title: Frost/Nixon (film)

text: Four days before the final interview, which will focus on Watergate, Frost receives a phone call from an inebriated Nixon. In a drunken rant, Nixon declares that they both know the final interview will make or break their careers. He compares himself to Frost, insisting that they both came from humble backgrounds and had to struggle to make it to the top of their fields, only to be knocked back down again. Frost gains new insight into his subject, while Nixon assures Frost that he will do everything in his power to emerge the victor of the final interview.


title: Frost/Nixon (film)

text: The conversation spurs Frost into action. For the next three days, he works relentlessly to prepare as Reston pursues a lead at the Federal Courthouse library in Washington.


title: Frost/Nixon (film)

text: As the final interview begins, Frost ambushes Nixon with damning transcripts of a conversation between Nixon and Charles Colson that Reston dug up in Washington. As his own team watches in horror from an adjoining room, Nixon admits that he did unethical things, adding, "When the President does it, that means it's not illegal." A stunned Frost is on the verge of inducing a confession when Brennan bursts in and stops the recording. After Nixon and Brennan confer, the interview resumes. Frost aggressively pursues his original line of questioning; Nixon admits that he participated in a cover-up and that he "let the American people down."


title: Frost/Nixon (film)

text: Some time after the interviews have aired, Frost and Cushing pay a farewell visit to Nixon at his villa. Frost thanks Nixon for the interviews and Nixon, graciously admitting defeat, thanks Frost in return and wishes him well. Frost gifts Nixon a pair of Italian shoes identical to the ones Frost wore during the interviews. In a private moment, Nixon asks about the night he drunkenly called Frost, implying that he has no recollection of the event. For the first time, Nixon addresses Frost by his first name. Nixon watches Frost and Cushing leave before placing the shoes on the villa's stone railing and solemnly looking out at the sunset.


title: Frost/Nixon (film)

text: A textual epilogue states that the interviews were wildly successful and that Nixon never escaped controversy until his death from a stroke in 1994.


title: Frost/Nixon (film)

text: Other figures and personalities depicted in the film include Tricia Nixon Cox, Michael York, Hugh Hefner, helicopter pilot Gene Boyer (as himself), Raymond Price, Ken Khachigian, Sue Mengers and Jay White as Neil Diamond. To prepare for his role as Richard Nixon, Frank Langella visited the Richard Nixon Presidential Library in Yorba Linda, California, and interviewed many people who had known the former president. On the set, the cast and crew addressed Langella as "Mr. President".


title: Frost/Nixon (film)

text: "Frost/Nixon" had its world premiere on October 15, 2008, as the opening film of the 52nd annual London Film Festival. It was released in three theaters in the United States on December 5 before expanding several times over the following weeks. It was released in the United Kingdom and expanded into wide status in the United States on January 23, 2009.


title: Frost/Nixon (film)

text: The film was released on DVD and Blu-ray on April 21, 2009. Special features include deleted scenes, the making of the film, the real interviews between Frost and Nixon, the Nixon Presidential Library and a feature commentary with Ron Howard.


title: Frost/Nixon (film)

text: "Frost/Nixon" had a limited release at three theaters on December 5, 2008, and grossed $180,708 in its opening weekend, ranking number 22. Opening wide at 1,099 theaters on January 23, 2009, the film grossed $3,022,250 at the box office in the United States and Canada, ranking number 16. The film's gross for Friday, January 30 was estimated the next day at $420,000. "Frost/Nixon" grossed an estimated $18,622,031 in the United States and Canada and $8,804,304 in other territories for a total of $27,426,335 worldwide, recouping its $25 million budget by a thin margin but making a loss when factoring in the significant promotional costs.


title: Frost/Nixon (film)

text: On review aggregation website Rotten Tomatoes, the film has an approval rating of 93% based on 258 reviews, with a weighted average score of 8.00/10. The site's critical consensus reads, ""Frost/Nixon" is weighty and eloquent; a cross between a boxing match and a ballet with Oscar worthy performances." Metacritic gives the film an average score of 80 out of 100, based on 38 critics, indicating "generally favorable reviews".


title: Frost/Nixon (film)

text: Critic Roger Ebert gave the film four stars, commenting that Langella and Sheen "do not attempt to mimic their characters, but to embody them." Peter Travers of "Rolling Stone" gave the film 3½ stars, saying that Ron Howard "turned Peter Morgan's stage success into a grabber of a movie laced with tension, stinging wit and potent human drama." Writing for "Variety", Todd McCarthy praised Langella's performance in particular, stating, "[B]y the final scenes, Langella has all but disappeared so as to deliver Nixon himself." René Rodríguez of "The Miami Herald" gave the film two stars and commented that the picture "pales in comparison to Oliver Stone's "Nixon" when it comes to humanizing the infamous leader" despite writing that the film "faithfully reenacts the events leading up to the historic 1977 interviews." Manohla Dargis of "The New York Times" said, "[S]tories of lost crowns lend themselves to drama, but not necessarily audience-pleasing entertainments, which may explain why "Frost/Nixon" registers as such a soothing, agreeably amusing experience, more palliative than purgative."


title: Frost/Nixon (film)

text: Both the film and the play take dramatic license with the on-air and behind-the-scene details of the Nixon interviews. Jonathan Aitken, one of Nixon's official biographers who spent much time with the former president at La Casa Pacifica, rebukes the film for its portrayal of a drunken Nixon making a late-night phone call as never having happened. Ron Howard discussed the scene on his feature commentary for the DVD release, pointing out it was a deliberate act of dramatic license, and while Frost never received such a phone call, "it was known that Richard Nixon, during ...the Watergate scandal, had occasionally made midnight phone calls that he couldn't very well recall the following day." Elizabeth Drew of the "Huffington Post" and author of "Richard M. Nixon" (2007) noted some inaccuracies, including a misrepresentation of the end of the interviews, the failure to mention the fact that Nixon received 20% of the profits from the interviews, and what she says are inaccurate representations of some of the characters. Drew points out a critical line in the movie that is particularly deceptive: Nixon admitted he "'...was involved in a 'cover-up,' as you call it.' The ellipsis is of course unknown to the audience, and is crucial: What Nixon actually said was, 'You're wanting me to say that I participated in an illegal cover-up. No!'"


title: Frost/Nixon (film)

text: According to a 2014 "Baltimore Sun" article by Jules Witcover, Nixon didn't admit his guilt until he was interviewed in 1983 by former White House aide Frank Gannon (played by Andy Milder in the film).


title: Frost/Nixon (film)

text: David Edelstein of "New York" wrote that the film overstates the importance of its basis, the Frost interviews, stating it "elevates the 1977 interviews Nixon gave (or, rather, sold, for an unheard-of $600,000) to British TV personality David Frost into a momentous event in the history of politics and media." Edelstein also noted that "with selective editing, Morgan makes it seem as if Frost got Nixon to admit more than he actually did." Edelstein wrote that the film "is brisk, well crafted, and enjoyable enough, but the characters seem thinner (Sheen is all frozen smiles and squirms) and the outcome less consequential."


title: Frost/Nixon (film)

text: Writing for the conservative "National Review", Fred Schwarz, who deemed the Frost/Nixon interviews "a notorious fizzle", commented that, the film "is an attempt to use history, assisted by plenty of dramatic license, to retrospectively turn a loss into a win. By all accounts, "Frost/Nixon" does a fine job of dramatizing the negotiations and preparation that led up to the interviews. And it’s hard to imagine Frank Langella, who plays a Brezhnev-looking Nixon, giving a bad performance. Still, the movie’s fundamental premise is just plain wrong." Though generally approving, critic Daniel Eagan notes that partisans on both sides have questioned the accuracy of the film's script.


title: Frost/Nixon (film)

text: Caroline Cushing Graham, in a December 2008 interview, noted that her first trip with Frost was to the Muhammad Ali fight in Zaire, and that the two had been together for more than five years prior to when the film shows the two meeting. She remembered Frost as feeling that he did a pretty good job on every interview, whereas the film depicts him feeling he did a poor job with the first two interviews. She added that while the movie shows Frost driving, in fact they were always chauffeured because he was always making notes for the work he was doing.


title: Frost/Nixon (film)

text: The film implies that Nixon and Frost were meeting for the first time in 1977 for these interviews, but in fact Frost had interviewed Nixon for U.S. network television in 1968 as part of a series of interviews of leading candidates in the Presidential election of that year.


title: Frost/Nixon (film)

text: Diane Sawyer, portrayed in the film in her role as one of Nixon's researchers, said in December 2008 that, "Jack Brennan is portrayed as a stern military guy," citing both the play and what she'd heard about the film version. "And he’s the funniest guy you ever met in your life, an irreverent, wonderful guy. So there you go. It's the movies."


title: Frost/Nixon (film)

text: An early scene in the film set on the southern shore of Sydney Harbour in 1974, with the Sydney Opera House as a backdrop, shows buildings adjacent to the iconic structure which did not exist until 1998.


title: Mastectomy

text: Mastectomy is the medical term for the surgical removal of one or both breasts, partially or completely. A mastectomy is usually carried out to treat breast cancer. In some cases, women believed to be at high risk of breast cancer have the operation as a preventive measure. Alternatively, some women can choose to have a wide local excision, also known as a lumpectomy, an operation in which a small volume of breast tissue containing the tumor and a surrounding margin of healthy tissue is removed to conserve the breast. Both mastectomy and lumpectomy are referred to as "local therapies" for breast cancer, targeting the area of the tumor, as opposed to systemic therapies, such as chemotherapy, hormonal therapy, or immunotherapy.


title: Mastectomy

text: The decision to perform a mastectomy is based on various factors, including breast size, the number of lesions, biologic aggressiveness of a breast cancer, the availability of adjuvant radiation, and the willingness of the patient to accept higher rates of tumor recurrences after lumpectomy and/or radiation. Outcome studies comparing mastectomy to lumpectomy with radiation have suggested that routine radical mastectomy surgeries will not always prevent later distant secondary tumors arising from micro-metastases prior to discovery, diagnosis, and operation. In most circumstances, there is no difference in both overall survival and breast cancer recurrence rate. While there are both medical and non-medical indications for mastectomy, the clinical guidelines and patient expectations for before and after surgery remain the same.


title: Mastectomy

text: Despite the increased ability to offer breast conservation techniques to those with breast cancer, certain groups may be better served by traditional mastectomy procedures including:


title: Mastectomy

text: Mastectomy has non-cancer medical uses as well, including cosmetic or reconstructive surgery. Men with gynecomastia may be eligible for mastectomy, but minimally invasive surgical techniques also exist. Transgender men may undergo a mastectomy as a gender-affirming surgery.


title: Mastectomy

text: Aside from the post-surgical pain and the obvious change in the shape of the chest and/or breast(s), possible side effects of a mastectomy include soreness, scar tissue as the site of the incision, short-term swelling, phantom breast pain (pain in the breast or tissue that has been removed), wound infection or bleeding, hematoma (buildup of blood in the wound), and seroma (buildup of clear fluid in the wound). If the lymph nodes are also removed, additional side effects such as lymphedema (swelling of the lymph nodes) may occur.


title: Mastectomy

text: Upper limb problems such as shoulder and arm pain, weakness and restricted movement are a common side effect after breast cancer surgery. According to research in the UK, an exercise programme started 7–10 days after surgery can reduce upper limb problems.


title: Mastectomy

text: Currently, there are several surgical approaches to mastectomy, and the type that a person decides to undergo (or whether they will decide instead to have a lumpectomy) depends on factors such as the size, location, and behavior of the tumor (if one is present), whether or not the surgery is prophylactic, and whether the person intends to undergo reconstructive surgery.


title: Mastectomy

text: Prior to undergoing the mastectomy, it is important to meet with the surgeon to discuss the relevant risks and benefits of receiving the surgery. Depending on the indication for mastectomy, there may be other options to address the clinical condition. One important consideration to discuss with the surgeon is whether breast reconstruction will occur and when this procedure will take place. One option is to have the reconstruction immediately after the mastectomy in the same surgery, whereas other patients opt for a subsequent surgery for reconstruction. This breast reconstruction surgery will be conducted by a plastic surgeon. In addition to the surgeon, a meeting with an anesthesiologist is pertinent in order to review the patient’s medical history and determine the plan of anesthesia.


title: Mastectomy

text: Leading up to the day of the surgery, there are various considerations that patients can be cognizant of to facilitate their recovery following surgery. As with other surgeries that may lead to appreciable blood loss, it is advised not to take aspirin or aspirin-containing products for 10 days before the surgery. The reason for this is to prevent the anti-coagulative function of aspirin and other blood thinners that would make it difficult to achieve coagulation during the surgery. In addition, it is important for patients to tell the doctor about any medications, vitamins, or supplements that they are taking because some substances could interfere with the surgery. It is also pertinent for patients to not eat or drink 8 to 12 hours before surgery, however, there may be specific pre-operative instructions given by each patient’s care team.


title: Mastectomy

text: Maintaining fitness and proper nutrition is also an important measure to consider prior to receiving a surgery because it has been shown that postoperative outcomes are improved in patients that exercise and maintain a healthy diet prior to surgery. In addition to nutrition and exercise, it is advised to reduce alcohol consumption and smoking. This concept of pre-rehabilitation is beneficial in mitigating post-operative complications and decreasing length of stay in the hospital. The rationale is that increasing a patient’s functional status prior to surgery will allow for a smoother and faster recovery in the postoperative setting.


title: Mastectomy

text: Recent research has indicated that mammograms should not be done with any increased frequency than the normal procedure in women undergoing breast surgery, including breast augmentation, mastopexy, and breast reduction.


title: Mastectomy

text: Prior to leaving the hospital, patients will typically be given a prescription for pain medication to ameliorate any pain or discomfort at the surgery site. Recognizing signs of a surgical site infection including fever, redness, swelling, or pus is one of responsibilities of the patient, as these signs will need to be reported to and assessed by a medical provider. In addition, signs of lymphedema due if lymph node removal is performed during mastectomy may be detected by the presence of heaviness, tightness, or fullness in the hand, arm, or axillary area region.


title: Mastectomy

text: Regarding return to activity, it is advised not to engage in strenuous activity or lift objects above 5 pounds for up to six weeks after a mastectomy at the discretion of the physician. However, it is common for a member of the medical team to provide home exercises designed to maintain arm and shoulder movement and flexibility. Walking is also highly encouraged and allowed immediately after surgery. Most patients who undergo a mastectomy can return to work and other regular physical activities in approximately 4 weeks after surgery.


title: Mastectomy

text: Patients will usually have a post-operative follow-up visit with their provider 1–2 weeks after surgery. The time at which a patient can start to wear a bra or reconstructive breast varies and is often at the discretion of the physician.


title: Mastectomy

text: Between 2005 and 2013, the overall rate of mastectomy increased 36 percent, from 66 to 90 per 100,000 adult women. The rate of hospital-based bilateral mastectomies (inpatient and outpatient combined) more than tripled, from 9.1 to 29.7 per 100,000 adult women, whereas the rate of unilateral mastectomies remained relatively stable at around 60 per 100,000 women. From 2005 to 2013, the rate of bilateral outpatient mastectomies increased more than fivefold and the inpatient rate nearly tripled. The rate of unilateral mastectomies nearly doubled in the outpatient setting but decreased 28 percent in the inpatient setting. By 2013, nearly half of all mastectomies were performed outpatient. However, there are concerns that these rising rates of mastectomies are most greatly seen in women with node-negative and noninvasive lesions, which are subsets of patients that do not require mastectomy.


title: Mastectomy

text: Mastectomy rates vary tremendously worldwide, as was documented by the 2004 'Intergroup Exemestane Study', an analysis of surgical techniques used in an international trial of adjuvant treatment among 4,700 females with early breast cancer in 37 countries. The mastectomy rate was highest in central and eastern Europe at 77%. The USA had the second highest rate of mastectomy with 56%, western and northern Europe averaged 46%, southern Europe 42% and Australia and New Zealand 34%.


title: Mastectomy

text: Breast surgery was first described 3000 years ago. In the earliest stages, breast tumors were treated with simple cauterization. Later, alternating incision and cauterization with complete removal of tumors was suggested by Leonides, one of the first breast oncologic surgeons recorded in history. Other surgeons recommended excision and cauterization only if the tumor could be removed completely; otherwise, avoiding surgery was recommended. Ambrose Pare (b. 1510), a well-known surgeon from Paris who was well-known for his experience treating soldiers who were injured, proposed a multi-tiered approach to breast surgery. While superficial cancers could be excised, more advanced cancers were managed through compression by lead plates to reduce blood supply to the tumor.


title: Mastectomy

text: In the 1500s, William Fabry (b.1560), a German surgeon known as the father of German surgery, created a device that compressed and fixed the base of the breast during mastectomy, which subsequently allowed for faster excision of the breast. Another technique developed during this time to improve efficiency of breast dissection was using ligatures to achieve anterior traction. Despite the development of these techniques, there were few mastectomies actually performed at the time due to lack of qualified surgeons and the high morbidity, mortality and disfigurement associated with the surgery.


title: Mastectomy

text: During the 1700s, large contributions in mapping lymph nodes for surgery were made by Pieter Camper (b. 1722) and Paolo Mascagni (b. 1752). Lymph node removal was advocated for in managing breast cancer. At this time, surgeries were still performed without proper aseptics and without anesthesia.


title: Mastectomy

text: In the 19th century, Seishu Hanaoka, a Japanese surgeon, performed the first surgery in the world under general anesthesia. Many more advancements in anesthesia and aseptic technique were made during this century. William Roentgen discovered x-rays in 1895, which radically shifted breast cancer treatment from a solely surgical approach to the multi-pronged approach employed today, including imaging, hormonal therapy, radiation, chemotherapy and immunotherapy.


title: Mastectomy

text: During the 20th century, progress was made towards skin-sparing mastectomies for treatment of breast cancer. Recent literature suggests that these procedures allow for improved aesthetic outcomes while also not increasing risk for local recurrence compared to conventional mastectomies.


title: Hironobu Sakaguchi

text: Originally intending to become a musician, he briefly studied electronics and programming, joining Square as a part-time employee, then later a full-time employee when Square became an independent company in 1986. He led the development of several titles before helping to create the original "Final Fantasy", which proved highly successful and cemented his status within the company. Following the financial failure of "", his debut as a film director, Sakaguchi withdrew from Square's management and eventually resigned in 2003. He continued his game career through Mistwalker, first co-developing projects through external partners and then smaller in-studio mobile projects.


title: Hironobu Sakaguchi

text: Born in Hitachi, Ibaraki Prefecture, Sakaguchi now lives in Hawaii where one of Mistwalker's offices is based. Several of his projects after "Final Fantasy III", including "Final Fantasy VII" and "The Spirits Within", were influenced by his thoughts on what happened after death following the loss of his mother in an accident at that time. He has also influenced the creation of several other Square Enix projects, such as "Kingdom Hearts". Sakaguchi has received several industry awards.


title: Hironobu Sakaguchi

text: Hironobu Sakaguchi was born on November 25, 1962, in Hitachi, a city in Ibaraki Prefecture. His parents were from Kyushu, and he would visit there frequently with his family in his childhood. Notable elements of his youth were finding rock samples in a local quarry and gathering a collection of polished stones from an interested geologist and reading through his mother's large library. He enjoyed playing piano in elementary school, and during high school played the folk guitar and formed an amateur band with some friends and classmates, almost getting himself expelled by selling homemade concert tickets. Having little to no interest in gaming at the time, his aim was to become a musician. He was a computer science major at Yokohama National University, becoming friends with fellow student Hiromichi Tanaka. Through Tanaka, Sakaguchi got access to an Apple II computer on which he played and fell in love with "", frequently skipping classes to play it.


title: Hironobu Sakaguchi

text: Sakaguchi's programming studies led him to desire an Apple II of his own. Since he could not afford one, he instead purchased a knockoff in the Akihabara district, which, although cheaper than an actual Apple II, was still expensive. Realizing that he needed funds to buy software for his computer, he began to seek a part-time job to earn the necessary income. In 1983 towards the end of their third university year, Sakaguchi and Tanaka were both looking for part-time work in the electronics sector. They found work with Square, a newly-formed subsidiary of electric power conglomerate Den-Yu-Sha, formed by Masafumi Miyamoto to tap into the emerging video game market. Sakaguchi applied as Square was a new company and not as exacting as other larger companies such as Namco and Konami. The interview was quite informal, and Sakaguchi was hired. At this point, Sakaguchi still dreamed of becoming a professional musician but felt that working for a company like Square would provide him with needed programming experience in the meantime.


title: Hironobu Sakaguchi

text: During his time at Square, he was first involved in an unlicensed adaptation of the television game show "Torin-ingen", then later became part of the team for "The Death Trap". There was no formal reassignment, only an informal restructure that led Sakaguchi to take on a more senior role. Following the success of "The Death Trap", he took a leave of absence from university work to continue his work with Square. His first few titles including "The Death Trap" were for PCs, with his first Nintendo Entertainment System title being "King's Knight" (1986). Many of these projects, while keeping the company afloat, were not hugely successful and focused on action titles that Sakaguchi disliked. He also earned the reputation of being a hard person to work under. When Square became an independent company in 1986, Sakaguchi was appointed as a full-time employee as Director of Planning and Development. That year, following the release and success of Enix's role-playing video game (RPG) "Dragon Quest" for the NES, he persuaded Miyamoto to allow production of an RPG by Square. This RPG drew inspiration from multiple fantasy titles of the time including "The Legend of Zelda", and Origin Systems's "Ultima" series.


title: Hironobu Sakaguchi

text: Sakaguchi described the production as happening in "fits and starts" with a small staff as most of the company did not believe in his project. Production gained momentum during its second half and getting help from Tanaka's team. Initially titled "Fighting Fantasy", he changed it to "Final Fantasy" to both avoid trademark conflicts with the roleplaying gamebook series of the same name and to represent his feelings at the time. Had the title failed commercially, he would have quit Square and completed his university education. In fact, "Final Fantasy" was an impressive commercial success, selling 400,000 units in Japan, and kickstarting the "Final Fantasy" series as Square's leading series. Sakaguchi acted as director for the four subsequent entries between 1988 and 1992. He was promoted in 1991 to the position of Executive Vice President, which eventually lessened his creative role in titles, with "Final Fantasy V" (1992) being his final directorial role with the "Final Fantasy" series. He notably spearheaded the creation of "Chrono Trigger" (1995), forming one third of the core "Dream Team" staff alongside "Dragon Quest" creator Yuji Horii, and "Dragon Ball" and "Dragon Quest" artist Akira Toriyama.


title: Hironobu Sakaguchi

text: He acted as the producer for the company's PlayStation debut "Final Fantasy VII" (1997), which proved to be a massive financial and critical success. The greater production requirements of the PlayStation meant Sakaguchi became less involved in the creative process, giving greater creative control to Yoshinori Kitase, who had worked with Sakaguchi beginning with "Final Fantasy V". For "Final Fantasy VIII" (1999), he took on the role of executive producer. He acted as producer for some projects including "Final Fantasy Tactics" (1997) and "Parasite Eve" (1998). He attempted to outsource development of a "Tactics" sequel, but the project was cancelled. For "Final Fantasy IX" (2000), a nostalgic entry which was mainly produced by a team in Honolulu, Hawaii where Sakaguchi was based at the time, he took a greater creative role including producing the title and writing the scenario.


title: Hironobu Sakaguchi

text: Through this period, Sakaguchi was creating a CGI feature film based on the "Final Fantasy" franchise called "", establishing the in-house studio Square Pictures in 1997 to produce it and other CGI projects. Sakaguchi intended "The Spirits Within", his debut as a film director, to be the first of multiple film projects, with lead Aki Ross becoming a virtual actress whose model would be reused in other CGI movies. Due to cost overruns towards the end of production to ensure staff payments amid rising production costs, "The Spirits Within" cost Square and co-producer Columbia Pictures $137 million. Sakaguchi himself made a cameo appearance in the film. The commitment to "The Spirits Within" contributed to his less involved role in the main "Final Fantasy" series. He also planned to create an interactive movie, further bridging the gap between films and video games.


title: Hironobu Sakaguchi

text: Upon its release in cinemas "The Spirits Within" grossed just over $85 million, being labelled a box office bomb. The film's failure damaged Square financially, delayed a planned merger between Square and Enix, and prompted the closure of Square Pictures. Following the failure of "The Spirits Within", Sakaguchi was left in a state of low morale, and he decided to step away from active involvement in Square's projects. He was also growing tired of his administrative role within Square. In 2001, Sakaguchi resigned from his position at Square along with two other senior executives, signing an agreement to be credited as executive producer on future "Final Fantasy" projects. While still receiving credits on "Final Fantasy", Sakaguchi spent the next three years at his home in Hawaii in a demoralized state, describing himself as "doing nothing" and feeling guilty over his lack of contributions to the industry. He eventually found the motivation to make a return to game production after talking with Toriyama and fellow artist Takehiko Inoue. Sakaguchi officially left Square in 2003, with his last major credited role being on "Final Fantasy X-2".


title: Hironobu Sakaguchi

text: Sakaguchi became reinvested in making games again after his talks with Toriyama and Inoue, deciding to create his own studio. His independent studio Mistwalker was established in 2004 with funding help from Microsoft, although the Mistwalker trademark had existed since 2001. Mistwalker is based between Honolulu, Hawaii, and Tokyo, Japan. By this point, Sakaguchi had settled in Honolulu, travelling between there and Japan for game production. Many early Mistwalker titles were for the Xbox 360, a console struggling to find commercial success in Japan. He did not choose the PlayStation 3 due to its difficult production architecture and earlier disagreements with Sony president Ken Kutaragi.


title: Hironobu Sakaguchi

text: Four titles were announced in 2005. These were "Blue Dragon" (2006), with whom he collaborated with Toriyama and spawned sequels and media spin-offs; "" (2007), a tactical RPG on which he worked with staff from "Final Fantasy XII"; "Lost Odyssey" (2007), which featured artwork from Inoue and narrative elements by Japanese novelist Kiyoshi Shigematsu; and "Cry On", a title intended to evoke emotion that was ultimately cancelled in 2008. Sakaguchi returned to a directorial role for "The Last Story" (2011), which he took as an opportunity to adjust his gameplay approach based on feedback from his earlier titles and to keep up with gaming trends of the time.


title: Hironobu Sakaguchi

text: Following "The Last Story", Sakaguchi wanted Mistwalker to focus on smaller mobile projects they could produce in-house on a small intimate team. His first project was "Party Wave", a surfing simulator inspired by Sakaguchi's love of surfing, though this ended up being a commercial disappointment. Inspired by this failure, Sakaguchi re-evaluated the mobile market and began production on "Terra Battle", a card-based RPG with a storyline inspired by western television. "Terra Battle" was a commercial success for the company, prompting Sakaguchi to make plans to release eight games over six years, leading up to his possible retirement age. He created a sequel and spin-off for "Terra Battle", and had been planning a third entry and a console version. Eventually all three "Terra Battle" entries were shut down for various reasons by 2019. During this period, Sakaguchi replayed "Final Fantasy VI" (1994), rediscovering his love of creating compelling worlds and narratives. With this in mind, he began development of "Fantasian" (2021), a two-part RPG modelled on "Final Fantasy VI" with Sakaguchi producing and writing the title for the Apple Arcade service. He created "Fantasian" on the principle that it might be his last major game project due to his advancing age.


title: Hironobu Sakaguchi

text: Sakaguchi liked the design of the original "Dragon Quest", attributing his admiration to its scale and ability to create a long adventure with progress saving on hardware other than PCs. This admiration directly inspired his wish to create "Final Fantasy". His favorite game genre is tactical role-playing, though he was unable to lead development on one until Mistwalker's "" (2007). His favorite non-RPG title is "", which shocked him at the time due to its narrative achievements while being graphically unimpressive compared to "Final Fantasy". Speaking about his design approach in 1994, Sakaguchi said he preferred to break away from established conventions rather than working to a pre-existing plan. He also stated a wish to have games surpass the visual and narrative impact of films, citing their interactive elements as an advantage.


title: Hironobu Sakaguchi

text: During his tenure with the "Final Fantasy" series, Sakaguchi pushed never to create sequels, which he dislikes as he thinks games should be complete experiences on their own. He also has mixed opinions on remakes, seeing the positive aspect of giving new players access to older titles, but preferring that developers focus on new projects. During his early projects, he needed to be careful about hardware limitations when thinking how many characters to include on-screen, but as consoles became more powerful he was able to create what he imagined. In 2004, he stated some of his best ideas for storyline or game design came while he was taking showers. Later he described his surfing hobby as a key influence, citing a mechanic in "The Last Story" as being inspired by waiting for a good wave.


title: Hironobu Sakaguchi

text: Sakaguchi compared creating a new game with watching the birth of a child. For the early "Final Fantasy" titles, Sakaguchi placed little focus on the narrative, with his early failures in the gaming market making him feel he was not suited for scenario writing. His perspective changed during production of "Final Fantasy III". A fire started at his family home and his mother Aki was killed, and by the time he reached home the fire had consumed the house. His deep emotions, combined with criticism of the series for its lack of narrative, caused him to reflect on what happened to people after death and prompted a greater focus on narrative. Sakaguchi's reflections on life and death would directly inspire elements within "Final Fantasy VII", and "The Spirits Within". Sakaguchi named the film's lead Aki Ross after his mother. This focus on themes of life and death continued into his work at Mistwalker.


title: Hironobu Sakaguchi

text: Within Square, Sakaguchi helped support the careers of several notable staff members including "SaGa" creator Akitoshi Kawazu, artist Tetsuya Nomura, "Xeno" creator Tetsuya Takahashi and "Tactics" creator Yasumi Matsuno. In an interview, Nomura credited a suggestion by Sakaguchi with the complex narrative direction taken by the "Kingdom Hearts" series, as Nomura's original plan was for a simple narrative to fit with the demographics of development partner Walt Disney. He also proposed the concept for what would become "Final Fantasy XI" (2002), which was led by "Mana" creator Koichi Ishii.


title: Hironobu Sakaguchi

text: Following his departure and the merger with Enix, the upper management of Square Enix apparently instructed its staff not to communicate with Sakaguchi, severing ties with him and apparently ostracising staff within the company who were identified as protegees including Matsuno. Sakaguchi disliked the creative direction "Final Fantasy" took under its new management, particularly citing "Final Fantasy XIII" as indicative of this change. In later years, the position softened on both sides, though Sakaguchi did not describe it as a true reconciliation. Kitase and Nomura described the aftermath, saying that Sakaguchi provided a singular creative voice for the company which was never reclaimed after he left.


title: Hironobu Sakaguchi

text: In May 2000, Sakaguchi received the Hall of Fame Award of the Academy of Interactive Arts & Sciences. At the 2015 Game Developers Choice Awards, Sakaguchi was given the Lifetime Achievement award. The event's general manager Meggan Scavio spoke positively of Sakaguchi's contribution to narratives in gaming, saying his work on "Final Fantasy" "helped pave the way for game stories that dealt with death, regret, and character development in a mature and significant way". At the 2017 CEDEC Awards, Sakaguchi was given the Special Award, with his extensive contributions to the gaming industry and strong following in Japan and overseas being cited as reasons.


title: Ateneo de Manila University

text: The Ateneo de Manila University (Filipino: "Pamantasang Ateneo de Manila"; Spanish: "Universidad Ateneo de Manila"), also known simply as the Ateneo de Manila, the Ateneo, or AdMU, is a private, research, basic and higher education institution founded and run by the Society of Jesus in Quezon City, Philippines. Founded in 1859 as "Escuela Municipal de Manila", it is the second-oldest Jesuit-administered institution of higher learning in Asia-Pacific.


title: Ateneo de Manila University

text: The university provides primary and secondary education as well as undergraduate and graduate instruction in the humanities, social sciences, natural sciences and engineering, and business. It offers professional degrees through the Graduate School of Business, the School of Government, the School of Medicine and Public Health, and the Ateneo Law School. Known for its Jesuit liberal arts tradition, the humanities are a key feature of Ateneo education at all levels of study. This is especially pronounced in the undergraduate level as the "Core Curriculum".


title: Ateneo de Manila University

text: In 2022, the "Times Higher Education World University Rankings" ranked the Ateneo as the top university in the Philippines, placing it in the 351–400 bracket of its 2023 global rankings list.


title: Ateneo de Manila University

text: On September 25, 1969, Fr. Pacifico Ortiz, SJ was installed as the first Filipino president of the Ateneo de Manila University. At this time, the Graduate School split into the Graduate School of Arts and Sciences and what would become the Graduate School of Business. In 1968, the Ateneo co-founded the Asian Institute of Management. The Ateneo college opened its doors to its first female students in 1973. The Graduate School of Arts and Sciences moved to Loyola Heights in 1976, and the Padre Faura campus closed in 1977 as the Graduate School of Business and the Law School moved to Salcedo Village, Makati. In February 1978, Ateneo opened what would become the Ateneo Computer Technology Center.


title: Ateneo de Manila University

text: As student activism rose in the 1970s following the proclamation of martial law by President Ferdinand Marcos, Ateneans took an active role along with student organizations from other colleges and universities. On February 11, 1986, alumnus and Antique Governor Evelio Javier was shot and killed. Two weeks later, Ateneans joined thousands of Filipinos in the People Power Revolution along EDSA to oust Marcos.


title: Ateneo de Manila University

text: In 1991, the School of Law became the first law school in the Philippines to confer the Juris Doctor degree. The Ateneo School of Government was established in 1996 and, in 1998, a campus housing the Ateneo Graduate School of Business, Ateneo School of Law, and the Ateneo School of Government, opened in Rockwell Center in Poblacion, Makati. The Science Education Complex was also completed in the Loyola Heights campus. In 2000, the School of Arts and Sciences, consisting of the undergraduate and graduate schools, was restructured into the Loyola Schools. In April 2002, the office of the university president established Pathways to Higher Education-Philippines, one of the university's outreach initiatives, with the help of the Ford and Synergeia Foundations. In July 2002, the Church of the Gesù was completed in the Loyola Heights campus. In 2003, the Ateneo entered into its partnership with Gawad Kalinga, its first formal, university-wide social action program.


title: Ateneo de Manila University

text: In July 2006, the Manuel V. Pangilinan Center for Student Leadership was completed. The Ateneo School of Medicine and Public Health facility in Ortigas Center, Pasig welcomed its first batch of professional students in June 2007. In 2008, work on a new Rizal Library facility began and a new set of university dormitories was inaugurated.


title: Ateneo de Manila University

text: The university is organised into twelve schools, which are grouped into three units: The Loyola Schools, includes the School of Humanities, School of Social Sciences, School of Science and Engineering, the John Gokongwei School of Management, and the newly-established Gokongwei Brothers School of Education and Learning Design; the Professional Schools, composed of the Ateneo School of Law, the Ateneo School of Medicine and Public Health, the Ateneo Graduate School of Business, and the Ateneo School of Government; and the basic education unit.


title: Ateneo de Manila University

text: The Ateneo Professional Schools are the professional education units of the Ateneo de Manila University, and comprises the following four schools: the Ateneo Graduate School of Business, which offers a variety of Master of Business Administration (MBA) concentrations, including a Master in Health Services Administration; the Ateneo School of Law, which confers the Juris Doctor (JD) and Master of Laws (LL.M.) degrees; the Ateneo School of Medicine and Public Health, opened in 2007, which offers an integrated Doctor of Medicine (M.D.) and MBA program; and the Ateneo School of Government, which confers the Master in Public Management and Ph.D. in Leadership Studies degrees. The Professional Schools also confer certificates for short courses.


title: Ateneo de Manila University

text: The Loyola Schools are the higher education units of the Ateneo de Manila University since 2000, that offers undergraduate and graduate degree programs in the arts and sciences and operates under the Vice President of the university. It is located at the university's Loyola Heights campus in Katipunan, Quezon City. It confers the Bachelor of Arts, Bachelor of Science, Bachelor of Fine Arts, Master of Arts, Master of Science, and Doctor of Philosophy degrees. It offers its Honors Programs only to the top 10-15% of university applicants and has a higher cut-off grade than the minimum graduation grade requirement in order to remain in the program.


title: Ateneo de Manila University

text: The Loyola Schools are currently composed of four schools, the School of Humanities (SOH), the John Gokongwei School of Management (JGSOM), the School of Science and Engineering (SOSE), and the School of Social Sciences (SOSS). A fifth school, the Gokongwei Brothers School of Education and Learning Design(GBSEALD), was established in 2021. The arrangement of the Loyola Schools academic buildings reflects its origin as a single College of Arts and Sciences.


title: Ateneo de Manila University

text: The Ateneo de Manila Grade School is the elementary education unit of the Ateneo de Manila University, and is the oldest unit in existence - Ateneo de Manila began as a government-run primary school during the Spanish colonial period. It is an all-boys school with a population of around 4,000 students. The school had implemented the K-12 program since the 2013–14 academic year. The Ateneo de Manila Grade School is one of the first elementary schools in the Philippines to adopt the Singapore math curriculum.


title: Ateneo de Manila University

text: The Ateneo de Manila Junior High School (AJHS) is the middle school unit of the Ateneo de Manila University, offering the seventh to tenth grades for male students. It was established in 2013 to comply with the Philippines' K-12 education system. It currently occupies the old Ateneo de Manila High School campus.


title: Ateneo de Manila University

text: The Ateneo de Manila Senior High School (ASHS) is the senior high school unit of the Ateneo de Manila University, offering the eleventh and twelfth grades since 2013 following the restructuring of the university's high school units in adherence with the national K-12 program. Since the 2016–17 academic year, the ASHS has accepted female students.


title: Ateneo de Manila University

text: The ASHS offers all four academic strands of the K-12 program (Accountancy and Business Management; Humanities and Social Sciences; Science and Technology, Engineering, and Mathematics; and the General Academic). The ASHS currently occupies the new senior high school building topped off in 2016.


title: Ateneo de Manila University

text: The Loyola School of Theology is a Jesuit school of theology and pastoral studies, run separately from but federated with the Ateneo de Manila University. It is located in the portion of the university's Loyola Heights campus that is geographically within Marikina. The school offers baccalaureate, licentiate, and doctoral ecclesiastical degrees in theology, as well as postgraduate degrees in theological studies, theology, and pastoral ministry from the Ateneo de Manila. It also supports the theology and religious education postgraduate programs of the Ateneo Loyola Schools' Department of Theology.


title: Ateneo de Manila University

text: Currently, the main Ateneo de Manila campus is located along Katipunan Avenue in Loyola Heights, Quezon City, Metro Manila. The campus houses the Loyola Schools and the university's grade school and high schools, as well as the Loyola School of Theology. Two other campuses, in Rockwell Center and Salcedo Village, both in Makati, house the university's professional schools of business, law, and government. A fourth facility in the Don Eugenio López Sr. Medical Complex in Ortigas Center, Pasig, houses its professional school of medicine and public health.


title: Ateneo de Manila University

text: The main campus is located in Loyola Heights, along the eastern side of Katipunan Avenue, and is south of and adjacent to the campus of Miriam College. The Grade School, High School, and Loyola Schools are located in the Ateneo's Loyola Heights campus. Beside the Grade School is the Henry Lee Irwin Theater, built in 1995 to house the school's formal events and productions. Complementing the old buildings of the Loyola Schools are the Science Education Complex, as well as the PLDT Convergent Technologies Center and the John Gokongwei School of Management Complex. In 2018, the university inaugurated the Areté, the Ateneo's "creative hub" for visual, practical, and performing arts. The Areté houses the Ateneo Art Gallery, the Hyundai Hall (a 900-seat theater), the Doreen Gamboa Fernandez Black Box Theater, the Unionbank Ubuntu Space, and spaces for the Ateneo's collaborative degree program with Le Cordon Bleu. The Department of Fine Arts and the Ateneo Institute for the Science and Art of Learning and Teaching also hold office in the Areté.


title: Ateneo de Manila University

text: Within the campus is the Rizal Library, the main university library. Also located here are numerous units and research centers affiliated with the Ateneo, such as the Institute of Social Order, Institute of Philippine Culture, Institute on Church and Social Issues, Asian Public Intellectuals Fellowships, the Jesuit Communications Foundation, the Jesuit Basic Education Commission, and others. Also situated here are the East Asian Pastoral Institute, Loyola School of Theology, and the San Jose Seminary, all Jesuit formation institutions federated with the Ateneo de Manila University. The Loyola Heights campus also hosts three scientific research institutions: the Philippine Institute of Pure and Applied Chemistry, the National Chemistry Instrumentation Center, and the Manila Observatory.


title: Ateneo de Manila University

text: Its athletic facilities include the Blue Eagle Gym, also known as the Loyola Center, standing at the southern end of the campus, and the Moro Lorenzo Sports Center (MLSC) on the northern end. The Blue Eagle Gym is one of the largest gymnasiums among the universities in Metro Manila while the MLSC is often used by the Philippines men's national basketball team as well as other professional teams for their training needs.


title: Ateneo de Manila University

text: The Church of the Gesù, completed in July 2002, stands on top of the Sacred Heart Hill and overlooks the rest of the campus. The school's chapels include the St. Stanislaus Kostka Chapel and the Chapel of the First Companions in the high school, the Chapel of the Immaculate Conception in the college complex's Gonzaga Hall, the chapel at the Loyola House of Studies, and the Chapel of the Holy Guardian Angels in the grade school, among others. Two parish churches outside the campus, the Our Lady of Pentecost Parish in Varsity Hills and the Santa María della Strada Parish in Diliman, are within walking distance from the campus.


title: Ateneo de Manila University

text: The Ateneo de Manila is also home to the largest Jesuit community in the Philippines, most of whom reside at the Jesuit Residence in the Loyola Heights campus. These Jesuits are involved in teaching, administration, and research within the university and its affiliated units.


title: Ateneo de Manila University

text: The Rockwell Center campus of the Ateneo de Manila University in Makati houses the School of Law, Graduate School of Business, School of Government, AGSB-BAP Institute of Banking, and the Ateneo Center for Continuing Education. It includes several research centers, a moot court facility, and the Ateneo Professional Schools Library.


title: Ateneo de Manila University

text: The Salcedo Village campus in Makati houses the different facilities of the former Ateneo Information Technology Institute and the Ateneo Center for Continuing Education. This facility formerly housed the Professional Schools prior to the completion of the Rockwell campus in 1998.


title: Ateneo de Manila University

text: The Ateneo School of Medicine and Public Health (ASMPH) at the Don Eugenio López Sr. Medical Complex in Ortigas Center, Pasig, opened its doors to its pioneering batch of students in June 2007. Beside the ASMPH is its partner hospital, The Medical City.


title: Ateneo de Manila University

text: The Ateneo de Manila University established the Ateneo Institute of Sustainability in 2013 to oversee the research and implementation of the university's sustainable development initiatives. In 2014, the Ateneo launched an electric jeepney shuttle system operated by Meralco within its Loyola Heights campus to replace motorized tricycles as the means of transporting students and personnel around the campus; the Ateneo was the first educational institution in the Philippines to implement an electric jeepney transportation system.


title: Ateneo de Manila University

text: Amidst the COVID-19 pandemic in 2020, the Ateneo de Manila University launched the Ateneo Blue Cloud (stylized AteneoBlueCloud), an online platform for the university's scholarly community. Branded as the university's "virtual campus", the Ateneo Blue Cloud hosts school curricula materials based around their framework of an "adaptive design for learning", wherein online, blended, and face-to-face educational programs are accessible by registered users.


title: Ateneo de Manila University

text: The Ateneo offers instruction in the primary, secondary, undergraduate, and graduate level. It is a research university that offers 48 undergraduate degrees, 93 graduate degrees, and 11 professional degrees. Dual/joint degree programs are offered within the schools and with other universities as well. Individual schools conduct separate admissions processes; admission into one unit in no way guarantees admission into another unit.


title: Ateneo de Manila University

text: As is common in the Philippines, the primary medium of instruction is English, with a few classes taught in Filipino. Aside from teaching and research, the Ateneo de Manila also engages in social outreach.


title: Ateneo de Manila University

text: A key feature of the Loyola Schools is a liberal arts undergraduate core curriculum, required for all undergraduates. It includes philosophy, English and Filipino literature, theology, history, various branches of social sciences, and a community service component. Undergraduate programs at the Loyola Schools are aimed at fostering student-centred learning. The Ateneo follows the semester hour system common in American universities. Most classes are held below 40 students and student discussion is encouraged.


title: Ateneo de Manila University

text: The United Kingdom-based Quacquarelli Symonds ranked the Ateneo de Manila University first among private universities and second among all universities in the Philippines in both its "World University Rankings" and "Asia Rankings" in 2022 (for its 2023 list). In the "QS Graduate Employability Rankings 2022", the Ateneo is ranked in the 151–160 bracket worldwide, making it the highest ranked Philippine university in terms of employability. The "Times Higher Education" placed Ateneo in the 101–200 bracket worldwide in its "Impact Rankings 2022" on the seven criteria of Sustainable Development Goals (SDGs), making it the highest among the Philippine institutions included. The university received an overall score of 83.7 out of 100, placing 48th worldwide in adhering to the clean water and sanitation SDG.


title: Ateneo de Manila University

text: CHED has designated the Ateneo with 11 Centers of Excellence and 6 Centers of Development. The 11 Centers of Excellence are in information technology, chemistry, mathematics, physics, English literature, philosophy, biology, psychology, sociology, business administration or management, and entrepreneurship. The 6 Centers of Development are in environmental science, communication, electronic engineering, history, Philippine literature, and political science.


title: Ateneo de Manila University

text: The Ateneo grade school and high school divisions have both garnered the highest level of accreditation from the Philippine Accrediting Association of Schools, Colleges and Universities and the Federation of Accrediting Agencies of the Philippines, making it one of the top schools in the country for primary and secondary education, as of 2017.


title: Ateneo de Manila University

text: The Legal Education Board has ranked the Ateneo School of Law as the best performing private law school in the Philippines and the second-best overall, behind the University of the Philippines College of Law, as of 2019. Graduates of the Ateneo Law School have an average grade of 89.2 in the Philippine Bar Examination and the school has produced 23 topnotchers in history, the second highest number of bar top notchers. For the 2020/2021 Bar Examinations, the Ateneo Law School had the highest passing rate for first-time examinees in the Philippine Bar Examination at 99.64%.


title: Ateneo de Manila University

text: The Professional Regulation Commission has ranked the Ateneo School of Medicine and Public Health as the best performing medical school in the Philippines in 2019, as the school had a 100% passing rate in the Philippine Physician Licensure Examination out of 133 examinees.


title: Ateneo de Manila University

text: The Ateneo School of Government has been ranked as one of the top public policy schools in terms of research publication output published in the Asia & the Pacific Policy Studies. It is the only Philippine institution listed in the first-ever ranking of public-policy schools in the Asia-Pacific region. ASOG was ranked 25th by citation impact factor, with a total of 7 citations scoring 1.17 impact factor.


title: Ateneo de Manila University

text: The Paris-based Eduniversal ranked the Ateneo Graduate School of Business as one of the top business schools in the country, ranking 3rd with 3 Palmes of Excellence in 2021.


title: Ateneo de Manila University

text: Some of the Ateneo de Manila's most active research hubs work in the fields of disaster risk reduction, prevention, and management; public education; human migration; and governance. The university houses several research centers, and has many links with industry partners, government agencies, and research networks. Some research centers, called auxiliary units, are established by the university board of trustees, while others are organized by individual schools or departments.


title: Ateneo de Manila University

text: The Ateneo has grounded its vision and mission in the Jesuit educational tradition of engaging with the world at large, leading the university to be involved in civic activities. Social engagement is a key part of Ateneo education, integrated into the curricula of practically all university programs. Social entrepreneurship is also integrated into many of its academic programs.


title: Ateneo de Manila University

text: The Ateneo's social projects include the Ateneo-Mangyan Project for Understanding and Development to help the Mangyan indigenous group from Mindoro, the "Bigay Puso" and the Christian Service and Involvement Program for grade school students, and the Damay Immersion and "Tulong Dunong" program for senior high school students. In college, social development is fostered by programs of the Office of Social Concern and Involvement, including house-builds with the poverty alleviation movement Gawad Kalinga and the Ateneo Labor Trials Program tied into junior philosophy classes. Student organizations and offices of the Loyola Schools also operate their own social involvement programs. At the Ateneo Professional Schools, programs and units like the Graduate School of Business' "Mulat-Diwa", the Leaders for Health Program, and the Law School's Human Rights Center and Legal Aid programs aim to form leaders. Other Ateneo initiatives include "Pathways to Higher Education" to help the underprivileged youth and the Ateneo Center for Educational Development which conducts national teacher and principal training programs.


title: Ateneo de Manila University

text: The centerpiece social program of the university is its university-wide social action program in partnership with Gawad Kalinga which has helped build communities and schools in Payatas, Quezon City, in many Nueva Ecija municipalities, and three villages in the Bicol Region. The Ateneo–Gawad Kalinga partnership has also driven Kalinga Luzon, a rehabilitation effort for the victims of Typhoon Nanmadol (Yoyong); GK Youth–Ateneo, one of the largest and most active student social program of the Ateneo; Kalinga Leyte, an ongoing program which aims to provide long-term rehabilitation for the victims of the Southern Leyte landslide and Typhoon Haiyan (Yolanda), and ongoing reconstruction efforts for typhoon-stricken Bicol.


title: Ateneo de Manila University

text: The Ateneo has international linkages with several universities, institutions, and organizations, particularly in Asia, Australasia, North and South America, and Europe. Through these cooperative efforts, the university hosts visiting faculty and research fellows from institutions abroad, and in turn, Ateneo faculty members also engage in teaching, research, and study in institutions abroad. International cooperation also includes active student exchange through Philippine immersion programs.


title: Ateneo de Manila University

text: Since 2008, the Global Leadership Program was started for students from three other Catholic Jesuit universities in East Asia aside from the Ateneo: Fu Jen Catholic University in Taiwan, Sogang University in South Korea, and Sophia University in Japan.


title: Ateneo de Manila University

text: The Ateneo has partnered with several international institutions in offering certain degree programs. In 2006, the Ateneo partnered with the Sun Yat-sen University in Guangzhou to establish a Confucius Institute. The institute has since offered Mandarin Chinese courses, a Masters in Teaching Chinese as a Foreign Language program, and a scholarship grant for students to study in Chinese colleges and universities. In 2018, the Ateneo Department of Information Systems and Computer Science has partnered with the Queen Mary University of London to offer a double master's degree. In 2019, the School of Management ventured with French culinary school Le Cordon Bleu to establish the Le Cordon Bleu Ateneo de Manila Institute at the Areté. The institute offers a bachelor's degree through the Loyola Schools as well as professional degrees at the Center for Continuing Education at the Graduate School of Business. Other universities that the Ateneo has partnered with to offer joint master's degree programs are the École supérieure des sciences commerciales d'Angers in France, Kyushu University in Japan, Regis University in Denver, and the University of Sydney in Australia.


title: Ateneo de Manila University

text: The Ateneo has also collaborated with the Nippon Foundation and the University for Peace (UPEACE) in Costa Rica to initiate the Asian Peacebuilders Scholarship program. Under the program, students train to become peacebuilding practitioners for a total of eight months at the Ateneo and nine months at UPEACE, eventually receiving a master's degree in Transdisciplinary and Social Development from the Ateneo Department of Political Science and a Master of Arts degree from UPEACE.


title: Ateneo de Manila University

text: The Ateneo de Manila University has several libraries. The university's largest library is the Rizal Library, part of the Loyola Schools, located in the Loyola Heights campus. The Ateneo Professional Schools Library is housed in the Professional Schools building at the Rockwell campus. The basic education units (Ateneo Grade School, Junior High School, and Senior High School) each have their own libraries; the East Asian Pastoral Institute and the Loyola School of Theology, both located inside the Loyola Heights campus, also have their own.


title: Ateneo de Manila University

text: The Rizal Library is divided into the Old Rizal Library and the New Rizal Library. The Old Rizal Library, which is known as the Special Collections Building, houses rare Filipiniana items including a permanent exhibit of José Rizal memorabilia, a collection of Trinidad Pardo de Tavera, the American Historical Collection containing materials related to the American colonial period, the Ateneo Library of Women's Writings, and other special collections and manuscripts by Filipino scholars, writers, and artists. The New Rizal Library, a five-storey building which opened in November 2009, houses the library's circulation section, the undergraduate and graduate reserve sections, the multimedia collection, the periodicals collection, the Japanese collection, online database access terminals, an information commons, and the Library's technical services facilities.


title: Ateneo de Manila University

text: The Professional Schools Library holds one of the largest collections of materials in the fields of law, business, and government in the Philippines.


title: Ateneo de Manila University

text: The Ateneo Art Gallery is housed in the Arts Wing of the Areté. The gallery is the first and only museum of modern art in the Philippines. The centerpiece of its collection is a selection of post-war art donated to the university by Spanish Filipino businessman Fernando Zóbel de Ayala.


title: Ateneo de Manila University

text: The University Archives are housed in the Rizal Library annex building. Since 1958, it has served as the central repository of non-current records of the administrative offices, academic departments, and student organizations. Among its collections are papers and documents from key university people, relics and personal effects of alumni, some archived publications, theses, and dissertations, as well as other materials such as maps, photographs, and art work.


title: Ateneo de Manila University

text: The Ateneo de Manila University is governed by a board of trustees currently chaired by Bernadine T. Siy. A central administration, led by President of the Ateneo de Manila University Roberto Yap, oversees key initiatives related to academics, international programs, university development and alumni relations, personnel, security, and other university-wide concerns. Yap succeeded Jose Ramon Villarin, on August 1, 2020.


title: Ateneo de Manila University

text: For purposes of administration, the Ateneo's schools are grouped into three units: Basic education (grade school, junior high school, and senior high school), Loyola Schools, and Professional Schools. Each unit is led by a vice president. Each basic education school is led by a principal (JHS and SHS) and a headmaster (grade school); meanwhile, the individual schools under the Loyola Schools and the Professional Schools are led by deans, who in turn oversee department chairs and program directors.


title: Ateneo de Manila University

text: There are also designated vice presidents for Social Development (overseeing social initiatives); University and Global Relations (overseeing internationalization and institutional linkages); Finance (who also acts as the University Treasurer); and Administration & Human Resources.


title: Ateneo de Manila University

text: The Ateneo de Manila has been granted autonomous status by the Philippine Commission on Higher Education (CHED) since 2001, granting it relative leeway in introducing new programs. It is one of few universities in the country to be granted this status, which recognizes a number of the university's programs and departments as Centers of Excellence, indicating the highest degree of instruction, research, and extension in the Philippines of their particular fields or courses, and Centers of Development, indicating "significant improvement" of instruction, research, and extension from the previous year. In 2005, the academic programs of the Loyola Schools were granted a Level IV accreditation—the highest possible level—from the CHED through the Federation of Accrediting Agencies of the Philippines (FAAP) and the Philippine Accrediting Association of Schools, Colleges and Universities (PAASCU). In 2011, it was also granted an Institutional Accreditation by the same body, the first time that both citations were awarded to a university simultaneously.


title: Ateneo de Manila University

text: There are currently over 50 accredited college student organizations in the Loyola Schools and 19 accredited student organizations in the Ateneo Law School.


title: Ateneo de Manila University

text: The Ateneo College Glee Club (ACGC) is the oldest extant university chorale in the Philippines. ACGC has participated in numerous international choral and choir competitions including the European Grand Prix for Choral Singing. The Ateneo Debate Society, the undergraduate debate organization of Ateneo, is the highest-ranking debate team in the Philippines and Asia, and often in the top 10 of the World University Debate Rankings since the mid-2000s, peaking at no. 7 in 2012. The Ateneo de Manila High School's "Dulaang Sibol", which began as the Ateneo High School Dramatics Society in 1955, is the oldest existing theater group in the Philippines.


title: 0.999...

text: In mathematics, 0.999... (also written as 0., in repeating decimal notation) denotes the repeating decimal consisting of an unending sequence of 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...); that is, the supremum of this sequence. This number is equal to1. In other words, "0.999..." is not "almost exactly" or "very, very nearly but not quite" rather, "0.999..." and "1" represent "exactly" the same number.


title: 0.999...

text: There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined. In other systems, 0.999... can have the same meaning, a different definition, or be undefined.


title: 0.999...

text: More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all positional numeral system representations regardless of base. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons—such as rigorous proofs relying on non-elementary techniques, properties, or disciplines—some people can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.


title: 0.999...

text: There is an elementary proof of the equation , which uses just the mathematical tools of comparison and addition of (finite) decimal numbers, without any reference to more advanced topics such as series, limits, formal construction of real numbers, etc. The proof, an exercise given by , is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so .


title: 0.999...

text: If one places 0.9, 0.99, 0.999, etc. on the number line, one sees immediately that all these points are to the left of 1, and that they get closer and closer to 1.


title: 0.999...

text: More precisely, the distance from 0.9 to 1 is , the distance from 0.99 to 1 is , and so on. The distance to 1 from the th point (the one with 9s after the decimal point) is .


title: 0.999...

text: Therefore, if 1 were not the smallest number greater than 0.9, 0.99, 0.999, etc., then there would be a point on the number line that lies between 1 and all these points. This point would be at a positive distance from 1 that is less than for every integer . In the standard number systems (the rational numbers and the real numbers), there is no positive number that is less than for all "n". This is (one version of) the Archimedean property, which can be proven to hold in the system of rational numbers. Therefore, 1 is the smallest number that is greater than all 0.9, 0.99, 0.999, etc., and so .


title: 0.999...

text: Part of what this argument shows is that there is a least upper bound of the sequence 0.9, 0.99, 0.999, etc.: a smallest number that is greater than all of the terms of the sequence. One of the axioms of the real number system is the completeness axiom, which states that every bounded sequence has a least upper bound. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one.


title: 0.999...

text: The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For the accuracy of the proof, the number , with nines after the decimal point, is denoted . Thus , , , and so on. As , with digits after the decimal point, the addition rule for decimal numbers implies


title: 0.999...

text: One has to show that 1 is the smallest number that is no less than all . For this, it suffices to prove that, if a number is not larger than 1 and no less than all , then . So let such that


title: 0.999...

text: This implies that the difference between and is less than the inverse of any positive integer. Thus this difference must be zero, and, thus ; that is


title: 0.999...

text: This proof relies on the fact that zero is the only nonnegative number that is less than all inverses of integers, or equivalently that there is no number that is larger than every integer. This is the Archimedean property, that is verified for rational numbers and real numbers. Real numbers may be enlarged into number systems, such as hyperreal numbers, with infinitely small numbers (infinitesimals) and infinitely large numbers (infinite numbers). When using such systems, notation 0.999... is generally not used, as there is no smallest number that is no less than all . (This is implied by the fact that implies ).


title: 0.999...

text: Many algebraic arguments have been provided, which suggest that formula_1 They are not mathematical proofs since they are typically based on the fact that the rules for adding and multiplying finite decimals extend to infinite decimals. This is true, but the proof is essentially the same as the proof of formula_1 So, all these arguments are essentially circular reasoning.


title: 0.999...

text: Nevertheless, the matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique. discusses the argument that, in elementary school, one is taught that , so, ignoring all essential subtleties, "multiplying" this identity by gives . He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation ." Most undergraduate mathematics majors encountered by Byers feel that while is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1. discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.


title: 0.999...

text: Students who did not accept the first argument sometimes accept the second argument, but, in Byers' opinion, still have not resolved the ambiguity, and therefore do not understand the representation for infinite decimals. , presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. , citing , also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature.


title: 0.999...

text: The same argument is also given by , who notes that skeptics may question whether is cancellable that is, whether it makes sense to subtract from both sides.


title: 0.999...

text: Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more 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" and one can neglect negatives, so a decimal expansion has the form


title: 0.999...

text: The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 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.


title: 0.999...

text: Since 0.999... is such a sum with "a" = 9 and common ratio "r" = , the theorem makes short work of the question:


title: 0.999...

text: The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle's textbook "An Introduction to Algebra" uses such an argument for geometric series to justify the same maneuver on 0.999... A 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.


title: 0.999...

text: A sequence ("x", "x", "x", ...) has a limit "x" if the distance |"x" − "x"| becomes arbitrarily small as "n" increases. The statement that 0.999... = 1 can itself be interpreted and proven as a limit:


title: 0.999...

text: The first two equalities can be interpreted as symbol shorthand definitions. The remaining equalities can be proven. The last step, that → 0 as "n" → ∞, is often justified by the Archimedean property of the real numbers. This limit-based attitude towards 0.999... is often put in more evocative but less precise terms. For example, the 1846 textbook "The University Arithmetic" explains, ".999 +, continued to infinity = 1, because every annexation of a 9 brings the value closer to 1"; the 1895 "Arithmetic for Schools" says, "when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small". Such heuristics are often incorrectly interpreted by students as implying that 0.999... itself is less than 1.


title: 0.999...

text: The series definition above is a simple way to define the real number named by a decimal expansion. A complementary approach is tailored to the opposite process: for a given real number, define the decimal expansion(s) to name it.


title: 0.999...

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


title: 0.999...

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


title: 0.999...

text: 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. So "b"."b""b""b"... is defined to be the unique number contained within all the intervals ["b", "b" + 1], ["b"."b", "b"."b" + 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.


title: 0.999...

text: The Nested Intervals Theorem is usually founded upon a more fundamental characteristic of the real numbers: the existence of least upper bounds or "suprema". To directly exploit these objects, one may define "b"."b""b""b"... to be the least upper bound of the set of approximants {"b", "b"."b", "b"."b""b", ...}. 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,


title: 0.999...

text: Some approaches explicitly define real numbers to be certain structures built upon the rational numbers, using axiomatic set theory. The natural numbers – 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 integers, and to further extend to ratios, giving the rational numbers. These number systems are accompanied by the arithmetic of addition, subtraction, multiplication, and division. More subtly, they include ordering, so that one number can be compared to another and found to be less than, greater than, or equal to another number.


title: 0.999...

text: The step from rationals to reals is a major extension. There are at least two popular ways to achieve this step, both published in 1872: Dedekind cuts and Cauchy sequences. Proofs that 0.999... = 1 which directly use these constructions are not found in textbooks on real analysis, where the modern trend for the last few decades has been to use an axiomatic analysis. Even when a construction is offered, it is usually applied towards proving the axioms of the real numbers, which then support the above proofs. However, several authors express the idea that starting with a construction is more logically appropriate, and the resulting proofs are more self-contained.


title: 0.999...

text: In the Dedekind cut approach, each real number "x" is defined as the "infinite set of all rational numbers less than x". In particular, the real number 1 is the set of all rational numbers that are less than 1. 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


title: 0.999...

text: Every element of 0.999... is less than 1, so it is an element of the real number 1. Conversely, all elements of 1 are rational numbers that can be written as


title: 0.999...

text: by the definition above, every element of 1 is also an element of 0.999..., and, combined with the proof above that every element of 0.999... is also an element of 1, the sets 0.999... and 1 contain the same rational numbers, and are therefore the same set, that is, 0.999... = 1.


title: 0.999...

text: The above approach to assigning a real number to each decimal expansion is due to an expository paper titled "Is 0.999 ... = 1?" by Fred Richman in "Mathematics Magazine", which is targeted at teachers of collegiate mathematics, especially at the junior/senior level, and their students. Richman notes that taking Dedekind cuts in any dense subset of the rational numbers yields the same results; in particular, he uses decimal fractions, for which the proof is more immediate. He also notes that typically the definitions allow


title: 0.999...

text: Another approach is to define a real number as the limit of a Cauchy sequence of rational numbers. This construction of the real numbers uses the ordering of rationals less directly. First, the distance between "x" and "y" is defined as the absolute value |"x" − "y"|, where the absolute value |"z"| is defined as the maximum of "z" and −"z", thus never negative. Then the reals are defined to be the sequences of rationals that have the Cauchy sequence property using this distance. That is, in the sequence ("x", "x", "x", ...), a mapping from natural numbers to rationals, for any positive rational "δ" there is an "N" such that |"x" − "x"| ≤ "δ" for all "m", "n" > "N". (The distance between terms becomes smaller than any positive rational.)


title: 0.999...

text: If ("x") and ("y") are two Cauchy sequences, then they are defined to be equal as real numbers if the sequence ("x" − "y") has the limit 0. Truncations of the decimal number "b"."b""b""b"... generate a sequence of rationals which is Cauchy; this is taken to define the real value of the number. Thus in this formalism the task is to show that the sequence of rational numbers


title: 0.999...

text: The definition of real numbers as Cauchy sequences was first published separately by Eduard Heine and Georg Cantor, also in 1872. 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.


title: 0.999...

text: Commonly in secondary schools' mathematics education, the real numbers are constructed by defining a number using an integer followed by a radix point and an infinite sequence written out as a string to represent the fractional part of any given real number. In this construction, the set of any combination of an integer and digits after the decimal point (or radix point in non-base 10 systems) is the set of real numbers. This construction can be rigorously shown to satisfy all of the real axioms after defining an equivalence relation over the set that defines 1 = 0.999... as well as for any other nonzero decimals with only finitely many nonzero terms in the decimal string with its trailing 9s version. With this construction of the reals, all proofs of the statement "1 = 0.999..." can be viewed as implicitly assuming the equality when any operations are performed on the real numbers.


title: 0.999...

text: One of the notions that can resolve the issue is the requirement that real numbers are densely ordered. Students are taking for granted that formula_3 is before formula_4 while this kind of intuitive ordering is better defined as purely lexicographical.


title: 0.999...

text: "... the ordering of the real numbers is recognized as a dense order. However, depending on the context, students can reconcile this property with the existence of numbers just before or after a given number (0.999... is thus often seen as the predecessor of 1)."


title: 0.999...

text: Dense order requires that there is a third real value strictly between formula_3 and formula_4, but there is none: we cannot change a single digit in either of the two to obtain such a number. If formula_3 and formula_4 are to represent real numbers they have to be equal. Dense ordering implies that if there is no new element strictly between two elements of the set, the two elements must be considered equal.


title: 0.999...

text: The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense.


title: 0.999...

text: Second, a comparable theorem applies in each radix or 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. In general, any terminating base "b" expression has a counterpart with repeated trailing digits equal to "b" − 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.


title: 0.999...

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


title: 0.999...

text: A more far-reaching generalization addresses the most general positional numeral systems. They too have multiple representations, and in some sense the difficulties are even worse. For example:


title: 0.999...

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


title: 0.999...

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


title: 0.999...

text: Marko Petkovšek has proven that for any positional system that names all the real numbers, the set of reals with multiple representations is always dense. He calls the proof "an instructive exercise in elementary point-set topology"; it involves viewing sets of positional values as Stone spaces and noticing that their real representations are given by continuous functions.


title: 0.999...

text: 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 numbers. Examples include:


title: 0.999...

text: 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 it can be proved that if a decimal of the form 0."b""b""b"... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadratic reciprocity.


title: 0.999...

text: Returning to real analysis, the base-3 analogue 0.222... = 1 plays a key role in a characterization of one of the simplest fractals, the middle-thirds Cantor set:


title: 0.999...

text: 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 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 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.


title: 0.999...

text: 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 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. 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.


title: 0.999...

text: 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 infinitesimals. There are many common contributing factors to the confusion:


title: 0.999...

text: These ideas are mistaken in the context of the standard real numbers, although some may be valid in other number systems, either invented for their general mathematical utility or as instructive counterexamples to better understand 0.999...


title: 0.999...

text: Many of these explanations were found by 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.


title: 0.999...

text: The elementary argument of multiplying 0.333... = by 3 can convince 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. 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... = using a supremum definition, but then insisted that 0.999... < 1 based on her earlier understanding of long division. Others still are able to prove that = 0.333..., but, upon being confronted by the fractional proof, insist that "logic" supersedes the mathematical calculations.


title: 0.999...

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


title: 0.999...

text: As part of Ed Dubinsky's APOS theory of mathematical learning, he 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 as a number in its own right and to dealing with the set of natural numbers as a whole.


title: 0.999...

text: With the rise of the Internet, debates about 0.999... have become commonplace on newsgroups and message boards, including many that nominally have little to do with mathematics. In the newsgroup , arguing over 0.999... is described as a "popular sport", and it is one of the questions answered in its FAQ. The FAQ briefly covers , multiplication by 10, and limits, and it alludes to Cauchy sequences as well.


title: 0.999...

text: A 2003 edition of the general-interest newspaper column "The Straight Dope" discusses 0.999... via and limits, saying of misconceptions,


title: 0.999...

text: A "Slate" article reports that the concept of 0.999... is "hotly disputed on websites ranging from "World of Warcraft" message boards to Ayn Rand forums". In the same vein, the question of 0.999... proved such a popular topic in the first seven years of Blizzard Entertainment's Battle.net forums that the company issued a "press release" on April Fools' Day 2004 that it is 1:


title: 0.999...

text: Although the real numbers form an extremely useful number system, the decision to interpret the notation "0.999..." as naming a real number is ultimately a convention, and Timothy Gowers argues in "Mathematics: A Very Short Introduction" that the resulting identity 0.999... = 1 is a convention as well:


title: 0.999...

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