staaten|USA]] war am 29. Juni 2007. Das Mobiltelefon wird in den USA ausschließlich in Kooperation mit dem amerikanischen Mobilfunkkonzern [[AT&T Wireless]] angeboten.<ref>[[Der Spiegel]]: [http://www.spiegel.de/netzwelt/mobil/0,1518,486421,00.html ''iPhone kommt am 29. Juni'' (4. Juni 2007)]</ref> In Europa wird das Gerät seit dem 9. November 2007 verkauft; zuerst in [[Deutschland]] und [[Vereinigtes Königreich|Großbritannien]], seit dem 28. November auch in [[Frankreich]] und seit dem 14. März 2008 in [[Österreich]] und [[Irland]]. In fast allen Ländern, in denen das weiterentwickelte [[Apple iPhone#Apple iPhone 3G|iPhone 3G]] verfügbar ist, begann der Verkauf parallel am 11. Juli 2008; darunter auch in der [[Schweiz]]. In Frankreich war die neue Geräteversion am 17. Juli 2008 erhältlich.

Das US-Magazin ''[[Time]]'' wählte das iPhone zur ''„Erfindung des Jahres 2007“''.<ref>[http://www.time.com/time/business/article/0,8599,1678581,00.html ''Invention Of the Year: The iPhone''] (www.time.com am 31. Oktober 2007, englisch)</ref> Die überproportionale Medienresonanz war jedoch nur zum Teil eine Reaktion auf die besonderen technischen Merkmale. Häufig wurden stattdessen der ''[[Lebensstil|Lifestyle]]''-Faktor des Geräts und die [[Apple iPhone#Juristische Auseinandersetzung|juristischen Auseinandersetzungen]] um die Vermarktung thematisiert.

== Geschichte ==
Bereits im Dezember 1999 hatte sich Apple die Internet-Domain ''iphone.org'' registrieren lassen.<ref>[http://www.mac-essentials.de/index.php/mac/article/18615/ ''Kurze Geschichte des »iPhone«-Gerüchtes''] (''Mac Essentials'' am 4. Januar 2007)</ref> Im September 2005 stellte Apple das [[Motorola]] ''ROKR E1'' vor, das als erstes Mobiltelefon mit [[iTunes]] synchronisiert werden konnte. Gleichzeitig wurde eine entsprechende Aktualisierung des Programms zur Verfügung gestellt, um den Austausch von Audio-Dateien zu ermöglichen. Offenbar waren die Apple-Designer bis 2005 mit der Entwicklung der iPod-Linie ausgelastet gewesen, so dass keine Kapazitäten für ein neues Produkt übrig blieben und eine strategische Partnerschaft mit Motorola eingegangen werden musste. Schon kurz darauf sickerte jedoch durch, dass [[Steve Jobs]] mit dem ''ROKR'' unzufrieden sei, weil es als Fremdprodukt nicht in die Designlinie der Apple-Produktpalette passte. Diese Einschätzung wurde im September 2006 von Apple durch den Entzug der ''ROKR''-Unterstützung bei ''iTunes'' bestätigt. Stattdessen wurde eine weitere ''iTunes''-Aktualisierung mit Unterstützung für ein noch unbekanntes Mobiltelefon veröffentlicht, das offensichtlich nicht nur Audio-, sondern auch Video- und Bilddateien wiedergeben können sollte. Dies führte in verschiedenen Medien zu weitreichenden Spekulationen über ein zu erwartendes Apple-Mobiltelefon, die bis zum Jahresende 2006 immer konkreter wurden.<ref>[http://www.pc-professionell.de/specials/sa2007062901820070703008.aspx ''Trotz Problemen: iPhone endlich da – Der lange Weg zum iPhone''] (''PC professionell'', Juni 2007)</ref> Vor der offiziellen Vorstellung mahnte Apple jedoch [[Blog]]ger ab, die bereits über ein iPhone berichtet hatten.<ref>[http://www.law-blog.de/349/apple-mahnt-blogger-wegen-iphone-berichterstattung-ab ''Apple mahnt Blogger wegen iPhone-Berichterstattung ab'']</ref> Am 9. Januar 2007 wurde der Öffentlichkeit ein erster Prototyp dieses Geräts auf der [[Macworld Conference & Expo]] in San Francisco vorgestellt.

=== Markteinführung ===
[[Bild:Waiting for iPhones NYC.jpg|thumb|left|Wartende iPhone-Käufer am 29. Juni 2007 vor einem Apple-Store in [[New York City]]]]

Apple kündigte Anfang Juni 2007 durch die Ausstrahlung eines TV-Werbespots den Verkaufsbeginn in den USA für den 29. Juni an. Dieser erfolgte exklusiv in rund 200 ''Apple-Stores'' und etwa 1800 Verkaufsstellen des Netzbetreibers [[AT&T Wireless]] (ehemals ''Cingular Wireless''). Die mit 4 [[Byte|Gigabyte]]-Speicher ausgestatteten Geräte wurden für [[United States Dollar|$]] 499, die 8 GB-Versionen für $ 599 angeboten. Laut dem Apple-Bericht für das dritte Quartal des [[Wirtschaftsjahr]]es<ref>Apple Inc. hat ein vom tatsächlichen Jahr abweichendes Wirtschaftsjahr, das jeweils am 1. Oktober beginnt</ref> wurden innerhalb der ersten zwei Verkaufstage rund 270.000 iPhones inklusive Zubehör abgesetzt.<ref>[http://www.apple.com/quicktime/qtv/earningsq307/ ''Apple – Quarter 3 – Financial Results''] (26. Juli 2007)</ref> Nach Einschätzungen von Analysten im August 2007 konnte Apple im dritten Jahresquartal mit rund 800.000 weiteren verkauften Geräten rechnen, was insgesamt die Apple-Zielvorgabe von einer Million seit Verkaufsstart bis Ende September leicht übertroffen hätte.<ref>[http://www.heise.de/newsticker/meldung/94731/from/rss09 ''Analyst: Apple verkauft mehr iPhones als erwartet''] (Meldung bei heise.de am 22. August 2007)</ref> Am 5. September 2007 verkündete Steve Jobs eine Preissenkung: Die 8 GB-Version kostet ab sofort nur $ 399, die 4 GB-Version wird nicht mehr angeboten. Verärgerten Erstkunden bot Jobs tags darauf auf der Apple-Webseite einen Gutschein über $ 100 an. Wer das 8 GB-Gerät innerhalb der zehn Tage vor Ankündigung der Preissenkung gekauft hatte, bekomme die volle Differenz von $ 200 erstattet. Im vierten Quartal des Wirtschaftsjahres 2006/2007 (bis 29. September 2007) wurden nach Angaben von Apple etwa 1,1 Millionen iPhones verkauft. Laut dem am 22. Oktober 2007 veröffentlichten Quartalsbericht seien damit insgesamt seit der Markteinführung rund 1,4 Millionen Exemplare abgesetzt worden.
[[Bild:IPhone_3G_Availability.svg|thumb|''Blau:'' Länder, in denen das iPhone im Handel erhältlich war; ''hellgrün:'' Markteinführung später geplant (Stand: Juli 2008)]]
In Europa wird das Gerät seit dem 9. November 2007 angeboten. Zu diesem Zeitpunkt begann der Vertrieb in Deutschland exklusiv über [[T-Mobile]] zum Preis von 399 Euro, gekoppelt mit einem Zwei-Jahres-Vertrag und einer Sperre, die auch nach Ablauf dieser Frist eine Nutzung des iPhones ausschließlich im T-Mobile-Netz erlaubt. Im Gegenzug beteiligt T-Mobile Apple an den monatlichen Umsätzen.<ref>[http://de.news.yahoo.com/dpa2/20070919/tbs-telekom-startet-verkauf-von-iphone-a-9806f5a.html ''Telekom startet Verkauf von iPhone am 9. November''] (dpa-Meldung vom 19. September 2007)</ref>

Am 19. November 2007 erließ das [[Landgericht Hamburg]] auf Antrag von [[Vodafone|Vodafone (D2)]] eine einstweilige Verfügung gegen den vertragsgebundenen Verkauf des iPhones ''(Details siehe [[Apple iPhone#Juristische Auseinandersetzung|Juristische Auseinandersetzung]])''. Am 21. November 2007 teilte T-Mobile mit, dass das Gerät ab sofort für 999 Euro auch ohne Vertragsbindung und SIM-Sperre verkauft werde, um die gerichtlichen Vorgaben zu erfüllen. In der 46. Kalenderwoche ausgelieferte iPhones würden nachträglich kostenlos entsperrt. Am Verkaufspreis von 399 Euro für ein vertragsgebundenes Gerät ändere sich jedoch nichts.<ref>[http://www.welt.de/webwelt/article1385122/T-Mobile_muss_iPhone_ohne_Vertrag_verkaufen.html „T-Mobile muss iPhone ohne Vertrag verkaufen“] ([[Die Welt]]-online am 21. November 2007)</ref> Am 4. Dezember 2007 hob das Landgericht Hamburg die einstweilige Verfügung wieder auf; der exklusive Verkauf mit Vertragsbindung und SIM-Sperre verstoße nicht gegen das Wettbewerbs- oder Kartellrecht. T-Mobile reagierte sofort und verkaufte das iPhone wieder ausschließlich mit Vertrag.<ref>[http://de.news.yahoo.com/afp/20071204/tbs-d-justiz-telekommunikation-mobil-unt-f41e315_2.html „Deutsche Telekom darf iPhone weiter exklusiv anbieten“] ([[Agence France-Press|AFP]]-Meldung vom 4. Dezember 2007)</ref>

Seit dem 9. November 2007 ist das ''iPhone'' auch in [[Vereinigtes Königreich|Großbritannien]] exklusiv über das Mobilfunk-Unternehmen ''[[Telefónica O2 Europe|O2]]'' und das Vertriebsnetz von ''Carphone Warehouse'' erhältlich. Der Kauf ist für 269 [[Pfund Sterling|£]] (rund 387 Euro) nur mit einer Vertragsbindung an O2 möglich.<ref>[http://de.news.yahoo.com/dpa2/20070918/tbs-o2-erhlt-britische-exklusivrechte-fr-0e25ddc_1.html „O2 erhält britische Exklusivrechte für iPhone“] (dpa-Meldung vom 18. September 2007)</ref> In [[Frankreich]] wird das Gerät seit dem 28. November 2007 verkauft. Hier hat die [[France Télécom|France-Télécom]]-Tochter [[Orange (Unternehmen)|Orange]] den Alleinvertrieb bekommen, muss das iPhone aber aufgrund eines französischen Gleichheitsgesetzes auch ohne [[SIM-Lock]] oder mit einer Option auf Freischaltung anbieten. Die Gerätepreise liegen hier zwischen 399 Euro (mit Vertragsbindung) und 749 Euro (inklusive 100 Euro Freischaltungsgebühr, ohne Vertrag).<ref>[http://de.news.yahoo.com/dpa2/20071128/tbs-france-tlcom-verkauft-iphone-ohne-ve-9806f5a_1.html „France Télécom verkauft iPhone ohne Vertrag für 649 Euro“] (dpa-Meldung vom 28. November 2007)</ref> Die Markteinführung in Asien und Australien ist für 2008 geplant. Das Schweizer Telekommunikations-Unternehmen [[Swisscom]] teilte im Mai 2008 mit, dass es das iPhone noch im selben Jahr anbieten werde.<ref>[http://www.heise.de/newsticker/Swisscom-bringt-das-iPhone-in-die-Schweiz--/meldung/107832 „Swisscom bringt das iPhone in die Schweiz“] (heise.de am 14. Mai 2008)</ref>

=== Apple iPhone 3G ===
Am 9. Juni 2008 kündigte Apple auf der [[Worldwide Developers Conference#WWDC 2008|WWDC 2008]] an, das iPhone 3G ab dem 11. Juli 2008 parallel in 21 Ländern auszuliefern, darunter Deutschland, Österreich und die Schweiz. Am 17. Juli begann der Verkauf auch in Frankreich. Zusätzlich zu [[Enhanced Data Rates for GSM Evolution|EDGE]] unterstützt die neue Geräteversion die Mobilfunkstandards [[Universal Mobile Telecommunications System|UMTS]]/[[High Speed Downlink Packet Access|HSDPA]].<ref>http://www.apple.com/de/iphone/features/wireless.html</ref> Außerdem ist die Standortbestimmung mittels [[Assisted Global Positioning System|A-GPS]] möglich.<ref>http://www.apple.com/de/iphone/features/gps.html</ref> Das iPhone mit 8 GB wird mit schwarzer Gehäuserückseite ausgeliefert, bei der 16 GB-Variante besteht eine Auswahl zwischen schwarzen und weißen Geräten. Dabei wird die rückseitige Hülle künftig aus [[Kunststoff]] hergestellt. Als Grund dafür wird eine bessere Empfangsleistung bei niedrigerem Energieverbrauch angegeben. Die Preise wurden gegenüber dem Vorgängermodell gesenkt; in den USA ist das iPhone 3G zum Beispiel für 199 USD (8 GB) bzw. 299 USD (16 GB) erhältlich<ref>[http://newsticker.sueddeutsche.de/list/id/53409 „Apple präsentiert rundum erneuertes iPhone“] ([[Süddeutsche Zeitung]] am 10. Juni 2008)</ref>, in Deutschland liegt der von T-Mobile subventionierte Preis je nach Tarifbindung für die 8 GB-Variante zwischen einem Euro und 169,95 Euro, beziehungsweise zwischen 19,95 Euro und 249,95 Euro für die 16 GB-Variante.<ref>[https://www.t-mobile.de/iphone/showTariffs.do iPhone-Tarife bei T-Mobile]</ref> Seit dem 1. Oktober 2008 sind neue Tarife sowie ein Prepaid-Paket (Xtra Nonstop Tarif) des iPhone 3G in der 8GB Version für 569,95€ erhältlich. In diesem Paket sind 100€ Startguthaben inklusive.<ref>http://www.t-mobile.de/shop/handy/0,4855,2963-_10112-0-1-0,00.html</ref>

=== Namenshistorie ===
Bereits 1996 brachte die Firma [[Infogear Technology]] ein Tischtelefon mit E-Mail-Client unter dem Namen ''iPhone'' heraus und ließ dies markenrechtlich schützen.<ref>[[Gadget]]: [http://www.streettech.com/archives_gadget/iPhone.html ''Infogear iPhone'']</ref> Das Unternehmen wurde im Jahr 2000 von [[Cisco Systems]] aufgekauft; damit ging auch das [[Markenrecht]] an ''iPhone'' in den USA auf Cisco über. Im Dezember 2006 gab dann Ciscos Tochterfirma [[Linksys]] die Markteinführung einer Produktfamilie von [[IP-Telefonie|VoIP]]-Telefonen unter der Bezeichnung ''iPhone'' bekannt.<ref>[[Süddeutsche Zeitung]]: [http://www.sueddeutsche.de/computer/artikel/459/95364/ ''iPhone kommt nicht von Apple – Das Kuckucks-Ei'' (18. Dezember 2006)]</ref> Am 9. Januar 2007 teilte Cisco Systems mit, dass man sich mit Apple in Verhandlungen bezüglich der Markenrechte befinde und dass eine baldige Einigung zu erwarten sei. Nach einer [[Klage|Markenrechtsklage]] Ciscos gegen Apple vom 10. Januar 2007<ref>MarketWatch: [http://www.marketwatch.com/News/Story/Story.aspx?guid=%7b86AE140E-D93B-44A9-BD45-5A5202832F85%7d&siteid=yhoo&dist=yhoo ''Cisco pushes Apple for permission to use iPhone trademark'' (9. Januar 2007)]</ref> erzielten Cisco und Apple am 22. Februar 2007 nach außergerichtlichen Verhandlungen eine Einigung, die vorsieht, dass beide Unternehmen den Markennamen weltweit nutzen dürfen.<ref>[[heise online]]: [http://www.heise.de/newsticker/meldung/85682 ''Cisco und Apple legen iPhone-Namensstreit bei'' (21. Februar 2007)]</ref> Im Gegenzug wird geprüft, ob eine [[Interoperabilität]] beider Firmen in den Bereichen [[Sicherheit]] sowie [[Kommunikation]] für [[Verbraucher]] und [[Geschäftskunde]]n möglich ist.<ref>[[International Data Group|ChannelPartner Online]]: [http://www.computerpartner.de/unternehmenundmaerkte/238201/index.html ''‚iPhone‘-Deal von Apple und Cisco: Was amerikanische Analysten dazu sagen'' (23. Februar 2007)]</ref>, <ref>The Mercury News: [http://www.mercurynews.com/mld/mercurynews/16761179.htm ''Cisco-Apple truce over iPhone murky on details of collaboration'' (22. Februar 2007)]</ref>

Neben Cisco Systems hat auch das kanadische Unternehmen [[Comwave Telecom]] aus Toronto die Markenrechte an ''iPhone'' für sich beansprucht. Comwave Telecom vertreibt seit 2004 einen [[IP-Telefonie|VoIP]]-Dienst inklusive eigenem Mobilgerät unter dieser Marke. Allerdings wurde von Comwave Telecom der [[Markenschutz]] in [[Kanada]] erst 2005 beantragt, aber das ''Canadian Intellectual Property Office'' (CIPO) hat bislang die Rechte an ''iPhone'' noch nicht vergeben.<ref>Telekom Presse: [http://www.telekom-presse.at/channel_mobile/news_26904.html ''Debatte um den Namen ''iPhone'' geht weiter'' (29. Januar 2007)]</ref>

In Deutschland betreibt auch die [[Freenet AG]] einen VoIP-Dienst unter dem Namen ''iPhone''.

== Technik ==
[[Bild:Iphone-comparison.png|thumb|Größenvergleich zwischen einem iPod (4G), iPhone und iPod Nano (1G, von unten nach oben)]]
Nach Angaben des [[Chief Executive Officer|CEO]] von Apple, [[Steve Jobs]], wurden während der Entwicklung des iPhones über 300 [[Patent]]e angemeldet. Welche dieser Patentanmeldungen (wobei im amerikanischen Sprachbereich auch [[Geschmacksmuster]] als ''[design] patents'' bezeichnet werden) auch zu Schutzrechten, insbesondere im europäischen Raum führen, unterliegt den jeweils maßgeblichen Prüfungsverfahren für diese Anmeldungen und dem vorhandenen Stand der Technik. Dem [[Europäisches Patentamt|Europäischen Patentamt]] liegt zumindest eine Patentanmeldung vor.<ref>[[Süddeutsche Zeitung]]: [http://www.sueddeutsche.de/wirtschaft/artikel/7/99907/2/ ''Das iPhone – Sonderbare Geschäfte'' (30. Januar 2007)]</ref>

=== Zulieferer und Fertigung ===
Die Federführung der Produktion des Bildschirms hat das deutsche [[SDAX]]-Unternehmen [[Balda]].<ref>pressetext.de: [http://www.pressetext.de/pte.mc?pte=070110016 ''Touchscreens für Apples iPhone: Balda-Aktie explodiert'' (10. Januar 2007)]</ref> Die Produktion selbst erfolgt unter anderem durch den Bildschirm-Hersteller ''TPK'' in der chinesischen Sonderwirtschaftszone [[Xiamen]].<ref>EE Times Europe: [http://www.eetimes.eu/germany/196802556 ''Infineon und Balda profitieren von Apples iPhone'' (10. Januar 2007)]</ref> Balda ist zu 50 Prozent Anteilseigner von ''TPK''. Die Fertigungsmaschine, mit der die neuartigen Touchscreens für das iPhone von Apple produziert und mit der auch sonstige Oberflächen von Kunststoffteilen für Mobiltelefone und tragbare Spielekonsolen veredelt werden können, wurde von Balda in Kooperation mit dem [[TecDAX]]-notierten [[Maschinenbau]]er [[Singulus]] entwickelt.<ref>[[ARD]]: [http://www.boerse.ard.de/content.jsp?key=dokument_208708 ''Singulus mitten im iPhone-Hypes'' (23. Januar 2007)]</ref> Als weiterer Touchscreen-Lieferant wird der [[japan]]ische Elektronikkonzern [[Sharp]] vermutet, der dies, ähnlich wie Balda, bisher jedoch nicht offiziell bestätigt hat.<ref>[[Handelsblatt]].com am 25. Juni 2007: [http://de.biz.yahoo.com/25062007/299/iphone-fehlen-teile.html ''Für das iPhone fehlen Teile'']</ref> Kurz vor der Markteinführung Ende Juni 2007 wurde berichtet, dass die Balda-Order erhöht worden sei, weil es bei Sharp Lieferverzögerungen bei den Touchscreens gebe. Balda denke deshalb über den Bau eines zweiten Werkes in Xiamen nach.<ref>[[Deutsche Presse-Agentur|dpa]]-Meldung vom 26. Juni 2007: [http://de.biz.yahoo.com/26062007/3/apple-erhoeht-order-balda-iphone-bildschirme.html ''Apple erhöht Order bei Balda für iPhone-Bildschirme'']</ref>

Der im iPhone verbaute [[NAND-Flash|NAND]]-[[Flash-Speicher]] stammt von den Firmen [[Hynix]], [[Intel]], [[Micron Technology|Micron]], [[Samsung Electronics]] und [[Toshiba]], mit denen Apple eine Übereinkunft bis 2010 vereinbart hat.<ref>[[Apple|Apple Inc]]: [http://www.apple.com/pr/library/2005/nov/21flash.html ''Apple Announces Long-Term Supply Agreements for Flash Memory'' (21. November 2005)]</ref> Die Hardware-Stücklisten- und Fertigungskosten des iPhones liegen Apple-Analysten zufolge bei $ 265,83 für das 8 GB-Modell und somit bei etwa 67 Prozent des Verkaufspreises. Die Fertigungskosten des Touchscreens sollen laut iSuppli bei rund $ 27 liegen.<ref>iSuppli Corporation: [http://www.isuppli.com/news/default.asp?id=7308&m=1&y=2007 ''Apple iPhone to Generate 50 Percent Margin, According to iSuppli’s Preliminary Analysis'' (18. Januar 2007)]</ref>
Ein weiterer Zulieferer ist [[Infineon]], wie die Bestückung der [[Leiterplatte]] zu erkennen gibt.<ref>[http://i.cmpnet.com/techonline/uth/iPhone1_High.jpg ''Inside Apple’s IPhone''] (Bild der iPhone-Platine)</ref>

=== Prozessor und Grafikprozessor ===
In dem Apple iPhone arbeiten ein 667 MHz [[ARM Limited|ARM]]-1176-Prozessor (getaktet mit 412 MHz / iPod touch mit 532 MHz)<ref>[http://www.arm.com/markets/mid/armpp.html ARM Powered Products]</ref>, <ref>[http://www.arm.com/products/CPUs/ARM1176.html ARM 1176]</ref>, sowie ein 120 MHz [[PowerVR Technologies#MBX|PowerVR-MBX]]-Grafikprozessor<ref>[http://www.imgtec.com/powervr/mbx.asp Imagination Technologies PowerVR MBX]</ref>, <ref>[http://www.heise.de/newsticker/3D-Spiele-und-Video-auf-dem-Handy--/meldung/25068 3D-Spiele und Video auf dem Handy]</ref> der bis zu vier Millionen Dreiecke pro Sekunde verarbeiten können soll. Die PowerVR-MBX-Technologie wird auch im [[Nokia Nseries#Nokia N95|N95]] von Nokia zur Darstellung von Spielen eingesetzt.

=== Bedienung ===
Das iPhone ist mit einem berührungsempfindlichen Bildschirm ([[Touchscreen]]) unter einer Abdeckung aus optischem [[Linse (Optik)|Glas]] ausgestattet, den Apple als ''[[Multi-Touch]]'' bezeichnet und der bis zu zwölf Berührungsimpulse gleichzeitig verarbeiten kann. Das iPhone verfügt lediglich über fünf physische Bedienelemente: die Home-Taste auf der Vorderseite, seitlich zwei Tasten zur Lautstärkeregelung, einen Schalter zum Stummschalten und oben die [[Sleep-Modus|Standby-Taste]]. Zum Schreiben wird an entsprechenden Stellen eine Tastatur eingeblendet, bei ausschließlicher Eingabemöglichkeit von Ziffern (beispielsweise beim Wählen einer Telefonnummer) wird ein Ziffernblock aufgerufen.

=== Sensoren ===
Das iPhone hat außerdem drei [[Sensor]]en. Durch einen Näherungssensor wird bei dem Gerät automatisch die Eingabefunktion sowie die Bildschirmbeleuchtung ausgeschaltet, wenn es an das Ohr gehalten wird. Die Anzeige wird ferner automatisch umgestellt, wenn das Gerät vertikal oder horizontal gehalten wird. Dazu dient ein Drei-Achs-[[Beschleunigungssensor]].<ref>http://www.st.com/stonline/products/literature/ds/12726/lis302dl.htm verbauter Beschleunigungssensor LIS302DL </ref> Über einen Helligkeitssensor kann die Bildschirmhelligkeit an die Lichtverhältnisse der Umgebung angepasst werden bzw. in dunklerer Umgebung durch Abdunkeln Energie gespart werden.

=== Mobilfunk und Vernetzung ===
[[Bild:IPhone SIM Slot.jpg|thumb|Das SIM-Kartenfach des iPhones mit eingelegter SIM-Karte]]
[[Bild:IPhoneBluetooth scale mm.jpg|thumb|upright 0.5|Das Bluetooth-Headset des iPhones.]]
==== iPhone (1. Generation) ====
Die Telefonie erfolgt bei der ersten Version des iPhones ausschließlich über [[Global System for Mobile Communications|GSM]] und Daten können per [[Enhanced Data Rates for GSM Evolution|EDGE]] empfangen werden.

Das Gerät ist ein sogenanntes ''[[Quadband]]''-Telefon und unterstützt die [[Global System for Mobile Communications#Verwendete Frequenzen|Frequenzen]] 850, 900, 1800 und 1900 [[Hertz (Einheit)|MHz]]. Des Weiteren kann es sich per [[Wireless Local Area Network|WLAN]] (802.11b/g) oder [[Bluetooth]] verbinden. Da sich das iPhone nicht an den Bluetooth-Standard ''Advanced Audio Distribution Profile'' ([[A2DP]]) hält, ist eine Zusammenarbeit mit entsprechenden Bluetooth-Stereo-Geräten wie Kopfhörern und Autoradioschnittstellen anderer Hersteller nicht möglich.<ref>Tecchannel [http://www.tecchannel.de/pc_mobile/news/484080/index.html Apple ignoriert Bluetooth-Standard]</ref> Auch zahlreiche andere Bluetooth-Profile sind nicht vorhanden oder aktiviert. Das iPhone der ersten Generation unterstützt keinen Breitbandzugang per [[Universal Mobile Telecommunications System|UMTS]] oder [[High Speed Downlink Packet Access|HSDPA]]. Auch die im November 2007 eingeführte Europa-Version verfügt nicht über UMTS. Aufgrund der im Vergleich zu den USA noch schlechten Netzabdeckung mit dem Übertragungsstandard EDGE werden die Kunden vielfach auf [[General Packet Radio Service|GPRS]] zurückgreifen müssen.<ref>[http://www.mactechnews.de/index.php?id=17788 ''Apple iPhone in Großbritannien mit O2-Vertrag für 269 Pfund''] (auf mactechnews.de am 17. September 2007)</ref> T-Mobile kündigte allerdings an, ''„bis Ende 2007 […] EDGE im gesamten GSM-Netz an[zu]bieten“''.<ref>[http://www.t-mobile.de/unternehmen/presse/pressemitteilungen/1,12219,19337-_,00.html Pressemitteilung vom 19. September 2007]</ref>
Das iPhone der ersten Generation verfügt über keinen [[Global Positioning System|GPS]]-Empfänger. Eine Ortsbestimmung (und Anzeige in Google Maps) ist dennoch möglich, sie basiert auf [[Triangulation]] der jeweils vom iPhone empfangenen Mobilfunkzellen sowie der Auswertung von bekannten WLAN Hotspots. In Städten ist die erreichbare Genauigkeit mitunter relativ hoch und kann unter 50 Meter liegen, eine Internetverbindung ist zur Positionsbestimmung wegen der Online-Standortabfragen der empfangenen Zellen zwingend notwendig.

==== iPhone 3G ====
Wie von Steve Jobs im September 2007 angekündigt, wurden die Kommunikationsfähigkeiten des iPhones mit der Einführung des ''iPhone 3G'' am 11. Juli 2008 erweitert.<ref>[http://www.pocket-lint.co.uk/news/news.phtml/10212/11236/apple-iphone-3g-coming-2008.phtml „Apple promises 3G iPhone for 2008“] auf pocket-lint.co.uk am 18. September 2007</ref>

Die bedeutendste Neuerung ist, dass das ''iPhone 3G'' auch UMTS-Mobilfunk unterstützt.<ref>http://www.chip.de/news/Apple-iPhone-2-ist-da-UMTS-und-GPS-an-Bord_31692914.html</ref> Außerdem ist die Verwendung von [[High Speed Downlink Packet Access|HSDPA]], einer Verbesserung von UMTS möglich, soweit dies netzseitig implementiert ist.

Mit dem eingebauten [[Global Positioning System|GPS]]-Modul ist es möglich, über "Google Maps" seine exakte Position zu bestimmen; vorausgesetzt wird eine aktive Internet-Verbindung zum Abrufen des Kartenmaterials. Drittanbieter wie [[TomTom]] haben bereits eine lauffähige Navigations-Software entwickelt, aber noch nicht veröffentlicht.
Viele im [[AppStore]] erhältliche Programme verwenden das GPS-Modul, z. B. um einen gelaufenen Weg nachzuvollziehen oder um die Geschwindigkeit zu ermitteln.

Einige Kritiker bemängeln, dass die GPS-Antenne (eine der zehn internen Antennen) zu klein sei, um eine genaue Ortung zuzulassen. Viele Tests berichten jedoch von punktgenauer Ortung.

==== Synchronisation ====
Über iTunes können Kontakte, Termine, E-Mails, Fotos und Musikstücke mit einem Computer synchronisiert werden. Seit Version 2.0 der Firmware, die gleichzeitig mit dem iPhone 3G veröffentlicht wurde, kann das iPhone auch über einen [[Microsoft Exchange]]-Server synchronisieren.<ref>http://cgi.cnet.de/alpha/artikel/handys/200806/iphone-3g-apple-integriert-umts-hsdpa-gps-exchange-und-push-e-mail.htm</ref> Außerdem ist die Synchronisation über den kostenpflichtigen Dienst [[MobileMe]] von Apple möglich.<ref>http://www.apple.com/de/mobileme/</ref>

=== Anrufbeantworter ===
Mit ''Visual Voicemail'' können aufgesprochene Nachrichten übersichtlich in einer Liste angezeigt werden. Das Anrufen des Anrufbeantworters entfällt so. Die Mobilfunkanbieter müssen für diesen Dienst Server mit Apple-Software betreiben. Im Dezember 2007 erhob die US-Firma ''Klausner Technologies'' Klage gegen Apple und AT&T wegen vermuteter widerrechtlicher Nutzung von zwei Patenten bei Visual Voicemail.<ref>[http://www.golem.de/0712/56353.html ''Patentklage gegen Visual Voicemail im iPhone''] (www.golem.de am 7. Dezember 2007)</ref>

=== Medienwiedergabe ===
Der Prozessor des iPhone ermöglicht das Abspielen von hochauflösenden komprimierten Videos im [[H.264]]-Standard, die auf die 480 × 320 Pixel des Bildschirms heruntergerechnet werden. [[Multimedia Messaging Service|MMS]] für Audio und Video wird nicht unterstützt.<ref>MacRumors.com: [http://www.macrumors.com/2007/06/08/iphone-details-from-atandts-sales-training-workbook/ ''iPhone Sales Training Workbook'' (8. Juni 2007)]</ref> Das Gerät besitzt sämtliche Funktionen des [[Apple iPod]] mit Video-Abspielmöglichkeit und die aus [[iTunes]] bekannten Animationen der CD-Titelbilder ''([[CoverFlow]])''. Bei der Wiedergabe von Videos wechselt die Ansicht in die Horizontale, wodurch man einen ''Widescreen''-Bildschirm erhält.

=== Headset ===
Das Standard-Headset des iPhones ist mit weißen Stereokopfhörern ausgestattet und verfügt über ein Mikrofon. Es verfügt darüber hinaus über einen sogenannter ''Klicker'' mit dem man in verschiedenen Situationen das iPhone steuern kann.

=== Digitalkamera ===
[[Bild:IPhone at Macworld (rear view).jpg|thumb|upright|Rückseite des iPhone mit Digitalkamera]]
Das Gerät verfügt über eine Zwei-[[Megapixel]]-[[Digitalkamera]], deren Linsenöffnung ohne Abdeckung in der Rückwand des Mobiltelefons ausgespart ist und die nur Einzelbilder aufnehmen kann.

=== Datenspeicher ===
Bei dem Speicher des iPhone handelt es sich um 8 GB beziehungsweise 16 GB großen [[NAND-Flash|NAND]]-[[Flash-Speicher]]. Ein Erweiterungs-Steckplatz für Speicherkarten ist nicht vorhanden. Die Größe des Arbeitsspeichers ([[Random Access Memory|RAM]]) beträgt 128 MB DRAM.

=== Energiespeicher ===
Die Akkukapazität des [[Lithium-Polymer-Akkumulator|Lithium-Polymer-Akkus]] genügt laut Hersteller für Telefongespräche von bis zu acht Stunden sowie sieben Stunden für Videofunktionen und sechs Stunden Surfen im Internet. Sie solle ferner ausreichen, um mit dem iPhone bis zu 24 Stunden Musik hören zu können. Die Akkukapazität im [[Bereitschaftsbetrieb|Standby-Modus]] betrage 250 Stunden.<ref>[http://www.apple.com/iphone/specs.html Technische Daten auf apple.com] (englisch)</ref> Nach ersten Tests beträgt die Sprechzeit über sieben Stunden, Musik kann man knapp über 22 Stunden lang hören, und bei Internetnutzung entlädt sich der Akku in etwas mehr als neun Stunden.<ref>[http://online.wsj.com/article/SB118289311361649057.html ''Testing Out the iPhone''] ([[The Wall Street Journal]], englisch)</ref>
Oftmals erreicht der Akku erst nach einigen Wochen Betrieb seine volle Leistungsfähigkeit, daher können die Laufzeiten bei Neugeräten kürzer sein.

Der eingebaute Akku kann vom Anwender nicht selbst ausgetauscht werden.<ref>[http://www.rp-online.de/public/article/aktuelles/digitale/handy/454757 Meldung] von rp-online 3. Juli 2007</ref> Apple selbst bietet den Wechsel für rund $ 86 (inklusive Versandkosten) innerhalb von drei Werktagen an; ein Verfahren, das von Verbraucherschützern scharf kritisiert wird.

=== Anschlüsse ===
Das iPhone verfügt über eine 30-polige Anschlussbuchse ''Dock-Connector'', die physikalisch weitgehend identisch mit der der iPods ist. Der Anschluss enthält Leitungen für Stereoton, Composit-Video ([[Fernsehsignal#FBAS-Signal|FBAS]]), Komponenten-Video (NTSC und PAL, eine Neuerung gegenüber früheren iPods, die einen S-Video Ausgang haben; für die Verwendung dieses Ausgangs mit typischen europäischen Fernsehern wird eine Wandlerbox zur Wandlung in ein SCART-RGB-Signal benötigt), eine [[serielle Schnittstelle]] (zum Beispiel zur Steuerung), [[Universal Serial Bus|USB]] und [[FireWire]]. Es ist nicht bekannt, ob einer der bisher beim iPod freien Kontakte des Dock-Connectors zum Anschluss einer externen Antenne dienen kann. Das iPhone verfügt nicht über einen speziellen Antennenanschluss. Auf der Oberseite des iPhone ist eine 3,5 Millimeter-[[Klinkenbuchse]] mit vier Kontakten (Stereoübertragung plus Zusatzleitung zum Anschluss des Mikrophons des Headsets oder eines ''TTY''-[[Schreibtelefon]]s für Gehörlose und Schwerhörige) untergebracht, in die [[Kopfhörer]] und [[Headset]]s eingesteckt werden können.

== Software ==
=== Betriebssystem ===
Als [[Betriebssystem]] dient ein angepasstes [[Mac OS X]] auf Flash-Speicher (laut ''[[The New York Times|New York Times]]'' 700 MB groß) ähnlich der Version für [[Apple Macintosh|Mac]]s, allerdings fällt der Funktionsumfang geringer aus.<ref>News York Times: [http://www.nytimes.com/2007/06/27/technology/circuits/27pogue.html?_r=3&pagewanted=1&hp&adxnnl=0&adxnnlx=1182895890-m2yYGaQuzXLdm1ZcCLu6fw&oref=slogin&oref=slogin ''The iPhone Matches Most of Its Hype'' (englisch; 27. Juni 2007)]</ref> Der [[Betriebssystemkern]] ''(Kernel)'' der Version 1.0 identifiziert sich als [[Darwin (Betriebssystem)|Darwin]] 9.0.0d1, was gemäß der von Apple seit Mac OS X 10.1 verwendeten [[Namenskonvention]] der Mac-OS-X-Betriebssystemversion [[Mac OS X#Mac OS X 10.5 (Leopard)|10.5 ''(Leopard)'']] entspricht. Das iPhone verfügt jedoch nicht über eine [[Java-Plattform]].

=== Bedienkonzept ===
Das Apple iPhone wird praktisch vollständig über den [[Multi-Touch]]-Bildschirm bedient, wobei die verwendete [[Grafische Benutzeroberfläche|grafische Benutzeroberfläche]] jeweils nur ein Programmfenster anzeigt. Der Bildschirm ist meist so aufgeteilt:
* Eine rund 4 mm breite Statusleiste am oberen Bildrand (Anzeige von Feldstärke, Provider, EDGE, Batteriestatus etc.)
* Darunter ein 1 cm breiter Menübereich mit Programmnamen und, sofern erforderlich, einigen wenigen Menüpunkten (etwa ein ''+'' zum Hinzufügen neuer Einträge, ein ''Bearbeiten'' zum Bearbeiten von Einträgen, oder ''Tag'', ''Monat'', ''Jahr'' zum Umschalten der Kalenderansichten)
* Darunter schließt sich das eigentliche Programmfenster an
* Gegebenenfalls findet sich unter dem Programmfenster noch eine weitere Menüleiste, mit der Programme in verschiedene Modi geschaltet werden können (beispielsweise die Uhr zwischen Weltzeit, Timer, Wecker und Stoppuhr)

Der Touchscreen wird mit verschiedenen Fingerbewegungen bedient:
* Tippen: Unterliegende Funktion wird ausgeführt
* Doppeltippen: Vergrößert/verkleinert einen Bildausschnitt
* Antippen und halten auf Texteingabefeldern: Es wird eine Lupenfunktion eingeblendet. Durch Bewegen des gedrückten Fingers kann man die Textmarke zu einer gewünschten Stelle bewegen.
* ''Wischen'' (Aufsetzen des Fingers in der Bewegung, Wischen über den Bildschirm, Loslassen): Dient dem [[Bildlauf|Rollen]] durch Listen (Adressen und Ähnliches; vertikale Wischbewegung) oder Ansichten (Wetterbilder, ''Cover Flow'', Webseiten; horizontales Wischen)
* Antippen und Bewegen: Verschiebt die Ansicht auf dem Bildschirm (etwa bei Landkarten oder HTML-Seiten)
* Greifen und Bewegen: Bei bestimmten Funktionen kann durch Antippen eines Objekts (zum Beispiel Anruferliste, Ortsliste für das Wetter) an einer bestimmten, durch drei horizontale Striche gekennzeichneten Stelle, das Objekt durch Bewegen des Fingers an eine andere Stelle der Liste verschoben werden.
* Spreizen und Zusammendrücken: Diese Geste wird mit zwei Fingern durchgeführt, die voneinander weg oder zueinander hin bewegt werden. Die Ansicht auf dem Bildschirm wird vergrößert oder verkleinert (etwa im Webbrowser, bei der Ansicht von Fotos und bei Landkarten).

Die Texteingabe erfolgt über eine auf dem Bildschirm eingeblendete [[QWERTY-Tastaturbelegung|QWERTZ-Tastatur]], wobei Umlaute durch Halten statt Tippen der Grundtaste und anschließendem Bewegen des Fingers zu dem gewünschten, nach kurzer Wartezeit eingeblendeten Umlaut eingegeben werden können. Der eingegebene Text wird in einem kleinen Feld am oberen Bildrand oder direkt über der Tastatur angezeigt. Die Genauigkeit der Eingabe hängt von der Größe der Finger und dem manuellen Geschick des Bedieners ab, wobei eine lernfähige Korrekturfunktion die Eingabe unterstützt.

=== Mitgelieferte Programme ===
{| class="prettytable float-right"
|+ Programme
|- align="center" class="hintergrundfarbe5"
! Art
! Name
|-
| [[Webbrowser]] || [[Safari (Browser)|Safari]]
|-
| [[E-Mail-Programm]] || [[Mail (Apple)|Apple Mail]] *
|-
| [[Kalendarium]] || [[Apple iCal]] (ohne ToDo)
|-
| Software-Quelle || ''App-Store'' **
|-
| Kartendienst || [[Google Maps]]
|-
| Notizen || ''Notes''
|-
| valign="top" | [[Dienstprogramm]]e ||
* [[YouTube]] Player<ref>[http://www.apple.com/pr/library/2007/06/20youtube.html ''YouTube Live on Apple TV Today; Coming to iPhone on June 29''] auf apple.com, 20. Juni 2007 (englisch)</ref>
* Taschenrechner
* Wetter
* Aktien
* ''Text'' (zum Verfassen von SMS-Nachrichten)
* Einstellungen (inklusive WiFi-Konfiguration)
|-
| colspan="2" | * Angepasste Version mit zusätzlicher ''Push''-[[Internet Message Access Protocol|IMAP]]- Mail-Funktion via [[Yahoo]]-Mail ** ''Seit Juli 2008''
|}

Eine Reihe von Programmen (siehe Tabelle) wird mit dem iPhone mitgeliefert. Trotz der Namensgleichheit mit entsprechenden Programmen für Mac OS X handelt es sich bei den Programmen des iPhone um veränderte Versionen, die beispielsweise über einen geringeren Funktionsumfang verfügen. Bei der Vorstellung des Geräts hatte Steve Jobs bereits angekündigt, dass die Software des iPhone weiterentwickelt werden soll und neue Versionen des Betriebssystems (vgl. oben) sowie weitere Programme von Apple angeboten werden sollen. Bislang beschränken sich Änderungen weitestgehend auf kleinere Korrekturen, die einzigen neu hinzugekommenen Funktionen sind zwei Programme zum Laden von Dateien aus dem [[iTMS]] über WLAN-Verbindungen.

=== Weitere Programme ===
In seiner [[Keynote]] zur Einführung des iPhone wies Steve Jobs auf [[Cocoa]] als Haupt-[[Programmierschnittstelle|API]] der iPhone-Software hin.

Auf der [[Worldwide Developers Conference|WWDC]] 2007 wurde zunächst bekanntgegeben, dass Dritthersteller keine klassischen Anwendungen für das iPhone schreiben können und dies stattdessen über [[Webapplikation]]en gelöst werden soll, die auf offenen Standards wie [[Ajax (Programmierung)|Ajax]] basieren und im Webbrowser [[Safari (Browser)|Safari]] angezeigt werden können. Von Safari aus können dann iPhone-eigene Anwendungen wie [[Google Maps]] oder die Telefonfunktion genutzt werden. Eine ebenfalls angekündigte Version von Safari für [[Microsoft Windows]] ermöglicht das Mac-OS-X-unabhängige Testen solcher Anwendungen für Entwickler aus der Windows-Welt. Durch das dabei verwendete, auch von [[Java (Programmiersprache)|Java]] bekannte [[Sandbox|Sandkasten]]-Prinzip soll das iPhone vor Fehlfunktionen und Manipulationen durch Software besser geschützt werden.

Am 17. Oktober 2007 stellte Steve Jobs (nach zahlreichen Protesten von Entwicklern) auf der Apple-Webseite als ''Hot News'' ein Entwicklungswerkzeug auch für native Anwendungen in Aussicht. Die Freigabe des [[Software Development Kit|SDK]]s werde bis Februar 2008 dauern, weil Apple versuche, zwei entgegengesetzte Ziele zu verwirklichen – eine offene Plattform für Entwickler zu bieten und gleichzeitig das iPhone vor Viren, Malware, Angriffe auf private Daten usw. zu schützen. Tatsächlich wurde das Kit jedoch bis Ende Februar nicht veröffentlicht; am 27. Februar 2008 kündigte Apple aber für den 6. März 2008 einen Pressetermin an, auf dem unter Anderem weitere Informationen über das SDK bekannt gegeben werden sollten.<ref>[http://www.macrumors.com/2008/02/27/iphone-ipod-touch-sdk-details-march-6th/ topright iPhone, iPod Touch SDK Details March 6th] (auf MacRumors.com)</ref> Auf dieser Veranstaltung wurde die Firmware-Version 2.0 vorgestellt, welche neben [[Microsoft Exchange Server]] und anderen Netzwerkprotokollen auch das iPhone SDK unterstützt. Die neue Firmware ist seit dem 11. Juli 2008 öffentlich verfügbar.

Apple entwickelte für das iPhone eine mobile Variante seines Cocoa-[[Framework]]s namens ''Cocoa touch''. Das SDK wird zusammen mit einer neuen Version von Apples [[integrierte Entwicklungsumgebung|integrierter Entwicklungsumgebung]] [[Xcode]] ausgeliefert. Darin enthalten ist auch ein iPhone-[[Emulator]], der es ermöglicht, die Anwendungen während der Entwicklungsphase auf dem [[Apple Macintosh|Mac]] zu testen. Der Vertrieb der Programme erfolgt in dem [[App Store]] über die gleichnamige iPhone-Anwendung oder [[iTunes]]. Die Entwickler können den Preis für ihre Software selbst festlegen, Apple nimmt jedoch 30 % davon als Provision. Während das SDK selbst kostenlos von Apples Entwicklerseiten bezogen werden kann, ist für die Veröffentlichung im ''App Store'' ein kostenpflichtiges Entwicklerkonto zum Preis von 99 (Standard) oder 299 US-Dollar (unternehmensinterne Anwendungen) erforderlich.<ref>[http://www.mactechnews.de/news/index.html?id=140491 MacTechNews: Das iPhone SDK]</ref>

In der Zwischenzeit haben Programmierer Verfahren etabliert, um Programme auf dem Gerät lauffähig zu machen, die nicht als Webapplikation ausgeführt sind (sogenannte native Applikationen). Nachdem es Hackern im Juli 2007 erstmalig gelungen war, ein [[Hallo-Welt-Programm]] auf dem iPhone auszuführen<ref>[http://www.rautemusik.fm/netzwelt/552-Erstmals-iPhone-Schutz-umgangen.html ''iPhone Hello World''] (auf [[RauteMusik|RauteMusik.FM]] am 31. Juli 2007)</ref>, lagen Mitte September 2007 bereits über 60 solcher Applikationen vor, einschließlich einem Installer und [[Doom]] (das ursprünglich mit [[NeXTStep]] entwickelt worden ist und seitdem traditionell jede [[Portierung]] dieses Betriebssystems mitvollzieht).<ref>[http://www.modmyiphone.com/nativeapps/the-list/ Liste nativer Programme] (auf ModMyiPhone.com am 18. September 2007, englisch)</ref> Apple steht diesen Programmieransätzen neutral gegenüber; stört also die Entwicklungen nicht, bemüht sich aber auch nicht, die Lauffähigkeit solcher Programme mit späteren Versionen des Betriebssystems zu erhalten.<ref>[http://www.gearlog.com/2007/09/apples_joswiak_we_dont_hate_ip.php#more Interview mit Greg Joswiak] (auf Gearlog.com am 11. September 2007, englisch)</ref>

== Kritik ==
=== Umweltverträglichkeit ===
[[Greenpeace]] kritisiert Apple für die im iPhone verwendeten Schadstoffe. Anders als bei modernen Mobiltelefonen üblich, fänden sich im iPhone noch schädliche Materialien wie [[Brom]] und [[Polyvinylchlorid]] (PVC). Greenpeace zeigt sich vor allem nach Steve Jobs’ offenem Brief ''A Greener Apple''<ref>[http://www.apple.com/hotnews/agreenerapple/ ''A Greener Apple''] (Offener Brief von Steve Jobs, englisch)</ref>, in dem er umweltfreundlichere Apple-Produkte angekündigt hatte, enttäuscht und fordert ein ''green iPhone''.<ref>[http://macnews.de/news/103576.html ''Greenpeace findet schädliche Materialien im iPhone''] (auf www.macnews.de am 13. Oktober 2007)</ref> Der US-Verband ''Bromine Science and Environmental Forum (BSEF)'' reagierte auf die Vorwürfe mit der Erklärung, dass die genannten Substanzen alle zur Verwendung zugelassen seien. Greenpeace habe zudem nicht nachweisen können, in welcher Konzentration und Zusammensetzung Brom im iPhone vorkomme, da die verwendeten Analysemethoden dafür nicht ausgereicht hätten.<ref>[http://www.heise.de/newsticker/meldung/97792 ''Chemische Industrie kritisiert Kritik an Apples iPhone''] (heise.de am 23. Oktober 2007)</ref>

Der US-Verbraucher- und Umweltschutzverband ''Center for Environmental Health (CEH)'' drohte Apple mit einer Klage. Nach Ansicht des CEH verstößt Apple mit dem iPhone gegen Abschnitt 65 des kalifornischen ''Safe Drinking Water and Toxic Enforcement Act of 1986'', der vorsieht, dass gefährliche Stoffe nicht ohne deutliche Warnung verwendet werden dürfen.<ref>[http://www.cehca.org/press-releases/eliminating-toxics/iphone-phthalates-violate-california-law/ ''iPhone Phthalates Violate California Law''] (Pressemitteilung des CEH, 15. Oktober 2007, englisch)</ref> Das ''CEH'' forderte Apple auf, für bisher gekaufte Geräte eine [[Rückrufaktion]] zu starten und zukünftig angebotene Geräte mit einem deutlichen Warnhinweis auf die schädlichen Substanzen zu versehen.

=== Sicherheit ===
Schon kurz nach dem Verkaufsstart wurden zahlreiche Sicherheitslücken in der [[Firmware]] des iPhone entdeckt. So wurden zum Beispiel zwei [[Kennwort|Passwörter]], mit denen zentrale Funktionen des iPhone gesichert sind, schnell entschlüsselt. Das Passwort ''dottie'' erlaubte es zum Beispiel, Programme mit [[Root-Account|Root-Rechten]] zu starten; ''alpine'' war ein weiteres entschlüsseltes Passwort.<ref>[http://www.theregister.co.uk/2007/07/03/iphone_hacking_progress ''iPhone Hacking Process''] (auf theregister.co.uk am 3. Juli 2007, englisch)</ref>

Sicherheitsexperten entdeckten zu Beginn beim iPhone eine weitere schwerwiegende Sicherheitslücke. Durch sie konnten persönliche Daten des Besitzers ausgespäht werden, etwa Telefonnummern.<ref>Welt Online: [http://www.welt.de/webwelt/article1048126/Hacker_haben_beim_iPhone_leichtes_Spiel.html Hacker haben beim iPhone leichtes Spiel] vom 23. Juli 2007</ref> Die neue Firmware-Version 1.0.1 am 30. Juli 2007 behob dieses Problem offenbar.<ref>[http://docs.info.apple.com/article.html?artnum=306173 Apple: ''About the security content of iPhone v1.0.1 Update'']</ref>

Das Fraunhofer-Institut für sichere Informationstechnologie (SIT) hat eine Sicherheitslücke im iPhone entdeckt, durch die eine Webseite das Mobiltelefon zur Anwahl einer beliebigen Telefonnummer veranlassen kann.<ref>Heise Online: [http://www.heise.de/newsticker/iPhone-telefoniert-ferngesteuert--/meldung/119222 iPhone telefoniert ferngesteuert] vom 20. November 2008</ref> Die neue Firmware-Version 2.2 vom 21. November 2008 behebt das Problem.

=== Empfangsprobleme ===
Eine größere Anzahl von Nutzern beklagt sich über schlechten Empfang und Verbindungsproblemen mit dem iPhone 3G. Ein Diskussionsforum bei Apple sowie weitere Foren weltweit diskutieren das Thema. Eine offizielle Erklärung von Apple gibt es zu diesem Thema nicht. <ref>[http://www.iphoneatlas.com/2008/09/22/iphone-3g-signal-strength-problems-persist-in-os-21/ iPhone 3G Signal Strength Problems Persist in OS 2.1]</ref> <ref>[http://news.cnet.com/8301-13579_3-10017657-37.html CNET readers share their iPhone 3G stories]</ref> <ref>[http://www.gearlog.com/2008/07/iphone_3g_reception_problems_y.php iPhone 3G Reception Problems? You're Not Alone]</ref>

=== Plagiatvorwurf ===
[[LG Electronics]] wirft Apple vor, das Design des iPhone vom ''Prada phone by LG'' kopiert zu haben. Dieses Mobiltelefon – ebenfalls mit Touchscreen und stiftloser Bedienung – wurde bereits im September 2006, rund vier Monate vor der ersten Präsentation des iPhone, vorgestellt und sein Design ausgezeichnet. Angesichts der langen Entwicklungszeiten eines neuen Produktes wie dem iPhone ist dies jedoch unwahrscheinlich.

== SIM-Lock ==
[[Bild:Entsperrtes iPhone.jpg|thumb|upright 0.7|Entsperrtes iPhone, eingebucht bei Vodafone Neuseeland]]
Das iPhone wird zur Zeit in den USA ausschließlich mit einem [[SIM-Lock]] verkauft. Es kann dadurch nur mit der bereits ab Werk mitgelieferten SIM-Karte des Mobilfunk-Betreibers AT&T verwendet werden. Ein Vertrag mit AT&T ist auch erforderlich für die Aktivierung der Funktionen des iPhones, die nicht in Zusammenhang mit der Telefon-Funktion stehen, etwa iPod, WLAN und Digitalkamera. Auch in Europa wird die Nutzung des iPhones zum Teil auf das Funknetz jeweils eines Anbieters pro Land beschränkt.

=== Offizielle Entsperrung ===
Die Gesetzeslage in einigen Ländern Europas verbietet einen SIM- und Netlock von Seiten der Provider/Anbieter. In folgenden Ländern kann das iPhone deshalb entsperrt erworben werden:
* Italien
* Belgien
* Frankreich (nach 6 Monaten)
* Tschechien
* Griechenland

In [[Hongkong]] vertreibt Apple das iPhone nebem dem offiziellen Provider [[Drei (Mobilfunkmarke)|3 Hong Kong]] auch über den eigenen Online-Store.<ref>[http://www.apple.com/hk/en/iphone/buy/ Apple Hong Kong: Where to buy iPhone]</ref>

=== Inoffizielle Entsperrungen ===
Am 3. Juli 2007 beschrieb der norwegische [[Content Scramble System|CSS]]- und mehrfache iTunes-[[Digitale Rechteverwaltung|DRM]]-Knacker [[Jon Lech Johansen]] in seinem Blog<ref>nanocr.eu: [http://nanocr.eu/2007/07/03/iphone-without-att/ ''iPhone Independence Day'' (3. Juli 2007)]</ref>, wie man die Funktionen des iPhones auch ohne AT&T-Vertrag nutzen kann.<ref>heise.de: [http://www.heise.de/newsticker/meldung/92188 ''iPhone-Aktivierung teilweise geknackt'' (4. Juli 2007)]</ref>

Am 14. August 2007 berichtete der weltweit populäre Technik-Blog ''Gizmodo'', dass sich das iPhone mit Hilfe einer sogenannten ''[[Turbo-SIM-Karte]]'' auch in anderen GSM-Netzen betreiben lasse.<ref>gizmodo.com: [http://www.gizmodo.com/gadgets/breaking/iphone-100-unlocked-using-turbo-sim-card-289148.php ''iPhone 100% Unlocked Using Turbo SIM Card'' (14. August 2007)]</ref> Am 6. September veröffentlichte die Webseite des ''APC Magazine'' (ehemals: ''Australian Personal Computer Magazine'')<ref>en.wikipedia.org: ''[[:en:Australian Personal Computer|APC im englischen Wikipedia]]''</ref> eine Zehn-Schritte-Anleitung zur Nutzung des iPhones in weltweit allen GSM-Netzen mit Hilfe einer Turbo-SIM-Karte.<ref>apcmag.com/: [http://apcmag.com/node/7122/ ''Hack the iPhone to work anywhere in the world'' (6. September 2007)]</ref>

Am 23. August 2007 gelang es dem US-Amerikaner George Hotz, ohne den Umweg über eine Turbo-SIM-Karte die Beschränkung der Nutzung seines iPhones auf das AT&T-Netz aufzuheben. In seinem Blog beschrieb er dazu ein Verfahren, das komplizierte Eingriffe in die Hardware erfordert.<ref>iphonejtag.blogspot.com: [http://iphonejtag.blogspot.com/2007/08/step-1.html ''Finding JTAG on the iPhone''] (iphonejtag.blogspot.com am 24. August 2007, englisch)</ref> Danach war es ihm möglich, sein iPhone im US-Netz von T-Mobile anzumelden.<ref>iphonejtag.blogspot.com: [http://bp2.blogger.com/_NJ4JFBfr1tY/Rs3ATDhDtyI/AAAAAAAAALI/BTZgOOSRN0Q/s320/finished.JPG ''Foto: iPhone T-Mobile'' (23. August 2007)]</ref>

Am 11. September 2007 gelang dem ''iPhone Dev Team'' die Programmierung einer Software-Netlock-Entsperrung für das iPhone, die anschließend frei im Internet zur Verfügung gestellt wurde. Damit war das iPhone weltweit in allen GSM-Netzen auch für iPhone-Besitzer nutzbar, die den technischen Aufwand bislang verfügbarer komplizierter Hacks gescheut haben.<ref>gizmodo.com: [http://gizmodo.com/gadgets/breaking/iphone-free-software-unlock-achieved-hacker-claims-+-details-coming-298825.php ''iPhone Free Software Unlock Confirmed! (Death Star Explodes)'' (11. September 2007)]</ref>, <ref>engadget.com: [http://www.engadget.com/2007/09/11/iunlock-released-the-first-free-open-source-iphone-sim-unlock/ ''iUnlock released: the first free, open source iPhone SIM unlock software'' (11. September 2007)]</ref>

Seit Erscheinen der Firmware-Version 1.1.1 am 27. September berichteten Besitzer von SIM-Lock-entsperrten Geräten nach einer Firmware-Aktualisierung über massive Probleme: Die neue Firmware-Version setzte die Entsperrungen zurück und das gesamte Gerät außer Betrieb.<ref>[http://www.macworld.com/news/2007/09/27/iphoneunlock/index.php ''Apple update disables unlocked iPhones''] (Macworld.com am 29. September 2007, englisch)</ref>
Besitzer eines iPhones mit Firmware-Version 1.1.1 konnten mit Hilfe der auf zahlreichen Webseiten publizierten Anleitungen die Firmware auf Version 1.0.2 zurücksetzen. Anschließend ließ sich das Gerät mit den bekannten ''Unlock''-Techniken entsperren.<ref>iphone-ticker.de: [http://www.iphone-ticker.de/2007/10/12/zuruck-zu-102-tool-zum-baseband-downgrade/ ''Zurück zu 1.0.2 - Tool zum Baseband Downgrade'' (12. Oktober 2007)]</ref>
Kurze Zeit später war auch dieser Umweg nicht mehr nötig; iPhones mit Firmware 1.1.1 konnten direkt und ohne Anschluss an einen Computer entsperrt und für beliebige [[SIM-Karte]]n freigeschaltet werden.<ref>[http://iphone.unlock.no/ ''Unlock your iPhone for free - no computer necessary!''] (iphone.unlock.no, undatiert, gefunden am 27. November 007, englisch)</ref>
Auch die am 9. November 2007 erschienene Firmware 1.1.2 wurde schon vor der offiziellen Veröffentlichung mit einem ''Jailbreak'' entsperrt, einem aufwendigen Vorbereitungsprozess für den Zugriff auf das Dateisystem. Vorher gespeicherte Daten gingen dabei allerdings verloren.<ref>[http://www.tuaw.com/2007/11/09/iphone-testing-the-1-1-2-jailbreak/ ''iPhone: Testing the 1.1.2 Jailbreak''] (www.tuaw.com am 9. November 2007, englisch)</ref>

Seit dem 20. Juli 2008 bietet die Software PwnageTool 2.0 des iPhone Dev Teams einen ''Jailbreak'' und ''Unlock'' für das iPhone der ersten Generation mit der neu erschienen iPhone-Firmware 2.0 an; die Nachfolgerversion PwnageTool 2.0.3.1 unterstützt sowohl das iPhone 3G (ohne Unlock) als auch die iPhone-Firmware 2.0.2. Es ist auch eine simple Variante der Software erschienen, mit dem Namen QuickPwn.

Apple behält sich das Recht vor, Garantieleistungen bei [[Autorisierung|unautorisierter]] Modifikation des Gerätes zu verweigern.

== Juristische Auseinandersetzung ==
Mitte November 2007 erwirkte T-Mobile-Konkurrent [[Vodafone|Vodafone (D2)]] beim [[Landgericht Hamburg]] eine [[Vorläufiger Rechtsschutz#Einstweilige Verfügung|einstweilige Verfügung]] gegen den Exklusivvertrieb des iPhones in Deutschland. Vodafone-Geschäftsführer [[Friedrich Joussen]] erklärte, man wolle nicht den Vertrieb an sich verbieten lassen, sondern nur den Verkauf des Gerätes ohne zwangsweise Vertragsbindung ermöglichen. Vodafone selbst hatte sich im Sommer 2007 aus den Vertragsverhandlungen mit Apple zurückgezogen, nachdem Apple eine Umsatzbeteiligung von rund einem Drittel verlangt hatte.<ref>[http://de.news.yahoo.com/dpa2/20071119/tbs-einstweilige-verfgung-gegen-iphone-v-9806f5a.html ''Einstweilige Verfügung gegen iPhone-Verkaufsmodell''] (dpa-Meldung vom 19. November 2007)</ref> T-Mobile kündigte am 20. November 2007 Einspruch an, das Unternehmen wolle sich aber bis zu einer endgültigen Entscheidung an die Auflagen der Verfügung halten.<ref>[http://de.news.yahoo.com/dpa2/20071120/tbs-telekom-wehrt-sich-gegen-einstweilig-9806f5a.html ''Telekom wehrt sich gegen Einstweilige Verfügung zu iPhone''] (dpa-Meldung vom 20. November 2007)</ref> In dieser heißt es unter anderem, dass das iPhone nicht mehr verkauft werden darf, ''„wenn es nur in Verbindung mit dem Abschluss eines Mobilfunkvertrages (…) mit einer Mindestvertragslaufzeit von 24 Monaten angeboten (…) wird.“''<ref>[http://www.faz.net/d/invest/meldung.aspx?id=63228167 ''Vodafone und T-Mobile streiten um iPhone vor Gericht''] ([[Dow Jones|Dow-Jones]]-Newswire-Meldung am 20. November 2007)</ref> Der Mobilfunk-Anbieter [[debitel]] legte ebenfalls am 20. November 2007 wegen der T-Mobile-Vertragsgestaltung Beschwerde bei der [[Bundesnetzagentur]] ein.<ref>[http://de.news.yahoo.com/dpa2/20071120/tbs-debitel-iphone-vertriebsmodell-verst-9806f5a.html ''debitel: iPhone-Vertriebsmodell verstößt gegen Mobilfunklizenz''] (dpa-Meldung am 20. November 2007)</ref> Bereits am Tag danach korrigierte T-Mobile sein Vertriebsmodell: Vertragsgebundene Geräte würden weiterhin für 399 Euro verkauft, zusätzlich bestehe jedoch die Möglichkeit, für 999 Euro ein iPhone ohne SIM-Lock zu erwerben, das in allen Mobilfunknetzen eingesetzt werden könne. Geräte, die in der 47. Kalenderwoche bereits verkauft wurden, könnten kostenlos entsperrt werden. Am 4. Dezember 2007 hob das Landgericht Hamburg seine einstweilige Verfügung wieder auf, anschließend erklärte T-Mobile, dass ab sofort wieder nur noch vertragsgebundene iPhones verkauft würden. Der Vertrieb der SIM-Lock-freien Geräte werde eingestellt.<ref>[http://de.news.yahoo.com/ap/20071204/tbs-t-mobile-darf-iphone-exklusiv-vermar-f8250da_2.html ''T-Mobile darf iPhone exklusiv vermarkten''] ([[Associated Press|AP]]-Meldung vom 4. Dezember 2007)</ref> Gegen das Urteil wurde innerhalb der einmonatigen Frist keine [[Berufung (Recht)|Berufung]] eingelegt; es wurde somit im Januar 2008 rechtskräftig.

== Einzelnachweise und Anmerkungen ==
<references />

== Weblinks ==
{{Commonscat|IPhone|iPhone}}
* [http://www.apple.com/de/iphone/ iPhone bei Apple.com]
* [http://www.t-mobile.de/iphone/technischeDetails.jsp iPhone-Spezifikationen in Deutschland] (bei [[T-Mobile]].de im September 2007)
* [http://www.macnotes.de/2007/10/12/test-iphone-8-gb/ Ausführlicher Testbericht] (macnotes.de)

{{Lesenswert}}

[[Kategorie:Apple]]
[[Kategorie:Smartphone]]

{{Link FA|pt}}

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IPhone (1. Generation)

2008-12-20T13:44:39Z

HereToHelp: /* Markteinführung */ Bild:IPhone_3G_Availability.svg Neue Bildshows ursprünglich, 3G, und geplante Verfügbarkeit.

IPhone

2008-12-20T13:44:39Z

HereToHelp: /* Markteinführung */ Bild:IPhone_3G_Availability.svg Neue Bildshows ursprünglich, 3G, und geplante Verfügbarkeit.

0,999…

2006-10-18T20:32:06Z

HereToHelp: /* Algebra proof */ organize

[[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]] which is [[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 a number of [[mathematical proof|proof]]s of this identity, which vary with their level of [[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. Their reasoning may be based on the 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 in the [[real number]]s, as 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 which 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. 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 terminating decimal expansion 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 zeroes 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]], 0.3333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999…. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…. But 3 × 1⁄3 equals 1, so 0.9999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Martin Gardner]] and [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

=== 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 fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. To see this, consider that subtracting 0.9999… from 9.9999… can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. 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, 0.9999…, 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"/>

Notice that this proof does not work with any other repeating digit, because it relies on the number ten to move the decimal place to the right, then ten minus one (nine) to remove the infinite decimal places. When any digit other than nine (or zero, since zero cannot repeat into infinity) is divided by nine, it produces an infinite string of that digit after the decimal place, which was the original premise.

== 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]], labelled 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. Those who accept the difference between a sequence of numbers and its limit might read "0.999…" as meaning the former rather than the latter.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
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 calculate 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. Two numbers (which are now sets) are [[equality (mathematics)|equal]] if and only if they have the same elements.

The step from rationals to reals is a huge extension, and there are at least two popular ways to achieve the 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 rather unique 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 |''z''| is 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 and/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 infinitesmals (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 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 way that 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 nothing more or less than 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>

During the definition of multiplication Richman 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 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 an [[April 1]], 2004 [[press conference]] 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 runner paradox, 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 after.<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.<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>

==Notes==
<div class="references-2column">
<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 |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{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]]
[[ja:0.999...が1に等しいことの証明]]
[[th:การพิสูจน์ว่า 0.999... เท่ากับ 1]]
[[zh:证明0.999...等于1]]

0,999…

2006-10-18T20:31:42Z

HereToHelp: /* Algebra proof */ explain

[[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]] which is [[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 a number of [[mathematical proof|proof]]s of this identity, which vary with their level of [[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. Their reasoning may be based on the 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 in the [[real number]]s, as 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 which 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. 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 terminating decimal expansion 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 zeroes 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]], 0.3333…, in which the digits repeat without end. This decimal yields a quick proof for 0.999…. Multiplication of 3 times 3 produces 9 in each digit, so 3 × 0.3333… equals 0.9999…. But 3 × 1⁄3 equals 1, so 0.9999… = 1.<ref name="CME">cf. with the binary version of the same argument in [[Martin Gardner]] and [[Silvanus P. Thompson]], ''Calculus made easy'', St. Martin's Press, New York, 1998. ISBN 0-312-18548-0.</ref>

=== 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 fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. To see this, consider that subtracting 0.9999… from 9.9999… can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. 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, 0.9999…, 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"/> Notice that this proof does not work with any other repeating digit, because it relies on the number ten to move the decimal place to the right, then ten minus one (nine) to remove the infinite decimal places. When any digit other than nine (or zero, since zero cannot repeat into infinity) is divided by nine, it produces an infinite string of that digit after the decimal place, which was the original premise.

== 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]], labelled 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. Those who accept the difference between a sequence of numbers and its limit might read "0.999…" as meaning the former rather than the latter.<ref>Tall and Schwarzenberger p.6; Tall 2001 p.221</ref>
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 calculate 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. Two numbers (which are now sets) are [[equality (mathematics)|equal]] if and only if they have the same elements.

The step from rationals to reals is a huge extension, and there are at least two popular ways to achieve the 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 rather unique 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 |''z''| is 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 and/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 infinitesmals (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 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 way that 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 nothing more or less than 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>

During the definition of multiplication Richman 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 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 an [[April 1]], 2004 [[press conference]] 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 runner paradox, 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 after.<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.<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>

==Notes==
<div class="references-2column">
<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 |title=Everything and more: a compact history of infinity |year=2003 |publisher=Norton |id=ISBN 0-393-00338-8}}
</div>

== External links==
{{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]]
[[ja:0.999...が1に等しいことの証明]]
[[th:การพิสูจน์ว่า 0.999... เท่ากับ 1]]
[[zh:证明0.999...等于1]]

Franklin Delano Roosevelt Memorial

2006-05-21T01:24:05Z

HereToHelp: /* Tidal Basin memorial */ captions

{{Infobox_protected_area | name = Franklin Delano Roosevelt Memorial
| iucn_category = V
| image = US_Locator_Blank.svg
| caption =
| locator_x = 240
| locator_y = 76
| location = [[Washington, D.C.]], [[United States|USA]]
| nearest_city =
| lat_degrees = 38
| lat_minutes = 53
| lat_seconds = 2
| lat_direction = N
| long_degrees = 77
| long_minutes = 2
| long_seconds = 40
| long_direction = W
| area = 7.50 acres (0.03 km²)
| established = [[May 2]], [[1997]]
| visitation_num = 2,852,565
| visitation_year = 2005
| governing_body = [[National Park Service]]
}}
The '''Franklin Delano Roosevelt Memorial''' is a [[United States Presidential Memorial]] built not only to the memory of [[President of the United States|President]] [[Franklin Delano Roosevelt]], but also to the era he represents.

==Tidal Basin memorial==
[[Image:FDR memorial.jpg|left|thumb|240px|FDR with his dog [[Fala]] on the [[Tidal Basin]] near the [[National Mall]].]]
Dedicated on [[May 2]], [[1997]], the monument traces twelve years of the [[History of the United States]] through a sequence of four outdoor roto one of FDR's terms of office. [[Sculpture]]s inspired by [[photograph]]s depict the 32nd President. At the very beginning of the memorial he is shown seated in a [[wheelchair]] much like the one he actually used. Other sculptures depict scenes from the Great Depression, such as listening to a [[Fireside chats|Fireside Chat]] on the radio and waiting in a bread line.

In his 1941 [[State of the Union Address]], as the nation contemplated the increasingly more inevitable prospect of being drawn into the war, President Roosevelt spelled out "[[Four Freedoms]]" as a reminder of what America must stand for. From the days of his first Presidential campaign during the [[Great Depression]], Roosevelt spoke directly to the people. "I pledge you, I pledge myself," he said in his 1932 acceptance speech, "to a new deal for the American people." Four years later, he proclaimed that "this generation of Americans has a rendezvous with destiny." Throughout his Presidency, 1933 - 1945, he addressed America by radio in what came to be known as [[fireside chats]]. Each idea, each phrase was underscored by courage and optimism that inspired the people he served.

[[Image:FDR Memorial wall.jpg|thumb|right|230px|A memorial wall engraved with the [[Four Freedoms]].]]
More than 50 years after Roosevelt's death, his own words call out from the walls of his memorial as if he were somehow present. Those who know FDR only as a historical figure recognize these words by their association with great and catastrophic events. For the many Americans who lived through the Roosevelt years, the words recall personal struggles and triumphs during 12 years that seemed like a lifetime.

The memorial was designed by [[Lawrence Halprin]], and includes sculptures and works by [[Leonard Baskin]], [[Neil Estern]], [[Robert Graham (sculptor)|Robert Graham]], [[Thomas Hardy (sculptor)|Thomas Hardy]], and [[George Segal (sculptor)|George Segal]]. The [[national memorial]] is part of [[National Mall and Memorial Parks]]. As an historic area managed by the [[National Park Service]], the memorial was administratively listed on the [[National Register of Historic Places]] on date of its establishment, [[May 2]], [[1997]].

==Original memorial==
The new memorial on the [[Tidal Basin]] was almost 50 years in the making. When plans for the memorial
stalled in the 1960's, a simple memorial was placed according to Roosevelt's expressed wishes:

<blockquote>
[P]laced in the center of that green plot [in front of the National Archives in Washington D.C. should be] ... a block about the size of this (putting his hand on his desk). I don't care what it is made of, whether limestone or granite or what not, but I want it to be plain, without any ornamentation, with the simple carving "In memory of...."
</blockquote>

Indeed, this simple memorial still rests at the North West corner of the National Archives grounds on Pennsylvania Ave.

==References==
* ''The National Parks: Index 2001–2003''. Washington: [[United States Department of the Interior|U.S. Department of the Interior]].

==See also==
* [[United States Presidential Memorial]]
* [[Roosevelt Island]], [[New York City]], planned site of another, as-yet unbuilt, memorial.

[[Image:FDR_chair.jpg|thumb|right|230px|Statue of FDR in his wheelchair]]
==External links==
* Official NPS website: [http://www.nps.gov/fdrm/ Franklin Delano Roosevelt Memorial]
* [http://www.journals.uchicago.edu/AmArt/journal/issues/v18n1/180132/180132.html The President's Two Bodies]
* [http://www.dcphototour.com/WashingtonDCMemorialPhotoTour/FDRMemorial.shtml Franklin Delano Roosevelt Memorial photo tour]
* [http://www.dcguide.com/attractions/memorials/fdr-memorial.html Franklin Delano Roosevelt Virtual Tour]

[[Category:1997 establishments]]
[[Category:Buildings and monuments honoring American Presidents|Roosevelt, Franklin Delano Memorial]]
[[Category:National Mall]]
[[Category:National Memorials of the United States]]
[[Category:Monuments and memorials in Washington, D.C.]]
[[Category:Registered Historic Places in the District of Columbia]]

[[he:אנדרטת פרנקלין דלנו רוזוולט]]

Nicht-destruktive Bearbeitung

2006-04-08T17:55:51Z

HereToHelp: syntax

'''Non-destructive editing''' is a form of editing signals where the original content is not modified in the course of editing - instead the edits themselves are edited.

A pointer-based playlist - effectively an [[Edit decision list|EDL]] - is used to keep track of edits. Each time the edited video is played back or accessed, it is reconstructed from the original video source and the EDL. Although this process is more computationally intensive than rendering each edit, changing the edits themselves can be almost instantaneous, and it prevents further generation loss as the video is edited.

{{compu-stub}}

Nicht-destruktive Bearbeitung

2006-04-08T17:55:35Z

HereToHelp: +stub

XQuartz

2005-12-18T03:14:58Z

HereToHelp:

[[Image:Apple_X11.png|left|80px]]
[[Image:X11_mac_gimp_app.jpg|thumb|200px|X11.app running rootless on [[Mac OS X]]. The screen shows [[GIMP]] running within the [[X window system]].]]
'''X11.app''' is [[Apple Computer]]'s implementation of the [[X Window System]] for [[Mac OS X]]. Apple's implementation of X11 is based on [[XFree86]] and adds support for hardware accelerated [[2D computer graphics|2D graphics]], hardware [[OpenGL]] acceleration and integration with [[Aqua (GUI)|Aqua]], the Mac OS X [[graphical user interface]] (GUI). X11.app was initially available as a downloadable public [[development stage|beta]] for [[Mac OS X v10.2]] and has been included as a standard package since [[Mac OS X v10.3]].

The source code for X11.app is available from Apple under the [[Apple Public Source License]]. [http://developer.apple.com/darwin/projects/X11/index.html]

==See also==
*[[XDarwin]]

==External links==
*[http://www.apple.com/macosx/features/x11/ X11.app website]

[[Category:Mac OS-only software made by Apple Computer|X11.app]]
[[Category:X servers]]

[[fr:Apple X11]]
[[it:Apple X11]]
[[zh:Apple X11]]

Gebet (Judentum)

2005-11-23T02:16:49Z

HereToHelp: added template

'''Jewish services''' are the prayers recited as part of observance of [[Judaism]]. These prayers, often with instructions and commentary, are found in the [[siddur]], the traditional Jewish prayer book.

The individual is required to pray three times daily and four times daily on the [[Shabbat|Sabbath]] and [[Jewish holiday]]s (five times on [[Yom Kippur]]). While prayer alone is valid, praying with a ''[[minyan]]'' (quorum of ten adult males) is considered ideal. Many synagogues (particularly [[Reform Judaism|Reform]] and [[Conservative Judaism|Conservative]], and large [[Orthodox Judaism|Orthodox]] ones) have a ''[[hazzan]]'' (cantor) who is a professional or lay-professional singer employed for the purpose of leading the congregation in prayer.

'''Daven''' is the originally exclusively Eastern [[Yiddish]] verb meaning "pray"; it is widely used by [[Ashkenazic]] [[Orthodox Judaism|Orthodox Jews]]. In [[Yinglish]], this has become the [[Anglicise]]d ''davening''. The origin of the word is obscure, but is thought by some to have come from Middle [[French language|French]] and by others to be derived from a [[Slavonic languages|Slavic]] word meaning "give". In Western [[Yiddish]], the term for "pray" is '''oren''', a word with clear roots in [[Romance languages]] — compare [[Spanish language|Spanish]] and [[Portuguese language|Portuguese]] "orar" and [[Latin]] "[[Oratorium]]".
[[Image:2nd century Rome gold goblet shows Jewish ritual objects.jpg|thumb|240px|Jewish ritual objects shown on a gold goblet found in Rome (2nd century CE)]]

==The prayers and their origins==
===Backgrounds===
{{Jew}}
There are three prayer services each day on weekdays. A fourth additional prayer service (called ''mussaf'', "additional"), is added on [[Shabbat]] (the Jewish [[Sabbath]]) and on major holidays. A fifth prayer (''ne'ilah''), is only recited on [[Yom Kippur]].

According to the [[Talmud]] (tractate [[Taanit]] 2a), prayer is a Biblical command: "''Your shall serve God with your whole heart'' ([[Deuteronomy]] 11:13) - What service is performed with the heart? This is prayer". The prayers are therefore referred to as ''Avodah sheba-Lev'' (service from in the heart). [[Maimonides]] ([[Mishneh Torah]], Laws of Prayer 1:1) likewise categorises prayer as a Biblical command, but states that the number of prayers or their times are not. This statement is relied upon by the authorities that hold that women, while being required to pray, only need to pray once a day (preferably in the morning), though they can, if they wish, pray all three daily prayers.

The [[Talmud]] (tractate Berachoth 26b) gives different reasons why there are three basic prayers.
# According to one sage, every one of the [[Patriarchs (Bible)|Patriarchs]] instituted one prayer: [[Abraham]] the morning, [[Isaac]] the afternoon and [[Jacob]] the evening prayers. This view is supported with Biblical quotes indicating that the Patriarchs prayed at the time mentioned.
# A second opinion states that each was instituted parallel to a sacrificial act in the [[Temple in Jerusalem]]: the morning ''Tamid'' [[korban|offering]] in the morning for the morning, the afternoon ''Tamid'' for the afternoon prayers and the overnight burning of the leftovers for the evening prayers.

Additional Biblical references suggest that [[David|King David]] and the prophet [[Daniel]] prayed three times a day. In [[Book of Psalms|Psalms]], David states: "Evening, morning and afternoon do I pray and cry, and He will hear my voice" (55:18). As in [[Book of Daniel|Daniel]]: "[...] his windows being open in his chamber toward Jerusalem, he kneeled upon his knees three times a day, and prayed, and gave thanks before his God, as he had done before" (6:11).

===Text and language===
Maimonides (Mishneh Torah, Laws of Prayer 1:4) relates that until the Babylonian exile, all Jews composed their own prayers. After the exile, however, the sages of the time (united in the [[Great Assembly]]) found the ability of the people insufficient to continue the practice, and they composed the main portions of the ''[[siddur]]'', such as the ''[[Amidah]]''. The language of the prayers, while clearly being from the Second Temple period, often employs Biblical idiom, and according to some authorities it should not contain rabbinic or Mishnaic idiom apart from in the sections of Mishnah that are featured (see Baer).

Prayer is done almost exclusively in [[Hebrew language|Hebrew]], but Jewish law allows for prayers to be said in any language that the person praying understands. [[Ashkenazi]] [[Orthodox Judaism|Orthodox]] synagogues use almost exclusively Hebrew, and use the local language only for sermons and directions; [[Conservative Judaism|Conservative]] synagogues use Hebrew for 75% to 100% of the service (depending on the local custom), and the rest is in the local language. [[Reform Judaism|Reform]] synagogues (usually called Temples in North America) use anywhere from 10% to 50% Hebrew; most of the service is in the local language. [[Sephardic]] customs vary, but [[Ladino language|Ladino]] or [[Portuguese language|Portuguese]] may be used in smaller or greater parts of the service even in the most traditional and [[Orthodox]] communities.

=== Quorum ===
''Main article: [[Minyan]]''

Prayer alone is considered acceptable, but prayer with a quorum of ten adults (a [[minyan]]) is considered "prayer with the community", and this is the most highly recommended form of prayer. Judaism has traditionally counted only men in the minyan for formal prayer, on the basis that one does not count someone who is not obligated to participate. Many [[Conservative Judaism|Conservative]] congregations have recently begun to count women in the minyan as well, although the determination of whether or not to do so is left to the individual congregation. Those Reform and Reconstructionist congregations that consider a minyan mandatory for communal prayer, count both men and women for a minyan.

===Concentration===
Proper concentration (''kavvanah'') is considered essential for prayer. There are only certain portions that are invalid ''a posteriori'' if they were recited without the required awareness. These are the first line of ''[[Shema Yisrael]]'' and the first of the nineteen benedictions of the ''[[Amidah]]''.

== Weekday prayer services ==
=== Shacharit: morning prayers ===
Various prayers are said upon arising; [[tzitzit]] (small garment with fringes) are donned at this time. The [[tallit]] (large prayer shawl) is donned before or during the actual prayer service, as are the [[tefillin]] (phylacteries); both are accompanied by blessings.

The service starts with the "morning blessings" (''birkot ha-shachar''), including blessings for the [[Torah]] (considered the most important ones). In Orthodox services this is followed by a series of readings from Biblical and rabbinic writings recalling the [[korban|offerings]] made in the [[Temple in Jerusalem]]. The section concludes with the "Rabbis' [[Kaddish]]" (''kaddish de-rabbanan'').

The next section of morning prayers is called ''Pesukei D'Zimrah'' ("verses of praise"), containing several [[psalms]] (100 and 145-150), and prayers (such as ''yehi chevod'') made from a tapestry of Biblical verses, followed by the Song at the Sea (Exodus, chapters 14 and 15).

Now begins ''Barechu'', the formal public call to prayer, and an expanded series of prayers embracing the main recitation of ''[[Shema Yisrael]]''. This is followed by the core of the prayer service, the ''[[Amidah]]'' or ''Shemoneh Esreh'', a series of 19 blessings. The next part of the service, is ''Tachanun'', supplications, which is omitted on days with a festive character (and by Reform services usually entirely).

On Mondays and Thursdays a [[Torah reading]] service is inserted. Concluding prayers and [[Aleinu]] then follow, with the Kaddish of the mourners generally after Aleinu.

=== Mincha: afternoon prayers ===
[[Sephardi]]m and [[Italki]]m start the Mincha with [[Psalm]] 84 and ''Korbanot'' (''[[Book of Numbers|Numbers]]'' 28:1-8), and usually continue with the ''Pittum hakketoret''. The opening section is concluded with ''[[Malachi]]'' 3:4. Prayers then continue as follows. ([[Ashkenazi]]m start here.) The ''Ashrei'', containing verses of [[Psalm]]s 84, 144, 115 and 141 and the entire [[Psalm]] 145, immediately followed by the ''Shemoneh Esreh'' (''Amidah''). This is followed by a shortened version of ''Tachanun'', supplications, and then the full Kaddish. Sephardim insert [[Psalm]] 67 or 93, followed by the Mourner's Kaddish. After this follows, in most modern rites, the ''[[Aleinu]]''. Ashkenazim then conclude with the Mourner's Kaddish.

=== Ma'ariv (or Arvit): evening prayers ===
This service begins with the ''Barechu'', the formal public call to prayer, and an expanded series of prayers relating to the ''Shema Yisrael''. This is followed by the ''Hashkiveinu'' ("Lay us down to sleep, Adonai, our God, in peace, raise us erect, our King, to life, and spread over us the shelter of Your peace.") (In the [[Ashkenazi]] ritual, a series of other blessings are added, which are made from a tapestry of biblical verses.) This is followed by the Half-Kaddish, and the ''Shemoneh Esreh'' (''Amidah''), bracketed with the full Kaddish. [[Sephardi]]m then repeat the ''Barechu'' and say the Mourner's Kaddish before concluding with the Aleinu. [[Ashkenazi]]m do not repeat the Barechu, but conclude with ''Aleinu'' followed by the Mourner's Kaddish.

In many congregations, the afternoon and evening prayers are recited back-to-back on a working day, to save people having attend synagogue twice. The [[Vilna Gaon]] discouraged this practice, and followers of his set of customs commonly wait until after nightfall to recite ''Ma'ariv''.

==Shabbat services==
=== Friday night services ===
[[Shabbat]] services begin on Friday evening with the weekday Mincha (see above), followed in some communities by the [[Song of Songs]], and then in most communities by the ''Kabbalat Shabbat'', the mystical prelude to Shabbat services composed by 17th century Kabbalists. This [[Hebrew language|Hebrew]] term literally means "Receiving the Sabbath".

It is, except for amongst many [[Italki]]m and Western [[Sephardi]]m, composed of six psalms, 95 to 99, and 29, representing the six week-days. Next comes the poem ''[[Lekha Dodi]]''. Composed by [[Solomon ha-Levi Alkabetz]] in the mid-[[16th Century|1500s]], it is based on the words of the [[Talmud]]ic sage Hanina: "Come, let us go out to meet the Queen Sabbath" (Talmud Shabbat 119a). Many add a study section here, including ''Bameh Madlikin'' and ''Amar ribbi El'azar'' and the concluding ''Kaddish deRabbanan''. Kabbalat Shabbat is concluded by ''Psalm 92'' (the recital of which constitutes men's acceptance of the current Shabbat with all its obligations) and ''Psalm 93'', and is then followed by the Maariv service.

The [[Shema Yisrael|Shema]] section of the Friday night service varies in some details from the weekday services — mainly in the different ending of the ''Hashkivenu'' prayer and the omission of the ''Barukh A. le'olam'' prayer in those traditions where this section is otherwise recited. In the [[Italki]] tradition, there are also different versions of the ''Ma'ariv 'aravim'' prayer (beginning ''asher killah'' on Friday nights) and the ''Ahavat 'olam'' prayer.

Most commemorate the Shabbat at this point with ''VeShameru'' (Ex. 31:16,17). The custom to recite the biblical passage at this point has its origins in the Lurianic [[Kabbalah]], and does not appear before the 16th century. It is therefore absent in traditions and prayer books less influenced by the Kabbalah (such as the [[Yemenite Jews|Yemenite]] ''[[Baladi]]'' tradition), or those that opposed adding additional readings to the siddur based upon the Kabbalah (such the [[Vilna Gaon]]).

The [[Amidah]] on Shabbat is abbreviated, and is read in full once. This is then followed by the [[hazzan]]'s mini-repetition of the Amidah, ''Magen Avot'', a digest of the seven benedictions. In some [[Ashkenazi]] [[Orthodox Judaism|Orthodox]] synagogues the second chapter of [[Mishnah]] tractate Shabbat, ''Bameh Madlikin'', is read at this point, instead of earlier. The service then follows with the [[Aleinu]]. [[Kiddush]] is recited in the synagogue in Ashkenazi and a few Sephardi communities. Most Sephardi and many Ashkenazi synagogues end with the singing of ''Yigdal'', a poetic adaptation of [[Maimonides]]' 13 principles of Jewish faith. Other Ashkenazi synagogues end with ''[[Adon Olam|Adon `olam]]'' instead.

=== Saturday morning: Shacharit ===
[[Shabbat]] morning prayers commence as on week-days. Of the hymns, Psalm 100 is omitted, its place being taken in the [[Ashkenazi]] tradition by Psalms 19, 34, 90, 91, 135, 136, 33, 92, 93. [[Sephardi]]c Jews maintain a different order, add several psalms and two religious poems. The ''Nishmat'' prayer is recited at the end of the ''Pesukei D'Zimrah''. The blessings before ''Shema'' are expanded, and include the hymn ''El Adon'', which is often sung communally.

The fourth intermediary benediction of the ''Shacharit [[Amidah]]'' begins with ''Yismah Mosheh''. The Torah scroll is taken out of the Ark, and the [[Parsha|weekly portion]] is [[Torah reading|read]], followed by the ''[[haftarah]]''.

After the [[Torah]] reading, three prayers for the community are recited. Two prayers starting with ''Yekum Purkan'', composed in Babylon in [[Aramaic language|Aramaic]], are similar to the subsequent ''Mi sheberakh'', a blessing for the leaders and patrons of the synagogue. The Sephardim omit much of the Yekum Purkan. Prayers are then recited (in most communities) for the government of the country, for peace, and for the State of [[Israel]].

=== Saturday morning additional service: Musaf ===
The ''Musaf'' service starts with the silent recitation of the ''[[Amidah]]''. It is followed by a second public recitation that includes an additional reading known as the ''Kedushah''. This is followed by the ''Tikanta Shabbat'' reading on the holiness of [[Shabbat]], and then by a reading from the biblical [[Book of Numbers]] about the sacrifices that used to be performed in the [[Temple in Jerusalem]]. Next comes ''Yismechu'', "They shall rejoice in Your sovereignty"; ''Eloheynu'', "Our God and God of our Ancestors, may you be pleased with our rest"; ''Retzei'', "Be favorable, our God, toward your people Israel and their prayer, and restore services to your Temple";

After the ''Amidah'' comes the full [[Kaddish]], followed by ''Ein ke'eloheinu''. In [[Orthodox Judaism]] this is followed by a reading from the [[Talmud]] on the sacrifices that used to be performed in the Temple in Jerusalem. These readings are usually omitted by [[Conservative Judaism|Conservative]] Jews, and are always omitted by [[Reform Judaism|Reform]] Jews.

The ''Musaf'' service culminates with the Rabbi's Kaddish, the ''[[Aleinu]]'', and then the Mourner's Kaddish. Some [[synagogue]]s conclude with the reading of ''An'im Zemirot'', "The Hymn of Glory".

[[United States|American]] Reform services omit the entire Musaf service.

=== Saturday afternoon: Mincha ===
Mincha commences with Psalm 145 and the prayer ''U'va le-Tziyon'', after which the first section of the next [[Parsha|weekly portion]] is [[Torah reading|read from the Torah scroll]]. The ''Amidah'' follows the same pattern as the other Shabbat ''Amidah'' prayers, with the middle blessing starting ''Attah Echad''.

After Mincha, during the winter Sabbaths (from [[Sukkot]] to [[Passover]]), ''Bareki Nafshi'' (Psalms 104, 120-134) is recited. During the summer Sabbaths (from Passover to [[Rosh Hashanah]]) chapters from the ''[[Avot]]'', one every Sabbath in consecutive order, are recited instead of ''Barekhi Nafshi''.

=== Saturday evening: Maariv ===
The week-day Maariv is recited on Sabbath evening, concluding with ''Vihi No'am'', ''Ve-Yitten Leka'', and ''[[Havdalah]]''.

== Services on Passover, Shavuot and Sukkot ==

The services for the three festivals of [[Pesach]] ("Passover"), [[Shavuot]] ("Feast of Weeks" or "Pentecost"), and [[Sukkot]] ("Feast of Tabenacles") are alike, except for interpolated references and readings for each individual festival. The preliminaries and conclusions of the prayers are the same as on Sabbath. The Amidah on these festivals only contains seven benedictions, with ''Attah Bechartanu'' as the main one. [[Hallel]] (communal recitation of Psalms 113-118) follows.

The Musaf service includes Mi-Pene Hata'enu, with reference to the special festival and [[Temple in Jerusalem|Temple]] sacrifices on the occasion. A blessing on the pulpit ("Dukan") is pronounced by the "[[Kohen|kohanim]]" (Jewish priests) during the Amidah (this occurs daily in Israel, but only on [[Pesach]], [[Shavuot]], [[Sukkot]], [[Rosh Hashanah]], and [[Yom Kippur]] in the [[diaspora]]). On week-days and Sabbath the priestly blessing is recited by the [[hazzan]] after the Modim ("Thanksgiving") prayer. (American Reform Jews omit the Musaf service.)

==Related customs==
Many Jews sway their body back and forth during prayer. This practice (referred to as ''shokeling'' in [[Yiddish language|Yiddish]]) is not mandatory, and in fact the [[kabbalah|kabbalist]] [[Isaac Luria]] felt that it should not be done. In contrast, the German Medieval authority ''[[Maharil]]'' (Rabbi Jacob Molin) linked the practice to a statement in the [[Talmud]] that the Mishnaic sage [[Rabbi Akiva]] would sway so forcefully that he ended up at the other side of the room when praying (Talmud tractate Berachot).

Money for ''[[tzedakah]]'' (charity) is given during the weekday morning and afternoon services in many communities.

== Guide on etiquette for visitors ==
In most synagogues or temples, it is considered a sign of respect for all male attendees to wear a head covering, usually a dress hat or ''[[yarmulke]]'' (''kipa''); the latter are usually provided near the front door. The yarmulkes may be provided by the families of the celebrants of special occasions; however, this is not always the case.

Orthodox and Conservative (also called [[Masorti]]) synagogues encourage all male attendees to cover their heads out of respect for God. Many Reform (or Progressive) temples do not require people to cover their heads, although some Reform Jews may choose to.

As might be expected, there are some things that a non-Jewish visitor should do during a Jewish religious service, and there are some things widely considered inappropriate:
* ''Seating'' - Orthodox synagogues will have separate seating by gender. Be certain to sit in the appropriate section. To avoid sitting in someone's "accustomed spot" (''maqom qavua`'' מקום קבוע), depending on the congregation especially, it is best to call ahead and have someone, probably the [[gabbai]], meet you beforehand.
* ''[[Prayer]]'' - A person who is not Jewish should not recite any of the blessings that deal with commandments given to, or blessings bestowed upon, Jews exclusively, such as the blessings recited by an ''oleh'' (the person who has an aliyah--see below), dealing with the distinction between Jews and non-Jews, and praising God for giving the [[Torah]] to Jews. Prayers in Orthodox synagogues and, oftentimes, in Conservative synagogues are entirely in Hebrew; other synagogues use local languages along with the Hebrew (often, English).
* ''Standing'' - Parts of the services are recited standing; visitors are expected to stand together with the congregation.
* ''Bowing'' - At certain points in the service, congregants bow; visitors who are following along and wish to bow should feel free to do so as well.
* ''[[Tallit]]'' (prayer shawl) - non-Jewish visitors should not don a ''tallit''.
* ''[[Tzeniut]]'' (modesty) - Appropriate dress for a house of worship is expected. When attending Orthodox synagogues, women will likely be expected to wear long sleeves (past the elbows), long skirts (past the knees), a high neckline (to the collar bone), and if married, to cover their hair. Men are expected to dress respectfully, short pants or sleeveless shirts are generally regarded as inappropriate. In some congregations, however, these expectations are more lax. The best course of action is for a visitor to inquire about dresscode expectations beforehand.
*''[[Aliyah]]'' - If a non-Jewish visitor is offered the honor of reading from the Torah, or to recite the blessings for the readings (called an ''‘aliyah'', a person so-honored is called an ''‘oleh''), they should (as discreetly and politely as decorum permits) inform the person inviting them for the honor, that they are not Jewish. In some Reform congregations, non-Jews may be permitted to receive this honor, but no assumption on the matter should be made. Inform the inviter, and let them determine whether or not the invitation stands.

==See also==
* [[Haftara]]
* [[Siddur]]
* [[Prayer]]
* [[List of Jewish Prayers and Blessings]]
* [[Torah reading]]

== References ==
* ''To Pray As a Jew'', Hayim Halevy Donin, Basic Books (ISBN 0465086330)
* ''Entering Jewish Prayer'', Reuven Hammer (ISBN 0805210229)
* ''Kavvana: Directing the Heart in Jewish Prayer'', Seth Kadish, Jason Aronson Inc. 1997.
* ''Or Hadash: A Commentary on Siddur Sim Shalom for Shabbat and Festivals'', Reuven Hammer, The [[Rabbinical Assembly]] and the [[United Synagogue of Conservative Judaism]]
* Rabbi S. Baer. ''Siddur Avodath Yisrael'' (newly researched text with commentary Yachin Lashon), 19th century.
* ''A Guide to Jewish Prayer'', Rabbi Adin Steinsaltz, Shocken Books (ISBN 0805241744)

== External links ==
* [http://ezra.mts.jhu.edu/~rabbiars/synagogue/synagogue.html Introduction to synagogue services]
* [http://www.simpletoremember.com/faqs/Mitzvos_Prayer_Religion.htm FAQs on Jewish Prayer]
* [http://www.jewfaq.org/liturgy.htm Outline of prayer services]
* [http://www.GoDaven.com GoDaven.com - The Worldwide Minyan Database]

{{JewishLifeCycle}}

[[Category:Prayer]]
[[Category:Jewish law and rituals]]
[[Category:Jewish liturgy|*]]